# Stability analysis of two-dimensional ideal flows with applications to
viscous fluids and plasmas

Diogo Arsénio  and  Haroune Houamed  New York University Abu Dhabi
Abu Dhabi
United Arab Emirates<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We are interested in the stability analysis of two-dimensional incompressible
inviscid fluids. Specifically, we revisit a known recent result on the
stability of Yudovich’s solutions to the incompressible Euler equations in
$L^{\infty}([0,T];H^{1})$ by providing a new approach to its proof based on
the idea of compactness extrapolation and by extending it to the whole plane.

This new method of proof is robust and, when applied to viscous models, leads
to a remarkable logarithmic improvement on the rate of convergence in the
vanishing viscosity limit of two-dimensional fluids. Loosely speaking, this
logarithmic gain is the result of the fact that, in appropriate high-
regularity settings, the smoothness of solutions to the Euler equations at
times $t\in[0,T)$ is strictly higher than their regularity at time $t=T$. This
“memory effect” seems to be a general principle which is not exclusive to
fluid mechanics. It is therefore likely to be observed in other setting and
deserves further investigation.

Finally, we also apply the stability results on Euler systems to the study of
two-dimensional ideal plasmas and establish their convergence, in strong
topologies, to solutions of magnetohydrodynamic systems, when the speed of
light tends to infinity. The crux of this asymptotic analysis relies on a fine
understanding of Maxwell’s system.

###### Key words and phrases:

Perfect incompressible two-dimensional fluids, Maxwell’s system, plasmas,
Yudovich’s theory, inviscid limit, singular limits.

###### Contents

  1. 1 Stability of the incompressible Euler system in the plane
    1. 1.1 Introduction and first main result
    2. 1.2 Overview of previous results
    3. 1.3 A new method of proof
    4. 1.4 Notation
    5. 1.5 Proof of Theorem 1.2
      1. 1.5.1 Convergence in $L^{\infty}_{t}L^{2}$
      2. 1.5.2 Convergence of low frequencies in $L^{\infty}_{t}\dot{H}^{1}$
      3. 1.5.3 Convergence of high frequencies in $L^{\infty}_{t}\dot{H}^{1}$
      4. 1.5.4 Justification of (1.15)
      5. 1.5.5 Justification of (1.17)
  2. 2 Rate of convergence of the inviscid limit in Yudovich’s class
    1. 2.1 Second main result
    2. 2.2 Proof of Theorem 2.1
      1. 2.2.1 Bounds on low frequencies
      2. 2.2.2 Bounds on high frequencies
      3. 2.2.3 Optimization of $\Theta_{\varepsilon}$
  3. 3 Two-dimensional incompressible plasmas
    1. 3.1 Third main result
    2. 3.2 Asymptotic analysis of Ampère’s equation
    3. 3.3 Proof of Theorem 3.2
      1. 3.3.1 Stability of the velocity field
      2. 3.3.2 Stability of the magnetic field.
  4. A Paradifferential calculus and the normal structure

##  1\. Stability of the incompressible Euler system in the plane

###  1.1. Introduction and first main result

We are interested in the stability of the two-dimensional incompressible Euler
equations

(1.1) $\begin{cases}\begin{aligned}
\partial_{t}u^{\varepsilon}+u^{\varepsilon}\cdot\nabla u^{\varepsilon}+\nabla
p^{\varepsilon}&=g^{\varepsilon},\\\ \operatorname{div}u^{\varepsilon}&=0,\\\
{u^{\varepsilon}}_{|_{t=0}}&=u_{0}^{\varepsilon},\end{aligned}\end{cases}$

in the limit $\varepsilon\rightarrow 0$, where
$u^{\varepsilon}=u^{\varepsilon}(t,x)$ are fluid velocity fields and
$g^{\varepsilon}=g^{\varepsilon}(t,x)$ are prescribed source terms, with
$(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{2}$, taking values in
$\mathbb{R}^{2}$. The pressures $p^{\varepsilon}$ are real-valued and also
depend on $t$ and $x$.

Formally, if the inputs $(g^{\varepsilon},u_{0}^{\varepsilon})$ converge to
some states $(g,u_{0})$, as $\varepsilon$ vanishes, then it is expected that
$u^{\varepsilon}$ approaches $u$, the solution of the same Euler equations

(1.2) $\begin{cases}\begin{aligned} \partial_{t}u+u\cdot\nabla u+\nabla
p&=g,\\\ \operatorname{div}u&=0,\\\
{u}_{|_{t=0}}&=u_{0}.\end{aligned}\end{cases}$

The celebrated Yudovich’s theorem allows us to formulate the rigorous weak
convergence result described in the following statement.

(Note that we employ below the classical notation $H^{s}$ to denote the
Sobolev space of functions $f\in L^{2}$ such that the Riesz potential
$|D|^{s}f$ also belongs to $L^{2}$, whereas $W^{1,p}$ denotes the Sobolev
space of functions $f\in L^{p}$ such that $\nabla f\in L^{p}$. Homogeneous
versions of these spaces are defined in a similar fashion.)

######  Theorem 1.1 ([9]).

Let $\varepsilon\in(0,1)$ and $T\in\mathbb{R}^{+}\cup\\{\infty\\}$. For any
initial data and source satisfying

$\displaystyle u_{0}^{\varepsilon}\in H^{1}(\mathbb{R}^{2}),$
$\displaystyle\operatorname{curl}u_{0}^{\varepsilon}\in
L^{\infty}(\mathbb{R}^{2}),$ $\displaystyle g^{\varepsilon}\in
L^{1}([0,T];H^{1}(\mathbb{R}^{2})),$
$\displaystyle\operatorname{curl}g^{\varepsilon}\in
L^{1}([0,T];L^{\infty}(\mathbb{R}^{2})),$

uniformly in $\varepsilon$, there is a unique solution $u^{\varepsilon}$ to
(1.1) on $[0,T]$ enjoying the bounds

$u^{\varepsilon}\in
C([0,T];H^{1}(\mathbb{R}^{2})),\quad\operatorname{curl}u^{\varepsilon}\in
L^{\infty}([0,T];L^{\infty}(\mathbb{R}^{2})),$

uniformly in $\varepsilon.$ Moreover, one has, for all $t\in[0,T]$, that

(1.3)
$\left\|\operatorname{curl}u^{\varepsilon}(t)\right\|_{L^{p}}\leq\left\|\operatorname{curl}u^{\varepsilon}_{0}\right\|_{L^{p}}+\left\|\operatorname{curl}g^{\varepsilon}\right\|_{L^{1}([0,t);L^{p})},\quad\text{for
all }p\in[1,\infty].$

Furthermore, if $(u_{0},g)$ denotes a weak limit, in the sense of
distributions, of the family
$(u^{\varepsilon}_{0},g^{\varepsilon})_{\varepsilon>0}$, then
$u^{\varepsilon}$ converges weakly, in the sense of distributions, to the
unique solution $u$ of the Euler system (1.2).

We are now interested in refining our understanding of the limit of
$u^{\varepsilon}$ to $u$, provided the inputs of (1.1) converge in a suitable
strong sense. Our first main result is featured in the following theorem,
which is a variant of Theorem 1 from [5] extended to an unbounded domain and
more general source terms.

######  Theorem 1.2.

Let $T\in\mathbb{R}^{+}$ and
$(u_{0}^{\varepsilon},g^{\varepsilon})_{\varepsilon>0}$ be a family of initial
data and source terms satisfying the assumptions of Theorem 1.1, which
converges to $(u_{0},g)$ strongly in $H^{1}(\mathbb{R}^{2})\times
L^{1}([0,T];H^{1}(\mathbb{R}^{2}))$, as $\varepsilon\rightarrow 0$. Then, it
holds that

$\lim_{\varepsilon\rightarrow
0}\sup_{t\in[0,T]}\left\|u^{\varepsilon}(t)-u(t)\right\|_{H^{1}(\mathbb{R}^{2})}=0,$

where $u^{\varepsilon}$ and $u$ are the unique solutions of (1.1) and (1.2),
respectively.

