# The gauge-invariant I-method for Yang-Mills

Cristian Gavrus<EMAIL_ADDRESS>
###### Abstract.

We prove global well-posedness of the $3d$ Yang-Mills equation in the temporal
gauge in $H^{\sigma}$ for $\sigma>\frac{5}{6}$.

Unlike related equations, Yang-Mills is not directly amenable to the method of
almost conservation laws (I-method) in its Fourier and global version. We
propose a modified energy which:

  1. (1)

Is gauge-invariant and easy to localize

  2. (2)

Provides local gauges which give control of local Sobolev norms (through an
Uhlenbeck-type lemma for fractional regularities)

  3. (3)

Is slightly smoother in time compared to the classical I-method energy for
related systems.

The spatial smoothing is realized via the Yang-Mills heat flow instead of the
multiplier $I$.

Due to the temporal condition and its finite speed of propagation, the local
gauge selection is compatible with recent initial data extension results.
Therefore, smoothened energy differences
$\mathcal{E}(t_{1},s)-\mathcal{E}(t_{0},s)$ can be partitioned into local
pieces whose (appropriately extended) bounds can be square summed. After
revealing the null structure within the trilinear integrals, these can be
estimated using known methods.

In an appendix we show how an invariant modified energy for Maxwell-Klein-
Gordon can extend previous results to regularities $\sigma>\frac{5}{6}$.

The author thanks Ben Dodson for suggesting the paper [22] and for discussions
on the I-method, and thanks Sung-Jin Oh for discussions on his Yang-Mills
papers. The author expresses gratitude to the John Hopkins University, where a
large part of this work was carried.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Local well-posedness for Yang-Mills
  4. 4 Modified local well-posedness
  5. 5 Yang-Mills Heat Flow solutions
  6. 6 Energy estimates for the curvature and the tension field
  7. 7 The local change of gauge: Proof of Proposition 1.10
  8. 8 Local differences of heat flows which coincide on a ball
  9. 9 Decompositions and estimates in DeTurck’s gauge
  10. 10 Trilinear estimates
  11. A Modified energy for Maxwell-Klein-Gordon

##  1\. Introduction

In this paper we consider the problem of low regularity global existence for
the initial value problem for the Yang-Mills equations on $\mathbb{R}^{3+1}$
with a non-abelian structural group $G$. We say that the connection
$A_{\alpha}:\mathbb{R}^{3+1}\to\mathfrak{g}$ solves the _Yang-Mills equation_
if

$\mathbf{D}^{\alpha}F_{\alpha\beta}=0$ (YM)

where

$F_{\alpha\beta}\vcentcolon=\partial_{\alpha}F_{\beta}-\partial_{\beta}F_{\alpha}+[A_{\alpha},A_{\beta}]\qquad\text{and}\qquad\mathbf{D}_{\alpha}B\vcentcolon=\partial_{\alpha}B+[A_{\alpha},B]$

denote the _curvature_ and the _covariant derivative_ , while $\mathfrak{g}$
is the associated Lie algebra. The background on Lie theory notions used is
presented in section 2.0.2.

Expanding, (YM) takes the form

$\partial^{\alpha}F_{\alpha\beta}+\left[A^{\alpha},\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}+\left[A_{\alpha},A_{\beta}\right]\right]=0$

An important feature of the (YM) equations is its _gauge invariance_. Given a
$G$ -valued potential $U$, (YM) is invariant under the following gauge
transformations

$A_{\alpha}\mapsto UA_{\alpha}U^{-1}-\partial_{\alpha}UU^{-1},\quad
F_{\alpha\beta}\mapsto UF_{\alpha\beta}U^{-1}.$ (1.1)

For this reason, the Cauchy problem cannot be well-posed until a choice of
gauge is made. If we impose the _Temporal gauge_ condition $A_{0}=0$ the
equations become

$\displaystyle\partial_{t}\text{div}A$
$\displaystyle=[\partial_{t}A_{j},A_{j}]$ (1.2) $\displaystyle\Box
A_{i}-\partial_{i}\text{div}A$
$\displaystyle=\partial_{j}[A_{i},A_{j}]+[\partial_{j}A_{i},A_{j}]+[A_{j},\partial_{i}A_{j}]-[A_{j},[A_{j},A_{i}]]$

These equations can be rewritten as (4.1) or (4.4).

The initial data sets consist of
$(\bar{A}_{i},\bar{E}_{i})=(A_{i},F_{0i})(t=0)$ for $i=1,2,3$. Note that in
the temporal gauge $F_{0i}=\partial_{t}A_{i}$. Due to the first equation in
(1.2), i.e. (YM) for $\beta=0$, initial data sets have to satisfy the
_constraint_ (or Gauss) _equation_

$\bar{\mathbf{D}}^{\ell}\bar{E}_{\ell}\vcentcolon=\partial^{\ell}\bar{E}_{\ell}+\left[\bar{A}^{\ell},\bar{E}_{\ell}\right]=0.$
(1.3)

The energy of a connection $A_{\alpha}$ at $t$ is defined by

${\bf
E}[F_{\alpha\beta}](t)\vcentcolon=\frac{1}{2}\sum_{\alpha<\beta}\int_{\mathbb{R}^{3}}(F_{\alpha\beta}(t),F_{\alpha\beta}(t))\,\mathrm{d}x$
(1.4)

and it is conserved for sufficiently smooth solutions of (YM).

###  1.1. Prior Yang-Mills results on $\mathbb{R}^{3+1}$

The early works of Segal [46], Choquet-Bruhat and Christodoulou [3], and
Eardley-Moncrief [16] establish local and global well-posedness with high
regularity (e.g. $H^{2}$).

Finite energy global well-posedness for (YM) was first proved by Klainerman
and Machedon in [27]:

######  Theorem 1.1 ([27]).

The Yang-Mills equation in the temporal gauge is globally well-posed for
finite energy initial data which is locally in $H^{1}\times L^{2}$.

The proof involves local Coulomb gauges based on Uhlenbeck’s lemma. Two other
proofs appear in [38] and [40]: one is based on the Yang-Mills heat flow for
gauge selection, the other proof rests on initial data surgery techniques. In
this work we need to combine these three methods together with some new ones.

Going below the energy regularity the first result was due to Tao [50]:

######  Theorem 1.2 ([50]).

Let $\sigma>\frac{3}{4}$. The Yang-Mills equations in the temporal gauge (1.2)
are locally well-posed on $[-1,1]\times\mathbb{R}^{3}$ for sufficiently small
initial data in $H^{\sigma}(\mathbb{R}^{3})\times
H^{\sigma-1}(\mathbb{R}^{3})$ satisfying (1.3).

Removing the smallness assumption by shrinking the time interval is a delicate
matter. This was obtained by Oh and Tătaru in [40] as a consequence of their
initial data excision and extension techniques for the Gauss equation (1.3):

######  Theorem 1.3 ([40]).

Let $\sigma>\frac{3}{4}$. The Yang-Mills equation in the temporal gauge is
locally well-posed for initial data satisfying (1.3) and in
$H^{\sigma}_{loc}(\mathbb{R}^{3})\times H^{\sigma-1}_{loc}(\mathbb{R}^{3})$,
on a sufficiently small time interval.

By a square summability argument one can formulate this result for global
Sobolev spaces as well (see section 3.4):

######  Corollary 1.4 (Local well-posedness in $H^{\sigma}\times
H^{\sigma-1}$).

Let $\sigma>\frac{3}{4}$. Suppose the initial data $(\bar{A}_{x},\bar{E}_{x})$
is in $H^{\sigma}(\mathbb{R}^{3})\times H^{\sigma-1}(\mathbb{R}^{3})$ and
satisfies (1.3). Then the local in time solution from Theorem 1.3 is in
$C_{t}H^{\sigma}\cap C_{t}^{1}H^{\sigma-1}$. For any $t$ in the time interval
the solution map $(\bar{A}_{x},\bar{E}_{x})\mapsto A[t]$ is locally Lipschitz
continuous on $H^{\sigma}\times H^{\sigma-1}$.

Persistence of higher regularity in $H^{\bar{\sigma}}\times
H^{\bar{\sigma}-1}$ holds too ($\bar{\sigma}\geq\sigma$). Solutions solve
(1.2) in the sense of distributions. Note that $[A,\partial
A]\in\dot{H}_{x}^{-1}\subset\mathcal{S}^{*}$ due to (2.21).

We remark that extending Theorem 1.2 (and thus also 1.3\- 1.4) to
$\sigma>\frac{1}{2}$ (which is the critical regularity) as in the case of
Maxwell-Klein-Gordon [36] remains an open question.

###  1.2. Main result

The purpose of this article is to obtain global solutions with regularities
below the energy:

######  Theorem 1.5.

Let $\sigma>\frac{5}{6}$. The Yang-Mills equation in the temporal gauge is
globally well-posed in $H^{\sigma}(\mathbb{R}^{3})\times
H^{\sigma-1}(\mathbb{R}^{3})$: the solutions in Corollary 1.4 extend to
$C_{t}H^{\sigma}\cap
C_{t}^{1}H^{\sigma-1}(\mathbb{R}_{t}\times\mathbb{R}^{3}).$

This result was motivated by [22] which proves a global result for the
Maxwell-Klein-Gordon (MKG) equations in the Coulomb gauge for regularities
$\sigma>\frac{\sqrt{3}}{2}$ . Indeed, results on Yang-Mills usually follow in
the footsteps of similar results for MKG, although the transition can often be
highly non-trivial. The paper [22] uses the I-method which proceeds by
inserting a smoothing Fourier multiplier operator $I=I_{N}$ (see (2.10)) into
the MKG Hamiltonian (Energy) $H$. The main technical step is showing that
$H[I\Phi(t)]$ is ”almost conserved” - i.e. it varies very slowly in time.
Letting $N\to\infty$ allows control of the size of the solution for long
times. See also: [2], [23], [24], [5], [6], [7], [8], [9], [45], [14], [19].
For an introduction to the I-method for the cubic wave equation we refer to
[44].

However, applying the usual I-method directly to (YM) is unlikely to work and
a new approach is needed for the modified energy. We discuss the reasons
below.

To obtain a global in time result for (YM) the (global or local) choices of
gauge are essential. In particular, the energy ${\bf E}[F](t)$ or modified
energies (such as ${\bf E}[IF](t)$) should provide control of (global or
local) Sobolev norms such as $\|\nabla_{t,x}A_{i}\|_{L^{2}}$ or
$\|I\nabla_{t,x}A_{i}\|_{L^{2}}$. We must consider:

  1. (1)

In general, the curvature $F_{ij}$ can control only the curl
$\partial_{i}A_{j}-\partial_{j}A_{i}$ part of $\nabla_{x}A_{i}$. For instance,
in the temporal gauge $F_{0i}=\partial_{t}A_{i}$ and ${\bf
E}[F](t)\approx\|\nabla_{x}A^{df}(t)\|_{L^{2}_{x}}^{2}+\|\partial_{t}A(t)\|_{L^{2}_{x}}^{2}$
so the energy gives no Sobolev control of the curl-free part
$\nabla_{x}A^{cf}$. Setting the Coulomb condition $\text{div}A_{x}=0$ would
make $A^{cf}=0$ automatically. However,

  2. (2)

Global Coulomb gauges cannot be imposed for large data when $G$ is non-abelian
[27], [17]. This is a fundamental difference to the MKG case [26], [22]. Local
Coulomb gauges were used instead in [27] where Uhlenbeck’s lemma provided
control of local $H^{1}$ norms.

  3. (3)

Unless one fixes a global gauge (and does all the analysis in that gauge),
setting the local gauges and patching the solutions together using finite
speed of propagation (in the temporal gauge) requires the gauge invariance of
the energy, [27], [40]. However,

  4. (4)

The classical I-method modified energy ${\bf E}[F[IA]](t)$ or ${\bf E}[IF](t)$
fails to be gauge-invariant or easy to localize, like the original energy.
This is due to the presence of the Fourier multiplier $I$, see (2.10).

To deal with these considerations we propose a modified energy which:

  1. I.

Is gauge invariant and fairly easy to localize, addressing (3),(4)

  2. II.

Addresses issues (1),(2) through Proposition 1.10 (which is an Uhlenbeck-type
lemma below $H^{1}$) providing gauge selections which give control of local
Sobolev norms, see Remark 1.11.

  3. III.

Is smoother in time compared to the classical I-method modified energy111 The
structure of the differentiated energy effectively shifts a derivative from
high to low frequency (Remark 1.14), slightly improving the numerology
compared to the MKG case in [22], allowing lower regularities
$\sigma>\frac{5}{6}$ rather than $\sigma>\frac{\sqrt{3}}{2}$, see Remark 1.14.

Instead of using the Fourier multiplier $I$ in the definition, the same degree
of spatial smoothing will be realized geometrically using the Yang-Mills heat
flow

$F_{si}=\mathbf{D}^{\ell}F_{\ell i}$ (cYMHF)

and

$F_{s\alpha}=\mathbf{D}^{\ell}F_{\ell\alpha}.$ (dYMHF)

introduced by Donaldson [15] and developed extensively by Sung-Jin Oh in [37],
[38] and in [39], [41] for the analysis of finite energy (YM), along with its
dynamic version (dYMHF), together with Oh’s related notion of caloric gauges
for (YM), paralleling the caloric gauges for Wave Maps [53] and Schrödinger
Maps [49], [1] introduced by Tao. The smoothing is realized in a parabolic
way, effectively implementing a covariant Littlewood-Paley decomposition, see
Remark 1.12.

As an alternative to II one could address (1),(2) by opting for global in
space caloric gauges like in [37], [38] instead of local ones. Unfortunately,
that approach requires integration in time which would worsen the numerology
and would also be more technically difficult to implement in the present
context.

If $A_{\alpha}:\mathbb{R}\times\mathbb{R}^{3}\times[0,s_{0}]\to\mathfrak{g}$
is a regular connection, using the bi-invariant inner product from (2.1), we
denote the energies at each $s\in[0,s_{0}]$ by

$\mathcal{E}(t,s)\vcentcolon=\frac{1}{2}\sum_{\alpha<\beta}\int_{\mathbb{R}^{3}}(F_{\alpha\beta},F_{\alpha\beta})(t,s)\,\mathrm{d}x$
(1.5)

######  Definition 1.6 (Modified energy).

Let $N^{2}s_{0}=1$. We define the following _modified energy_

$\mathcal{IE}(t)\vcentcolon=\sup_{s\in[0,s_{0}]}(N^{2}s)^{1-\sigma}\mathcal{E}(t,s)+\int_{0}^{s_{0}}(N^{2}s)^{1-\sigma}\mathcal{E}(t,s)\frac{\,\mathrm{d}s}{s}$
(1.6)

The similarity of $\mathcal{IE}(t)$ with the classical modified energy ${\bf
E}[F[IA]](t)$ is discussed in Remark 1.13 based on the heat flow evolution
relation to Littlewood-Paley theory (Remark 1.12).

The key to the proof of the main result is the following:

######  Proposition 1.7 (Almost conservation law).

Let $\sigma>\frac{5}{6}$. Let $A_{t,x}$ be a global regular solution to (YM)
in the temporal gauge. Suppose $A_{t,x,s}$ solves (dYMHF) on
$\mathbb{R}\times\mathbb{R}^{3}\times[0,s_{0}]$. Let $t_{0}$ be a time such
that

$\mathcal{IE}(t_{0})\leq 2\eta^{2},\qquad\qquad\|A[t_{0}]\|_{H^{\sigma}\times
H^{\sigma-1}}\leq M_{0}$

Then for any $t\in[t_{0},t_{1}]$, $t_{1}=t_{0}+1$ and any $s\in[0,s_{0}]$,
$s_{0}=N^{-2}$ one has

$(N^{2}s)^{1-\sigma}\left|\mathcal{E}(t,s)-\mathcal{E}(t_{0},s)\right|\lesssim\frac{s^{\frac{1}{4}-}}{(Ns^{\frac{1}{2}})^{(1-\sigma)}}\eta^{2}\leq\frac{\eta^{2}}{N^{\frac{1}{2}-}}$
(1.7)

as well as

$\|A[t]\|_{H^{\sigma}\times H^{\sigma-1}}\leq C(M_{0}).$ (1.8)

In particular,
$\left|\mathcal{IE}(t)-\mathcal{IE}(t_{0})\right|\lesssim\eta^{2}/N^{\frac{1}{2}-}$.

The error $1/N^{\frac{1}{2}-}$ corresponds to the condition
$\sigma>\frac{5}{6}$, which appears to be the limit of this method in the
current form.

######  Remark 1.8.

A feature of the proof is that we do not prove the almost-conservation law for
the whole solution at once (this is due to the fact that we only control the
solution in local gauges). Instead, we partition
$\mathcal{E}(t_{1},s)-\mathcal{E}(t_{0},s)$ into local pieces (see (1.31)) and
then we square sum the estimates obtained for the (appropriately extended)
local solutions.

Finally, let

$w_{\alpha}(s)\vcentcolon=\mathbf{D}^{\beta}F_{\alpha\beta}(s)$ (1.9)

be the Yang-Mills tension field, which is a measure of the failure of
$A_{\alpha}(s)$ to satisfy the (YM) equation for $s>0$, since the heat flow
and (YM) do not exactly commute. The gauge transformation (1.1) makes
$w_{\alpha}\mapsto Uw_{\alpha}U^{-1}$.

###  1.3. Overview of the paper and methods

We now outline the key results which play a role in the proof and their
organization.

Section 2 introduces the basic definitions, notations, properties and
preliminaries.

Section 3 is devoted to local well-posedness for (YM). It begins by addressing
$H^{\sigma}\times H^{\sigma-1}$ approximations by regular initial data sets
satisfying (1.3). Then in Prop. 3.2 we discuss square summability for
fractional local Sobolev spaces
$\|u\|_{H^{s}}^{2}\simeq\sum_{j}\|u\|_{W^{s,2}(B_{j})}^{2}$. We continue by
recalling initial data extension results from [40] before reviewing Theorem
1.3 and discussing Corollary 1.4.

Section 4 develops space-time control of temporal solutions in: hyperbolic
$X^{s,b}$ spaces for the divergence free part $A^{df}={\bf P}A$ and in
$X^{r,\theta}_{\tau=0}=H_{x}^{r}H^{\theta}_{t}$ spaces for the curl free part
$A^{cf}={\bf P}^{\perp}A$. As revealed in [50], small solutions can enjoy
$\frac{1}{4}$ more regularity for $A^{cf}$ after a suitable change of gauge.
Locally in time, by Proposition 4.1 one will have $IA^{df}\in
X^{1,\frac{3}{4}+},\ IA^{cf}\in H_{x}^{1+\frac{1}{4}}H^{\frac{1}{2}+}_{t}$
provided one can somehow make the solution small.

Section 5 is reserved for the Yang-Mills heat flow in deTurck’s and caloric
gauges including: well-posedness in $I^{-1}H^{1}$ and $H^{\rho}$, control of
the modified energy (1.6) from the initial data and the change between the two
gauges as in [37] (deTurck’s trick).

A bound on the modified energy $\mathcal{IE}(t)$ allows us to control higher
covariant derivatives of $F_{\alpha\beta}(s)$ in $L^{2}$. This is proved in
Proposition 6.3 using covariant energy estimates for parabolic equations. The
same is true for $w_{x}(s)$, see Section 6. To implement Remark 1.8, we show
that these bounds can be localized while maintaining square summability:

######  Proposition 1.9.

Let $A_{t,x,s}$ be a regular solution to (YM) and (dYMHF) on an interval
$J\times\mathbb{R}^{3}\times[0,s_{0}]$ satisfying $\quad\mathcal{IE}(t)\leq
C\eta^{2}\ll 1\quad\forall\ t\in J.$

Let $(B_{j})_{j\in\mathcal{J}}$ be a finitely overlapping covering of
$\mathbb{R}^{3}$ by balls of radius $\simeq 1$.

Then there exist square summable coefficients $(c_{j})_{j\in\mathcal{J}}$
associated to $(B_{j})_{j\in\mathcal{J}}$ with

$\displaystyle\|(Ns^{\frac{1}{2}})^{1-\sigma}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}F_{\alpha\beta}(t,s)\|_{L^{\infty}_{s}L^{2}_{x}\cap
L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}([0,s_{0}]\times B_{j})}\leq c_{j}$
(1.10)
$\displaystyle\|(N^{2}s)^{1-\sigma}s^{\frac{1}{4}}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}w_{x}(t,s)\|_{L^{\infty}_{s}L^{2}_{x}\cap
L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}([0,s_{0}]\times B_{j})}\leq c_{j}$
(1.11)

holding for all $t\in J$, $j\in\mathcal{J}$, $0\leq m\leq 8$ and

$\sum_{j\in\mathcal{J}}c_{j}^{2}\lesssim\eta^{2}.$ (1.12)

Once these coefficients are defined, section 7 shows that on each ball we can
define local gauges using:

######  Proposition 1.10.

Let $B$ be a ball of radius $\simeq 1$ and $N^{2}s_{0}=1$.

(1) Let $A_{x,s}$ be a smooth solution to (cYMHF) in the caloric gauge
$A_{s}=0$ on $B\times[0,s_{0}]$. There is a sufficiently small $\delta>0$ such
that if the curvature $F_{ij}$ obeys

$\sup_{0\leq m\leq
3}\sup_{s\in[0,s_{0}]}(Ns^{\frac{1}{2}})^{1-\sigma}\|(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}F(s)\|_{L^{2}(B)}\leq\delta$
(1.13)

then there exists a spatial field $U:B\to G$, $\ U=U(x)$such that the
transformation

$\tilde{A}_{i}=UA_{i}U^{-1}-\partial_{i}UU^{-1}$ (1.14)

satisfies $\tilde{A}_{i}(s_{0})\in\delta H^{1}(B)$ and

$\delta^{-1}\tilde{A}_{i}(\cdot,0)\in H^{1}(B)+N^{\sigma-1}H^{\sigma}(B)$
(1.15)

(2) Consider a smooth solution $A_{t,x,s}$ of (dYMHF) on $[t_{0},t_{1}]\times
B\times[0,s_{0}]$ in the temporal-caloric gauge
$A_{0}(t,x,0)=0,A_{s}(t,x,s)=0$ for which the curvature
$F_{\alpha\beta}(t_{0})$ obeys (1.13). Apply Part (1) to $A_{x}(t_{0},\cdot)$
and use $U=U(x)$ to make the transformation (1.14) on $[t_{0},t_{1}]\times B$.
Then in addition to (1.15) one also has

$\delta^{-1}\partial_{t}\tilde{A}_{i}(t_{0},\cdot,0)\in
L^{2}(B)+N^{\sigma-1}H^{\sigma-1}(B)$ (1.16)

######  Remark 1.11.

This Proposition can be thought of as an Uhlenbeck-type lemma for fractional
regularities below $L^{2}$. It essentially says that if
$\|IF\|_{L^{2}(B)}<\delta$ then one can find gauge-equivalent $\tilde{A}\sim
A,\ \tilde{E}\sim E$ for which $\|I\tilde{A}\|_{H^{1}(B)}\lesssim\delta$ and
$\|I\tilde{E}\|_{L^{2}(B)}\lesssim\delta$.