######  Remark 1.1.

Observe in the above theorem that the convergence holds in the whole space
$\mathbb{R}^{2}$. Moreover, employing the Gagliardo–Nirenberg and Hölder
inequalities, note that one can show the convergence of the velocity in
$L^{\infty}_{t}(L^{\infty}\cap W^{1,p})$, for all $p\in[2,\infty)$.
Accordingly, Theorem 1.2 provides a convergence result in spaces similar to
those given in Theorem 1.1.

The novelty of our work lies in the method of proof of Theorem 1.2, given in
Section 1.5, below, and in applications of this method in Sections 2 and 3,
later on, where we give our second and third main results in Theorems 2.1 and
3.2, respectively.

It is to be emphasized that our approach also allows us to extend Theorem 1.2
to an inviscid limit. More precisely, considering the Navier–Stokes system

(1.4) $\begin{cases}\begin{aligned}
\partial_{t}u^{\varepsilon}+u^{\varepsilon}\cdot\nabla
u^{\varepsilon}-\varepsilon\Delta u^{\varepsilon}+\nabla
p^{\varepsilon}&=g^{\varepsilon},\\\ \operatorname{div}u^{\varepsilon}&=0,\\\
{u^{\varepsilon}}_{|_{t=0}}&=u_{0}^{\varepsilon},\end{aligned}\end{cases}$

we have the next result, whose proof is a straighforward adapation of the
proof of Theorem 1.2.

######  Theorem 1.3.

Let $T\in\mathbb{R}^{+}$ and
$(u_{0}^{\varepsilon},g^{\varepsilon})_{\varepsilon>0}$ be a family of initial
data and source terms satisfying the assumptions of Theorem 1.1, which
converges to $(u_{0},g)$ strongly in $H^{1}(\mathbb{R}^{2})\times
L^{1}([0,T];H^{1}(\mathbb{R}^{2}))$, as $\varepsilon\rightarrow 0$. Then, it
holds that

$\lim_{\varepsilon\rightarrow
0}\sup_{t\in[0,T]}\left\|u^{\varepsilon}(t)-u(t)\right\|_{H^{1}(\mathbb{R}^{2})}=0,$

where $u^{\varepsilon}$ and $u$ are the unique solutions of (1.4) and (1.2),
respectively.

Note that there are several recent results establishing the strong inviscid
limit of solutions of (1.4) (see [4, 5, 10], for instance). Therefore, let us
now clarify the novelty in our result as well as the key differences with
other works.

###  1.2. Overview of previous results

We first note that the convergence in $L^{\infty}_{t}L^{2}(\mathbb{R}^{2})$ of
the velocity $u^{\varepsilon}$ to a solution of Euler’s equations is standard
and follows from classical stability estimates based on the uniform bounds
$\operatorname{curl}u^{\varepsilon}\in L^{\infty}_{t,x}$ first established by
Yudovich [9].

The corresponding convergence in $L^{\infty}_{t}\dot{H}^{1}(\Omega)$, with
$\Omega=\mathbb{T}^{2}$, has been established much more recently by
Constantin, Drivas and Elgindi in [5]. Their method of proof consists in
approaching both systems (1.2) and (1.4) by two linear transport equations
with regularized inputs, where the solutions are advected by the velocities
$u$ and $u^{\varepsilon}$, respectively.

Hence, the problem of proving the convergence of the velocity in
$L^{\infty}_{t}\dot{H}^{1}(\Omega)$ (or equivalently the vorticity in
$L^{\infty}_{t}L^{2}(\Omega)$) is then reduced to proving a stability result
for the regularized systems. In [5], in order to study the stability of the
vortices in $L^{\infty}_{t}L^{2}$ in the asymptotic regime
$\varepsilon\rightarrow 0$, the authors perform an $L^{2}$-energy estimate in
the vorticity formulation of the regularized linear systems. They are then
able to close the estimates by relying on two essential lemmas.

The first one [5, Lemma 1] is, roughly speaking, a variation of the
John–Nirenberg inequality, which states that, for any $f\in
BMO(\mathbb{R}^{2})$ and any compact domain $K\subset\mathbb{R}^{2}$, there
exists $\beta>0$ and $C>0$ such that

$\int_{K}\exp(\beta|f(x)|)dx\leq C.$

In [5], the authors utilize a variant of the preceding inequality for
$f=\nabla u$, which belongs to $BMO$ as soon as $\nabla\times u$ belongs to
$L^{\infty}$.

The second crucial ingredient in the proof given in [5] establishes a loss
estimate for the regularized vortices in

$L^{\infty}\left([0,T];\dot{W}^{1,p(t)}\right),$

where $p(t)$ is a decreasing continuous function of time. However, this bound
is not uniform with respect to the regularizing parameter in the inputs.

Although the proof from [5] hinges on the boundedness of the torus
$\mathbb{T}^{2}$, we emphasize that a careful and suitable adaptation of that
proof would lead to similar results in the whole plane $\mathbb{R}^{2}$.
Rather than rigorously justifying the extension of the arguments from [5] to
the whole domain $\mathbb{R}^{2}$, we choose to present hereafter a different
approach, which does not rely on properties of $BMO$ functions or time-
dependent Sobolev norms. Furthermore, this new approach will then play a key
role in the proofs of the important applications of Sections 2 and 3, below.

Finally, we also refer to [4, 10] for different proofs using techniques from
the celebrated work of DiPerna and Lions [7] on renormalized solutions of
general transport equations.

###  1.3. A new method of proof

Our proof of Theorem 1.2 establishes the $L^{\infty}_{t}\dot{H}^{1}$ stability
of velocity fields without studying the equations satisfied by the difference
$u^{\varepsilon}-u$ in $L^{\infty}_{t}\dot{H}^{1}$.

Instead, we exploit the convergence of velocities in $L^{\infty}_{t}L^{2}$,
which, as previously emphasized, is a classical consequence of the stability
estimates established by Yudovich [9], in combination with a time-dependent
version of the following simple but essential extrapolation lemma. This result
provides a useful criterion which allows us to recover strong compactness
properties in endpoint functional settings. In particular, in the notation of
the lemma, we will be using the functional spaces $\dot{H}^{s_{0}}=L^{2}$ and
$\dot{H}^{s_{1}}=\dot{H}^{1}$.

######  Lemma 1.4 (Compactness Extrapolation Lemma).

Fix the dimension $d\geq 1$. Let $s_{0}<s_{1}$ be two real numbers and
$(u^{\varepsilon})_{\varepsilon\in(0,1]}$ be a family of bounded functions in
$\dot{H}^{s_{0}}\cap\dot{H}^{s_{1}}(\mathbb{R}^{d})$, uniformly in
$\varepsilon\in(0,1]$. Further assume that

$u^{\varepsilon}\to u\quad\text{in }\dot{H}^{s_{0}}(\mathbb{R}^{d}).$

Then, it holds that

$u^{\varepsilon}\to u\quad\text{in }\dot{H}^{s_{1}}(\mathbb{R}^{d})$

if and only if

$\lim_{\varepsilon\rightarrow
0}\left\|\mathds{1}_{|D|\geq\Theta_{\varepsilon}}u^{\varepsilon}\right\|_{\dot{H}^{s_{1}}}=0,$

for some $\Theta_{\varepsilon}>0$ satisfying

$\lim_{\varepsilon\rightarrow
0}\Theta_{\varepsilon}=\infty\quad\text{and}\quad\lim_{\varepsilon\rightarrow
0}\Theta_{\varepsilon}^{s_{1}-s_{0}}\left\|u^{\varepsilon}-u\right\|_{\dot{H}^{s_{0}}}=0.$