Of course, the multiplier $I$ is not defined on a domain, so the remark above
is simply a heuristic. However, once one has the bounds (1.15), (1.16) it is
possible to use an extension result (Proposition 3.5) to extend $\tilde{A}$ to
the whole space $\mathbb{R}^{3}$ such that $I\tilde{A}\in\delta
H^{1},I\partial_{t}\tilde{A}\in\delta L^{2}$ while still maintaing the Gauss
equation (1.3).

Once we restrict a solution to a ball $B$ and perform the extension procedure,
one obtains two heat flows: one for the original solution and one for the new
one, which coincide at $s=0$ on $B$. The need arises to compare them locally
for parabolic times $s>0$. Due to infinite speed of parabolic propagation,
they will not coincide, but one can obtain difference bounds of type
$\|\chi[F(s)-F^{\prime}(s)]\|_{L^{2}(B)}\ll s^{M}$. This is proved by
Proposition 8.1 to which Section 8 is devoted.

Eventually, the result is reduced to the space-time bound (1.32) which
represents a trilinear estimate proved in Proposition 10.1. Building upon the
decompositions and estimates from Section 9 (in particular the main Fourier
bilinear component of $w$ identified in [39]), the leading term of the
trilinear integral is revealed in Section 10 to consist of a null form
interaction with favorable frequency weights arising from the ”heat-wave
commutator” $w$, see (1.40) and Remark 1.14.

In the Appendix we formulate the invariant modified energy for Maxwell-Klein-
Gordon and show how it can be used to increase the range of regularities for
which global well-posedness is known to hold, from $\sigma>\frac{\sqrt{3}}{2}$
in [22] to $\sigma>\frac{5}{6}$.

###  1.4. Reduction of Theorem 1.5 to the almost conservation law Prop. 1.7

Consider initial data sets of size
$\|(\bar{A}_{x},\bar{E}_{x})\|_{H^{\sigma}\times H^{\sigma-1}}\leq M$. Fix an
arbitrary time $T>0$. By the well-posedness statement in Corollary 1.4, it
suffices to prove

$\|A[T]\|_{H^{\sigma}\times H^{\sigma-1}}\leq C_{M,T}<\infty$ (1.17)

for all such solutions which exist on $[0,T]$. Applying Corollary 1.4 again
together with the approximation statement in Proposition 3.1 implies that it
suffices to show (1.17) for regular solutions $A\in
C^{\infty}_{t}H^{\infty}_{x}(\mathbb{R}\times\mathbb{R}^{3})$, which are known
to be global.

We will choose a large number $N=N(M,T)$ to be specified later, with
$T\ll_{M}N^{0+}$ as usual with the I-method. Rescale the solution by
$A^{\lambda}(t,x)\vcentcolon=\lambda^{-1}A(t/\lambda,x/\lambda)$ where we
choose $\lambda,N$ such that the rescaled initial data satisfies

$\|(\bar{A}^{\lambda},\bar{E}^{\lambda})\|_{\dot{H}^{\sigma}\times\dot{H}^{\sigma-1}}\lesssim\lambda^{\frac{1}{2}-\sigma}M\ll\frac{\eta}{N^{1-\sigma}},\qquad\|\bar{A}^{\lambda}\|_{\dot{H}^{\frac{1}{2}+}}\lesssim\lambda^{0-}M\ll
1,$ (1.18)

so $\lambda,N$ are related by

$\lambda=N^{\frac{1-\sigma}{\sigma-\frac{1}{2}}}\left(\frac{CM}{\eta}\right)^{\frac{1}{\sigma-\frac{1}{2}}}$
(1.19)

and $\eta\ll 1$ is a constant chosen small enough to overpower a number of
universal constants in different estimates. Now we need to obtain a bound on
$A^{\lambda}[\lambda T]$.

We extend $A^{\lambda}_{t,x}(t,x)$ to $A^{\lambda}_{t,x,s}(t,x,s)$ by the heat
flow using Theorem 5.9. We obtain a regular solution $A^{\lambda}\in
C^{\infty}_{t,s}H^{\infty}_{x}([0,\lambda T]\times\mathbb{R}^{3}\times[0,1])$
to (dYMHF) in the caloric gauge $A^{\lambda}_{s}=0$, i.e.

$\partial_{s}A_{\alpha}^{\lambda}=D^{\ell}F_{\ell\alpha}^{\lambda},\qquad
A_{\alpha}^{\lambda}(t,x,s=0)=A_{\alpha}^{\lambda}(t,x)$

Let $s_{0}=N^{-2}$. Denote the energies (1.5) determined by
$A_{\alpha}^{\lambda}$ at $(t,s)$ by $\mathcal{E}(t,s)$ and consider the
modified energy (1.6). Begin with the initial data:

By Corollary 5.7 with $\varepsilon\ll\eta/N^{1-\sigma}$ and (1.18) we obtain
$\mathcal{IE}(0)\leq\eta^{2}$.

From this and $O(\lambda T)$ application of Proposition 1.7 we obtain
$\mathcal{IE}(t)\leq 2\eta^{2}$ for all $t\in[0,\lambda T]$ and
$\|A^{\lambda}[\lambda T]\|_{H^{\sigma}\times H^{\sigma-1}}<\infty$ provided
that

$\lambda T\ll N^{\frac{1}{2}-},\quad\text{i.e.}\quad
T\ll_{M}N^{\frac{1}{2}-\frac{1-\sigma}{\sigma-1/2}-}=N^{0+}.$ (1.20)

It is seen that it is possible to choose $N,\lambda$ satisfying (1.18),
(1.19), (1.20) if $\sigma>\frac{5}{6}$. Undoing the scaling we obtain (1.17).

###  1.5. Proof of the almost conservation Proposition 1.7

By a continuity argument one can assume that $\mathcal{IE}(t)\leq 3\eta^{2}$
holds for all $t\in[t_{0},t_{1}]$.

Without loss of generality we prove (1.7) for $t=t_{1}$.

By Theorem 5.9 we may assume the caloric gauge condition $A_{s}(t,x,s)=0$.

We note that by using Bianchi’s identity (2.3) and the Leibniz rule (2.2) one
can write the differentiated energy at $s$ as

$\frac{d}{dt}\mathcal{E}(t,s)=\sum_{\ell=1}^{3}\int_{\mathbb{R}^{3}}(w_{\ell},F_{0\ell})(s)\,\mathrm{d}x$

where $w_{\ell}$ is given by (1.9) and therefore

$\mathcal{E}(t_{1},s)-\mathcal{E}(t_{0},s)=\sum_{\ell=1}^{3}\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}(w_{\ell}(s),F_{0\ell}(s))\,\mathrm{d}x\,\mathrm{d}t$
(1.21)

The strategy we use consists of localizing this integral and estimating it in
appropriate local gauges, crucially using the fact that (1.21) is gauge-
invariant.

Let $(B_{j}^{(1)})_{j\in\mathcal{J}}$ be a finitely overlapping covering of
$\mathbb{R}^{3}$ with balls of radius $1$ centered at
$(y_{j})_{j\in\mathcal{J}}$. Let $(\chi_{j})_{j\in\mathcal{J}}$ be a spatial
partition of unity with ${\mathrm{supp}}\chi_{j}\subset B_{j}^{(1)}$.

Let $(B_{j})_{j\in\mathcal{J}}$ consist of balls with same centers
$(y_{j})_{j\in\mathcal{J}}$ but radius $2$, which are also finitely
overlapping. Let $(\mathcal{D}_{j})_{j\in\mathcal{J}}$ be domains of
dependency with bases $(B_{j})_{j\in\mathcal{J}}$, which are truncated cones

$\mathcal{D}_{j}=\\{(t,x)\ |\ t-t_{0}+\left|x-y_{j}\right|<2,\
t\in[t_{0},t_{1}]\\}$

which form a covering of $[t_{0},t_{1}]\times\mathbb{R}^{3}$. Note that
$[t_{0},t_{1}]\times B_{j}^{(1)}\subset\mathcal{D}_{j}$.

By Proposition 1.9 we define square summable coefficients
$(c_{j})_{j\in\mathcal{J}}$ associated to $(B_{j})_{j\in\mathcal{J}}$ such
that (1.10)-(1.11) and (1.12) hold.

For each $j\in\mathcal{J}$ we invoke Propositions 1.10 on $B_{j}$ with
$\delta=c_{j}$ which provide spatial $U^{(j)}:B_{j}\to G$ and corresponding
transformations on $[t_{0},t_{1}]\times B_{j}\times[0,s_{0}]$:

$A_{\alpha}^{(j)}\vcentcolon=U^{(j)}A_{\alpha}U^{(j)-1}-\partial_{\alpha}U^{(j)}U^{(j)-1},\qquad
F_{\alpha\beta}^{(j)}=U^{(j)}F_{\alpha\beta}U^{(j)-1}$ (1.22)

with

$\|A^{(j)}(t_{0},\cdot,s_{0})\|_{H^{1}(B_{j})}\lesssim c_{j}$ (1.23)

and

$\|A^{(j)}(t_{0},\cdot,0)\|_{H^{1}(B_{j})+N^{\sigma-1}H^{\sigma}(B_{j})}+\|\partial_{t}A^{(j)}(t_{0},\cdot,0)\|_{L^{2}(B_{j})+N^{\sigma-1}H^{\sigma-1}(B_{j})}\lesssim
c_{j}$

Since the $U^{(j)}(x)$ are independent of $t$ and $s$, the $A_{i}^{(j)}$’s
remain smooth solutions of (YM) and (dYMHF) on $[t_{0},t_{1}]\times
B_{j}\times[0,s_{0}]$ in the temporal and caloric gauges
$A_{0}^{(j)}(t,x,0)=0,\ A_{s}^{(j)}(t,x,s)=0$. Due to gauge-invariance the
bounds (1.10)-(1.11) hold for $F_{\alpha\beta}^{(j)},w^{(j)}$ too (replacing
$c_{j}$ by $Cc_{j}$ if needed) and, moreover

$\chi_{j}(w_{\ell}(s),F_{0\ell}(s))=\chi_{j}(w^{(j)}_{\ell}(s),F^{(j)}_{0\ell}(s)).$
(1.24)

We now use Proposition 3.5 to extend
$(A^{(j)}(t_{0},\cdot,0),\partial_{t}A^{(j)}(t_{0},\cdot,0))$ from $B_{j}$ to
a regular $(\bar{a}^{(j)},\bar{e}^{(j)})$ on $\mathbb{R}^{3}$ in such a way
that the constrain equation (1.3) holds and we have the bound (see Lemma 2.1)

$\|(I\bar{a}^{(j)},I\bar{e}^{(j)})\|_{H^{1}\times
L^{2}}\simeq\|\bar{a}^{(j)}\|_{H^{1}+N^{\sigma-1}H^{\sigma}}+\|\bar{e}^{(j)}\|_{L^{2}+N^{\sigma-1}H^{\sigma-1}}\lesssim
c_{j}$

Let $\bar{A}^{(j)}$ be the regular Yang-Mills solution in the temporal gauge
on $[t_{0},t_{1}]\times\mathbb{R}^{3}$ with initial data
$(\bar{A}^{(j)}(t_{0}),\partial_{t}\bar{A}^{(j)}(t_{0}))=(\bar{a}^{(j)},\bar{e}^{(j)})$.
By Proposition 4.1 Part (1):

$\|I\bar{A}^{(j)}[t]\|_{L^{\infty}_{t}(H^{1}\times
L^{2})([t_{0},t_{1}]\times\mathbb{R}^{3})}\lesssim c_{j}$ (1.25)

Due to finite speed of propagation we have that

$A^{(j)}(\cdot,\cdot,0)_{\mkern 1.0mu\vrule height=6.02777pt\mkern
2.0mu\mathcal{D}_{j}}=\bar{A}^{(j)}_{\mkern 1.0mu\vrule height=6.02777pt\mkern
2.0mu\mathcal{D}_{j}}$ (1.26)

From this and (1.25) one has

$\|A^{(j)}[t_{1}]_{\mkern 1.0mu\vrule height=6.02777pt\mkern
2.0mus=0}\|_{(H^{1}+N^{\sigma-1}H^{\sigma})(B_{j}^{(1)})\times(L^{2}+N^{\sigma-1}H^{\sigma-1})(B_{j}^{(1)})}\lesssim
c_{j}$ (1.27)

We now extend $\bar{A}^{(j)}$ from $[t_{0},t_{1}]\times\mathbb{R}^{3}$ to
$[t_{0},t_{1}]\times\mathbb{R}^{3}\times[0,s_{0}]$ as the solution of the
(dYMHF) in the caloric gauge $\bar{A}^{(j)}_{s}=0$ given by Theorem 5.9. By
Proposition 5.10 and Lemma 2.1

$\|\bar{A}^{(j)}_{i}\|_{L^{\infty}_{t}L^{\infty}_{s}(H^{1}+N^{\sigma-1}H^{\sigma})}\lesssim
c_{j}$ (1.28)

From Corollary 5.2 we control the modified energy of $\bar{A}^{(j)}$ as
$\mathcal{IE}[\bar{F}_{\alpha\beta}^{(j)}](t)\lesssim c_{j}^{2}$ for all
$t\in[t_{0},t_{1}]$. From Propositions 6.3 and 6.4 we control higher
derivatives of $\bar{F}^{(j)}$ and $\bar{w}^{(j)}$, which restricted to the
ball $B_{j}$ provide (1.10)-(1.11) for $\bar{F}^{(j)}$ and $\bar{w}^{(j)}$ as
well. These bounds together with (1.28) and (1.26) allows us to apply
Proposition 8.1 to conclude, for all $s\in[0,s_{0}]$, that

$\displaystyle(Ns^{\frac{1}{2}})^{9(1-\sigma)}\|\chi_{j}[F^{(j)}(t,s)-\bar{F}^{(j)}(t,s)]\|_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\lesssim s^{2}c_{j}$ (1.29)
$\displaystyle(Ns^{\frac{1}{2}})^{8(1-\sigma)}\|\chi_{j}[w_{\ell}^{(j)}(t,s)-\bar{w}_{\ell}^{(j)}(t,s)]\|_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\lesssim s^{\frac{5}{4}}c_{j}$ (1.30)

By (1.21) and (1.24)

$\mathcal{E}(t_{1},s)-\mathcal{E}(t_{0},s)=\sum_{j\in\mathcal{J}}\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\chi_{j}(x)(w^{(j)\ell}(s),F^{(j)}_{0\ell}(s))\,\mathrm{d}x\,\mathrm{d}t$
(1.31)

Using (1.29), (1.30) and (1.10), (1.11) with $m=0$ for $\bar{F}^{(j)}$ and
$w^{(j)}$ to bound

$\chi_{j}(w^{(j)}_{\ell},F^{(j)}_{0\ell}-\bar{F}^{(j)}_{0\ell})(s)\qquad\text{and}\qquad\chi_{j}(w^{(j)}_{\ell}-\bar{w}^{(j)}_{\ell},\bar{F}^{(j)}_{0\ell})(s)$

in $L^{1}_{t}L^{1}_{x}$, together with (1.12) allows us to reduce (1.7) to
summing

$(N^{2}s)^{1-\sigma}\left|\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\chi_{j}(x)(\bar{w}^{(j)\ell}(s),\bar{F}^{(j)}_{0\ell}(s))\,\mathrm{d}x\,\mathrm{d}t\right|\lesssim\frac{s^{\frac{1}{4}-}}{(Ns^{\frac{1}{2}})^{(1-\sigma)}}c_{j}^{2}$
(1.32)

We now use Proposition 4.1 Part (2) to place $\bar{A}^{(j)}(t_{0},\cdot,0)$ in
the Coulomb gauge (only at $t=t_{0}$), as this results in the improved
regularity from (4.8) for the curl-free part. We make this transformation on
$[t_{0},t_{1}]\times\mathbb{R}^{3}\times[0,s_{0}]$ using a spatial
$\bar{U}^{(j)}(x)$, so we maintain a regular caloric solution which at $s=0$
is temporal, but now at $s=0$ satisfying (4.7), (4.8) with
$\varepsilon=c_{j}$. Finally we make another change of gauge (5.29) to
deTurck’s gauge using Proposition 5.10 and Remark 5.11 which leaves the
solution invariant at $s=0$. Due to gauge invariance (1.32) now follows from
Proposition 10.1.

####  1.5.1. Proof of the Sobolev bound (1.8)

We apply Proposition 5.12 to obtain
$\|A_{i}(t_{0},\cdot,s_{0})\|_{H^{\sigma}}\leq C_{0}(M_{0})$, as we have
(5.30) from Proposition 6.3. Without loss of generality we prove (1.8) for
$t=t_{1}$.

Applying Lemma 2.3 for (1.22) at $s=s_{0},\ t=t_{0}$, using (1.23) we obtain

$\|\nabla U^{(j)}\|_{H^{\sigma}(B_{j})}\lesssim d_{j}(1+C_{0}(M_{0}));\qquad
U^{(j)},U^{(j)-1}\in C_{1}(M_{0})X_{\sigma}(B_{j})$ (1.33)

where we denote

$d_{j}=c_{j}+\|A_{i}(t_{0},\cdot,s_{0})\|_{H^{\sigma}(B_{j})},\qquad\sum_{j}d_{j}^{2}\lesssim
C_{0}(M_{0})^{2}+1$

The last bound follows from Proposition 3.2. From (1.27) we have

$\|A^{(j)}(t_{1})\|_{H^{\sigma}(B_{j}^{(1)})}+\|\partial_{t}A^{(j)}(t_{1})\|_{H^{\sigma-1}(B_{j}^{(1)})}\lesssim
c_{j}$

Combining this with (1.33), from (2.26) and (1.22) at $s=0,\ t=t_{1}$ we
obtain

$\|A(t_{1})\|_{H^{\sigma}(B_{j}^{(1)})}+\|\partial_{t}A(t_{1})\|_{H^{\sigma-1}(B_{j}^{(1)})}\lesssim_{M_{0}}c_{j}+d_{j}.$

Applying Proposition 3.2 again we conclude $\|A[t_{1}]\|_{H^{\sigma}\times
H^{\sigma-1}}\leq C(M_{0})<\infty$. ∎

###  1.6. Heuristic remarks and other works

######  Remark 1.12 (Smoothing heuristics and connection to Littlewood-Paley
theory).

The heat flows solutions to (cYMHF), (dYMHF) satisfy the covariant parabolic
equations (6.1). Therefore, in the first approximation one can think of the
nonlinear $F_{\alpha\beta}(s)$ as roughly $F_{\alpha\beta}(s)\approx
e^{s\Delta}F_{\alpha\beta}(0)$. Expanding (cYMHF) it is seen that in general,
the evolution of $A_{i}(s)$ is only degenerately parabolic. With an
appropriate choice of gauge (De Turck $A_{s}=\partial^{\ell}A_{\ell}$) one can
make the $A_{i}$ equations genuinely parabolic, specifically (5.1). Then the
approximation $A_{i}(s)\approx e^{s\Delta}A_{i}(0)$ is still expected.

The Gaussian multipliers $e^{s\Delta}$ make the bulk of the frequencies to
become concentrated to $\left|\xi\right|\lesssim s^{-\frac{1}{2}}$. In
connection to Littlewood-Paley projections this can be stated as
$e^{s\Delta}\approx P_{\leq k(s)}$ and $\partial_{x}e^{s\Delta}\approx
2^{k(s)}P_{k(s)}$ where $2^{k(s)}=s^{-\frac{1}{2}}$. Since $F_{\alpha\beta}$
contains derivatives, in a certain gauge one heuristically has $F(s)\approx
P_{k(s)}F(0)$.

Now to decompose $A_{i}(0)$ one can set the caloric condition $A_{s}=0$,
turning (cYMHF) into $\partial_{s}A_{i}=\mathbf{D}^{\ell}F_{\ell i}$, which
integrates to

$A_{i}(s)=A_{i}(s_{0})+\int_{s}^{s_{0}}s\mathbf{D}^{\ell}F_{i\ell}\frac{\,\mathrm{d}s}{s}$
(1.34)

Since $\,\mathrm{d}k\simeq-\,\mathrm{d}s/s$, $2^{2k}s=1$ and taking
$s_{0}=N^{-2}$ one can interpret (1.34) as a continuous Littlewood-Paley
decomposition

$A(0)\approx P_{\leq N}A(0)+\int_{\log
N}^{\infty}2^{-k}P_{k}F(0)\,\mathrm{d}k,\qquad P_{k}A_{i}(0)\approx
s\mathbf{D}^{\ell}F_{i\ell}(s)$ (1.35)

######  Remark 1.13 (Comparison of $\mathcal{IE}(t)$ with the classical
I-method modified energy).

The I-method suggests applying the Fourier multiplier $I$ inside the energy
and trying to prove almost conservation. This is explained in detail in [22]
for the closely related MKG and in [44] for the simpler cubic wave equation.

The $I$ operator is defined in (2.10). Roughly speaking

${\bf E}[F[IA]](t)\approx{\bf
E}[IF](t)\approx\|P_{<N}F\|_{L^{2}}^{2}+N^{2(1-\sigma)}\sum_{2^{k}\geq
N}\|\left|D\right|^{\sigma-1}P_{k}F\|_{L^{2}}^{2}$ (1.36)

Recall the definition of $\mathcal{IE}(t)$ and $\mathcal{E}(t,s)$ in (1.6),
(1.5). In light of Remark 1.12,

$\mathcal{E}(t,s_{0})\approx\|P_{<N}F\|_{L^{2}}^{2},\quad\int_{0}^{s_{0}}(N^{2}s)^{1-\sigma}\mathcal{E}(t,s)\frac{\,\mathrm{d}s}{s}\approx\sum_{2^{k}\geq
N}\Big{(}\frac{N}{2^{k}}\Big{)}^{2(1-\sigma)}\|P_{k}F\|_{L^{2}}^{2}$ (1.37)

where we denote $F=F_{\alpha\beta}(0)$. Since $s^{\frac{1}{2}}\approx
2^{-k}\approx\left|D\right|^{-1}$ the similarity between (1.36) and (1.37) is
clear, suggesting $\mathcal{IE}(t)$ could serve as a substitute to ${\bf
E}[F[IA]](t)$.

######  Remark 1.14 (Comparison of almost conservation laws).

We have claimed above in III that $\mathcal{IE}(t)$ enjoys better almost
conservation properties than ${\bf E}[F[IA]](t)$ or $H[I\Phi](t)$ \- the
latter being the Maxwell-Klein-Gordon (MKG) classical I-method modified energy
from [22] ($\Phi=(A_{x},\phi)$), while the former would be the YM I-method
modified energy in the Coulomb gauge (if it could be implemented for large
data). We consider ${\bf E}[F[IA]](t)$ and $H[I\Phi](t)$ as essentially the
same, since we can ignore the $[A_{i},A_{j}]$ commutators because they have
better regularity.