###### Proof.

The proof straightforwardly follows from the observations that

$\left\|\mathds{1}_{|D|\leq\Theta_{\varepsilon}}(u^{\varepsilon}-u)\right\|_{\dot{H}^{s_{1}}}\leq\Theta_{\varepsilon}^{s_{1}-s_{0}}\left\|u^{\varepsilon}-u\right\|_{\dot{H}^{s_{0}}},$

the fact that the right-hand side above vanishes due to the assumptions on
$\Theta_{\varepsilon}$, and the convergence

$\lim_{\varepsilon\rightarrow
0}\left\|\mathds{1}_{|D|\geq\Theta_{\varepsilon}}u\right\|_{\dot{H}^{s_{1}}}=0,$

for any fixed $u\in\dot{H}^{s_{1}}$. ∎

Thus, by virtue of the compactness criterion given in the previous lemma, our
work will be reduced to the control of some high frequencies of the
vorticities $\omega^{\varepsilon}\overset{\rm
def}{=}\operatorname{curl}u^{\varepsilon}$ in $L^{\infty}_{t}L^{2}$. This is
crucial and will be achieved by performing an $L^{2}$-energy estimate on
dyadic blocks of $\omega^{\varepsilon}$ which is compatible with the nonlinear
structure of the equations and relies on the remarkable identities established
in Lemma 1.6, later on.

###  1.4. Notation

Before moving on to the proofs of our theorems, allow us to clarify some
elements of notation that we are about to use.

We denote by $C>0$ any universal constant that is independent of the main
variables of the given problem. Accordingly, we use the inequality $A\lesssim
B$ when there exists a constant $C>0$ independent of $A$ and $B$ such that
$A\leq CB$. In general, the constant $C$ is allowed to change from one line to
the next.

Moreover, for any bounded function $m:\mathbb{R}^{2}\rightarrow\mathbb{R}$,
the Fourier multiplier operator $m(D)$ is defined by

$m(D)f\overset{\rm
def}{=}\mathcal{F}^{-1}\big{(}m(\xi)\mathcal{F}f(\xi)\big{)},$

for any tempered distribution $f\in\mathcal{S}^{\prime}(\mathbb{R}^{2})$,
where $\mathcal{F}$ and $\mathcal{F}^{-1}$ denote the Fourier transform and
its inverse, respectively.

Finally, the commutator between two operators $Q$ and $S$ is denoted by
$[Q,S]$ and is defined by the relation

$[Q,S]\omega\overset{\rm def}{=}QS\omega-SQ\omega,$

for any suitable $\omega$.

###  1.5. Proof of Theorem 1.2

We proceed in several steps:

  1. (1)

First, in Section 1.5.1, we recall the classical ideas leading to the
convergence $u^{\varepsilon}\to u$ in $L^{\infty}_{t}L^{2}$.

  2. (2)

Then, in Section 1.5.2, we discuss the convergence of some suitable low
frequencies of vorticities with a simple estimate based on the convergence of
velocities in $L^{\infty}_{t}L^{2}$.

  3. (3)

Finally, in Section 1.5.3, in the spirit of the Compactness Extrapolation
Lemma (Lemma 1.4), we identify and control the remaining high frequencies of
vorticities, thereby establishing their convergence and completing the proof
of Theorem 1.2.

  4. (4)

The remaining sections, i.e., Sections 1.5.4 and 1.5.5, are dedicated to
essential technical results which are justified separately, for the sake of
clarity.

####  1.5.1. Convergence in $L^{\infty}_{t}L^{2}$

The stability of the two-dimensional Euler equations in $L^{\infty}_{t}L^{2}$
is classical and follows from the uniqueness methods employed in [9]. We also
refer to [5, Lemma 4] for a different proof in the torus which can be adapted
to the whole plane.

More specifically, it is possible to show that if

$\widetilde{u}^{\varepsilon}\overset{\rm
def}{=}u^{\varepsilon}-u\quad\text{and}\quad\widetilde{g}^{\varepsilon}\overset{\rm
def}{=}g^{\varepsilon}-g$

are small in the sense that

$\left\|\widetilde{u}^{\varepsilon}_{0}\right\|_{L^{2}}+\left\|\widetilde{g}^{\varepsilon}\right\|_{L^{1}([0,T];L^{2})}<C_{*}e^{-\exp(C_{*}T)},$

where

(1.5) $C_{*}\overset{\rm
def}{=}C\sup_{\varepsilon>0}\left(\left\|(\omega_{0},\omega^{\varepsilon}_{0})\right\|_{L^{2}\cap
L^{\infty}}+\left\|(\operatorname{curl}g,\operatorname{curl}g^{\varepsilon})\right\|_{L^{1}([0,T];L^{2}\cap
L^{\infty})}\right),$

for some universal constant $C>0$, then one has the stability estimate

(1.6)
$\displaystyle\left\|\widetilde{u}^{\varepsilon}\right\|_{L^{\infty}([0,T];L^{2})}\lesssim_{C_{*},T}\Big{(}\left\|\widetilde{u}^{\varepsilon}_{0}\right\|_{L^{2}}+\left\|\widetilde{g}^{\varepsilon}\right\|_{L^{1}([0,T];L^{2})}\Big{)}^{\exp(-C_{*}T)}.$

For the sake of completeness, we now outline the idea leading to (1.6) by
following the modern proof given in [2, Section 7.3.3]. To that end, we begin
by observing that $\widetilde{u}^{\varepsilon}$ and
$\widetilde{g}^{\varepsilon}$ solve the system

$\begin{cases}\begin{aligned}
\partial_{t}\widetilde{u}^{\varepsilon}+u^{\varepsilon}\cdot\nabla\widetilde{u}^{\varepsilon}+\nabla\widetilde{p}^{\varepsilon}&=-\widetilde{u}^{\varepsilon}\cdot\nabla
u+\widetilde{g}^{\varepsilon},\\\
\operatorname{div}\widetilde{u}^{\varepsilon}&=0,\\\
{\widetilde{u}}^{\varepsilon}_{|_{t=0}}&=u_{0}^{\varepsilon}-u_{0}.\end{aligned}\end{cases}$

Then, performing a standard $L^{2}$-energy estimate yields, for any
$t\in[0,T]$,

$\displaystyle\frac{1}{2}\frac{d}{dt}\left\|\widetilde{u}^{\varepsilon}(t)\right\|_{L^{2}}^{2}\leq\left|\int_{\mathbb{R}^{2}}\nabla
u:(\widetilde{u}^{\varepsilon}\otimes\widetilde{u}^{\varepsilon})(t,x)dx\right|+\left\|\widetilde{g}^{\varepsilon}(t)\right\|_{L^{2}}\left\|\widetilde{u}^{\varepsilon}(t)\right\|_{L^{2}}.$

Therefore, by Hölder’s inequality and the classical Biot–Savart estimate (see
[2, Section 7.1.1])

$\left\|\nabla u\right\|_{L^{q}}\leq Cq\left\|\omega\right\|_{L^{q}},$

where $q=q(t)\in(2,\infty)$ is allowed to depend on $t\in[0,T]$, we infer that

$\displaystyle\frac{1}{2}\frac{d}{dt}\left\|\widetilde{u}^{\varepsilon}(t)\right\|_{L^{2}}^{2}$
$\displaystyle\leq\left\|\nabla
u(t)\right\|_{L^{q}}\left\|\widetilde{u}^{\varepsilon}(t)\right\|_{L^{\frac{2q}{q-1}}}^{2}+\left\|\widetilde{g}^{\varepsilon}(t)\right\|_{L^{2}}\left\|\widetilde{u}^{\varepsilon}(t)\right\|_{L^{2}}$
$\displaystyle\leq Cq\left\|\omega(t)\right\|_{L^{2}\cap
L^{\infty}}\left\|\widetilde{u}^{\varepsilon}(t)\right\|_{L^{2}}^{2-\frac{2}{q}}\left\|\widetilde{u}^{\varepsilon}(t)\right\|_{L^{\infty}}^{\frac{2}{q}}+\left\|\widetilde{g}^{\varepsilon}(t)\right\|_{L^{2}}\left\|\widetilde{u}^{\varepsilon}(t)\right\|_{L^{2}}.$