The dominant part of $H[I\Phi](t_{1})-H[I\Phi](t_{0})$ is proved in [22,
(106),(124)] to be

$\sum_{k,k_{i}}\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}\partial_{t}I\Phi_{k},I{\bf
N}(\Phi_{k_{1}},\Phi_{k_{2}})-{\bf
N}(I\Phi_{k_{1}},I\Phi_{k_{2}})\big{)}\,\mathrm{d}x\,\mathrm{d}t$ (1.38)

where ${\bf N}$ is a null form defined in (2.8), (2.9). Consider the more
difficult case $N<2^{k_{1}}<2^{k_{2}}\simeq 2^{k}$, in which the commutator in
(1.38) does not provide any cancelation and can be effectively written as

$\frac{1}{2^{k_{\min}+k_{\max}}}\left(\frac{N}{2^{k}}\right)^{2(1-\sigma)}\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}\partial_{t}\Phi_{k},{\bf
N}(\nabla_{x,t}\Phi_{k_{1}},\nabla_{x,t}\Phi_{k_{2}})\big{)}\,\mathrm{d}x\,\mathrm{d}t$
(1.39)

On the other hand, $\mathcal{E}(t_{1},s)-\mathcal{E}(t_{0},s)$ can be written
as (1.21), of which (10.5) is the leading part. The similarity to (1.38)
begins by noting that $w$ is also an essentially bilinear term with some
cancelations, representing a ”heat-wave commutator”.

In (10.14), (10.15) it is revealed that the main contribution to
$\mathcal{IE}(t_{1})-\mathcal{IE}(t_{0})$ comes roughly from terms

$(N^{2}s)^{1-\sigma}\sum_{k,k_{i}}\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}\partial_{t}A_{k}^{df},{\bf
N}(\partial_{t}A^{df}_{k_{1}},\partial_{t}A^{df}_{k_{2}})\big{)}\,\mathrm{d}x\,\mathrm{d}t\times
weight$ (1.40)

The weight in (1.40) is from (10.15) and in the same case
$N<2^{k_{1}}<2^{k_{2}}\simeq 2^{k}$ it becomes concentrated at $2^{k}\simeq
s^{-\frac{1}{2}}$ turning (1.40) into (with the notation $\Phi=A^{df}$ )

$\frac{1}{2^{2k_{\max}}}\left(\frac{N}{2^{k}}\right)^{2(1-\sigma)}\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{3}}\big{(}\partial_{t}\Phi_{k},{\bf
N}(\nabla_{x,t}\Phi_{k_{1}},\nabla_{x,t}\Phi_{k_{2}})\big{)}\,\mathrm{d}x\,\mathrm{d}t$
(1.41)

Comparing (1.41) with (1.38)-(1.39) shows that the structure of
$\mathcal{IE}(t_{1})-\mathcal{IE}(t_{0})$ effectively shifts a derivative from
a high frequency to low frequency, compared to
$H[I\Phi](t_{1})-H[I\Phi](t_{0})$. This translates into a bound of
$O(N^{-\frac{1}{2}})$ in (1.7) compared to $O(N^{\frac{1}{2}-\sigma})$ in [22,
(45)], leading to a result for lower regularities $\sigma>\frac{5}{6}$ rather
than $\sigma>\frac{\sqrt{3}}{2}$.

####  1.6.1. Other works

Local well-posedness in the Lorenz gauge is given in [48], [55]. Global
existence for critical power Yang-Mills-Higgs equations at the energy
regularity is proved in [21]. Global existence of solutions on globally
hyperbolic Lorentzian manifolds, under a higher regularity assumption, is
considered in [4] following the classical approach of [16].

We now mention notable results in $4+1$-dimensions, which is the energy-
critical case, beginning with [31]. The global result for small energy was
established in [33] building upon the high-dimensional case in [32]. The large
data theory culminated with the proof of the Threshold Conjecture in [42].

##  2\. Preliminaries

####  2.0.1. Space-time

We will work on the Minkowski space $\mathbb{R}^{1+3}$ equipped with the
Minkowski metric, according to which indices will be raised and lowered. We
adopt the Einstein summation convention. Greek letters run over the space-time
variables, while latin indices are used only for spatial variables.

We use $A[t]$ as a short-hand for the pair $(A(t),\partial_{t}A(t))$.

####  2.0.2. Lie theory

Let $G$ be a Lie group and $\mathfrak{g}$ its associated Lie algebra. We
assume the existence of a bi-invariant inner product
$(\cdot,\cdot):\mathfrak{g}\times\mathfrak{g}\to[0,\infty)$. This means

$([A,B],C)=(A,[B,C])\quad\text{or
equivalently},\quad(UAU^{-1},UBU^{-1})=(A,B)$ (2.1)

for all $A,B,C\in\mathfrak{g}$, $U\in G$. The Leibniz rule

$\partial_{\alpha}(B,C)=\left(\mathbf{D}_{\alpha}B,C\right)+\left(B,\mathbf{D}_{\alpha}C\right)$
(2.2)

holds, as well as Bianchi’s identity

$\mathbf{D}_{\alpha}F_{\beta\gamma}+\mathbf{D}_{\beta}F_{\gamma\alpha}+\mathbf{D}_{\gamma}F_{\alpha\beta}=0.$
(2.3)

Covariant derivatives are commuted using

$\mathbf{D}_{\alpha}\mathbf{D}_{\beta}-\mathbf{D}_{\beta}\mathbf{D}_{\alpha}=[F_{\alpha\beta},\cdot]$
(2.4)

The covariant laplacian is denoted by
$\Delta_{A}\vcentcolon=\mathbf{D}^{\ell}\mathbf{D}_{\ell}$.

Due to (2.1), the inner-product can be used to define the $L^{p}$ norms of
$\mathfrak{g}$ -valued functions in a gauge-invariant way.

For concreteness, it is common to take $G$ to be a matrix group such as
$SO(n,\mathbb{R})$ or $SU(n,\mathbb{C})$, with the associated Lie Algebras
$\mathfrak{so}(n,\mathbb{R})$ and $\mathfrak{su}(n)$. In these cases the Lie
bracket is $[A,B]=AB-BA$ given by matrix multiplication, while the bi-
invariant scalar products are $(A,B)=\textrm{tr}(AB^{T})$, respectively
$(A,B)=\textrm{tr}(AB^{*})$.

####  2.0.3. Regular functions

A function $f$ on $\mathbb{R}^{3}$ (such as an initial data $\bar{A}_{i}$) is
called _regular_ if $f\in H^{\infty}(\mathbb{R}^{3})\vcentcolon=\bigcap_{n\geq
0}H^{n}(\mathbb{R}^{3})$. Solutions $A_{\alpha}(t,x)$ or $A_{i}(x,s)$ defined
on intervals are called regular if $A_{\alpha}\in
C^{\infty}_{t}H^{\infty}_{x}(I\times\mathbb{R}^{3})$ , respectively $A_{i}\in
C^{\infty}_{s}H^{\infty}_{x}(\mathbb{R}^{3}\times J)$. Similarly for
$A_{\alpha}\in C^{\infty}_{t,s}H^{\infty}_{x}(I\times\mathbb{R}^{3}\times J)$.
Gauge transformations are called regular if $\nabla U,\nabla U^{-1}\in
H^{\infty}(\mathbb{R}^{3})$.

####  2.0.4. Boundedness notations

We use $A\lesssim B$ to denote $A\leq CB$ and $A\ll B$ to denote $AC\leq B$,
where $C$ is a large constant. We use $A+$ or $A-$ to denote $A+\varepsilon$
and $A-\varepsilon$ where $0<\varepsilon\ll 1$ is a large number which can
depend on $\sigma>0$. We denote $\left\langle
x\right\rangle^{2}=1+\left|x\right|^{2}$.

For injective operators $T$ on $X$ we denote by $TX$ the image space with norm
$\|T^{-1}u\|_{X}$. The embedding $X\subseteq Y$ as an estimate denotes
$\|u\|_{Y}\lesssim\|u\|_{X}$. The mapping $T:X\to Y$ as an estimate means
$\|Tu\|_{Y}\lesssim\|u\|_{X}$. We say $u\in\delta X$ if
$\|u\|_{X}\lesssim\delta$. Product estimates $X\times Y\to Z$ denote
$\|uv\|_{Z}\lesssim\|u\|_{X}\|v\|_{Y}$. The space of sums $X+Y$ is endowed
with the norm $\|u\|_{X+Y}\vcentcolon=\inf\\{\|u_{1}\|_{X}+\|u_{2}\|_{Y}\ |\
u=u_{1}+u_{2}\\}$.

####  2.0.5. Fourier operators

The Fourier transform of $f$ is denoted by $\hat{f}$ or $\mathcal{F}f$.
Littlewood-Paley projections are denoted by $P_{k}$. Whether the frequencies
$\left|\xi\right|\lesssim 1$ are included in $P_{0}$ or not will be clear from
the context. Moreover, $\tilde{P}_{k}$ denotes a similar multiplier such that
$\tilde{P}_{k}P_{k}=P_{k}$.

Linear and bilinear operators which are invariant to spatial translations are
understood either through their spatial kernels, or as multipliers through
their Fourier symbols. For instance: the smoothing $I$ operator is defined by
(2.10), fractional derivative operators $\left|D\right|^{p}$, $\left\langle
D\right\rangle^{p}$ are defined through their symbols $\left|\xi\right|^{p}$,
$\left\langle\xi\right\rangle^{p}$.

When the fields $A,B$ are $\mathfrak{g}$-valued a bilinear operator with
symbol $m^{ij}(\xi_{1},\xi_{2})$ takes the form

$\displaystyle\mathcal{M}(A,B)(x)$
$\displaystyle=\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}e^{ix\cdot(\xi_{1}+\xi_{2})}m^{ij}(\xi_{1},\xi_{2})[\hat{A}_{i}(\xi_{1}),\hat{B}_{j}(\xi_{2})]\,\mathrm{d}\xi_{1}\,\mathrm{d}\xi_{2}$
(2.5)
$\displaystyle=\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}K^{ij}(x-y_{1},x-y_{2})[A_{i}(y_{1}),B_{j}(y_{2})]\,\mathrm{d}y_{1}\,\mathrm{d}y_{2}$

where $m(\xi_{1},\xi_{2})=\hat{K}(\xi_{1},\xi_{2})$.

A linear or bilinear multiplier operator is called disposable when its kernel
is a function or measure with bounded mass. Minkowski’s inequality insures
that disposable operators are bounded on translation-invariant normed spaces.

####  2.0.6. Leray projections and null forms

Denote by ${\bf P}$ and ${\bf P}^{\perp}$ the projections on divergence free,
respectively curl-free vector fields:

${\bf
P}_{j}A\vcentcolon=\Delta^{-1}\partial^{\ell}(\partial_{\ell}A_{j}-\partial_{j}A_{\ell}),\quad{\bf
P}_{j}^{\perp}A\vcentcolon=(1-{\bf
P}_{j})A=\Delta^{-1}\partial_{j}\text{div}A.$ (2.6)

Recall the classical identities ([26], [27])

$\displaystyle{\bf
P}_{j}\left(\phi\nabla_{x}\varphi\right)=\Delta^{-1}\nabla^{i}Q_{ij}\left(\phi,\varphi\right)$
(2.7) $\displaystyle{\bf
P}A^{i}\partial_{i}\phi=Q_{ij}\left(\Delta^{-1}\nabla^{i}A^{j},\phi\right)$

which follow from the definitions by simple computations.

By ${\bf N}(f,g)$ we denote a null form, meaning a linear combination of

$\Delta^{-1}\nabla^{i}Q_{ij}(f,g),\quad\text{and}\quad
Q_{ij}(\Delta^{-1}\nabla^{i}f,g)\quad\text{or}\quad$ (2.8)

where

$Q_{ij}(f,g)\vcentcolon=\partial_{i}f\partial_{j}g-\partial_{j}f\partial_{i}g$
(2.9)

In the context of $\mathfrak{g}$-valued fields (2.9) is replaced by (4.3).

####  2.0.7. The $I$ multiplier

Fix a number $N\gg 1$ and let $m(\xi)$ and $I$ be a smooth positive radial
symbol and the associated Fourier multiplier operator such that

$m(\xi)=\begin{cases}1,\qquad\qquad\left|\xi\right|\leq N\\\
\big{(}\frac{N}{\left|\xi\right|}\big{)}^{1-\sigma},\quad\left|\xi\right|\geq
2N\end{cases};\qquad\widehat{If}(\xi)\vcentcolon=m(\xi)\hat{f}(\xi)$ (2.10)

The $I$ operator smoothens the high frequencies and acts as the identity on
relatively lower frequencies. It has an integrable kernel. It is
straightforward to verify

######  Lemma 2.1.

One has $\quad\ If\in H^{1}\ \iff\ f\in H^{1}+N^{\sigma-1}H^{\sigma}\quad$ and

$\qquad\qquad\quad Ig\in L^{2}\ \ \iff\ \ g\in
L^{2}+N^{\sigma-1}H^{\sigma-1}=L^{2}+N^{\sigma-1}\dot{H}^{\sigma-1}$

with equivalent norms.

The advantage of these second formulations is that they can be directly
localized to a domain.

####  2.0.8. Covariant Gagliardo-Nirenberg inequalities and elliptic equations

We recall the following _diamagnetic_ , or _Kato’s_ , _inequality_. For a
proof see, for example [38, Lemma 4.2] or [54, Lemma 3.9]. One has

$\left|\partial_{x}\left|\varphi\right|\right|\leq\left|\mathbf{D}_{x}\varphi\right|$

for regular $\mathfrak{g}$-valued functions $\varphi$ on $\mathbb{R}^{3}$. As
a consequence one has the following covariant Gagliardo-Nirenberg and Sobolev
inequalities on $\mathbb{R}^{3}$

$\displaystyle\|\varphi\|_{L^{p}}\lesssim\|\varphi\|_{L^{2}}^{1-\alpha}\left\|\mathbf{D}_{x}\varphi\right\|_{L^{2}}^{\alpha},\qquad\alpha=3(2^{-1}-p^{-1})$
(2.11)
$\displaystyle\|\varphi\|_{L^{3}}\lesssim\|\varphi\|_{L^{2}}^{\frac{1}{2}}\left\|\mathbf{D}_{x}\varphi\right\|_{L^{2}}^{\frac{1}{2}}$
$\displaystyle\|\varphi\|_{L^{6}}\lesssim\left\|\mathbf{D}_{x}\varphi\right\|_{L^{2}}$
$\displaystyle\|\varphi\|_{L^{\infty}}\lesssim\left\|\mathbf{D}_{x}\varphi\right\|_{L^{2}}^{\frac{1}{2}}\left\|\mathbf{D}_{x}^{2}\varphi\right\|_{L^{2}}^{\frac{1}{2}}$

If $\varphi$ is defined on a unit ball $B$ instead, one has

$\|\varphi\|_{L^{p}(B)}\lesssim\|\varphi\|_{L^{2}(B)}^{1-\alpha}\|\mathbf{D}_{x}\varphi\|_{L^{2}(B)}^{\alpha}+\|\varphi\|_{L^{2}(B)},\qquad\qquad\qquad\frac{1}{p}=\frac{\alpha}{6}+\frac{1-\alpha}{2}.$
(2.12)

$\|\varphi\|_{L^{\infty}(B)}\lesssim\big{(}\|\mathbf{D}_{x}\varphi\|_{L^{2}(B)}+\|\varphi\|_{L^{2}(B)}\big{)}^{\frac{1}{2}}\big{(}\|\mathbf{D}_{x}^{2}\varphi\|_{L^{2}(B)}+\|\mathbf{D}_{x}\varphi\|_{L^{2}(B)}\big{)}^{\frac{1}{2}}+\|\varphi\|_{L^{2}(B)}$
(2.13)

By the argument in [39, Theorem 4.1] one has

######  Proposition 2.2.

Assume $A\in\dot{H}^{\frac{1}{2}}(\mathbb{R}^{3})$. Then the covariant
elliptic equation $\Delta_{A}B=F$ is solved by the $\mathfrak{g}$-valued
function $B$ with the bound

$\|B\|_{\dot{H}^{\rho}}\lesssim\|F\|_{\dot{H}^{\rho-2}},\qquad\qquad\rho\in(0,2).$

###  2.1. Sobolev spaces

The global Sobolev spaces $H^{s}=H^{s}(\mathbb{R}^{n})$ and
$\dot{H}^{s}=\dot{H}^{s}(\mathbb{R}^{n})$ are defined using the Fourier
transform, for any $s\in\mathbb{R}$, by

$\|u\|_{H^{s}}\vcentcolon=\left\|\left\langle\xi\right\rangle^{s}\hat{u}\right\|_{L^{2}(\mathbb{R}^{n})},\quad\quad\|u\|_{\dot{H}^{s}}\vcentcolon=\left\||\xi|^{s}\hat{u}\right\|_{L^{2}(\mathbb{R}^{n})}$
(2.14)

We denote $H^{\infty}(\mathbb{R}^{n})\vcentcolon=\bigcap_{m\geq
0}H^{m}(\mathbb{R}^{n})$. For $s>0$ one has $H^{s}=L^{2}\cap\dot{H}^{s}$ while
for $s<0$ one has $H^{s}=L^{2}+\dot{H}^{s}$. By duality one may identify
$H^{-s}$ with $(H^{s})^{*}$.

We next define Sobolev spaces $W^{s,2}(D)$ (also denoted $H^{s}(D)$) when $D$
is either a ball $B\subset\mathbb{R}^{n}$ of radius $\simeq 1$ or
$D=\mathbb{R}^{n}$. When $s\in\\{0,1\\}$ we use the classical definition. When
$s\in(0,1)$ we use the norm

$\|u\|_{W^{s,2}(D)}^{2}\vcentcolon=\|u\|_{L^{2}(D)}^{2}+\left|u\right|_{\dot{W}^{s,2}(D)}^{2}$
(2.15)

where $\left|\cdot\right|_{\dot{W}^{s,2}(D)}$ denotes the Gagliardo seminorm
defined by

$\left|u\right|_{\dot{W}^{s,2}(D)}^{2}\vcentcolon=\int_{D}\int_{D}\frac{\left|u(x)-u(y)\right|^{2}}{\left|x-y\right|^{n+2s}}\,\mathrm{d}x\,\mathrm{d}y$
(2.16)

Due to (2.17) we will use the notations $H^{s}(D)$ and $W^{s,2}(D)$ (resp.
$\dot{H}^{s}(D)$ and $\dot{W}^{s,2}(D)$) interchangeably for both $D=B$ and
$D=\mathbb{R}^{n}$.

We denote $W^{s,2}_{0}(B)$ the closure of $C_{c}^{\infty}(B)$ in the norm
$\left|\cdot\right|_{W^{s,2}(B)}$. Of course
$W^{s,2}_{0}(\mathbb{R}^{n})=W^{s,2}(\mathbb{R}^{n})$.

When $s\in(-1,0)$ we define $W^{s,2}(D)$, as a subspace of distributions, by
duality

$W^{s,2}(D)\vcentcolon=W^{-s,2}_{0}(D)^{*}.$

We recall some properties from [13]. For $D=\mathbb{R}^{n}$ the two
definitions (2.14), (2.15) coincide

$\|u\|_{H^{s}}\simeq\|u\|_{W^{s,2}(\mathbb{R}^{n})},\quad\quad\|u\|_{\dot{H}^{s}}\simeq\left|u\right|_{\dot{W}^{s,2}(\mathbb{R}^{n})}$
(2.17)

If $B$ is a ball, then any $u\in W^{s,2}(B)$ has an extension $\tilde{u}\in
W^{s,2}(\mathbb{R}^{n})$ with comparable norm

$\|u\|_{W^{s,2}(B)}\simeq\|\tilde{u}\|_{W^{s,2}(\mathbb{R}^{n})}\simeq\inf\\{\|\bar{u}\|_{W^{s,2}(\mathbb{R}^{n})}\
|\ \ \bar{u}\restriction_{B}=u\\}.$

If $\psi\in C^{0,1}(B)$, $0\leq\psi\leq 1$ then

$\|\psi u\|_{W^{s,2}(B)}\lesssim\|u\|_{W^{s,2}(B)},\qquad s\in(0,1).$ (2.18)

####  2.1.1. Sobolev multiplication laws

We recall from [11], [52] that: if $s_{1}+s_{2}>s+\frac{3}{2}$,
$s_{1}+s_{2}\geq 0$ and $s\leq\min(s_{1},s_{2})$ then one has

$\|fg\|_{H^{s}}\lesssim\|f\|_{H_{x}^{s_{1}}}\|g\|_{H_{x}^{s_{2}}}.$ (2.19)

In a similar spirit, one has the homogeneous version [37, Lemma 3.2]

$\|fg\|_{\dot{H}^{s}}\lesssim\|f\|_{\dot{H}_{x}^{s_{1}}}\|g\|_{\dot{H}_{x}^{s_{2}}}.$
(2.20)

provided $s_{1}+s_{2}=s+\frac{3}{2}$ and $-s,s_{1},s_{2}<\frac{3}{2}$.

For $r>3/4$ we will use the version [22, Lemma 2.1]:

$\|fg\|_{\dot{H}_{x}^{-1}}\lesssim\|f\|_{H_{x}^{r}}\|g\|_{H_{x}^{r-1}}.$
(2.21)

One also has

$\|I(fg)\|_{\dot{H}^{-1}}\lesssim\|If\|_{H^{1}}\|Ig\|_{L^{2}}$ (2.22)

which follows immediately from (2.21) with $r=\sigma$

####  2.1.2. Norms for gauge transformations

We now define some further norms. Let
$X_{\sigma}=\dot{H}^{1}\cap\dot{H}^{1+\sigma}\cap L^{\infty}(\mathbb{R}^{3})$,
and the similar definition on a ball, i.e.

$\|U\|_{X_{\sigma}}\vcentcolon=\|\nabla
U\|_{H^{\sigma}(\mathbb{R}^{3})}+\|U\|_{L^{\infty}(\mathbb{R}^{3})},\quad\|U\|_{X_{\sigma}(B)}\vcentcolon=\|\nabla
U\|_{H^{\sigma}(B)}+\|U\|_{L^{\infty}(B)}.$

Similarly one defines

$\|V\|_{X}\vcentcolon=\|I\nabla
V\|_{H^{1}(\mathbb{R}^{3})}+\|V\|_{L^{\infty}(\mathbb{R}^{3})}$ (2.23)

One has the following algebra and product estimates

$\displaystyle X_{\sigma}\times X_{\sigma}\to X_{\sigma},\qquad$
$\displaystyle H^{\sigma}\times X_{\sigma}\to H^{\sigma},\qquad$
$\displaystyle H^{\sigma-1}\times X_{\sigma}\to H^{\sigma-1}$ (2.24)
$\displaystyle X\times X\to X,\qquad$ $\displaystyle I^{-1}H^{1}\times X\to
I^{-1}H^{1},\qquad$ $\displaystyle I^{-1}L^{2}\times X\to I^{-1}L^{2}$ (2.25)

These are proved using the Littlewood-Paley trichotomy, (2.24) is [50, Eq.
(10)]. By an extension argument, (2.24) holds on domains as well

$\|UV\|_{X_{\sigma}(B)}\lesssim\|U\|_{X_{\sigma}(B)}\|V\|_{X_{\sigma}(B)},\quad\|fU\|_{H^{r}(B)}\lesssim\|f\|_{H^{r}(B)}\|U\|_{X_{\sigma}(B)}$
(2.26)

for $r\in\\{\sigma,\sigma-1\\}$.

One may obtain $X_{\sigma}(B)$ control for the gauge transformation between
two $H^{\sigma}(B)$ fields. This is [56, Lemma 1.2] for integer regularities
and it easily generalizes.

######  Lemma 2.3.

Let $U:B\to G$ be such that $A_{i}=U\tilde{A}_{i}U^{-1}-\partial_{i}UU^{-1}$
on $B$. If $M=\|A_{i}\|_{H^{\sigma}(B)}+\|\tilde{A}_{i}\|_{H^{\sigma}(B)}$,
then $\|U\|_{X_{\sigma}(B)}-1\lesssim\|\nabla U\|_{H^{\sigma}(B)}\lesssim
M(1+M)$.

###### Proof.

The $L^{\infty}(B)$ bound is automatic, so we prove
$\|\partial_{i}U\|_{H^{\sigma}(B)}\lesssim M(1+M)$. Write
$\partial_{i}U=U\tilde{A}_{i}-A_{i}U$, which implies
$\|U\|_{W^{1,p}(B)}\lesssim 1+M$ where $p$ is the Sobolev exponent for
$H^{\sigma}$. We use

$\|Vg\|_{H^{\sigma}(\mathbb{R}^{3})}\lesssim\|V\|_{\dot{W}^{1,p}\cap
L^{\infty}(\mathbb{R}^{3})}\|g\|_{H^{\sigma}(\mathbb{R}^{3})}\ \Rightarrow\
\|Uf\|_{H^{\sigma}(B)}\lesssim\|U\|_{{W^{1,p}(B)}}\|f\|_{H^{\sigma}(B)}$

The first inequality follows from Littewood-Paley theory, while the second
follows from the first by an extension argument. Apply this inequality for
$U\tilde{A}_{i}$ and $A_{i}U$ to obtain
$\|\partial_{i}U\|_{H^{\sigma}(B)}\lesssim M(1+M)$. ∎

###  2.2. $X^{s,b}$ spaces

We outline the main properties, for more details we refer to [47], [30]. The
$X^{s,b}$ spaces associated to the wave equation are defined by the norm

$\|u\|_{X^{s,b}}=\|\left\langle\xi\right\rangle^{s}\left\langle\left|\tau\right|-\left|\xi\right|\right\rangle^{b}\mathcal{F}u(\tau,\xi)\|_{L^{2}_{\tau,\xi}}$

where $\mathcal{F}u$ denotes the space-time Fourier transform on
$\mathbb{R}\times\mathbb{R}^{3}$.

Following [50] we also use the $X^{r,\theta}_{\tau=0}=H^{r}_{x}H^{\theta}_{t}$
spaces adapted to solutions of the equation $\partial_{t}v=F$, with norm

$\|v\|_{X^{r,\theta}_{\tau=0}}=\|\left\langle\xi\right\rangle^{r}\left\langle\tau\right\rangle^{\theta}\mathcal{F}v(\tau,\xi)\|_{L^{2}_{\tau,\xi}}$

For a bounded interval $J$ and $X=X^{s,b}$ or $X^{r,\theta}_{\tau=0}$ one
defines the restricted space $X_{J}$ consisting of functions or tensors $u$
defined on $J\times\mathbb{R}^{3}$ for which the following norm is finite:

$\|u\|_{X_{J}}=\inf\\{\|v\|_{X}\ ;\ v_{\mkern 1.0mu\vrule
height=6.02777pt\mkern 2.0muJ\times\mathbb{R}^{3}}=u\ \\}$

To say $u\in X$ on $J\times\mathbb{R}^{3}$ means $u\in X_{J}$, i.e. it has an
extension $v\in X$ to $\mathbb{R}\times\mathbb{R}^{3}$.

One has the well-known embeddings

$X^{s,\frac{1}{2}+}\subseteq L^{\infty}_{t}H^{s}_{x},\qquad
X^{r,\frac{1}{2}+}_{\tau=0}\subseteq L^{\infty}_{t}H^{r}_{x}.$

Moreover, one has the embedding

$X^{0,0+}\subset L^{2+}_{t}L^{2}_{x}$ (2.27)

which follows by interpolation between $X^{0,0}=L^{2}_{t}L^{2}_{x}$ and
$X^{0,\frac{1}{2}+}\subset L^{\infty}_{t}L^{2}_{x}$.

Multiplications by cutoff functions interact well with these spaces:

######  Lemma 2.4.

Denote
$\|u\|_{\mathcal{X}^{1,b}}\vcentcolon=\|u\|_{X^{1,b}}+\|\partial_{t}u\|_{X^{0,b}}$
and let $\chi$ be a bump function. Then, for $0\leq b\leq 1$ one has $\|\chi
u\|_{\mathcal{X}^{1,b}}\lesssim\|u\|_{\mathcal{X}^{1,b}}$

###### Proof.

It is easily seen that an equivalent norm for $\mathcal{X}^{1,\theta}$ is

$\|\left\langle\left|\tau\right|+\left|\xi\right|\right\rangle\left\langle\left|\tau\right|-\left|\xi\right|\right\rangle^{b}\mathcal{F}u(\tau,\xi)\|_{L^{2}_{\tau,\xi}}$

When $b=0$, respectively $b=1$ this is equivalent to

$\|u\|_{L^{2}_{t,x}}+\|\nabla_{t,x}u\|_{L^{2}_{t,x}},\quad\text{respectively}\quad\|u\|_{L^{2}_{t,x}}+\|\nabla_{t,x}u\|_{L^{2}_{t,x}}+\|\Box
u\|_{L^{2}_{t,x}}$

and then the property follows from the chain rule. The general case follows
from these by complex interpolation, see [18, Lemma 1.4]. ∎

####  2.2.1. $X^{s,b}$ and $X^{r,\theta}_{\tau=0}$ energy estimates

Suppose $t_{0}\in J$ with $\left|J\right|\lesssim 1$ and
$b,\theta\in(\frac{1}{2},1)$, $s,r\in\mathbb{R}$. If $\Box u=F$ then

$\|u\|_{X^{s,b}_{J}}+\|\partial_{t}u\|_{X^{s-1,b}_{J}}\lesssim\|u[t_{0}]\|_{H^{s}\times
H^{s-1}}+\|F\|_{X^{s-1,b-1}_{J}}$ (2.28)

If $\partial_{v}=G$ then

$\|v\|_{X^{r,\theta}_{\tau=0},J}\lesssim\|v(t_{0})\|_{H^{r}}+\|G\|_{X^{r,\theta-1}_{\tau=0},J}$
(2.29)

####  2.2.2. Strichartz estimates

We recall

$X^{s,\frac{1}{2}+}\subseteq L_{t}^{q}L_{x}^{r}$ (2.30)

when $q\in(2,\infty]$, $s\geq 0$ and
$\frac{1}{q}+\frac{1}{r}\leq\frac{1}{2},\quad\frac{1}{q}+\frac{3}{r}\geq\frac{3}{2}-s.$
Moreover, we will use the following Bilinear Strichartz estimate (see e.g.
[31]):

$D^{-\left(\frac{1}{2}-\right)}\left(X^{\frac{1}{4}\cdot\frac{1}{2}+}\times
X^{\frac{1}{4},\frac{1}{2}+}\right)\subseteq L_{t}^{2}L_{x}^{2}$ (2.31)

##  3\. Local well-posedness for Yang-Mills

###  3.1. Approximation by regular initial data sets

We begin with a $H^{\sigma}\times H^{\sigma-1}$ approximation result,
recalling that initial data sets for Yang-Mills have to satisfy the Gauss
equation (1.3) in order to be admissible. The similar statement for
$H^{1}\times L^{2}$ appears in [27, Proposition 1.2].

######  Proposition 3.1.

Any initial data $(A_{i},E_{i})\in H^{\sigma}\times H^{\sigma-1}$ satisfying
the constraint equation (1.3) can be approximated in $H^{\sigma}\times
H^{\sigma-1}$ by a sequence of regular initial data $(A_{i}^{n},E_{i}^{n})\in
H^{\infty}\times H^{\infty}$ satisfying the constraint equation (1.3).
Moreover, $A_{i}^{n}$ can be chosen compactly supported.

###### Proof.

Begin with a sequence of smooth compactly supported sequence
$(A^{n},\tilde{E}^{n})$ converging to $(A,E)$ in $H^{\sigma}\times
H^{\sigma-1}$. Denote the covariant derivatives by
$D^{n}\vcentcolon=\partial+[A^{n},\cdot]$. Using (1.3) and writing, for any
$f\in H^{1-\sigma}\dot{H}^{1}=\dot{H}^{1}\cap\dot{H}^{2-\sigma}$

$\langle D^{n,i}\tilde{E}^{n}_{i},f\rangle=\langle
E_{i}-\tilde{E}^{n}_{i},\partial^{i}f\rangle+\langle[A^{n}-A,\tilde{E}^{n}],f\rangle+\langle[A,\tilde{E}^{n}-E],f\rangle$

one obtains $D^{n,i}\tilde{E}^{n}_{i}\to 0$ in
$H^{\sigma-1}\dot{H}^{-1}=\dot{H}^{-1}+\dot{H}^{\sigma-2}$ after using
$H^{\sigma}\times H^{1-\sigma}\dot{H}^{1}\to H^{1-\sigma}$, which is a version
of (2.19).

Define $E^{n}_{i}=\tilde{E}^{n}_{i}+D^{n}_{i}\phi^{n}$ which will satisfy
(1.3) provided we choose

$\Delta_{A^{n}}\phi^{n}=-D^{n,i}\tilde{E}^{n}_{i}$

It remains to show $D^{n}_{i}\phi^{n}\to 0$ in
$H^{\sigma-1}=L^{2}+\dot{H}^{\sigma-1}$. By Proposition 2.2 we obtain
$\phi^{n}\to 0$ in $\dot{H}^{1}+\dot{H}^{\sigma}$ so it remains to use the
following easy version of (2.19)

$\|[A^{n},\phi^{n}]\|_{H^{\sigma-1}}\lesssim\|A^{n}\|_{H^{\sigma}}\|\phi^{n}\|_{\dot{H}^{1}+\dot{H}^{\sigma}}\to
0$

∎

###  3.2. Square summability of fractional Sobolev spaces

We will need the following square-summability property of $W^{s,2}$ norms.

######  Proposition 3.2.

Let $(B_{j})_{j}$ be a finitely overlapping uniform covering of
$\mathbb{R}^{n}$ by balls of radius $\simeq 1$ and let $s\in(-1,1)$. Then, for
any $u\in W^{s,2}(\mathbb{R}^{n})$ one has

$\sum_{j}\|u\|_{W^{s,2}(B_{j})}^{2}\simeq\|u\|_{W^{s,2}(\mathbb{R}^{n})}^{2}.$
(3.1)

###### Proof.

When $s=0$ the property is obvious. Suppose first that $s\in(0,1)$. Clearly,
from (2.16)

$\sum_{j}\left|u\right|_{\dot{W}^{s,2}(B_{j})}^{2}\lesssim\left|u\right|_{\dot{W}^{s,2}(\mathbb{R}^{n})}^{2}.$

Conversely, by splitting the $\mathbb{R}^{n}\times\mathbb{R}^{n}$ integral in
(2.16) into regions where there exists $j$ such that both $x,y\in B_{j}$ and
regions where $|x-y|>\delta$ (where we bound $|u(x)-u(y)|\leq|u(x)|+|u(y)|$
and use the $L^{2}(B_{j})$ norms), we obtain

$|u|_{\dot{W}^{s,2}\left(\mathbb{R}^{n}\right)}^{2}\lesssim\sum_{j}\|u\|_{W^{s,2}\left(B_{j}\right)}^{2}.$

Now assume $s\in(-1,0)$. Let $(\psi_{j})_{j}$ be a partition of unity
associated to $(B_{j})_{j}$.

Let $u$ be a tempered distribution and let $\phi$ be a Schwartz function.
Using (2.18) and what we already proved

$\displaystyle|\langle u,\phi\rangle|$
$\displaystyle\lesssim\sum_{j}\|u\|_{W^{s,2}\left(B_{j}\right)}\|\psi_{j}\phi\|_{W^{-s,2}_{0}(B_{j})}$
$\displaystyle\lesssim\big{(}\sum_{j}\|u\|_{W^{s,2}\left(B_{j}\right)}^{2}\big{)}^{\frac{1}{2}}\|\phi\|_{W^{-s,2}(\mathbb{R}^{n})}$

This shows RHS (3.1) $\lesssim$ LHS (3.1). Conversely,

$\displaystyle LHS\eqref{sqSumWs2}^{\frac{1}{2}}$
$\displaystyle=\sup_{\|c\|_{\ell^{2}}\leq
1}\sum_{j}c_{j}\|u\|_{W^{s,2}(B_{j})}=\sup_{\|c\|_{\ell^{2}}\leq
1}\sum_{j}\sup_{\|\phi_{j}\|_{W^{-s,2}_{0}(B_{j})}\leq 1}c_{j}\langle
u,\phi_{j}\rangle$
$\displaystyle\lesssim\sup_{\|\phi\|_{W^{-s,2}(\mathbb{R}^{n})}\leq 1}\langle
u,\phi\rangle=\|u\|_{W^{s,2}(\mathbb{R}^{n})}=RHS\eqref{sqSumWs2}^{\frac{1}{2}}$

∎

###  3.3. Initial data extension

We recall the following result from [40], which we state only for a ball $B$
of radius $\simeq 1$.

######  Theorem 3.3 ([40] \- Theorem 5.4.).

Let $\rho\in(\frac{1}{2},\frac{5}{2})$ and let $(\bar{A}_{x},\bar{E}_{x})\in
H^{\rho}(B)\times H^{\rho-1}(B)$ satisfy the Gauss equation (1.3). If
$\|\bar{A}_{x}\|_{H^{\frac{1}{2}}(B)}\leq\varepsilon$ for a sufficiently small
$\varepsilon>0$, then there exist $(\tilde{A}_{x},\tilde{E}_{x})\in
H^{\rho}(\mathbb{R}^{3})\times H^{\rho-1}(\mathbb{R}^{3})$ which satisfies the
Gauss equation (1.3), which coincides with $(\bar{A}_{x},\bar{E}_{x})$ on $B$
such that

$\|(\tilde{A}_{x},\tilde{E}_{x})\|_{H^{\rho}\times
H^{\rho-1}}\lesssim\|(\bar{A}_{x},\bar{E}_{x})\|_{H^{\rho}(B)\times
H^{\rho-1}(B)}$

and such that the map
$(\bar{A}_{x},\bar{E}_{x})\mapsto(\tilde{A}_{x},\tilde{E}_{x})$ is locally
Lipschitz continuous. If $(\bar{A}_{x},\bar{E}_{x})$ is smooth then so is
$(\tilde{A}_{x},\tilde{E}_{x})$.

The proof is based from the following solvability result for the inhomogeneous
Gauss equation.

######  Proposition 3.4 ([40] \- Prop. 4.2.).

Suppose $a$ satisfies
$\|a\|_{\dot{H}^{\frac{1}{2}}(\mathbb{R}^{3})}\leq\epsilon_{*}$. Let $K$ be a
convex domain. If $\epsilon_{*}>0$ is small enough there exists a solution
operator $T_{a}$ for the equation $\mathbf{D}^{\ell}e_{\ell}=h$ with the
following properties:

(1)
$\left\|T_{a}h\right\|_{\dot{W}^{r,p}}\lesssim_{K,r,p}\|h\|_{\dot{W}^{r-1,p}}$
holds for $2\leq p<\infty$ and $1-\frac{3}{p}<r<\frac{3}{2}$.

(2) If $h=0$ in $\lambda K$, then $T_{a}h=0$ in $\lambda K$.

(3) If $a$ and $h$ are smooth, so is $T_{a}h$.

In the proof of the almost conservation law we need the following version of
Theorem 3.3. The proof carries over almost verbatim from Theorem 5.4. from
[40]. We present the argument for the sake of completeness.

######  Proposition 3.5.

Suppose $(a,e)$ satisfy the constraint equation (1.3) on a ball
$B\subset\mathbb{R}^{3}$ of radius $\simeq 1$ and

$(a,e)\in(H^{1}+N^{\sigma-1}H^{\sigma})(B)\times(L^{2}+N^{\sigma-1}H^{\sigma-1})(B)$

Suppose $\|a\|_{H^{\frac{1}{2}}(B)}\leq\epsilon$ for $\epsilon>0$ small
enough. Then there exist $(\bar{a},\bar{e})$ solving (1.3) on $\mathbb{R}^{3}$
which coincide with $(a,e)$ on $B$ with

$\|\bar{a}\|_{H^{1}+N^{\sigma-1}H^{\sigma}}+\|\bar{e}\|_{L^{2}+N^{\sigma-1}H^{\sigma-1}}\lesssim\|a\|_{H^{1}+N^{\sigma-1}H^{\sigma}(B)}+\|e\|_{L^{2}+N^{\sigma-1}H^{\sigma-1}(B)}.$

If $(a,e)$ are smooth, then $(\bar{a},\bar{e})$ are regular.

###### Proof.

Decompose $a=a^{1}+N^{\sigma-1}a^{\sigma}$, $e=e^{0}+N^{\sigma-1}e^{\sigma-1}$
where $a^{1}\in H^{1}(B)$, $a^{\sigma}\in H^{\sigma}(B)$, $e^{0}\in L^{2}(B)$,
$e^{\sigma-1}\in H^{\sigma-1}(B)$. One begins by applying a (universal)
extension operator to $a^{1},a^{\sigma},e^{0},e^{\sigma-1}$ obtaining
$\bar{a}^{1},\bar{a}^{\sigma},\bar{e}^{\prime 0},\bar{e}^{\prime\sigma-1}$ on
$\mathbb{R}^{3}$ with comparable norms. Let
$\bar{a}=\bar{a}^{1}+N^{\sigma-1}\bar{a}^{\sigma}$,
$\bar{e}^{\prime}=\bar{e}^{\prime 0}+N^{\sigma-1}\bar{e}^{\prime\sigma-1}$.

Since the constraint equation (1.3) is violated in general outside $B$ by
$(\bar{a},\bar{e}^{\prime})$, one must correct $\bar{e}^{\prime}$ by adding a
term. Let $h^{-1}=(\mathbf{D}^{\bar{a}})^{\ell}\bar{e}^{\prime 0}_{\ell}\in
H^{-1}$ and
$h^{\sigma-2}=(\mathbf{D}^{\bar{a}})^{\ell}\bar{e}^{\prime\sigma-1}_{\ell}\in
H^{\sigma-2}$ be the errors of the Gauss equations, which are supported
outside $B$. Now apply Proposition 3.4 to obtain $d^{0}=-T_{\bar{a}}h^{-1}$
and $d^{\sigma-1}=-T_{\bar{a}}h^{\sigma-2}$ which solve
$(\mathbf{D}^{\bar{a}})^{\ell}d^{0}_{\ell}=-h^{-1}$ and
$(\mathbf{D}^{\bar{a}})^{\ell}d^{\sigma-1}_{\ell}=-h^{\sigma-2}$ and satisfy
$\|d^{0}\|_{L^{2}}\lesssim\|\bar{e}^{\prime 0}\|_{L^{2}}$,
$\|d^{\sigma-1}\|_{H^{\sigma-1}}\lesssim\|\bar{e}^{\prime\sigma-1}\|_{H^{\sigma-1}}$.

Now let $d=d^{0}+N^{\sigma-1}d^{\sigma-1}$ and define
$\bar{e}=\bar{e}^{\prime}+d$. Then $(\bar{a},\bar{e})$ satisfy the conditions.
∎

###  3.4. Review of Theorem 1.3 and proof of Corollary 1.4

For the sake of completeness and to set up the notation we review the argument
for Theorem 1.3 from [40, Theorem 1.27]. The key technical tool is their
initial data surgery result in Theorem 3.3 ([40, Theorem 5.4]).

We begin with an initial data set $(\bar{A}_{x},\bar{E}_{x})\in
H^{\sigma}(\mathbb{R}^{3})\times H^{\sigma-1}(\mathbb{R}^{3})$ satisfying
(1.3). It suffices to consider $\frac{3}{4}<\sigma<1$ and to obtain a solution
for positive times.

Rescale the initial data by
$\bar{A}^{\lambda}(x)=\lambda^{-1}\bar{A}(x/\lambda)$ ,
$\bar{E}^{\lambda}(x)=\lambda^{-2}\bar{E}(x/\lambda)$ so that

$\|(\bar{A}^{\lambda},\bar{E}^{\lambda})\|_{\dot{H}^{\sigma}\times\dot{H}^{\sigma-1}}\lesssim\lambda^{\frac{1}{2}-\sigma}\|(\bar{A},\bar{E})\|_{\dot{H}^{\sigma}\times\dot{H}^{\sigma-1}}\ll\varepsilon_{*}$

where $\varepsilon_{*}>0$ is chosen small enough to be able to apply Theorem
1.2 and Theorem 3.3 to obtain modified initial data sets as follows.

Let $(B_{j})_{j}$ be a finitely overlapping uniform covering of
$\mathbb{R}^{3}$ by balls of radius $2$ centered at $(y_{j})_{j}$ such that
the domains of dependency $(\mathcal{D}_{j})_{j}$ with bases $(B_{j})_{j}$
form a covering of $[0,1]\times\mathbb{R}^{3}$. These are truncated cones
$\mathcal{D}_{j}\vcentcolon=\\{(t,x)\ |\ t+\left|x-y_{j}\right|<2,\
t\in[0,1]\\}.$

One may apply Theorem 3.3 for each $B_{j}$ to obtain an extension
$(\bar{A}^{(j)},\bar{E}^{(j)})$ to $\mathbb{R}^{3}$ satisfying the constraint
equation (1.3) and which coincide with $(\bar{A}^{\lambda},\bar{E}^{\lambda})$
on $B_{j}$ with the bounds

$\|(\bar{A}^{(j)},\bar{E}^{(j)})\|_{H^{\sigma}\times
H^{\sigma-1}}\lesssim\|(\bar{A}^{\lambda},\bar{E}^{\lambda})\|_{W^{\sigma,2}\times
W^{\sigma-1,2}(B_{j})}\ll\varepsilon_{*}.$

The map
$(\bar{A}^{\lambda},\bar{E}^{\lambda})\mapsto(\bar{A}^{(j)},\bar{E}^{(j)})$ is
locally Lipschitz on these spaces.

To each $(\bar{A}^{(j)},\bar{E}^{(j)})$ one applies Theorem 1.2 (i.e. Theorem
1.1. in [50]) to obtain a Yang-Mills solution $A^{(j)}$ in the temporal gauge
on $[0,1]\times\mathbb{R}^{3}$ satisfying

$\|A^{(j)}[t]\|_{L^{\infty}(H^{\sigma}\times
H^{\sigma-1})([0,1]\times\mathbb{R}^{3})}\lesssim\|(\bar{A}^{\lambda},\bar{E}^{\lambda})\|_{W^{\sigma,2}\times
W^{\sigma-1,2}(B_{j})}$ (3.2)

One obtains a solution $A^{\lambda}$ on $[0,1]\times\mathbb{R}^{3}$ defined by
each $A^{(j)}$ restricted to $\mathcal{D}_{j}$ since any two of these
solutions coincide on their common domain as a consequence of finite speed of
propagation in the temporal gauge.