Thus, by virtue of the control (1.3) and the Gagliardo–Nirenberg interpolation
inequality

$\left\|\widetilde{u}^{\varepsilon}(t)\right\|_{L^{\infty}}\lesssim\left\|\widetilde{u}^{\varepsilon}(t)\right\|_{L^{2}}^{\frac{1}{2}}\left\|\widetilde{\omega}^{\varepsilon}(t)\right\|_{L^{\infty}}^{\frac{1}{2}},$

we find that

$\displaystyle\frac{1}{2}\left\|\widetilde{u}^{\varepsilon}(t)\right\|_{L^{2}}^{2}$
$\displaystyle\leq\frac{1}{2}\left\|\widetilde{u}^{\varepsilon}_{0}\right\|_{L^{2}}^{2}+C\int_{0}^{t}q(\tau)\left\|\omega(\tau)\right\|_{L^{2}\cap
L^{\infty}}\left\|\widetilde{u}^{\varepsilon}(\tau)\right\|_{L^{2}}^{2-\frac{1}{q(\tau)}}\left\|\widetilde{\omega}^{\varepsilon}(\tau)\right\|_{L^{\infty}}^{\frac{1}{q(\tau)}}d\tau$
$\displaystyle\quad+\int_{0}^{t}\left\|\widetilde{g}^{\varepsilon}(\tau)\right\|_{L^{2}}\left\|\widetilde{u}^{\varepsilon}(\tau)\right\|_{L^{2}}d\tau$
$\displaystyle\leq\frac{1}{2}\left\|\widetilde{u}^{\varepsilon}_{0}\right\|_{L^{2}}^{2}+\int_{0}^{t}q(\tau)C_{*}^{1+\frac{1}{q(\tau)}}\left\|\widetilde{u}^{\varepsilon}(\tau)\right\|_{L^{2}}^{2-\frac{1}{q(\tau)}}d\tau$
$\displaystyle\quad+\left(\int_{0}^{t}\left\|\widetilde{g}^{\varepsilon}(\tau)\right\|_{L^{2}}d\tau\right)^{2}+\frac{1}{4}\left\|\widetilde{u}^{\varepsilon}\right\|_{L^{\infty}([0,t);L^{2})}^{2},$

where $C_{*}$ is defined by (1.5).

Now, introducing the continuous function

$f^{\varepsilon}(t)\overset{\rm
def}{=}C_{*}^{-1}\left\|\widetilde{u}^{\varepsilon}\right\|_{L^{\infty}([0,t);L^{2})},$

the preceding estimate can be recast as

$\frac{1}{4}\big{(}f^{\varepsilon}(t)\big{)}^{2}\leq\frac{1}{2}\big{(}f^{\varepsilon}(0)\big{)}^{2}+C_{*}\int_{0}^{t}q(\tau)\big{(}f^{\varepsilon}(\tau)\big{)}^{2-\frac{1}{q(\tau)}}d\tau+\left(C_{*}^{-1}\int_{0}^{t}\left\|\widetilde{g}^{\varepsilon}(\tau)\right\|_{L^{2}}d\tau\right)^{2}.$

Therefore, further assuming that

$f^{\varepsilon}(t)\leq 1$

on a time interval $[0,t_{*}]$, for some $t_{*}\in(0,T]$, and setting

$q(t)\overset{\rm
def}{=}2\log\left(\frac{e}{\big{(}f^{\varepsilon}(t)\big{)}^{2}}\right),$

we deduce that

$\big{(}f^{\varepsilon}(t)\big{)}^{2}\leq
2\big{(}f^{\varepsilon}(0)\big{)}^{2}+C_{0}C_{*}\int_{0}^{t}\log\left(\frac{e}{f^{\varepsilon}(\tau)^{2}}\right)\big{(}f^{\varepsilon}(\tau)\big{)}^{2}d\tau+4\left(C_{*}^{-1}\int_{0}^{t}\left\|\widetilde{g}^{\varepsilon}(\tau)\right\|_{L^{2}}d\tau\right)^{2},$

where we employed the observation that

$q(t)\big{(}f^{\varepsilon}(t)\big{)}^{2-\frac{1}{q(t)}}\leq
C_{0}\log\left(\frac{e}{\big{(}f^{\varepsilon}(t)\big{)}^{2}}\right)\big{(}f^{\varepsilon}(t)\big{)}^{2},$

for some $C_{0}\geq 1$.

At last, by employing Osgood’s lemma [2, Lemma 3.4] (see also [8, Lemma A.1],
which can be applied directly to the present case), we arrive at the desired
bound

$\frac{1}{eC_{*}}\left\|\widetilde{u}^{\varepsilon}\right\|_{L^{\infty}([0,t_{*}];L^{2})}\leq\Big{(}\frac{\sqrt{2}}{eC_{*}}\left\|\widetilde{u}^{\varepsilon}_{0}\right\|_{L^{2}}+\frac{2}{eC_{*}}\left\|\widetilde{g}^{\varepsilon}\right\|_{L^{1}([0,t_{*}];L^{2})}\Big{)}^{\exp(-C_{0}C_{*}t_{*})}.$

In particular, assuming that

$\frac{\sqrt{2}}{eC_{*}}\left\|\widetilde{u}^{\varepsilon}_{0}\right\|_{L^{2}}+\frac{2}{eC_{*}}\left\|\widetilde{g}^{\varepsilon}\right\|_{L^{1}([0,T];L^{2})}<e^{-\exp(C_{0}C_{*}T)}\leq
1$

leads to the estimate

$\frac{1}{eC_{*}}\left\|\widetilde{u}^{\varepsilon}\right\|_{L^{\infty}([0,t_{*}];L^{2})}\leq\Big{(}\frac{\sqrt{2}}{eC_{*}}\left\|\widetilde{u}^{\varepsilon}_{0}\right\|_{L^{2}}+\frac{2}{eC_{*}}\left\|\widetilde{g}^{\varepsilon}\right\|_{L^{1}([0,T];L^{2})}\Big{)}^{\exp(-C_{0}C_{*}T)}<e^{-1}.$

Then, a classical continuation argument allows us to deduce that $t_{*}$, in
the left-hand side above, can be chosen to be equal to $T$. This establishes
the stability estimate (1.6), by possibly redefining $C_{*}$ up to a
multiplicative universal constant.

####  1.5.2. Convergence of low frequencies in $L^{\infty}_{t}\dot{H}^{1}$

As in the proof of Lemma 1.4, it is readily seen that a direct use of
Plancherel’s theorem gives that

$\left\|\mathds{1}_{|D|\leq\Theta_{\varepsilon}}(\omega^{\varepsilon}-\omega)\right\|_{L^{\infty}([0,T];L^{2})}\leq\Theta_{\varepsilon}\left\|u^{\varepsilon}-u\right\|_{L^{\infty}([0,T];L^{2})},$

for any $\Theta_{\varepsilon}>0$. In particular, if

(1.7) $\lim_{\varepsilon\rightarrow
0}\Theta_{\varepsilon}=\infty\quad\text{and}\quad\lim_{\varepsilon\rightarrow
0}\Theta_{\varepsilon}\left\|u^{\varepsilon}-u\right\|_{L^{\infty}([0,T];L^{2})}=0,$

we deduce that

(1.8) $\lim_{\varepsilon\rightarrow
0}\left\|\phi(\Theta_{\varepsilon}^{-1}D)(\omega^{\varepsilon}-\omega)\right\|_{L^{\infty}([0,T];L^{2})}=0,$

for any compactly supported cutoff function $\phi\in
L^{\infty}(\mathbb{R}^{2})$.

Moreover, if $\phi$ is assumed to be supported away from the origin, then the
assumptions on the behavior of $\Theta_{\varepsilon}$ allow us to obtain that

(1.9) $\lim_{\varepsilon\rightarrow
0}\left\|\phi(\Theta_{\varepsilon}^{-1}D)\omega\right\|_{L^{\infty}([0,T];L^{2})}=0.$

This convergence hinges upon the time continuity of Yudovich’s solutions.
Indeed, let us suppose, by contradiction, that

$\left\|\phi(\Theta_{\varepsilon_{k}}^{-1}D)\omega\right\|_{L^{\infty}([0,T];L^{2})}\geq\delta,$

for some sequence $\varepsilon_{k}\to 0$, as $k\to\infty$, and some constant
$\delta>0$. Then, by continuity, writing

$\left\|\phi(\Theta_{\varepsilon_{k}}^{-1}D)\omega\right\|_{L^{\infty}([0,T];L^{2})}=\left\|\phi(\Theta_{\varepsilon_{k}}^{-1}D)\omega(t_{k})\right\|_{L^{2}},$

for some suitable $t_{k}\in[0,T]$, and assuming, by compactness, that
$t_{k}\to t$, we see that the control

$\left\|\phi(\Theta_{\varepsilon_{k}}^{-1}D)\omega(t_{k})\right\|_{L^{2}}\leq\left\|\phi\right\|_{L^{\infty}}\left\|\omega(t_{k})-\omega(t)\right\|_{L^{2}}+\left\|\phi(\Theta_{\varepsilon_{k}}^{-1}D)\omega(t)\right\|_{L^{2}}$

implies that

$\left\|\phi(\Theta_{\varepsilon_{k}}^{-1}D)\omega\right\|_{L^{\infty}([0,T];L^{2})}\to
0,$

which is impossible. It follows that (1.9) holds true.

We will make use of the preceding convergence properties with the particular
choice of cutoff function $\phi(D)=\mathds{1}_{M\leq|D|\leq N}$, for any
$0<M<N$, which yields

$\lim_{\varepsilon\rightarrow
0}\left\|\mathds{1}_{M\Theta_{\varepsilon}\leq|D|\leq
N\Theta_{\varepsilon}}\omega\right\|_{L^{\infty}([0,T];L^{2})}=0,$

and

(1.10) $\displaystyle\lim_{\varepsilon\rightarrow
0}\left\|\mathds{1}_{M\Theta_{\varepsilon}\leq|D|\leq
N\Theta_{\varepsilon}}\omega^{\varepsilon}\right\|_{L^{\infty}([0,T];L^{2})}=0,$

for any choice of parameters $\Theta_{\varepsilon}$ satisfying (1.7).

####  1.5.3. Convergence of high frequencies in $L^{\infty}_{t}\dot{H}^{1}$

We first introduce some notation. Let

$\mathcal{C}\overset{\rm
def}{=}\left\\{\xi\in\mathbb{R}^{2}:\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\right\\},\qquad\mathcal{B}\overset{\rm
def}{=}\left\\{\xi\in\mathbb{R}^{2}:|\xi|\leq\frac{4}{3}\right\\},$

and consider smooth radial functions $\varphi\in\mathcal{D}(\mathcal{C})$ and
$\psi\in\mathcal{D}(\mathcal{B})$ with

$0\leq\varphi(\xi),\psi(\xi)\leq 1,\quad\text{for all }\xi\in\mathbb{R}^{2},$

and

$\psi(\xi)=1,\quad\text{for all }|\xi|\leq 1.$

Furthermore, it is possible to ensure that the family of functions

$\varphi_{j}(\cdot)\overset{\rm
def}{=}\varphi(2^{-j}\cdot)\in\mathcal{D}(2^{j}\mathcal{C}),\quad
j\in\mathbb{Z},$

provides us with partitions of unity

$1=\sum_{j\in\mathbb{Z}}\varphi_{j}(\xi)=\sum_{j\geq
0}\varphi_{j}(\xi)+\psi(\xi),\quad\text{for all }\xi\in\mathbb{R}^{2}.$

In this case, the corresponding convolution operators

(1.11) $S_{0}^{\varepsilon}\overset{\rm
def}{=}\psi(\Theta_{\varepsilon}^{-1}D),\qquad\Delta_{j}^{\varepsilon}\overset{\rm
def}{=}\varphi_{j}(\Theta_{\varepsilon}^{-1}D),$

where $\Theta_{\varepsilon}>0$ is any parameter satisfying (1.7), satisfy that

$S_{0}^{\varepsilon}+\sum_{j\geq 0}\Delta_{j}^{\varepsilon}=\mathrm{Id}.$

Next, further introducing the Fourier multiplier operators

$\sqrt{S_{0}^{\varepsilon}}\overset{\rm
def}{=}\sqrt{\psi(\Theta_{\varepsilon}^{-1}D)}\qquad\text{and}\qquad\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\overset{\rm
def}{=}\sqrt{1-\psi(\Theta_{\varepsilon}^{-1}D)},$

and observing that

$\left(\sum_{j\geq 0}\varphi_{j}^{\varepsilon}(\xi)\right)^{2}\leq\sum_{j\geq
0}\varphi_{j}^{\varepsilon}(\xi),$

it holds, for any given $f\in L^{2}$, that

(1.12)
$\left\|(\mathrm{Id}-S_{0}^{\varepsilon})f\right\|_{L^{2}}\leq\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}f\right\|_{L^{2}},$

which will come in handy later on.

Now, observe that (1.9) yields

$\lim_{\varepsilon\rightarrow
0}\left\|(\mathrm{Id}-S_{0}^{\varepsilon})\omega\right\|_{L^{\infty}([0,T];L^{2})}=0.$

Therefore, by (1.8), we conclude that the convergence

$\lim_{\varepsilon\rightarrow
0}\left\|\omega^{\varepsilon}-\omega\right\|_{L^{\infty}([0,T];L^{2})}=0$

is equivalent to

(1.13) $\lim_{\varepsilon\rightarrow
0}\left\|(\mathrm{Id}-S_{0}^{\varepsilon})\omega^{\varepsilon}\right\|_{L^{\infty}([0,T];L^{2})}=0,$

which can be interpreted as a time-dependent version of the compactness
criterion given in Lemma 1.4.

Thus, in order to prove Theorem 1.2, there only remains to establish (1.13).
To that end, we first recall that $\omega^{\varepsilon}$ solves the transport
equation

(1.14)
$\partial_{t}\omega^{\varepsilon}+u^{\varepsilon}\cdot\nabla\omega^{\varepsilon}=\operatorname{curl}g^{\varepsilon}.$

Then, formally taking the inner product of (1.14) with $\sum_{j\geq
0}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}$ and using the divergence-free
condition of $u^{\varepsilon}$, we find that

(1.15)
$\displaystyle\frac{1}{2}\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega^{\varepsilon}(t)\right\|_{L^{2}}^{2}$
$\displaystyle=\frac{1}{2}\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega^{\varepsilon}_{0}\right\|_{L^{2}}^{2}+\int_{0}^{t}\int_{\mathbb{R}^{2}}\operatorname{curl}g^{\varepsilon}(\mathrm{Id}-S_{0}^{\varepsilon})\omega^{\varepsilon}(\tau,x)dxd\tau$
$\displaystyle\quad-\underbrace{\int_{0}^{t}\int_{\mathbb{R}^{2}}u^{\varepsilon}\cdot\nabla\left(\sum_{j\leq-1}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}\right)\sum_{j\geq
0}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}(\tau,x)dxd\tau}_{\mathcal{\overset{\rm
def}{=}J}(t)}.$

For the sake of completeness, we provide a rigorous justification of (1.15) in
Section 1.5.4, below.