We apply the square summability estimate (3.1) at the initial time and at an
arbitrary time $t\in[0,1]$, together with (3.2) obtaining

$\displaystyle\|A^{\lambda}[t]\|_{H^{\sigma}\times H^{\sigma-1}}^{2}$
$\displaystyle\simeq\sum_{j}\|A^{(j)}[t]\|_{W^{\sigma,2}\times
W^{\sigma-1,2}(\mathcal{D}_{j}\cap\\{t\\}\times\mathbb{R}^{3})}^{2}$
$\displaystyle\lesssim\sum_{j}\|(\bar{A}^{\lambda},\bar{E}^{\lambda})\|_{W^{\sigma,2}\times
W^{\sigma-1,2}(B_{j})}^{2}\simeq\|(\bar{A}^{\lambda},\bar{E}^{\lambda})\|_{H^{\sigma}\times
H^{\sigma-1}}^{2}.$

Undoing the scaling one obtains a solution $A\in C_{t}H^{\sigma}\cap
C_{t}^{1}H^{\sigma-1}([0,\lambda^{-1}]\times\mathbb{R}^{3})$ given by
$A(t,x)=\lambda A^{\lambda}(\lambda t,\lambda x)$.

If $(\bar{A}^{\prime},\bar{E}^{\prime})\in H^{\sigma}\times
H^{\sigma-1}(\mathbb{R}^{3})$ is another initial data set satisfying (1.3) one
has

$\|A^{(j)}[\cdot]-A^{\prime(j)}[\cdot]\|_{L^{\infty}(H^{\sigma}\times
H^{\sigma-1})}\lesssim\|(\bar{A}^{\lambda}-\bar{A}^{\prime\lambda},\bar{E}^{\lambda}-\bar{E}^{\prime\lambda})\|_{W^{\sigma,2}\times
W^{\sigma-1,2}(B_{j})}$ (3.3)

By repeating the same square summability argument for differences using (3.3)
and undoing the scaling one obtains Lipschitz dependence

$\|A[t]-A^{\prime}[t]\|_{H^{\sigma}\times
H^{\sigma-1}}\lesssim\|(\bar{A},\bar{E})-(\bar{A}^{\prime},\bar{E}^{\prime})\|_{H^{\sigma}\times
H^{\sigma-1}}.$

The Lipschitz constant depends, of course, on the size of the initial data.

##  4\. Modified local well-posedness

In this section we derive control of space-time norms that will be needed for
the trilinear estimate (1.32).

Let $A_{i}$ be a solution to (YM) in the temporal gauge $A_{0}=0$. We use the
Leray projections (2.6) to split $A={\bf P}A+{\bf P}^{\perp}A$ into a
divergence-free part and a curl-free part. From the definition of ${\bf
P}^{\perp}$ and the first equation in (1.2), and by applying ${\bf P}$ to the
second equation in (1.2) one obtains

$\displaystyle\partial_{t}{\bf P}^{\perp}A$
$\displaystyle=\Delta^{-1}\nabla[\partial_{t}A_{j},A_{j}]$ (4.1)
$\displaystyle\Box\ {\bf P}A$ $\displaystyle=-2{\bf
P}[A_{j},\partial_{j}A]+{\bf P}[A_{j},\nabla A_{j}]+{\bf
P}[A,\partial_{j}A_{j}]-{\bf P}[A_{j},[A_{j},A]]$

Following [50] one proceeds to uncovering the null structure in (4.1) as
follows.

Recall the definition of a null form from (2.8), (2.9). For Lie algebra valued
fields, ${\bf N}(A,B)$ is understood as a linear combination of

$\Delta^{-1}\nabla^{i}Q_{ij}(A_{\ell},B_{k}),\quad\text{and}\quad
Q_{ij}(\Delta^{-1}\nabla^{i}A_{\ell},B_{k})\quad$ (4.2)

where

$Q_{ij}(C,D)\vcentcolon=[\partial_{i}C,\partial_{j}D]-[\partial_{j}C,\partial_{i}D]$
(4.3)

Denote $A^{df}={\bf P}A$ and $A^{cf}={\bf P}^{\perp}A$. Using the identities
(2.7) one can write the system (4.1) as

$\displaystyle\partial_{t}A^{cf}$
$\displaystyle=\nabla^{-1}\mathcal{O}(\partial_{t}A,A)$ (4.4)
$\displaystyle\Box A^{df}$ $\displaystyle={\bf
N}(A^{df},A^{df})+\mathcal{O}(A^{df},\nabla A^{cf})+\mathcal{O}(A^{cf},\nabla
A^{df})+\mathcal{O}(A^{cf},\nabla A^{cf})+\mathcal{O}(A^{3})$

where $\mathcal{O}$ denote (matrix-valued) bilinear or trilinear expressions
in the values of the inputs.

Well-posedness in $I^{-1}H^{1}\times I^{-1}L^{2}$ holds for this system.

######  Proposition 4.1.

(1) Let $A_{x}(t,x)$ be a global regular solution to (YM) in the temporal
gauge $A_{0}=0$ such that

$\|IA_{x}(t_{0})\|_{H^{1}}+\|I\partial_{t}A_{x}(t_{0})\|_{L^{2}}\leq\varepsilon$

Let $t_{1}\leq t_{0}+1$. Then, assuming $\varepsilon$ is small enough, one has

$\|IA_{x}[t]\|_{L^{\infty}_{t}(H^{1}\times
L^{2})([t_{0},t_{1}]\times\mathbb{R}^{3})}\leq C\varepsilon$ (4.5)

(2) Moreover, there exists a spatial field $U:\mathbb{R}^{3}\to G$, $U=U(x)$
such that the change of gauge

$A_{i}\mapsto UA_{i}U^{-1}-\partial_{i}UU^{-1}$ (4.6)

achieves $\text{div}A_{x}(t_{0})=0$, the bound (4.5) and

$\displaystyle IA^{df}\in\varepsilon X^{1,\frac{3}{4}+},$ $\displaystyle
I\partial_{t}A^{df}\in\varepsilon X^{0,\frac{3}{4}+}$ (4.7) $\displaystyle
IA^{cf}\in\varepsilon X_{\tau=0}^{1+\frac{1}{4},\frac{1}{2}+},$ $\displaystyle
P_{k}I^{2}\partial_{t}A^{cf}\in\varepsilon L^{\infty}_{t}H^{\frac{1}{2}}$
(4.8)

hold on $[t_{0},t_{1}]\times\mathbb{R}^{3}$, where $A^{df}={\bf P}A_{x}$,
$A^{cf}={\bf P}^{\perp}A_{x}$.

This proposition follows from the local well-posedness theory developed in
[50].

###  4.1. Proof of Proposition 4.1

Step 1. We begin by discussing the change of gauge that sets
$A^{cf}(t_{0})=0$, i.e. the Coulomb gauge $\text{div}A_{x}(t_{0})=0$, only at
the initial time. Without this transformation (4.8) could not have more
regularity than the initial data.

The following procedure is adapted here to $I^{-1}H^{1}$ from the $H^{s}$ case
in [50, Section 3] (arxiv version), attributed there to Mark Keel. Suppose

$\|IA_{x}(t_{0})\|_{H^{1}}\leq\varepsilon,\qquad\|IA^{cf}(t_{0})\|_{H^{1}}\leq\delta$

for $\delta\leq\varepsilon\ll 1$. Write $A^{cf}(t_{0})=\nabla V$ for a
$\mathfrak{g}$-valued field $V$ with $\|V\|_{X}\lesssim\delta$ where the $X$
norm is defined in (2.23) and obeys the algebra and product estimates (2.25).

Applying (4.6) with $U\vcentcolon=\exp(V)$ one obtains a new temporal (because
$V$ is independent of $t$) solution $\tilde{A}$, with initial data written as

$\tilde{A}(t_{0})=e^{V}A^{df}(t_{0})e^{-V}+\big{(}e^{V}\nabla
V-\nabla(e^{V})\big{)}e^{-V}$

and

$e^{V}A^{df}(t_{0})e^{-V}=A^{df}(t_{0})+\big{(}e^{V}A^{df}(t_{0})e^{-V}-A^{df}(t_{0})\big{)}$

Based on these equalities, Taylor expansions and (2.25) one obtains that

$\|I\tilde{A}(t_{0})\|_{H^{1}}\leq\varepsilon+C\varepsilon\delta,\qquad\|I\tilde{A}^{cf}(t_{0})\|_{H^{1}}\leq
C\varepsilon\delta,\qquad\|U-1\|_{X}\lesssim\delta$

Iterating this procedure, one obtains in the limit a new temporal solution,
which by abuse of notation we still denote by $A_{x}$, such that
$A^{cf}(t_{0})=0$. The change of gauge (4.6) is done with a $U$ with
$\|U-1\|_{X}\lesssim\varepsilon$. The new solution has

$\|IA_{x}(t_{0})\|_{H^{1}}+\|I\partial_{t}A_{x}(t_{0})\|_{L^{2}}\lesssim\varepsilon$
(4.9)

and (4.5) follows using (2.25) if we prove

$\|IA_{x}[t]\|_{L^{\infty}_{t}(H^{1}\times
L^{2})([t_{0},t_{1}]\times\mathbb{R}^{3})}\lesssim\varepsilon$ (4.10)

for the new solution. Thus it remains to prove the bounds in part (2) of Prop.
4.1.

Step 2. We now consider a (YM) solution in the temporal gauge satisfying (4.9)
with $A^{cf}(t_{0})=0$. The system of equation solved by $IA$ is written
schematically as

$\Box
IA^{df}=I\mathcal{N},\qquad\partial_{t}IA^{cf}=I\nabla^{-1}(A\partial_{t}A)$
(4.11) $\mathcal{N}\vcentcolon={\bf N}(A^{df},A^{df})+A^{df}\nabla
A^{cf}+A^{cf}\nabla A^{df}+A^{cf}\nabla A^{cf}+A^{3}$

For $J=[t_{0},t_{1}]$ let

$\|IA\|_{\mathcal{X}}\vcentcolon=\left\|IA^{df}\right\|_{X_{J}^{1,\frac{3}{4}+}}+\left\|I\partial_{t}A^{df}\right\|_{X^{0,\frac{3}{4}+}_{J}}+\left\|IA^{cf}\right\|_{X_{\tau=0,J}^{1+\frac{1}{4},\frac{1}{2}+}}$

One obtains a-priori estimates using the following bounds:

$\displaystyle\left\|I{\bf
N}(A_{1},A_{2}\right\|_{X^{0,-\frac{1}{4}+}}\lesssim\left\|IA_{1}\right\|_{X^{1,\frac{3}{4}+}}\left\|IA_{2}\right\|_{X^{1,\frac{3}{4}+}}$
$\displaystyle\left\|I(A_{1}\nabla
A_{2})\right\|_{X^{0,-\frac{1}{4}+}}+\left\|I(A_{2}\nabla
A_{1})\right\|_{X^{0,-\frac{1}{4}+}}\lesssim\left\|IA_{1}\right\|_{X^{1,\frac{3}{4}+}}\left\|IA_{2}\right\|_{X_{\tau=0}^{1+\frac{1}{4},\frac{1}{2}+}}$
$\displaystyle\left\|I(A_{1}\nabla
A_{2})\right\|_{X^{0,-\frac{1}{4}+}}\lesssim\left\|IA_{1}\right\|_{X_{\tau=0}^{1+\frac{1}{4},\frac{1}{2}+}}\left\|IA_{2}\right\|_{X_{\tau=0}^{1+\frac{1}{4},\frac{1}{2}+}}$
$\displaystyle\left\|I(A_{1}A_{2}A_{3})\right\|_{X^{0,-\frac{1}{4}+}}\lesssim\prod_{i=1}^{3}\min\big{(}\left\|IA_{i}\right\|_{X^{1,\frac{3}{4}+}},\left\|IA_{i}\right\|_{X_{\tau=0}^{1+\frac{1}{4},\frac{1}{2}+}}\big{)}$

and

$\displaystyle\left\|I\nabla^{-1}\left(A_{1}\partial_{t}A_{2}\right)\right\|_{X_{\tau=0}^{1+\frac{1}{4},-\frac{1}{2}+}}\lesssim\left\|IA_{1}\right\|_{X^{1,\frac{3}{4}+}}\left\|IA_{2}\right\|_{X^{1,\frac{3}{4}+}}$
$\displaystyle\left\|I\nabla^{-1}\left(A_{1}\partial_{t}A_{2}\right)\right\|_{X_{\tau=0}^{1+\frac{1}{4},-\frac{1}{2}+}}+\left\|I\nabla^{-1}\left(A_{2}\partial_{t}A_{1}\right)\right\|_{X_{\tau=0}^{1+\frac{1}{4},-\frac{1}{2}+}}\lesssim\left\|IA_{1}\right\|_{X^{1,\frac{3}{4}+}}\left\|IA_{2}\right\|_{X_{\tau=0}^{1+\frac{1}{4},\frac{1}{2}+}}$
$\displaystyle\left\|I\nabla^{-1}\left(A_{1}\partial_{t}A_{2}\right)\right\|_{X_{\tau=0}^{1+\frac{1}{4},-\frac{1}{2}+}}\lesssim\left\|IA_{1}\right\|_{X_{\tau=0}^{1+\frac{1}{4},\frac{1}{2}+}}\left\|IA_{2}\right\|_{X_{\tau=0}^{1+\frac{1}{4},\frac{1}{2}+}}$

holding for any $A_{1},A_{2},A_{3}$ on $\mathbf{R}^{3+1}$.

These estimates are stated and proved in [50, Eq (15)-(21)] without the $I$
multipliers and with
$X^{s,\frac{3}{4}+},X^{s-1,-\frac{1}{4}+},X_{\tau=0}^{s+\frac{1}{4},\frac{1}{2}+},X_{\tau=0}^{s+\frac{1}{4},-\frac{1}{2}+}$
instead of $X^{1,\frac{3}{4}+}$, $X^{0,-\frac{1}{4}+}$,
$X_{\tau=0}^{1+\frac{1}{4},\frac{1}{2}+}$,
$X_{\tau=0}^{1+\frac{1}{4},-\frac{1}{2}+}$ for any $s>\frac{3}{4}$. The
version stated here follows immediately from those by doing basic Fourier
decompositions. For details see the interpolation lemma in [8, Lemma 12.1].

Combining these estimates with the energy-type inequalities (2.28), (2.29) for
$X^{s,b}$, $X_{\tau=0}^{r,\theta}$ spaces and $A^{cf}(t_{0})=0$ one obtains

$\|IA\|_{\mathcal{X}}\lesssim\varepsilon+\|IA\|_{\mathcal{X}}^{2}+\|IA\|_{\mathcal{X}}^{3}$

from which one obtains $\|IA\|_{\mathcal{X}}\lesssim\varepsilon$ (if
$\varepsilon$ is small enough) by a continuity argument in $t_{1}$. This
proves the first three bounds in (4.7), (4.8). In particular,

$\|IA^{df}[t]\|_{L^{\infty}_{t}(H^{1}\times
L^{2})}+\|IA^{cf}\|_{L^{\infty}_{t}H^{1+\frac{1}{4}}}\lesssim\varepsilon$

From (4.11), using the inequality (2.22), for any $t\in[t_{0},t_{1}]$ one
obtains

$\|I\partial_{t}A^{cf}\|_{L^{2}}\lesssim\|IA\|_{H^{1}}\big{(}\|I\partial_{t}A^{df}\|_{L^{2}}+\|I\partial_{t}A^{cf}\|_{L^{2}}\big{)}\lesssim\varepsilon^{2}+\varepsilon\|I\partial_{t}A^{cf}\|_{L^{2}}$

Absorbing the last term to the LHS completes the proof of (4.10).

In fact, we can improve the previous inequality at high frequencies - we
obtain the last bound in (4.8) by the Littlewood-Paley trichotomy for the
equation $\partial_{t}A^{cf}=\nabla^{-1}(A\partial_{t}A)$ from the $H^{1}$ and
$L^{2}_{x}$ bounds already obtained in (4.10) and using Bernstein’s inequality
for the low frequency (input or output). ∎

####  4.1.1. Improved bounds for $\partial_{t}A^{cf}$

######  Proposition 4.2.

Let $A_{x}$ be the solution from Proposition 4.1 Part (2) on
$[t_{0},t_{1}]\times\mathbb{R}^{3}$. Then

$\displaystyle\|P_{k}\partial_{t}A^{cf}\|_{L^{2}_{t}L^{\infty}_{x}}\lesssim\varepsilon
2^{(\frac{1}{2}+)k}\left\langle\frac{2^{k}}{N}\right\rangle^{2(1-\sigma)}$
(4.12)
$\displaystyle\|P_{k}\partial_{t}A^{cf}\|_{L^{2}_{t}L^{2}_{x}}\lesssim\varepsilon
2^{-\frac{3}{4}k}\left\langle\frac{2^{k}}{N}\right\rangle^{2(1-\sigma)}$
(4.13)
$\displaystyle\|P_{k}\partial_{t}A\|_{L^{2}_{t}L^{\infty}_{x}}\lesssim\varepsilon
2^{(1+)k}\left\langle\frac{2^{k}}{N}\right\rangle^{1-\sigma}$ (4.14)

###### Proof.

For (4.12), (4.13) we use the equation

$\partial_{t}A^{cf}=\nabla^{-1}(A^{df}\partial_{t}A^{df}+A^{df}\partial_{t}A^{cf}+A^{cf}\partial_{t}A^{df}+A^{cf}\partial_{t}A^{cf})$

and perform a Littlewood-Paley decomposition of the RHS.

Proof of (4.12) First consider the case when the output is a high frequency
(low-high or high-low cases). When and at least one of the inputs is $df$ \-
place that input in $L^{2+}L^{\infty}$ using Strichartz (2.30) and the other
term in $L^{\infty}L^{\infty}$ (using Bernstein and $L^{\infty}L^{2}$). When
both inputs are $cf$ use $L^{\infty}L^{\infty}$ norms (and Bernstein with
$L^{\infty}L^{2}$), based on (4.8).

In the case of high-high-to-low frequency interactions we use Bernstein first.
When both inputs are $cf$ \- estimate both inputs in $L^{\infty}L^{2}$ using
(4.8) and the output in $L^{2}L^{1}$. When both inputs are $df$ use the
bilinear Strichartz estimate (2.31) to bound the output in $L^{2}L^{2}$.

Now assume one of the high-high inputs is $df$ and one is $cf$. For
$A^{df}\partial_{t}A^{cf}$ estimate the output in $L^{2}L^{2}$, placing the
$df$ term in $L^{2+}L^{\infty}$ and the $cf$ term in $L^{\infty}L^{2}$. For
$A^{cf}\partial_{t}A^{df}$ estimate the output in $L^{2}L^{\frac{8}{5}}$,
placing $A^{cf}$ in $L^{\infty}L^{2}$ and $\partial_{t}A^{df}$ in $L^{2}L^{8}$
(by interpolating between $L^{2}L^{\infty}$ and energy - $L^{2}L^{2}$). This
is done to ensure adequate summation even when $\sigma$ is close to
$\frac{5}{6}$.

Proof of (4.14) This inequality follows from (4.12) and from (4.7) together
with Strichartz (2.30) for $L^{2+}L^{\infty}$ and Hölder in $t$.

Proof of (4.13) For high-high-to-low frequency interactions we argue exactly
the same as for (4.12), only the power obtained from Bernstein’s inequality
changes by a factor of $2^{-\frac{3}{2}k}$. The same is true for low-high and
high-low $A^{cf}\partial_{t}A^{cf}$ interactions.

For the remaining high-low interactions place $A$ (high frequency) in
$L^{\infty}L^{2}$ and $\partial_{t}A$ (low frequency) in $L^{2}L^{\infty}$
using (4.14).

For low-high interactions, place $A^{df}$ (low) in $L^{2}L^{\infty}$ and
$\partial_{t}A$ (high) in $L^{\infty}L^{2}$. It remains to deal with
$A^{cf}_{k_{1}}\partial_{t}A_{k}^{df}$ for $k_{1}\leq k$. This term is
responsible for the $2^{-\frac{3}{4}k}$ factor in (4.13), rather than $2^{-k}$
(which can probably be fully obtained). Place $A^{cf}_{k_{1}}$ in
$L^{\infty}L^{12}$ using (4.8) and Bernstein and place
$\partial_{t}A_{k}^{df}$ in $L^{\infty}L^{\frac{12}{5}}$ using Bernstein
again. ∎

##  5\. Yang-Mills Heat Flow solutions

This section is devoted to the Yang-Mills heat flow. We consider deTurck’s and
caloric gauges including: well-posedness in $I^{-1}H^{1}$ and $H^{\rho}$,
control of the modified energy (1.6) from the initial data and the change
between the two gauges.

###  5.1. Local well-posedness in DeTurck’s gauge

In the covariant (cYMHF) equation, setting the DeTurck gauge condition
$A_{s}=\partial^{\ell}A_{\ell}$ one obtains the parabolic equation for $A_{i}$

$(\partial_{s}-\Delta)A_{i}=2\left[A^{\ell},\partial_{\ell}A_{i}\right]-\left[A^{\ell},\partial_{i}A_{\ell}\right]+\left[A^{\ell},\left[A_{\ell},A_{i}\right]\right]$
(5.1)

In short, this equation can be written schematically as
$(\partial_{s}-\Delta)A=A\nabla A+A^{3}$.

One can separate the linear and nonlinear parts of $A_{i}$:

$A_{i}(s)=e^{s\Delta}A_{i}(0)+A^{bil}_{i}(s)$ (5.2)

where the bilinear term is written schematically as

$A^{bil}_{i}(s_{1})\vcentcolon=\int_{0}^{s_{1}}e^{(s_{1}-s)\Delta}(A\nabla
A+A^{3})\,\mathrm{d}s$ (5.3)

and will enjoys more favorable estimates, such as (5.14).

This equation is locally well-posed in $I^{-1}H^{1}$:

######  Proposition 5.1.

Suppose the initial data satisfies $I\bar{A}_{i}\in\varepsilon H^{1}$ on
$\mathbb{R}^{3}$ and $\varepsilon>0$ is small enough. Then there exists a
solution to (5.1) on $\mathbb{R}^{3}\times[0,1]$ with $A_{i}(s=0)=\bar{A}_{i}$
which satisfies

$IA_{i}\in\varepsilon L^{\infty}_{s}H^{1}(\mathbb{R}^{3}\times[0,1])$ (5.4)

and is the unique limit of regular solutions.