We show now how to estimate $\mathcal{J}(t)$. To that end, we split it into a
sum of three terms

(1.16) $\displaystyle\mathcal{J}_{1}(t)$ $\displaystyle\overset{\rm
def}{=}\int_{0}^{t}\int_{\mathbb{R}^{2}}u^{\varepsilon}\cdot\nabla\left(\Delta_{-1}^{\varepsilon}\omega^{\varepsilon}(\tau)\right)\Delta_{0}^{\varepsilon}\omega^{\varepsilon}(\tau,x)dxd\tau,$
$\displaystyle\mathcal{J}_{2}(t)$ $\displaystyle\overset{\rm
def}{=}\int_{0}^{t}\int_{\mathbb{R}^{2}}u^{\varepsilon}\cdot\nabla\left(\sum_{j\leq-2}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}\right)\sum_{j\geq
0}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}(\tau,x)dxd\tau,$
$\displaystyle\mathcal{J}_{3}(t)$ $\displaystyle\overset{\rm
def}{=}\int_{0}^{t}\int_{\mathbb{R}^{2}}u^{\varepsilon}\cdot\nabla\left(\Delta_{-1}^{\varepsilon}\omega^{\varepsilon}\right)\sum_{j\geq
1}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}(\tau,x)dxd\tau,$

which we control separately.

The term $\mathcal{J}_{1}$ is the most difficult to estimate. Specifically,
one can show that

(1.17) $|\mathcal{J}_{1}(t)|\lesssim\sum_{i=-1}^{0}\int_{0}^{t}\left\|\nabla
u^{\varepsilon}(\tau)\right\|_{L^{2}}\left\|\Delta_{i}^{\varepsilon}\omega^{\varepsilon}(\tau)\right\|_{L^{4}}^{2}d\tau.$

For the sake of clarity, we defer the justification of this estimate to
Section 1.5.5, below.

Then, observe that

$|\mathcal{J}_{1}(t)|\lesssim\left\|\omega^{\varepsilon}\right\|_{L^{\infty}([0,t];L^{2})}\left\|\omega^{\varepsilon}\right\|_{L^{\infty}([0,t];L^{\infty})}\sum_{i=-1}^{0}\left\|\Delta_{i}^{\varepsilon}\omega^{\varepsilon}\right\|_{L^{1}([0,t];L^{2})},$

which follows from Hölder’s inequality and the Biot–Savart estimate.

As for $\mathcal{J}_{2}$ and $\mathcal{J}_{3}$, we begin with exploiting the
support localizations

$\displaystyle\operatorname{supp}\mathcal{F}\left(\sum_{j\leq-2}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}\right)$
$\displaystyle=\left\\{\xi\in\mathbb{R}^{2}:|\xi|\leq\frac{2\Theta_{\varepsilon}}{3}\right\\},$
$\displaystyle\operatorname{supp}\mathcal{F}\left(\sum_{j\geq
0}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}\right)$
$\displaystyle=\left\\{\xi\in\mathbb{R}^{2}:|\xi|\geq\frac{3\Theta_{\varepsilon}}{4}\right\\},$

to deduce that

$\mathcal{F}\left(\sum_{j\leq-2}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}\sum_{j\geq
0}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}\right)\subset\left\\{\xi\in\mathbb{R}^{2}:|\xi|\geq\frac{\Theta_{\varepsilon}}{12}\right\\}.$

Similarly, the fact that

$\displaystyle\operatorname{supp}\mathcal{F}\left(\Delta_{-1}^{\varepsilon}\omega^{\varepsilon}\right)$
$\displaystyle\subset\left\\{\xi\in\mathbb{R}^{2}:|\xi|\leq\frac{4\Theta_{\varepsilon}}{3}\right\\},$
$\displaystyle\operatorname{supp}\mathcal{F}\left(\sum_{j\geq
1}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}\right)$
$\displaystyle\subset\left\\{\xi\in\mathbb{R}^{2}:|\xi|\geq\frac{3\Theta_{\varepsilon}}{2}\right\\},$

entails that

$\mathcal{F}\left(\Delta_{-1}^{\varepsilon}\omega^{\varepsilon}\sum_{j\geq
1}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}\right)\subset\left\\{\xi\in\mathbb{R}^{2}:|\xi|\geq\frac{\Theta_{\varepsilon}}{6}\right\\}.$

Accordingly, we infer that

$\displaystyle\mathcal{J}_{2}(t)$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{2}}\left(\mathds{1}_{|D|\geq\frac{1}{12}\Theta_{\varepsilon}}u^{\varepsilon}\right)\cdot\nabla\left(\sum_{j\leq-2}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}\right)\sum_{j\geq
0}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}(\tau,x)dxd\tau,$

and

$\displaystyle\mathcal{J}_{3}(t)$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{2}}\left(\mathds{1}_{|D|\geq\frac{1}{6}\Theta_{\varepsilon}}u^{\varepsilon}\right)\cdot\nabla\left(\Delta_{-1}^{\varepsilon}\omega^{\varepsilon}\right)\sum_{j\geq
1}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}(\tau,x)dxd\tau.$

Consequently, since $u^{\varepsilon}$ above is localized in its high
frequencies, it holds that

(1.18) $\displaystyle|\mathcal{J}_{2}(t)|$
$\displaystyle\lesssim\int_{0}^{t}\left\|\mathds{1}_{|D|\geq\frac{1}{12}\Theta_{\varepsilon}}\nabla
u^{\varepsilon}(\tau)\right\|_{L^{2}}\left\|(S_{0}^{\varepsilon}-\Delta_{-1}^{\varepsilon})\omega^{\varepsilon}(\tau)\right\|_{L^{\infty}}\left\|(\mathrm{Id}-S_{0}^{\varepsilon})\omega^{\varepsilon}(\tau)\right\|_{L^{2}}d\tau$
$\displaystyle\lesssim\int_{0}^{t}\left\|\mathds{1}_{|D|\geq\frac{1}{12}\Theta_{\varepsilon}}\omega^{\varepsilon}(\tau)\right\|_{L^{2}}\left\|\omega^{\varepsilon}(\tau)\right\|_{L^{\infty}}\left\|(\mathrm{Id}-S_{0}^{\varepsilon})\omega^{\varepsilon}(\tau)\right\|_{L^{2}}d\tau,$

and, in a similar fashion, that

(1.19) $\displaystyle|\mathcal{J}_{3}(t)|$
$\displaystyle\lesssim\int_{0}^{t}\left\|\mathds{1}_{|D|\geq\frac{1}{6}\Theta_{\varepsilon}}\nabla
u^{\varepsilon}(\tau)\right\|_{L^{2}}\left\|\Delta_{-1}^{\varepsilon}\omega^{\varepsilon}(\tau)\right\|_{L^{\infty}}\left\|(\mathrm{Id}-S_{0}^{\varepsilon}-\Delta_{0}^{\varepsilon})\omega^{\varepsilon}(\tau)\right\|_{L^{2}}d\tau$
$\displaystyle\lesssim\int_{0}^{t}\left\|\mathds{1}_{|D|\geq\frac{1}{12}\Theta_{\varepsilon}}\omega^{\varepsilon}(\tau)\right\|_{L^{2}}\left\|\omega^{\varepsilon}(\tau)\right\|_{L^{\infty}}\Big{(}\left\|\Delta_{0}^{\varepsilon}\omega^{\varepsilon}(\tau)\right\|_{L^{2}}+\left\|(\mathrm{Id}-S_{0}^{\varepsilon})\omega^{\varepsilon}(\tau)\right\|_{L^{2}}\Big{)}d\tau.$