This provides a solution to (cYMHF) with $A_{s}=\partial^{\ell}A_{\ell}$. If
the initial data is regular then also the solution is. Moreover,

$A_{s}\in\varepsilon L^{1}_{s}L^{\infty}_{x},\qquad\nabla A_{s}\in\varepsilon
L^{1}_{s}L^{3}_{x},\qquad A_{s}\in\varepsilon L^{1}_{s}H^{\frac{3}{2}+}$ (5.5)

and for any $s_{1},s_{2}\in[0,1]$, one has

$\int_{s_{1}}^{s_{2}}I\Delta A_{s}\,\mathrm{d}s\in\varepsilon L^{2}_{x}$ (5.6)

###### Proof.

This is proved using standard methods, so we only sketch the proof. One sets
up a contraction argument or Picard iteration, looking for a fixed point for
the functional

$\Phi
A_{i}(s_{1})=e^{s_{1}\Delta}\bar{A}_{i}+\int_{0}^{s_{1}}e^{(s_{1}-s)\Delta}(A\nabla
A+A^{3})\,\mathrm{d}s$

in a suitable space $S$ where one has the estimates

$\displaystyle\|\int_{0}^{s_{1}}e^{(s_{1}-s)\Delta}(A\nabla
B)\,\mathrm{d}s\|_{S}$ $\displaystyle\lesssim\|A\|_{S}\|B\|_{S}$
$\displaystyle\|\int_{0}^{s_{1}}e^{(s_{1}-s)\Delta}(A_{1}A_{2}A_{3}))\,\mathrm{d}s\|_{S}$
$\displaystyle\lesssim\|A_{1}\|_{S}\|A_{2}\|_{S}\|A_{3}\|_{S}$

One may use the space

$\|A\|_{S}\vcentcolon=\|\|\left\langle
2^{2k}s\right\rangle^{10}\|P_{k}IA(s)\|_{H^{1}}\|_{L^{\infty}_{s}}\|_{\ell^{2}_{k\in\mathbb{N}}}$
(5.7)

to check these estimates using basic Littewood-Paley theory.

One may also use the following parabolic energy estimates, [37, Proposition
3.12]

$\displaystyle\|\int_{0}^{s_{1}}e^{(s_{1}-s)\Delta}\mathcal{N}(s)\,\mathrm{d}s\|_{\tilde{S}}\lesssim\|s\mathcal{N}\|_{L^{1}_{\frac{\,\mathrm{d}s}{s}}\cap
L^{2}_{\frac{\,\mathrm{d}s}{s}}(L^{2}_{x})}$ (5.8)
$\displaystyle\|f\|_{\tilde{S}}\vcentcolon=\|f\|_{L^{\infty}_{s}L^{2}_{x}}+\|s^{\frac{1}{2}}\partial_{x}f\|_{L^{\infty}_{s}\cap
L^{2}_{\frac{\,\mathrm{d}s}{s}}(L^{2}_{x})}+\|s\partial_{x}^{2}f\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}}$

used in the proof of [37, Proposition 5.2] which is the corresponding well-
posedness statement for $\dot{H}^{1}$. Here one writes the equation as

$(\partial_{s}-\Delta)I\left\langle D_{x}\right\rangle A=I\left\langle
D_{x}\right\rangle(A\nabla A+A^{3})$ (5.9)

One defines $\|A\|_{S}\vcentcolon=\|I\left\langle D_{x}\right\rangle
A\|_{\tilde{S}}$. The needed estimates to check for this case of
$I^{-1}H^{1}$, as opposed to $\dot{H}^{1}$, become

$\displaystyle\|I\left\langle D\right\rangle(f\nabla g)\|_{L^{2}}$
$\displaystyle\lesssim\|If\|_{H^{\frac{3}{2}+(1-\sigma)+}}\|Ig\|_{H^{2}}$
(5.10) $\displaystyle\|I\left\langle
D\right\rangle(f_{1}f_{2}f_{3})\|_{L^{2}}$
$\displaystyle\lesssim\|If_{1}\|_{H^{2-\frac{2}{3}\sigma}}\|If_{2}\|_{H^{2-\frac{2}{3}\sigma}}\|If_{3}\|_{H^{2-\frac{2}{3}\sigma}}$

which follow easily from (2.19) or from Littewood-Paley, by splitting into
different cases depending on whether the inputs have frequencies higher or
lower than $N$.

By similar arguments for the differentiated equation one obtains persistence
of regularity. We refer to [37, Proposition 5.2] for more details.

The bounds (5.5) follow from the $\|A\|_{S}$ bound and the definition of
$A_{s}=\partial^{\ell}A_{\ell}$.

Denoting by $\mathcal{N}$ the nonlinear RHS of (5.1), for our solution, the
bounds above show

$\|I\left\langle
D_{x}\right\rangle\mathcal{N}\|_{L^{1}_{s}L^{2}_{x}}\lesssim\varepsilon^{2}$
(5.11)

We use this to prove (5.6). The equation for $IA_{s}$ is of the type
$(\partial_{s}-\Delta)IA_{s}=I\partial_{x}\mathcal{N}$. Integrating the
equation and taking $L^{2}_{x}$ norms one obtains

$\|\int_{s_{1}}^{s_{2}}I\Delta
A_{s}\,\mathrm{d}s\|_{L^{2}_{x}}\leq\|IA_{s}(s_{1})\|_{L^{2}_{x}}+\|IA_{s}(s_{2})\|_{L^{2}_{x}}+\|I\partial_{x}\mathcal{N}\|_{L^{1}_{s}L^{2}_{x}}\lesssim\varepsilon.$

∎

As a consequence one obtains gauge-invariant bounds for the curvature, such as
control of the modified energy (1.6).

######  Corollary 5.2.

Suppose $A_{i}(t,x)$ is a regular solution of the (YM) equation in the
temporal gauge on $[t_{0},t_{1}]\times\mathbb{R}^{3}$ with
$(IA_{i},IF_{0i})\in\varepsilon L^{\infty}_{t}(H^{1}\times
L^{2})([t_{0},t_{1}]\times\mathbb{R}^{3})$.

Suppose $A_{\alpha}(t,x,s)$ solves (dYMHF) on
$[t_{0},t_{1}]\times\mathbb{R}^{3}\times[0,s_{0}]$ in DeTurck’s gauge
$A_{s}=\partial^{\ell}A_{\ell}$, where $s_{0}=N^{-2}$. Then, assuming
$\varepsilon\ll 1$, for any $t\in[t_{0},t_{1}]$ one has

$\|(Ns^{\frac{1}{2}})^{1-\sigma}F_{\alpha\beta}(s)\|_{L^{\infty}_{s}L^{2}_{x}}+\|(Ns^{\frac{1}{2}})^{1-\sigma}F_{\alpha\beta}(s)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}}\lesssim\varepsilon$
(5.12)

As a consequence, for (1.6) one has the gauge-invariant bound
$\mathcal{IE}(t)\lesssim\varepsilon^{2}$ for all $t\in[t_{0},t_{1}]$.

###### Proof.

Fix a time $t\in[t_{0},t_{1}]$. For $A_{i}$ we apply Proposition 5.1 and so it
obeys the estimates established in that proof.

We begin with $F_{ij}(s)$. First note that
$\|[A_{i},A_{j}](s)\|_{L^{2}_{x}}\lesssim\varepsilon^{2}$ since from (5.4) and
Sobolev embedding we have $A_{i}(s)\in\varepsilon L^{4}_{x}$. From the
$\|I\left\langle D_{x}\right\rangle A\|_{\tilde{S}}$ norm and interpolation
one obtains
$(Ns^{\frac{1}{2}})^{1-\sigma}\|\partial_{i}A_{j}(s)\|_{L^{2}_{x}}\lesssim\varepsilon$
and the same for $\partial_{j}A_{i}$ which concludes the first term in (5.12)
for $F_{ij}$. For the second term we note that from control of (5.7) we have
$\|P_{k}A_{i}(s)\|_{H^{1}}\lesssim\alpha_{k}\left\langle\frac{2^{k}}{N}\right\rangle^{1-\sigma}\left\langle
2^{2k}s\right\rangle^{-10}$ for all $k\geq 0$, $s\in[0,s_{0}]$ for some
sequence with
$\|(\alpha_{k})\|_{\ell^{2}_{k\in\mathbb{N}}}\lesssim\varepsilon$. We bound

$\|(Ns^{\frac{1}{2}})^{1-\sigma}\partial_{j}A_{i}(s)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}}^{2}\lesssim\sum_{0\leq
k\leq
N}\int_{0}^{s_{0}}(N^{2}s)^{1-\sigma}\alpha_{k}^{2}\frac{\,\mathrm{d}s}{s}+\sum_{k\geq
N}\alpha_{k}^{2}\int_{0}^{s_{0}}\frac{(2^{2k}s)^{1-\sigma}}{\left\langle
2^{2k}s\right\rangle^{20}}\frac{\,\mathrm{d}s}{s}$

which is $\lesssim\varepsilon^{2}$.

Now consider $F_{0i}(s)$, which solves the covariant equation (6.1). Expanding
it and canceling the $[A_{s},F]$ term with the $[\partial^{\ell}A_{\ell},F]$
term one obtains equation (9.3), which can be written in the form (5.17) for
$B=F_{0i}$. As before, the solution is obtained by Picard iteration. One may
use the parabolic energy estimate (5.8) to estimate the solution of the
equation

$(\partial_{s}-\Delta)IB=I(A\nabla B+\nabla AB+A^{2}B),\qquad
IB(0)\in\varepsilon L^{2}$

in the norm $\|B\|_{I^{-1}\tilde{S}}=\|IB\|_{\tilde{S}}$. The estimates we
need now are

$\displaystyle\|I(A\nabla B)\|_{L^{2}}+\|I(B\nabla A)\|_{L^{2}}$
$\displaystyle\lesssim\|IA\|_{H^{\frac{3}{2}+(1-\sigma)+}}\|IB\|_{H^{1+}}$
$\displaystyle\|I(A^{2}B)\|_{L^{2}}$
$\displaystyle\lesssim\|IA\|_{H^{2-\sigma}}^{2}\|IB\|_{H^{2-\sigma}}$

which follow from (2.19), (2.20) or from Littewood-Paley, by splitting into
frequencies higher or lower than $N$. For $A$ we already have control of the
norms from the proof of Proposition 5.1. This provides
$\|IF_{0i}\|_{\tilde{S}}\lesssim\varepsilon$. One obtains control in the norm
$\|\|\left\langle
2^{2k}s\right\rangle\|P_{k}IF_{0i}(s)\|_{L^{2}}\|_{L^{\infty}_{s}}\|_{\ell^{2}_{k}}$
as well. Then we conclude exactly like for $F_{ij}$. ∎

We record some $L^{\infty}_{x}$ bounds separately.

######  Lemma 5.3.

Suppose $A_{i}$ obeys the conditions of Proposition 5.1 uniformly on an
interval $I\times\mathbb{R}^{3}\times[0,s_{0}]$. Then

$\|A_{i}(s)\|_{L^{\infty}_{t}L^{\infty}_{x}}\lesssim\varepsilon
s^{-\frac{1}{4}}/(Ns^{\frac{1}{2}})^{1-\sigma}$ (5.13)
$\|A^{bil}_{i}(s)\|_{L^{\infty}_{t}L^{\infty}_{x}}\lesssim\varepsilon/(N^{2}s)^{1-\sigma}.$
(5.14)

###### Proof.

Fix a time $t$. The first bound (5.13) follows immediately from the bounds in
Proposition 5.1. Since
$(s_{1}-s)^{\frac{3}{4}}e^{(s_{1}-s)\Delta}:L^{2}_{x}\to L^{\infty}_{x}$, we
can bound using (5.13)

$\displaystyle\|A^{bil}_{i}(s_{1})\|_{L^{\infty}_{x}}$
$\displaystyle\lesssim\int_{0}^{s_{1}}(s_{1}-s)^{-\frac{3}{4}}\|A\|_{L^{\infty}_{x}}\|\nabla
A\|_{L^{2}_{x}}+\|A\|_{L^{\infty}_{x}}^{3}\,\mathrm{d}s$
$\displaystyle\lesssim\int_{0}^{s_{1}}(s_{1}-s)^{-\frac{3}{4}}s^{-\frac{1}{4}}/(Ns^{\frac{1}{2}})^{2(1-\sigma)}+s^{-\frac{3}{4}}/(Ns^{\frac{1}{2}})^{3(1-\sigma)}\,\mathrm{d}s.$

which proves (5.14) after integration. ∎

######  Remark 5.4.

The same argument as in Proposition 5.1 proves well-posedness of (5.1) in
$\dot{H}^{\rho}$ for any $\frac{1}{2}<\rho<1$. Instead of (5.9) one considers
$(\partial_{s}-\Delta)\left|D\right|^{\rho}A=\left|D\right|^{\rho}(A\nabla
A+A^{3})$ together with the norm
$\|A\|_{S^{\rho}}\vcentcolon=\|\left|D\right|^{\rho}A\|_{\tilde{S}}$. The
estimates (5.10) are replaced by

$\displaystyle\|\left|D\right|^{\rho}(f\nabla
g)\|_{L^{2}}\lesssim\|f\|_{\dot{H}^{\frac{3}{2}}\cap
L^{\infty}}\|g\|_{\dot{H}^{1+\rho}}$ (5.15)
$\displaystyle\|\left|D\right|^{\rho}(f_{1}f_{2}f_{3})\|_{L^{2}}\lesssim\|f_{1}\|_{\dot{H}^{1+\frac{\rho}{3}}}\|f_{2}\|_{\dot{H}^{1+\frac{\rho}{3}}}\|f_{3}\|_{\dot{H}^{1+\frac{\rho}{3}}}$

See (2.20). Control in the norm $\|\|\left\langle
2^{2k}s\right\rangle\|P_{k}A(s)\|_{\dot{H}^{\rho}}\|_{L^{\infty}_{s}}\|_{\ell^{2}_{k}}$
holds as well. One obtains

######  Proposition 5.5.

Let $\frac{1}{2}<\rho<1$. Suppose the initial data satisfies
$\bar{A}_{i}\in\varepsilon\dot{H}^{\rho}(\mathbb{R}^{3})$ and $\varepsilon>0$
is small enough. Then there exists a solution to (5.1) on
$\mathbb{R}^{3}\times[0,1]$ with $A_{i}(s=0)=\bar{A}_{i}$ which is in
$S^{\rho}$ and is the unique limit of regular solutions. In particular, one
has

$\sup_{s\in[0,1]}s^{\frac{\rho_{1}-\rho}{2}}\|A_{i}(s)\|_{\dot{H}^{\rho_{1}}}+\|s^{\frac{\rho_{1}-\rho}{2}}A_{i}(s)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}\dot{H}^{\rho_{1}}}\lesssim\varepsilon\qquad\forall\
\rho_{1}\in(\rho,\rho+1].$ (5.16)

Similarly to Proposition 5.5 one has a statement for the equation satisfied by
$B_{i}=F_{0i}$ in DeTurck’s gauge:

######  Proposition 5.6.

Let $-\frac{1}{2}<\gamma<0$. Suppose the connection coefficients $A_{i}$
satisfy the conditions in Proposition 5.5 and Remark 5.4 on
$\mathbb{R}^{3}\times[0,1]$. Suppose the initial data satisfies
$\bar{B}_{i}\in\varepsilon\dot{H}^{\gamma}(\mathbb{R}^{3})$ and
$\varepsilon>0$ is small enough. Then the schematic equation

$(\partial_{s}-\Delta)B=A\nabla B+B\nabla A+A^{2}B$ (5.17)

has a unique solution on $\mathbb{R}^{3}\times[0,1]$ with
$B_{i}(s=0)=\bar{B}_{i}$ and satisfies

$\sup_{s\in[0,1]}s^{\frac{\gamma_{1}-\gamma}{2}}\|B_{i}(s)\|_{\dot{H}^{\gamma_{1}}}+\|s^{\frac{\gamma_{1}-\gamma}{2}}B_{i}(s)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}\dot{H}^{\gamma_{1}}}\lesssim\varepsilon\qquad\forall\
\gamma_{1}\in(\gamma,\gamma+1].$

###### Proof.

As before, the solution is obtained by Picard iteration. One uses the
parabolic energy estimate (5.8) to estimate the solution of the equation

$(\partial_{s}-\Delta)\left|D\right|^{\gamma}B=\left|D\right|^{\gamma}(A\nabla
B+\nabla AB+A^{2}B)$

in the norm
$\|B\|_{S^{\gamma}}\vcentcolon=\|\left|D\right|^{\gamma}B\|_{\tilde{S}}$.
Recall (2.20). The estimates we need are

$\displaystyle\|A\nabla B\|_{\dot{H}^{\gamma}}+\|B\nabla
A\|_{\dot{H}^{\gamma}}$ $\displaystyle\lesssim\|A\|_{\dot{H}^{\frac{3}{2}}\cap
L^{\infty}}\|B\|_{\dot{H}^{1+\gamma}}$
$\displaystyle\|A^{2}B\|_{\dot{H}^{\gamma}}$
$\displaystyle\lesssim\|A\|_{\dot{H}^{1}}^{2}\|B\|_{\dot{H}^{1+\gamma}}$

One obtains control in the norm $\|\|\left\langle
2^{2k}s\right\rangle\|P_{k}B(s)\|_{\dot{H}^{\gamma}}\|_{L^{\infty}_{s}}\|_{\ell^{2}_{k}}$
as well. ∎

Now we can obtain estimates for the curvature at the initial time $t=0$.

######  Corollary 5.7.

Suppose $A_{i}(t,x)$ is a regular solution to the Yang-Mills equation in the
temporal gauge on $[-1,1]\times\mathbb{R}^{3}$ with initial data
$(\bar{A}_{i},\bar{E}_{i})\in\varepsilon(\dot{H}^{\sigma}\times\dot{H}^{\sigma-1})(\mathbb{R}^{3})$
and $\bar{A}_{i}\in\dot{H}^{\frac{1}{2}+}(\mathbb{R}^{3})$.

Suppose $A_{\alpha}(t,x,s)$ solves (dYMHF) on
$[-1,1]\times\mathbb{R}^{3}\times[0,1]$ in DeTurck’s gauge
$A_{s}=\partial^{\ell}A_{\ell}$. Then, assuming $\varepsilon\ll 1$, at $t=0$
one has

$\|s^{\frac{1-\sigma}{2}}F_{\alpha\beta}(s)\|_{L^{\infty}_{s}L^{2}_{x}}+\|s^{\frac{1-\sigma}{2}}F_{\alpha\beta}(s)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}}\lesssim\varepsilon$
(5.18)

As a consequence, at $t=0$, one has the gauge-invariant version

$\sup_{s\in[0,1]}s^{1-\sigma}\int_{\mathbb{R}^{3}}(F_{\alpha\beta},F_{\alpha\beta})(s)\,\mathrm{d}x+\int_{0}^{1}s^{1-\sigma}\int_{\mathbb{R}^{3}}(F_{\alpha\beta},F_{\alpha\beta})(s)\,\mathrm{d}x\frac{\,\mathrm{d}s}{s}\lesssim\varepsilon^{2}.$

###### Proof.

Consider $F_{ij}(s)$ first. For $\partial_{i}A_{j}$ and $\partial_{j}A_{i}$ we
use Proposition (5.5) with $\rho=\sigma$ and $\rho_{1}=1$. For $[A_{i},A_{j}]$
we use that from (5.16) and applying again Proposition (5.5) with
$\rho=\frac{1}{2}+$ one has

$s^{\frac{1-\sigma}{2}}s^{-\frac{\varepsilon^{\prime}}{2}}\|A_{i}(s)\|_{\dot{H}^{1-\varepsilon^{\prime}}}\lesssim\varepsilon,\qquad\|A_{i}(s)\|_{\dot{H}^{\frac{1}{2}+}}\lesssim
1.$

The bounds for $[A_{i},A_{j}]$ follow from this with Hölder and Sobolev
embeddings $\dot{H}^{1-}\times\dot{H}^{\frac{1}{2}+}\subseteq L^{6-}\times
L^{3+}\subseteq L^{2}.$

Now consider $F_{0i}(s)$, which solves the covariant equation (6.1). Expanding
it and canceling the $[A_{s},F]$ term with the $[\partial^{\ell}A_{\ell},F]$
term one obtains equation (9.3), which can be written as (5.17). Then (5.18)
is concluded by Proposition 5.6 with $\gamma=\sigma-1$ and $\gamma_{1}=0$. ∎

The next goal is to switch to caloric gauges which satisfy $A_{s}=0$. This is
accomplished by a change of gauge (DeTurck’s trick) obtained by solving the
ODE (5.19), see Proposition 5.10.

######  Lemma 5.8.

Let $A_{i},A_{s}=\partial^{\ell}A_{\ell}$ be given by Proposition 5.1. Let
$U(s)$ be the solution of the following ODE

$\partial_{s}U=UA_{s},\qquad U(s=0)=1$ (5.19)

Then one has

$U-1\in\varepsilon
L^{\infty}_{s}L^{\infty}_{x},\qquad\nabla_{x}U\in\varepsilon
L^{\infty}_{s}L^{2}_{x},\qquad I\Delta U\in\varepsilon
L^{\infty}_{s}L^{2}_{x}$ (5.20)

and $U^{-1}$ obeys the same bounds. If $A_{i}$ is a regular solution then $U$
is a regular gauge transformation.

###### Proof.

Write the integral form

$U(s_{1})=1+\int_{0}^{s_{1}}UA_{s}\,\mathrm{d}s$ (5.21)

The first two bounds in (5.20) follow from the first two bounds in (5.5) using
Gronwall’s inequality. Similarly one shows $U-1\in\varepsilon
L^{\infty}_{s}H^{\frac{3}{2}+}$.

The third bound in (5.20) is more delicate. We use an argument from [27,
Theorem 7] and [37, Appendix B]. One writes

$I\Delta U(s_{1})=I\int_{0}^{s_{1}}\Delta UA_{s}+2\nabla U\nabla A_{s}+U\Delta
A_{s}\,\mathrm{d}s$

The first two terms in the integral are handled by Gronwall’s inequality and
by

$\displaystyle\|I(fg)\|_{L^{2}}\lesssim\|If\|_{L^{2}}\|g\|_{H^{\frac{3}{2}+}}$
(5.22)
$\displaystyle\|I(fg)\|_{L^{2}}\lesssim\|If\|_{H^{1}}\|g\|_{H^{\frac{1}{2}+}}$
(5.23)

for $f=\Delta U,\ g=A_{s}$, respectively $f=\nabla U,\ g=\nabla A_{s}$,
provided we can also bound the third term.

For the third term one plugs in $U(s)$ from (5.21) and by Fubini’s theorem
write

$\int_{0}^{s_{1}}I\Delta
A_{s}\,\mathrm{d}s+I\int_{0}^{s_{1}}U(s^{\prime})A_{s}(s^{\prime})\left(\int_{s^{\prime}}^{s_{1}}\Delta
A_{s}(s)\,\mathrm{d}s\right)\,\mathrm{d}s^{\prime}$

Bound this in $L^{2}_{x}$ using (5.6) for the first integral, then use (5.22)
for $g=U(s^{\prime})A_{s}(s^{\prime})$ and $f=\int_{s^{\prime}}^{s_{1}}\Delta
A_{s}(s)\,\mathrm{d}s\in\varepsilon I^{-1}L^{2}$ by (5.6). Note that
$UA_{s}\in\varepsilon L^{1}_{s^{\prime}}H^{\frac{3}{2}+}$ by using the algebra
property for $H^{\frac{3}{2}+}$. ∎

###  5.2. Caloric gauge solutions

Setting the caloric condition $A_{s}=0$ in the covariant (cYMHF) and the
dynamic (dYMHF) Yang-Mills heat flows one obtains the initial value problems

$\partial_{s}A_{i}=\mathbf{D}^{\ell}F_{\ell i},\qquad A_{i}(s=0)=\bar{A}_{i}$
(5.24) $\partial_{s}A_{\alpha}=\mathbf{D}^{\ell}F_{\ell\alpha},\qquad
A_{\alpha}(s=0)=\bar{A}_{\alpha}$ (5.25)

######  Theorem 5.9 (Existence and uniqueness for regular solutions, [38],
[43]).

(1) Consider the initial value problem (5.24), i.e. the (cYMHF) in the caloric
gauge $A_{s}=0$, with regular initial data $\bar{A}_{i}\in H^{\infty}_{x}$ (in
particular having finite magnetic energy). Then there exists a unique global
regular solution $A_{i}\in
C^{\infty}_{s}H^{\infty}_{x}(\mathbb{R}^{3}\times[0,\infty))$.

(2) Consider the initial value problem (5.25), i.e. the (dYMHF) in the caloric
gauge with regular initial data $\bar{A}_{\alpha}\in
C^{\infty}_{t}H^{\infty}_{x}(I\times\mathbb{R}^{3})$ where $I$ is a time
interval. Then there exists a unique regular solution $A_{\alpha}\in
C^{\infty}_{t,s}H^{\infty}_{x}(I\times\mathbb{R}^{3}\times[0,1])$.

For part (1) see [38, Corollary 6.7] for the statement and proof in the form
stated here, of the similar result due to [43] in a different context.

Part (2) is also from [38]. Like in the first part, no smallness assumption is
required. We sketch the details of part (2) from [38, Thm 6.10, Lemma 6.9] and
[37, Step 1 in Thm 4.8, Prop. 5.7, Lemma 6.1] for the sake of completeness:

The spatial components $A_{i}$ satisfy (5.24) and are obtained from part (1),
so it remains to determine $A_{0}$. This can be done from the equation
$\partial_{s}A_{0}=\mathbf{D}^{\ell}F_{\ell 0}$ if one can determine what
$F_{\ell 0}$ should be, knowing that they have to solve the linear system with
smooth coefficients

$(\partial_{s}-\Delta_{A})B_{\ell}=2[F_{\ell}^{\ j},B_{j}],\qquad
B_{\ell}(0)=\bar{F}_{\ell 0}$

Proposition 5.7 in [37] provides a unique regular solution $B$, which is used
to define regular $A_{0}$ from the initial data $\bar{A}_{0}$ by integrating
$\partial_{s}A_{0}=\mathbf{D}^{\ell}B_{\ell}$. Then one obtains curvatures
$F_{\ell 0}=\partial_{\ell}A_{0}-\partial_{t}A_{\ell}+[A_{\ell},A_{0}]$ which
turn out to be equal to $B_{\ell}$, as guaranteed by [37, Lemma 6.1].
Therefore, the solution satisfies (5.25). Uniqueness is proved in [38, Lemma
6.9].

Now we transfer $I^{-1}H^{1}$ bounds from DeTurck’s gauge to the Caloric gauge
under a smallness assumption.

######  Proposition 5.10.

Let $A_{i}$ be the unique regular solution of (5.24) obtained in Theorem 5.9
from an initial data $\bar{A}_{i}$ assumed to be regular and to satisfy
$\|I\bar{A}_{i}\|_{H^{1}}\leq\varepsilon$. Then, if $\varepsilon$ is small
enough, one has

$\|IA_{i}\|_{L^{\infty}_{s}H^{1}(\mathbb{R}^{3}\times[0,1])}\lesssim\varepsilon.$
(5.26)

Moreover, $A_{i}$ is equivalent, using a regular gauge transformation, to

$A_{i}^{\prime}=U^{-1}A_{i}U-\partial_{i}U^{-1}U,\qquad U(s=0)=1$ (5.27)

the regular solution to (cYMHF) in DeTurck’s gauge
$A^{\prime}_{s}=\partial^{\ell}A^{\prime}_{\ell}$ with initial data
$\bar{A}_{i}$ obtained in Proposition 5.1.

The proof follows the strategy of [37, Theorem 5.1] and consists of a version
of _DeTurck’s trick_ [12], [15], see [37, Remark 5.9].

###### Proof.

Let $A^{\prime}$ be the regular solution to (cYMHF) in DeTurck’s gauge
$A^{\prime}_{s}=\partial^{\ell}A^{\prime}_{\ell}$ with initial data
$\bar{A}_{i}$ obtained in Proposition 5.1 and which satisfies
$IA_{i}^{\prime}\in\varepsilon
L^{\infty}_{s}H^{1}(\mathbb{R}^{3}\times[0,1])$. Making the gauge
transformation

$\tilde{A}_{i}=UA_{i}^{\prime}U^{-1}-\partial_{i}UU^{-1},\qquad\tilde{A}_{s}=UA_{s}^{\prime}U^{-1}-\partial_{s}UU^{-1}$
(5.28)

on $\mathbb{R}^{3}\times[0,1]$ one obtains a caloric solution $\tilde{A}_{i},\
\tilde{A}_{s}=0$ with $\tilde{A}_{i}(s=0)=\bar{A}_{i}$ provided $U(s)$ solves
the following ODE pointwise in $x$

$\partial_{s}U=UA_{s}^{\prime},\qquad U(s=0)=1$

Since $A_{s}^{\prime}$ is regular, also $U$, and therefore $\tilde{A}_{i}$
are. From the uniqueness part in Theorem 5.9 we deduce that
$\tilde{A}_{i}=A_{i}$.

By Lemma 5.8, $U$ and $U^{-1}$ satisfy the bounds (5.20), and therefore
$U-1,U^{-1}-1\in\varepsilon L^{\infty}_{s}X$ where $X$ is defined in (2.23).
Using (5.28), $IA_{i}^{\prime}\in\varepsilon L^{\infty}_{s}H^{1}$ and the
product estimate (2.25) one obtains (5.26). ∎

######  Remark 5.11.

Suppose $A_{\alpha}(t,x,s)$ is the the unique regular solution of (5.25)
obtained in Theorem 5.9 Part(2) on an interval
$[t_{0},t_{1}]\times\mathbb{R}^{3}\times[0,1]$ from regular and temporal
initial data $\bar{A}_{\alpha}\in
C^{\infty}_{t}H^{\infty}_{x}([t_{0},t_{1}]\times\mathbb{R}^{3})$,
$\bar{A}_{0}=0$ satisfying
$\|I\bar{A}_{i}\|_{L^{\infty}_{t}H^{1}}\leq\varepsilon$. Then we may apply
Proposition 5.10 for each $t\in[t_{0},t_{1}]$ obtaining a regular $U(t,x,s)$
with $U(t,x,s=0)=1$ and (5.26) holds uniformly in $t$. In addition to (5.27)
one has

$A_{\alpha}^{\prime}=U^{-1}A_{\alpha}U-\partial_{\alpha}U^{-1}U,\quad
F_{\alpha\beta}^{\prime}=U^{-1}F_{\alpha\beta}U,\qquad U(t,s=0)=1$ (5.29)

on $[t_{0},t_{1}]\times\mathbb{R}^{3}\times[0,1]$ where $A_{\alpha}^{\prime}$
is the solution to (dYMHF) in DeTurck’s gauge, with (temporal) initial data
$\bar{A}_{\alpha}$.

Finally, one has improved well-posedness in Sobolev spaces conditional on the
smallness of the (almost conserved) modified energy.

######  Proposition 5.12.

Suppose $A_{i}$ is the (caloric) regular solution of the problem (5.24) from
Theorem 5.9 (1) for regular initial data $\bar{A}_{i}$ with
$\|\bar{A}_{i}\|_{H^{\sigma}}\leq M_{0}$ and

$\sup_{0\leq m\leq
3}\sup_{s\in[0,s_{0}]}(Ns^{\frac{1}{2}})^{1-\sigma}\|(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}F(s)\|_{L^{2}(\mathbb{R}^{3})}\ll
1$ (5.30)

Then one has $\|A_{i}(s)\|_{H^{\sigma}}\leq C(M_{0})$ for all $s\in[0,s_{0}]$,
$s_{0}=N^{-2}$.

###### Proof.

First note that we have local wellposedness for the equation in DeTurck’s
gauge (5.1). This means a (regular) solution $\bar{A}_{i}(s)$ to (5.1) exists
on $[0,s^{*}]$ with $\bar{A}_{i}(0)=\bar{A}_{i}$,
$\|\bar{A}_{i}(s)\|_{H^{\sigma}}\lesssim M_{0}$ for all $s\in[0,s^{*}]$ with
$s^{*}\simeq M_{0}^{-p_{\sigma}}$ where $p_{\sigma}=\frac{2}{\sigma-1/2}$.
This is proved by the argument in Proposition 5.1 and Remark 5.4, one just
replaces (5.15) by their homogeneous version and one uses the smallness of the
interval instead of smallness of the initial data.

We next consider the ODE (5.19) for $\bar{A}$, i.e.
$\partial_{s}U=U\bar{A}_{s}$, $U(0)=1$ where
$\bar{A}_{s}=\partial^{\ell}\bar{A}_{\ell}$, which defines the gauge
transformation from DeTurck’s gauge to the caloric gauge. The $A_{i}$ and
$\bar{A}_{i}$ are related by

$A_{i}=U\bar{A}_{i}U^{-1}-\partial_{i}UU^{-1},\qquad
A_{s}=U\bar{A}_{s}U^{-1}-\partial_{s}UU^{-1}=0$

Now (5.20) is replaced by: $U\in C(M_{0})L^{\infty}_{s}L^{\infty}_{x}$ and
$\nabla_{x}U\in C(M_{0})L^{\infty}_{s}H^{\sigma}$ on $[0,s^{*}]$, and the same
for $U^{-1}$. We denote by $C(M_{0})$ a constant that can change from line to
line. By (2.24) one obtains $\|A_{i}(s)\|_{H^{\sigma}}\lesssim C(M_{0})$ for
$s\in[0,s^{*}]$.

It remains to extend this bound to $[0,s_{0}]$ using (5.30). Let $p\in[4,6]$
be the Sobolev exponent such that $H^{\sigma}\subset L^{p}$. From (5.24) and
the covariant Gagliardo-Nirenberg inequality (2.11) one has
$\|\partial_{s}A_{i}(s)\|_{L^{p}}\lesssim
s^{-1}(Ns^{\frac{1}{2}})^{\sigma-1}$. Integrating one has

$\|A_{i}(s)\|_{L^{p}}\lesssim\|A_{i}(s^{*})\|_{H^{\sigma}}+(Ns^{*\frac{1}{2}})^{\sigma-1}\lesssim
C(M_{0}),\qquad s\in[s^{*},s_{0}].$

Now we let $1/q=1/2-1/p$, $q\in[3,4]$ and bound

$\displaystyle\|\partial_{s}A_{i}(s)\|_{H^{1}}$
$\displaystyle\lesssim\|\mathbf{D}_{x}F(s)\|_{L^{2}}+\|\mathbf{D}_{x}^{2}F(s)\|_{L^{2}}+\|A_{i}(s)\|_{L^{p}}\|\mathbf{D}_{x}F(s)\|_{L^{q}}$
$\displaystyle\lesssim
s^{-1}(Ns^{\frac{1}{2}})^{\sigma-1}+C(M_{0})s^{-1}(Ns^{\frac{1}{2}})^{\sigma-1}$

Integrating again one has

$\|A_{i}(s)\|_{H^{\sigma}}\leq\|A_{i}(s^{*})\|_{H^{\sigma}}+\int_{s^{*}}^{s}\|\partial_{s}A_{i}(s^{\prime})\|_{H^{1}}\,\mathrm{d}s^{\prime}\lesssim
C(M_{0})(Ns^{*\frac{1}{2}})^{\sigma-1}\lesssim C(M_{0})$

for all $s\in[s^{*},s_{0}]$. ∎

##  6\. Energy estimates for the curvature and the tension field

The goal of this section is to obtain weighted bounds for higher covariant
derivatives for the curvature $F_{\alpha\beta}(s)$ and the tension field
$w_{x}(s)$, first globally in space and then locally - with square summability
of the local bounds, proving Proposition 1.9. All estimates in this section
are covariant, independent of the choice of gauge.

Let $A_{t,x,s}$ be a regular solution to (dYMHF) on an interval
$I\times\mathbb{R}^{3}\times[0,s_{0}]$.

Then the curvature satisfies

$\left(\mathbf{D}_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)F_{\alpha\beta}=2[F_{\alpha}^{\
\ell},F_{\ell\beta}]$ (6.1)

This is obtained from Bianchi’s identity (2.3), commuting the covariant
derivatives by (2.4) and applying Bianchi’s identity again.

For higher covariant derivatives one computes (see [38, Section 5] for
details):

$\left(\mathbf{D}_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)\mathbf{D}_{x}^{(m)}F=\sum_{i=0}^{m}\mathcal{O}\left(\mathbf{D}_{x}^{(i)}F,\mathbf{D}_{x}^{(m-i)}F\right)$
(6.2)

Now let $w_{k}(s)$ be the Yang-Mills tension field defined in (1.9). It
satisfies

$\left(\mathbf{D}_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)w_{k}=2\left[F_{k}^{\ell},w_{\ell}\right]+2\left[F^{0\ell},\mathbf{D}_{k}F_{0\ell}+2\mathbf{D}_{\ell}F_{k0}\right]$
(6.3)

See [39, Eq. (8.8)], [37, Appendix A]. For higher covariant derivatives one
has

$\left(\mathbf{D}_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)\mathbf{D}_{x}^{(m)}w_{k}=\sum_{i=0}^{m}\mathcal{O}\left(\mathbf{D}_{x}^{(i)}F,\mathbf{D}_{x}^{(m-i)}w_{x}\right)+\mathcal{O}\left(\mathbf{D}_{x}^{(i)}F,\mathbf{D}_{x}^{(m+1-i)}F\right)$
(6.4)

###  6.1. Covariant energy estimates and the comparison principle

Suppose $G$ solves the covariant heat equation

$\left(\mathbf{D}_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)G=\mathcal{N}$

on an interval $\mathbb{R}^{3}\times[s_{1},s_{2}]$ and let $p>0$. Then one has

$\displaystyle\sup_{s\in[s_{1},s_{2}]}s^{2p}\int_{\mathbb{R}^{3}}(G,G)(s)\,\mathrm{d}x+\int_{s_{1}}^{s_{2}}s^{2p}\int_{\mathbb{R}^{3}}(\mathbf{D}^{\ell}G,\mathbf{D}_{\ell}G)(s)\,\mathrm{d}x\,\mathrm{d}s\lesssim$
$\displaystyle
s_{1}^{2p}\int_{\mathbb{R}^{3}}(G,G)(s_{1})\,\mathrm{d}x+p\int_{s_{1}}^{s_{2}}s^{2p-1}\int_{\mathbb{R}^{3}}(G,G)(s)\,\mathrm{d}x\,\mathrm{d}s+\left(\int_{s_{1}}^{s_{2}}s^{p}\|\mathcal{N}(s)\|_{L^{2}_{x}}\,\mathrm{d}s\right)^{2}$

or, in short, denoting $J=[s_{1},s_{2}]$,

$\|s^{p}G\|_{L^{\infty}_{s}L^{2}_{x}(J)}+\|s^{p}s^{\frac{1}{2}}\mathbf{D}_{x}G\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}(J)}\lesssim
s_{1}^{p}\|G(s_{1})\|_{L^{2}_{x}}+p\|s^{p}G\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}(J)}+\|s^{p}\mathcal{N}\|_{L^{1}_{s}L^{2}_{x}(J)}$
(6.5)

This is obtained by applying the Fundamental theorem of calculus on an
interval to the function
$\varphi(s)=\frac{1}{2}s^{2p}\int_{\mathbb{R}^{3}}(G,G)(s)\,\mathrm{d}x$,
using the equation, integrating by parts, applying Cauchy-Schwarz and
absorbing a term into the LHS. See [38, Lemma 4.5] for more details.

As a consequence of the diamagnetic-type inequality ([38, Lemma 4.6], [54,
Lemma 3.9])

$(\partial_{s}-\Delta)\left|G\right|\leq\left|\mathcal{N}\right|$

and the Duhamel principle, [38, Corollary 4.7] formulates the comparison
principle:

######  Lemma 6.1.

The following inequality holds point-wise

$\left|G(s)\right|\leq
e^{s\Delta}\left|G(0)\right|+\int_{0}^{s}e^{(s-s^{\prime})\Delta}\left|(\mathbf{D}_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell})G(s^{\prime})\right|\,\mathrm{d}s^{\prime}.$

We will also use the following estimate for Duhamel terms

######  Lemma 6.2 ( [38]).

For $p<\frac{3}{4}$ and $s_{0}\leq 1$ the following estimate holds on
$[0,s_{0}]\times\mathbb{R}^{3}$:

$\|s_{1}^{p}\int_{0}^{s_{1}}e^{(s_{1}-s)\Delta}\mathcal{N}(s)\,\mathrm{d}s\|_{L^{2}_{\frac{\,\mathrm{d}s_{1}}{s_{1}}}L^{2}_{x}}\lesssim\|s^{p+\frac{1}{4}}\mathcal{N}(s)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{1}_{x}}$

See [38, Lemma 4.8].

###  6.2. Higher covariant derivatives bounds for $F$ and $w$

Denote

$\alpha^{m}(s^{\prime})\vcentcolon=\|(Ns^{\frac{1}{2}})^{1-\sigma}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}F_{\alpha\beta}(s)\|_{L^{\infty}_{s}L^{2}_{x}([0,s^{\prime}]\times\mathbb{R}^{3})}$
$\beta^{m}(s^{\prime})\vcentcolon=\|(Ns^{\frac{1}{2}})^{1-\sigma}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}F_{\alpha\beta}(s)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}([0,s^{\prime}]\times\mathbb{R}^{3})}$

######  Proposition 6.3.

Let $A_{t,x,s}$ be a regular solution to (dYMHF) on an interval
$J\times\mathbb{R}^{3}\times[0,s_{0}]$. Fix a time $t\in J$ and denote
$F(s)=F(t,s)$. Let $\eta\ll 1$ and suppose

$\alpha^{0}(s_{0})+\beta^{0}(s_{0})\lesssim\eta$

Then, for $N^{2}s_{0}=1$ and for all $0\leq m\leq 10$ one has

$\alpha^{m}(s_{0})+\beta^{m+1}(s_{0})\lesssim_{m}\eta$

###### Proof.

Denote by $\mathcal{N}^{m}$ the RHS of (6.2) and let

$\gamma^{m}(s^{\prime})\vcentcolon=\|(Ns^{\frac{1}{2}})^{1-\sigma}s^{\frac{m}{2}}\|\mathcal{N}^{m}(s)\|_{L^{2}_{x}}\|_{L^{1}_{s}[0,s^{\prime}]}$

Applying (6.5) to (6.2) with $p=\frac{m}{2}+\frac{1-\sigma}{2}$ on $[0,s]$ and
multiplying by $N^{1-\sigma}$ one obtains

$\alpha^{m}(s)+\beta^{m+1}(s)\lesssim\beta^{m}(s)+\gamma^{m}(s)$

First let $m=0$. Using Holder’s inequality and (2.11) one has

$\|[F,F](s)\|_{L^{2}_{x}}\lesssim\|F(s)\|_{L^{2}_{x}}^{\frac{1}{2}}\|\mathbf{D}_{x}F(s)\|_{L^{2}_{x}}^{\frac{3}{2}}$

which implies that one can bound

$\beta^{1}(s)\lesssim\eta+\gamma^{1}(s)\lesssim\eta+\eta^{\frac{1}{2}}\frac{s^{1/4}}{(Ns^{\frac{1}{2}})^{1-\sigma}}\left(\beta^{1}(s)\right)^{\frac{3}{2}}$

Since $\lim_{s\to 0}\beta^{1}(s)=0$ one can use a continuity argument to
obtain $\beta^{1}(s_{0})\lesssim\eta$. Then one proceeds by induction on $m$,
similarly bounding

$\alpha^{m}(s)+\beta^{m+1}(s)\lesssim\beta^{m}(s)+\gamma^{m}(s)\lesssim\eta+\eta^{\frac{1}{2}}\frac{s^{1/4}}{(Ns^{\frac{1}{2}})^{1-\sigma}}\left(\beta^{m+1}(s)\right)^{\frac{3}{2}}$

and concluding like before. ∎

Now we turn to the Yang-Mills tension field $w_{x}$.

######  Proposition 6.4.

Given (YM) solutions $A_{t,x,s}$ and $F_{\alpha\beta}$ satisfying the
assumptions from Proposition 6.3, for
$w_{k}(s)\vcentcolon=\mathbf{D}^{\alpha}F_{\alpha k}(s)$ and $0\leq m\leq 8$
one has

$\|(N^{2}s)^{1-\sigma}s^{\frac{1}{4}}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}w_{x}(s)\|_{L^{\infty}_{s}L_{x}^{2}\cap
L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}([0,s_{0}]\times\mathbb{R}^{3})}\lesssim\eta^{2}$
(6.6)

Moreover,

$(N^{2}s)^{1-\sigma}\|w_{x}(s)\|_{L^{1}_{x}}\lesssim
s^{\frac{1}{2}}\eta\qquad\forall s\in[0,s_{0}].$ (6.7)

###### Proof.

Denote by $\delta_{m}$ and $\rho_{m}$ the $L^{\infty}_{s}L_{x}^{2}$,
respectively the $L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}$ norms from (6.6).
Note that $\mathbf{D}_{x}^{m}w(s=0)=0$ because $F$ is assumed to solve the
Yang-Mills equation at $s=0$.

We begin by showing $\rho_{0}\lesssim\eta^{2}$.

We use equation (6.3), the fact that $w_{x}(s=0)=0$, the Comparison principle
in Lemma 6.1, Lemma 6.2 with $p=\frac{1}{4}+1-\sigma$ and the following
estimates

$\|(N^{2}s)^{1-\sigma}s^{\frac{1}{2}}[F,w_{x}]\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{1}_{x}}\lesssim\|(Ns^{\frac{1}{2}})^{1-\sigma}F\|_{L^{\infty}_{s}L^{2}_{x}}\|s^{\frac{1}{4}}(Ns^{\frac{1}{2}})^{\sigma-1}\|_{L^{\infty}_{s}}\rho_{0}\lesssim\eta
s_{0}^{\frac{1}{4}}\rho_{0}$
$\|(N^{2}s)^{1-\sigma}s^{\frac{1}{2}}[F,\mathbf{D}_{x}F]\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{1}_{x}}\lesssim\|(Ns^{\frac{1}{2}})^{1-\sigma}F\|_{L^{\infty}_{s}L^{2}_{x}}\|(Ns^{\frac{1}{2}})^{1-\sigma}s^{\frac{1}{2}}\mathbf{D}_{x}F\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}}\lesssim\eta^{2}$

to obtain

$\rho_{0}\lesssim\eta s_{0}^{\frac{1}{4}}\rho_{0}+\eta^{2}.$

and then we absorb the small $\rho_{0}$ term to the left side since $\eta\ll
1$.