Therefore, the bounds

$\left\|\Delta_{0}^{\varepsilon}\omega^{\varepsilon}(\tau)\right\|_{L^{2}}\leq\left\|\mathds{1}_{\frac{1}{12}\Theta_{\varepsilon}\leq|D|\leq\frac{8}{3}\Theta_{\varepsilon}}\omega^{\varepsilon}(\tau)\right\|_{L^{2}}$

and

$\displaystyle\left\|\mathds{1}_{|D|\geq\frac{1}{12}\Theta_{\varepsilon}}\omega^{\varepsilon}(\tau)\right\|_{L^{2}}^{2}$
$\displaystyle=\left\|\mathds{1}_{\frac{1}{12}\Theta_{\varepsilon}\leq|D|<\frac{4}{3}\Theta_{\varepsilon}}\omega^{\varepsilon}(\tau)\right\|_{L^{2}}^{2}+\left\|\mathds{1}_{|D|\geq\frac{4}{3}\Theta_{\varepsilon}}\omega^{\varepsilon}(\tau)\right\|_{L^{2}}^{2}$
$\displaystyle\leq\left\|\mathds{1}_{\frac{1}{12}\Theta_{\varepsilon}\leq|D|\leq\frac{8}{3}\Theta_{\varepsilon}}\omega^{\varepsilon}(\tau)\right\|_{L^{2}}^{2}+\left\|(\mathrm{Id}-S_{0}^{\varepsilon})\omega^{\varepsilon}(\tau)\right\|_{L^{2}}^{2},$

lead to

$\displaystyle|\mathcal{J}_{2}(t)|+|\mathcal{J}_{3}(t)|$
$\displaystyle\lesssim\left\|\omega^{\varepsilon}\right\|_{L^{\infty}([0,t];L^{2}\cap
L^{\infty})}^{2}\left\|\mathds{1}_{\frac{1}{12}\Theta_{\varepsilon}\leq|D|\leq\frac{8}{3}\Theta_{\varepsilon}}\omega^{\varepsilon}\right\|_{L^{1}([0,t];L^{2})}$
$\displaystyle\quad+\int_{0}^{t}\left\|\omega^{\varepsilon}(\tau)\right\|_{L^{\infty}}\left\|(\mathrm{Id}-S_{0}^{\varepsilon})\omega^{\varepsilon}(\tau)\right\|_{L^{2}}^{2}d\tau.$

Finally, gathering all estimates on $\mathcal{J}_{1}$, $\mathcal{J}_{2}$ and
$\mathcal{J}_{3}$, employing the control

$\left\|\omega^{\varepsilon}\right\|_{L^{\infty}([0,T];L^{2}\cap
L^{\infty})}\leq\sup_{\varepsilon>0}\left(\left\|\omega^{\varepsilon}_{0}\right\|_{L^{2}\cap
L^{\infty}}+\left\|\operatorname{curl}g^{\varepsilon}\right\|_{L^{1}([0,T];L^{2}\cap
L^{\infty})}\right)\lesssim C_{*},$

where $C_{*}$ is defined in (1.5), and using (1.12), we arrive at the bound

$|\mathcal{J}(t)|\lesssim
C_{*}^{2}\left\|\mathds{1}_{\frac{1}{12}\Theta_{\varepsilon}\leq|D|\leq\frac{8}{3}\Theta_{\varepsilon}}\omega^{\varepsilon}\right\|_{L^{1}([0,T];L^{2})}+C_{*}\int_{0}^{t}\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega^{\varepsilon}(\tau)\right\|_{L^{2}}^{2}d\tau,$

for any $t\in[0,T]$.

Further incorporating this estimate into (1.15), we deduce that

$\displaystyle\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega^{\varepsilon}(t)\right\|_{L^{2}}^{2}$
$\displaystyle\leq\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega^{\varepsilon}_{0}\right\|_{L^{2}}^{2}+C_{*}\left\|(\mathrm{Id}-S_{0}^{\varepsilon})\operatorname{curl}g^{\varepsilon}\right\|_{L^{1}([0,T];L^{2})}$
$\displaystyle\quad+C_{*}^{2}\left\|\mathds{1}_{\frac{1}{12}\Theta_{\varepsilon}\leq|D|\leq\frac{8}{3}\Theta_{\varepsilon}}\omega^{\varepsilon}\right\|_{L^{1}([0,T];L^{2})}+C_{*}\int_{0}^{t}\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega^{\varepsilon}(\tau)\right\|_{L^{2}}^{2}d\tau.$

At last, applying Grönwall’s lemma yields that

$\displaystyle\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega^{\varepsilon}\right\|_{L^{\infty}([0,T];L^{2})}^{2}$
$\displaystyle\lesssim_{C_{*}}\bigg{(}\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega^{\varepsilon}_{0}\right\|_{L^{2}}^{2}+\left\|\mathds{1}_{\frac{1}{12}\Theta_{\varepsilon}\leq|D|\leq\frac{8}{3}\Theta_{\varepsilon}}\omega^{\varepsilon}\right\|_{L^{1}([0,T];L^{2})}$
$\displaystyle\qquad+\left\|(\mathrm{Id}-S_{0}^{\varepsilon})\operatorname{curl}g^{\varepsilon}\right\|_{L^{1}([0,T];L^{2})}\bigg{)}\exp\left(C_{*}T\right).$

Clearly, in view of (1.10), the right-hand side above vanishes in the limit
$\varepsilon\rightarrow 0$. We therefore conclude that (1.13) holds true,
thereby completing the proof of Theorem 1.2. ∎

####  1.5.4. Justification of (1.15)

Note that, in order to establish (1.15), we have taken advantage of the formal
cancellation

$\int_{\mathbb{R}^{2}}u^{\varepsilon}(\tau,x)\cdot\nabla\left(\sum_{j\geq
0}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}(\tau,x)\right)\left(\sum_{j\geq
0}\Delta_{j}^{\varepsilon}\omega^{\varepsilon}(\tau,x)\right)dx=0,\quad\text{for
all }\tau\in[0,T],$

despite the fact that the integral above is not well-defined, for
$\omega^{\varepsilon}(t)$ only belongs to Lebesgue spaces. Here, we show that
(1.15) can be justified without relying on the above identity.

To that end, we first observe that

$\int_{\mathbb{R}^{2}}\operatorname{div}(u^{\varepsilon}\omega^{\varepsilon})S_{0}^{\varepsilon}\omega^{\varepsilon}(\tau,x)dx=-\int_{\mathbb{R}^{2}}u^{\varepsilon}\cdot\nabla(S_{0}^{\varepsilon}\omega^{\varepsilon})(\mathrm{Id}-S_{0}^{\varepsilon})\omega^{\varepsilon}(\tau,x)dx,$

for any $\tau\in[0,T]$. Accordingly, by taking the inner product of the
transport equation (1.14) with $S_{0}^{\varepsilon}\omega^{\varepsilon}$, we
obtain, for any $t\in[0,T]$, that

$\displaystyle\frac{1}{2}\left\|\sqrt{S_{0}^{\varepsilon}}\omega^{\varepsilon}(t)\right\|_{L^{2}}^{2}$
$\displaystyle=\frac{1}{2}\left\|\sqrt{S_{0}^{\varepsilon}}\omega^{\varepsilon}_{0}\right\|_{L^{2}}^{2}+\int_{0}^{t}\int_{\mathbb{R}^{2}}\operatorname{curl}g^{\varepsilon}S_{0}^{\varepsilon}\omega^{\varepsilon}(\tau,x)dxd\tau$
$\displaystyle\quad+\int_{0}^{t}\int_{\mathbb{R}^{2}}u^{\varepsilon}\cdot\nabla(S_{0}^{\varepsilon}\omega^{\varepsilon})(\mathrm{Id}-S_{0}^{\varepsilon})\omega^{\varepsilon}(\tau,x)dxd\tau.$