The same estimate for $[F,\mathbf{D}_{x}F]$, but with a different factor of
$s$ and bounding in $L^{1}_{s}$, together with the Comparison principle in
Lemma 6.1 will prove (6.7). Denoting
$M_{1}\vcentcolon=\|w_{x}\|_{L^{\infty}_{s}L_{x}^{1}[0,s_{1}]}$, now one
absorbs the following term to the LHS:

$\|[F,w_{x}]\|_{L^{1}_{s}L^{1}_{x}}\lesssim
M_{1}\int_{0}^{s_{1}}\|F(s)\|_{L^{\infty}_{x}}\,\mathrm{d}s\lesssim
M_{1}s_{1}^{\frac{1}{4}}/(N^{2}s)^{1-\sigma}\eta\ll M_{1}.$ (6.8)

Now we proceed by induction. We assume we have showed
$\rho_{i}\lesssim\eta^{2}$ for all $0\leq i\leq m$ and we prove

$\delta_{m}+\rho_{m+1}\lesssim\rho_{m}+\|(N^{2}s)^{1-\sigma}s^{\frac{1}{4}}s^{\frac{m}{2}}\mathcal{N}_{m}(s)\|_{L^{1}_{s}L^{2}_{x}}\lesssim\eta^{2}$

where $\mathcal{N}_{m}(s)$ is the RHS of (6.4). The first inequality is the
energy estimate (6.5). We will use

$\|\mathbf{D}_{x}^{i}F\|_{L^{\infty}_{x}}\lesssim\left\|\mathbf{D}^{i+1}_{x}F\right\|_{L_{x}^{2}}^{1/2}\left\|\mathbf{D}^{i+2}_{x}F\right\|_{L_{x}^{2}}^{1/2}$

which follows from (2.11). We bound the first component of
$\mathcal{N}_{m}(s)$

$\|(N^{2}s)^{1-\sigma}s^{\frac{1}{4}}s^{\frac{m}{2}}\|\mathbf{D}_{x}^{i}F\|_{L^{\infty}}\|\mathbf{D}_{x}^{m-i}w_{x}\|_{L^{2}}\|_{L^{1}_{s}}\lesssim\eta\|s^{\frac{1}{4}}(Ns^{\frac{1}{2}})^{\sigma-1}\|_{L^{\infty}_{s}}\rho_{m-i}\lesssim\eta^{3}s_{0}^{\frac{1}{4}}$

and for the second component we may assume that $i\leq m+1-i$ and estimate

$\|(N^{2}s)^{1-\sigma}s^{\frac{1}{4}}s^{\frac{m}{2}}\|\mathbf{D}_{x}^{i}F\|_{L^{\infty}}\|\mathbf{D}_{x}^{m+1-i}F\|_{L^{2}}\|_{L^{1}_{s}}\lesssim\|(Ns^{\frac{1}{2}})^{1-\sigma}(s^{\frac{1}{2}}\mathbf{D}_{x})^{i+1}F\|^{1/2}_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}}\times$
$\|(Ns^{\frac{1}{2}})^{1-\sigma}(s^{\frac{1}{2}}\mathbf{D}_{x})^{i+2}F\|^{1/2}_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}}\|(Ns^{\frac{1}{2}})^{1-\sigma}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m+1-i}F\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}}\lesssim\eta^{2}$

∎

###  6.3. Defining the balls coefficients - Proof of Proposition 1.9

Fix a time $t\in J$ and denote $F(s)=F(t,s)$. It is easily seen that the proof
is uniform in $t$ . If one would only have $L^{2}$-type norms in (1.10),
(1.11) such as $L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}$ or $L^{2}_{x}$
instead of $L^{\infty}_{s}L^{2}_{x}$ the square summability would be obvious,
because by Proposition 6.3 we control all higher derivatives of $F$ on
$[0,s_{0}]\times\mathbb{R}^{3}$:

$\|(Ns^{\frac{1}{2}})^{1-\sigma}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}F(s)\|_{L^{\infty}_{s}L^{2}[0,s_{0}]}+\|(Ns^{\frac{1}{2}})^{1-\sigma}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m+1}F(s)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}[0,s_{0}]}\lesssim\eta$
(6.9)

and similarly for $w$, due to Proposition 6.4 we have

$\|(N^{2}s)^{1-\sigma}s^{\frac{1}{4}}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}w_{x}(s)\|_{L^{\infty}_{s}L_{x}^{2}\cap
L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}([0,s_{0}]\times\mathbb{R}^{3})}\lesssim\eta^{2}$
(6.10)

To deal with the $L^{\infty}_{s}L^{2}_{x}$ we use the Energy estimates from
section 6.1.

Consider a collection of spatial bump functions $(\chi_{j})_{j\in\mathcal{J}}$
with finitely overlapping supports which satisfy
$1\lesssim\sum_{j\in\mathcal{J}}\chi_{j}^{2}(x)$. We construct square summable
coefficients $(c_{j})_{j\in\mathcal{J}}$ satisfying

$\displaystyle\alpha_{j}^{m}\vcentcolon=\|\chi_{j}(Ns^{\frac{1}{2}})^{1-\sigma}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}F(s)\|_{L^{\infty}_{s}L^{2}_{x}([0,s_{0}]\times\mathbb{R}^{3})}\leq
c_{j}$ (6.11)
$\displaystyle\beta_{j}^{m}\vcentcolon=\|\chi_{j}(Ns^{\frac{1}{2}})^{1-\sigma}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}F(s)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}([0,s_{0}]\times\mathbb{R}^{3})}\leq
c_{j}$ (6.12)

and

$\displaystyle\delta_{j}^{m}\vcentcolon=\|\chi_{j}(N^{2}s)^{1-\sigma}s^{\frac{1}{4}}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}w_{x}(s)\|_{L^{\infty}_{s}L^{2}_{x}([0,s_{0}]\times\mathbb{R}^{3})}\leq
c_{j}$ (6.13)
$\displaystyle\rho_{j}^{m}\vcentcolon=\|\chi_{j}(N^{2}s)^{1-\sigma}s^{\frac{1}{4}}(s^{\frac{1}{2}}\mathbf{D}_{x})^{m}w_{x}(s)\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}L^{2}_{x}([0,s_{0}]\times\mathbb{R}^{3})}\leq
c_{j}$ (6.14)

for all $0\leq m\leq 8$ and then we take each $\chi_{j}\equiv 1$ on $B_{j}$ to
obtain the conclusion.

It suffices to show

$\sum_{j\in\mathcal{J}}(\alpha_{j}^{m})^{2}\lesssim\eta^{2},\quad\sum_{j\in\mathcal{J}}(\beta_{j}^{m})^{2}\lesssim\eta^{2},\quad\sum_{j\in\mathcal{J}}(\delta_{j}^{m})^{2}\lesssim\eta^{2},\quad\sum_{j\in\mathcal{J}}(\rho^{m})^{2}\lesssim\eta^{2}.$
(6.15)

Note that this is automatically true for the $\beta_{j}^{m}$’s and
$\rho_{j}^{m}$’s so it remains to focus on the $\alpha_{j}^{m}$’s and
$\delta_{j}^{m}$’s.

From (6.1) we obtain the parabolic equation of $\chi_{j}F$

$\left(\mathbf{D}_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)\chi_{j}F=\mathcal{N}_{j}\vcentcolon=2\chi_{j}[F,F]-\Delta\chi_{j}F-2\partial_{\ell}\chi_{j}\mathbf{D}^{\ell}F$

while from (6.3) we obtain the equation of $\chi_{j}w_{x}$

$\left(\mathbf{D}_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)\chi_{j}w_{x}=\mathcal{N}_{j}^{\prime}\vcentcolon=2\chi_{j}\left[F,w_{x}\right]+2\chi_{j}\left[F,\mathbf{D}_{x}F\right]-\Delta\chi_{j}w_{x}-2\partial_{\ell}\chi_{j}\mathbf{D}^{\ell}w_{x}$

By (6.5) on $[0,s_{0}]$, (2.11), (6.9) and Holder in $\frac{\,\mathrm{d}s}{s}$
we obtain

$\displaystyle\alpha_{j}^{0}$
$\displaystyle\lesssim\beta_{j}^{0}+\|(Ns^{\frac{1}{2}})^{1-\sigma}\mathcal{N}_{j}(s)\|_{L^{1}_{s}L^{2}_{x}[0,s_{0}]}$
$\displaystyle\lesssim\beta_{j}^{0}+\|(Ns^{\frac{1}{2}})^{1-\sigma}(\|\mathbf{D}F\|_{L^{2}_{x}}^{\frac{1}{2}}\|\mathbf{D}^{2}F\|_{L^{2}_{x}}^{\frac{1}{2}}\|\chi_{j}F\|_{L^{2}_{x}}+\|\Delta\chi_{j}F\|_{L^{2}_{x}}+\|\nabla\chi_{j}\mathbf{D}F\|_{L^{2}_{x}})\|_{L^{1}_{s}[0,s_{0}]}$
$\displaystyle\lesssim\beta_{j}^{0}+\eta\beta_{j}^{0}\|s^{\frac{1}{4}}(Ns^{\frac{1}{2}})^{\sigma-1}\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}[0,s_{0}]}+\tilde{\beta}_{j}^{0}\|s\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}[0,s_{0}]}+\bar{\beta}_{j}^{1}\|s^{\frac{1}{2}}\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}[0,s_{0}]}$
$\displaystyle\lesssim\beta_{j}^{0}+\tilde{\beta}_{j}^{0}+\bar{\beta}_{j}^{1}$

where $\tilde{\beta}_{j}^{m}$ and $\bar{\beta}_{j}^{m}$ (respectively
$\tilde{\rho}_{j}^{m}$ and $\bar{\rho}_{j}^{m}$ ) are defined to be the terms
in (6.12) (respectively (6.14)) with $\chi_{j}$ replaced by $\Delta\chi_{j}$
and $\nabla\chi_{j}$, which are also square summable, thus proving (6.15) for
the $\alpha_{j}^{0}$’s. Similarly

$\displaystyle\delta_{j}^{0}$
$\displaystyle\lesssim\rho_{j}^{0}+\|(N^{2}s)^{1-\sigma}s^{\frac{1}{4}}\mathcal{N}_{j}^{\prime}(s)\|_{L^{1}_{s}L^{2}_{x}[0,s_{0}]}$
$\displaystyle\lesssim\rho_{j}^{0}+\|(N^{2}s)^{1-\sigma}s^{\frac{1}{4}}(\|\mathbf{D}F\|_{L^{2}_{x}}^{\frac{1}{2}}\|\mathbf{D}^{2}F\|_{L^{2}_{x}}^{\frac{1}{2}}\|\chi_{j}w\|_{L^{2}_{x}}+\|\mathbf{D}F\|_{L^{2}_{x}}^{\frac{1}{2}}\|\mathbf{D}^{2}F\|_{L^{2}_{x}}^{\frac{1}{2}}\|\chi_{j}\mathbf{D}F\|_{L^{2}_{x}}$
$\displaystyle\qquad\
+\|\Delta\chi_{j}w\|_{L^{2}_{x}}+\|\nabla\chi_{j}\mathbf{D}w\|_{L^{2}_{x}})\|_{L^{1}_{s}[0,s_{0}]}$
$\displaystyle\lesssim\rho_{j}^{0}+\eta\rho_{j}^{0}\|s^{\frac{1}{4}}(Ns^{\frac{1}{2}})^{\sigma-1}\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}[0,s_{0}]}+\eta\beta^{1}_{j}+\tilde{\rho}_{j}^{0}\|s\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}[0,s_{0}]}+\bar{\rho}_{j}^{1}\|s^{\frac{1}{2}}\|_{L^{2}_{\frac{\,\mathrm{d}s}{s}}[0,s_{0}]}$
$\displaystyle\lesssim\rho_{j}^{0}+\beta^{1}_{j}+\tilde{\rho}_{j}^{0}+\bar{\rho}_{j}^{1}$

This proves (6.15) for $m=0$.

The same argument holds for $m\geq 1$. By (6.2) the equations for higher
derivatives of $F$ are
$\left(\mathbf{D}_{s}-\mathbf{D}^{\ell}\mathbf{D}_{\ell}\right)(\chi_{j}\mathbf{D}_{x}^{(m)}F)=\mathcal{N}_{j}^{m}$
where

$\mathcal{N}_{j}^{m}\vcentcolon=\chi_{j}\sum_{k=0}^{m}\mathcal{O}\left(\mathbf{D}_{x}^{(k)}F,\mathbf{D}_{x}^{(m-k)}F\right)-\Delta\chi_{j}\mathbf{D}_{x}^{(m)}F-\nabla\chi_{j}\mathbf{D}_{x}^{(m+1)}F$

Applying (6.5) with $p=\frac{m}{2}+\frac{1-\sigma}{2}$, multiplying by
$N^{1-\sigma}$ and doing the same manipulations one obtains

$\alpha_{j}^{m}\lesssim\beta_{j}^{m}+\eta\sum_{k=0}^{m}\beta_{j}^{k}+\tilde{\beta}_{j}^{m}+\bar{\beta}_{j}^{m+1}$

which is square-summable. Similarly one writes the equation of higher
covariant derivatives of $\chi_{j}w_{x}$ following from (6.4) and estimate

$\delta_{j}^{m}\lesssim\rho_{j}^{m}+\eta\sum_{k=0}^{m}\rho_{j}^{k}+\eta\sum_{k=1}^{m+1}\beta_{j}^{k}+\tilde{\rho}_{j}^{m}+\bar{\rho}_{j}^{m+1}$

which concludes the proof.

##  7\. The local change of gauge: Proof of Proposition 1.10

This section is devoted to the proof of Proposition 1.10, which is an
Uhlenbeck-type lemma for fractional regularities below $L^{2}$, as mentioned
in Remark 1.11. Unlike the previous section, which consisted of only gauge-
invariant bounds, the goal here is the opposite: to be as specific as possible
with the choice of gauge in order to obtain Sobolev bounds for $A_{i}$,
$\partial_{t}A_{i}$ from the curvature bounds.

We begin by recalling the classical result of Uhlenbeck [56], which played a
crucial role in the proof of finite energy global well-posedness [27] stated
in Theorem 1.1.

######  Theorem 7.1 (Uhlenbeck’s lemma [56]).

Let $A_{x}:B\to\mathfrak{g}$ have curvature $F_{ij}$ bounded in $L^{2}(B)$.
There exists a $\delta>0$ such that if

$\|F\|_{L^{\frac{3}{2}}(B)}<\delta$

then there exists $U:B\to G$ such that the transformation

$\tilde{A}_{i}=UA_{i}U^{-1}-\partial_{i}UU^{-1}$ (7.1)

satisfies $\partial^{\ell}\tilde{A}_{\ell}=0$ in $B$,
$x^{\ell}\tilde{A}_{\ell}=0$ on $\partial B$ and

$\|\tilde{A}\|_{H^{1}(B)}\lesssim\|F\|_{L^{2}(B)}.$

After smoothing the curvature $F_{\alpha\beta}$ using the Yang-Mills heat
flow, we will apply Uhlenbeck’s lemma at $s=s_{0}$ to first obtain an $H^{1}$
bound.

By elliptic regularity, one obtains additional bound on $\tilde{A}$:

######  Lemma 7.2.

Suppose $\tilde{A}_{i}$ is like in Theorem 7.1 and assume the (gauge-
invariant) bounds

$\|(s_{0}^{\frac{1}{2}}\tilde{\mathbf{D}}_{x})^{m}\tilde{F}\|_{L^{2}(B)}\leq\delta$

for $0\leq m\leq 2$ and some $s_{0}\ll 1$. Then

$s_{0}^{\frac{1}{4}}\|\tilde{A}_{i}\|_{L^{\infty}(B)}+s_{0}^{\frac{3}{4}}\|\partial_{x}\tilde{A}_{i}\|_{L^{\infty}(B)}\lesssim\delta.$

###### Proof.

First note that $s_{0}^{\frac{1}{2}}\partial_{x}\tilde{F}\in\delta L^{2}(B)$.
Indeed, by (2.12) one has $s_{0}^{\frac{1}{2}}\tilde{F}\in\delta L^{6}(B)$ and
then $s_{0}^{\frac{1}{2}}[\tilde{A},\tilde{F}]\in\delta L^{2}(B)$.

From $\partial^{\ell}\tilde{A}_{\ell}=0$ one obtains the elliptic equation

$\Delta\tilde{A_{j}}=\partial^{i}\tilde{F}_{ij}-\partial^{i}[\tilde{A}_{i},\tilde{A}_{j}].$
(7.2)

We want to show $s_{0}^{\frac{1}{2}}[\tilde{A},\nabla\tilde{A}]\in\delta
L^{2}(B)$. We begin with some weaker bounds, like
$[\tilde{A},\nabla\tilde{A}]\in\delta L^{\frac{3}{2}}(B)$ which follows from
Hölder and Sobolev embedding. Then $s_{0}^{\frac{1}{2}}\tilde{A}_{j}\in\delta
W^{2,\frac{3}{2}}(B)\subset\delta L^{p}(B)$ for all $p<\infty$. This implies
$s_{0}^{\frac{1}{2}}[\tilde{A},\nabla\tilde{A}]\in\delta L^{2-}(B)$ and
$s_{0}^{\frac{1}{2}}\Delta\tilde{A}\in\delta L^{2-}(B)$ from which, together
with $\tilde{A}\in\delta W^{1,2}(B)\subset\delta L^{6}(B)$ and Gagliardo-
Nirenberg inequalities one obtains $s_{0}^{\frac{1}{4}+}\tilde{A}\in\delta
L^{\infty}(B)$. Now we obtain
$s_{0}^{\frac{1}{2}}[\tilde{A},\nabla\tilde{A}]\in\delta L^{2}(B)$ from which
$s_{0}^{\frac{1}{2}}\Delta\tilde{A}\in\delta L^{2}(B)$ and similarly conclude
$s_{0}^{\frac{1}{4}}\tilde{A}\in\delta L^{\infty}(B)$.

For the second inequality note that $s_{0}\partial_{x}^{2}\tilde{F}\in\delta
L^{2}(B)$. Differentiating (7.2) one obtains an elliptic equation for
$\partial_{x}\tilde{A}_{j}$. Then we conclude by a similar argument as before,
we omit the details. ∎

###  7.1. Proof of Proposition 1.10 Part (1)

Step 1. (Change of gauge) By integrating the equation
$\partial_{s}A_{i}=\mathbf{D}^{\ell}F_{\ell i}$ one obtains

$A_{i}(s)=\int_{s}^{s_{0}}\mathbf{D}^{\ell}F_{i\ell}(s^{\prime})\,\mathrm{d}s^{\prime}+A_{i}(s_{0}),\qquad
s\in[0,s_{0}].$

Since $Ns^{\frac{1}{2}}_{0}=1$ we have $\|F(s_{0})\|_{L^{2}(B)}\leq\delta$ and
we can apply Uhlenbeck’s lemma (Theorem 7.1) at $s=s_{0}$ to obtain $U:B\to G$
such that after the transformation (7.1) one has
$\|\tilde{A}_{i}(s_{0})\|_{H^{1}(B)}\lesssim\delta$. Write

$\displaystyle\tilde{A}_{i}(s)$
$\displaystyle=\int_{s}^{s_{0}}U\mathbf{D}^{\ell}F_{i\ell}U^{-1}\,\mathrm{d}s^{\prime}+UA_{i}(s_{0})U^{-1}-\partial_{i}UU^{-1}$
$\displaystyle=\int_{s}^{s_{0}}\tilde{\mathbf{D}}^{\ell}\tilde{F}_{i\ell}(s^{\prime})\,\mathrm{d}s^{\prime}+\tilde{A}_{i}(s_{0})$

For $\tilde{A}_{i}(s_{0})$ we already have an $H^{1}(B)$ bound so it remains
to estimate the integral in $H^{\sigma}(B)$, and specifically in
$\dot{H}^{\sigma}(B)$, since integrating in $L^{2}(B)$ is straightforward.

Step 2. (Replace covariant derivatives by ordinary derivatives)

From Lemma 7.2 applied to $\tilde{A}_{i}(s_{0})$ we obtain
$s_{0}^{\frac{1}{4}}\tilde{A}_{i}(s_{0}),s_{0}^{\frac{3}{4}}\partial_{x}\tilde{A}_{i}(s_{0})\in\delta
L^{\infty}(B)$.

From this and from
$(Ns^{\prime\frac{1}{2}})^{1-\sigma}s^{\prime\frac{5}{4}}\tilde{\mathbf{D}}^{\ell}\tilde{F}_{i\ell}(s^{\prime})\in\delta
L^{\infty}(B)$ \- which follows from (2.13) one obtains

$(Ns^{\frac{1}{2}})^{1-\sigma}s^{\frac{1}{4}}(s^{\frac{1}{2}}\partial_{x})^{m}\tilde{A}_{i}(s)\in\delta
L^{\infty}(B).$ (7.3)

for $m=0$ and $s\in[0,s_{0}]$ . This allows us to obtain

$(Ns^{\frac{1}{2}})^{1-\sigma}\|(s^{\frac{1}{2}}\partial_{x})^{m}\tilde{F}_{ij}(s)\|_{L^{2}(B)}\lesssim\delta,\qquad
s\in[0,s_{0}]$ (7.4)

for $m=1$ and also (7.3) for $m=1$ which in turn implies (7.4) also for $m=2$.

Step 3. (Estimate the integral term) We first consider

$\tilde{A}_{i}^{main}\vcentcolon=\int_{0}^{s_{0}}\partial^{\ell}\tilde{F}_{i\ell}(s^{\prime})\,\mathrm{d}s^{\prime}$

Consider an extension $\bar{F}$ of $\tilde{F}$ from $B\times[0,s_{0}]$ to
$\mathbb{R}^{3}\times[0,s_{0}]$ such that

$(Ns^{\frac{1}{2}})^{1-\sigma}\|(s^{\frac{1}{2}}\partial_{x})^{m}\bar{F}(s)\|_{L^{2}(\mathbb{R}^{3})}\lesssim\delta,\qquad
0\leq m\leq 2.$ (7.5)

for $s\in[0,s_{0}]$. This implies

$(Ns^{\frac{1}{2}})^{1-\sigma}s^{\frac{1}{2}}\|P_{k}\partial_{x}\bar{F}(s)\|_{\dot{H}^{\sigma}}\lesssim\frac{2^{k\sigma}}{\left\langle
s^{\frac{1}{2}}2^{k}\right\rangle}\alpha_{k}$

for some $\|(\alpha_{k})\|_{\ell^{2}}\lesssim\delta$. Indeed, use (7.5) with
$m=1$ for low frequencies with $s2^{2k}<1$ and use (7.5) with $m=2$ for high
frequencies such that $s2^{2k}>1$. Now

$\displaystyle
N^{2(1-\sigma)}\|\tilde{A}_{i}^{main}\|_{\dot{H}^{\sigma}(B)}^{2}$
$\displaystyle\lesssim