On the other hand, we know, for any $t\in[0,T]$, that

$\displaystyle\frac{1}{2}\left\|\omega^{\varepsilon}(t)\right\|_{L^{2}}^{2}$
$\displaystyle=\frac{1}{2}\left\|\omega^{\varepsilon}_{0}\right\|_{L^{2}}^{2}+\int_{0}^{t}\int_{\mathbb{R}^{2}}\operatorname{curl}g^{\varepsilon}\omega^{\varepsilon}(\tau,x)dxd\tau.$

Consequently, we see that (1.15) follows from the combination of the two
preceding identities with

$\left\|\omega^{\varepsilon}(t)\right\|_{L^{2}}^{2}=\left\|\sqrt{S_{0}^{\varepsilon}}\omega^{\varepsilon}(t)\right\|_{L^{2}}^{2}+\left\|\sqrt{(\mathrm{Id}-S_{0}^{\varepsilon})}\omega^{\varepsilon}(t)\right\|_{L^{2}}^{2},$

which completes its justification.∎

####  1.5.5. Justification of (1.17)

Here, we give a complete proof of estimate (1.17) on $\mathcal{J}_{1}$, which
follows directly from an application of Lemma 1.7, below. We proceed in
several steps. First, in Lemma 1.5, we summarize the fundamental properties of
the partition of unity introduced in the preceding steps. Then, in Lemma 1.6,
we establish a crucial identity which is a natural consequence of the
localization of frequencies in the nonlinear advection term
$u\cdot\nabla\omega$. Finally, we combine the results of Lemmas 1.5 and 1.6 to
deduce Lemma 1.7.

######  Lemma 1.5.

The dyadic frequency decomposition operators given in (1.11) satisfy the
identities

$\displaystyle\Delta_{i}^{\varepsilon}\Delta_{j}^{\varepsilon}=0,$ (1.20)
$\displaystyle\Delta_{i}^{\varepsilon}=\Delta_{i}^{\varepsilon}\sum_{k\in\\{i,i\pm
1\\}}\Delta_{k}^{\varepsilon},$ (1.21)
$\displaystyle\left(\Delta_{i}^{\varepsilon}+\Delta_{i+1}^{\varepsilon}\right)\Delta_{i}^{\varepsilon}\Delta_{i+1}^{\varepsilon}=\Delta_{i}^{\varepsilon}\Delta_{i+1}^{\varepsilon},$

for any $i,j\in\mathbb{Z}$, with $|i-j|\geq 2$.

Furthermore, we have that

(1.22)
$\operatorname{supp}{\mathcal{F}\left(\Delta_{i}^{\varepsilon}f\Delta_{i+1}^{\varepsilon}\Delta_{i+2}^{\varepsilon}g\right)}\subset\left\\{|\xi|\geq\frac{\Theta_{\varepsilon}2^{i}}{3}\right\\},$

for any tempered distributions $f$ and $g$.

###### Proof.

This is a consequence of the localization of the support of the function
$\varphi_{j}$, which defines the dyadic block $\Delta_{j}^{\varepsilon}$. We
refer to [2, Proposition 2.10] for the proof of the first three identities.

As for (1.22), it follows from the fact that

$\operatorname{supp}\left(\varphi_{i+1}^{\varepsilon}\varphi_{i+2}^{\varepsilon}\right)\subset\left\\{\xi\in\mathbb{R}^{2}:3\cdot
2^{i}\leq\frac{|\xi|}{\Theta_{\varepsilon}}\leq\frac{8}{3}\cdot
2^{i+1}\right\\},$

which implies that

$\operatorname{supp}\varphi_{i}^{\varepsilon}+\operatorname{supp}\left(\varphi_{i+1}^{\varepsilon}\varphi_{i+2}^{\varepsilon}\right)\subset\left\\{\xi\in\mathbb{R}^{2}:\frac{|\xi|}{\Theta_{\varepsilon}}\geq\frac{2^{i}}{3}\right\\},$

thereby completing the proof. ∎

######  Lemma 1.6.

Let $u$ be a divergence-free vector field in $L^{2}(\mathbb{R}^{2})$ and
$\omega$ be a real-valued function in $L^{2}(\mathbb{R}^{2})$. Then, for any
$j\in\mathbb{Z}$, it holds that

$\displaystyle\int_{\mathbb{R}^{2}}u\cdot\nabla\Delta_{j}^{\varepsilon}\omega\Delta_{j+1}^{\varepsilon}\omega
dx=\mathcal{I}_{1}+\mathcal{I}_{2},$

where

$\displaystyle\mathcal{I}_{1}$ $\displaystyle\overset{\rm
def}{=}\int_{\mathbb{R}^{2}}\left[\Delta_{j+1}^{\varepsilon},u\cdot\nabla\right]\Delta_{j}^{\varepsilon}\omega\left(\Delta_{j}^{\varepsilon}+\Delta_{j+1}^{\varepsilon}\right)\omega
dx$
$\displaystyle\quad-\frac{1}{2}\int_{\mathbb{R}^{2}}\left[\Delta_{j}^{\varepsilon}\Delta_{j+1}^{\varepsilon},u\cdot\nabla\right]\left(\Delta_{j}^{\varepsilon}+\Delta_{j+1}^{\varepsilon}\right)\omega\left(\Delta_{j}^{\varepsilon}+\Delta_{j+1}^{\varepsilon}\right)\omega
dx,$

and

$\mathcal{I}_{2}\overset{\rm
def}{=}\int_{\mathbb{R}^{2}}\mathds{1}_{|D|>\frac{\Theta_{\varepsilon}2^{j}}{3}}u\cdot\nabla\Delta_{j}^{\varepsilon}\omega\Delta_{j+1}^{\varepsilon}\Delta_{j+2}^{\varepsilon}\omega
dx.$

###### Proof.

For the sake of simplicity, we assume that $j=0$. The general case
$j\in\mathbb{Z}$ follows from a similar argument.

We begin by utilizing identity (1.20) to write that

$\displaystyle\int_{\mathbb{R}^{2}}u\cdot\nabla\Delta_{0}^{\varepsilon}\omega\Delta_{1}^{\varepsilon}\omega=\int_{\mathbb{R}^{2}}u\cdot\nabla\Delta_{0}^{\varepsilon}\omega\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\Delta_{1}^{\varepsilon}\omega+\int_{\mathbb{R}^{2}}u\cdot\nabla\Delta_{0}^{\varepsilon}\omega\Delta_{1}^{\varepsilon}\Delta_{2}^{\varepsilon}\omega\overset{\rm
def}{=}\mathcal{K}_{1}+\mathcal{K}_{2}.$

Then, by virtue of (1.22), the expression $\mathcal{K}_{2}$ can be recast as

$\mathcal{K}_{2}=\int_{\mathbb{R}^{2}}\mathds{1}_{|D|>\frac{\Theta_{\varepsilon}}{3}}u\cdot\nabla\Delta_{0}^{\varepsilon}\omega\Delta_{1}^{\varepsilon}\Delta_{2}^{\varepsilon}\omega,$

which is precisely $\mathcal{I}_{2}$.

As for $\mathcal{K}_{1}$, we split it into two parts

$\mathcal{K}_{1}=\int_{\mathbb{R}^{2}}\left[\Delta_{1}^{\varepsilon},u\cdot\nabla\right]\Delta_{0}^{\varepsilon}\omega^{\varepsilon}\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega+\int_{\mathbb{R}^{2}}u\cdot\nabla\Delta_{0}^{\varepsilon}\Delta_{1}^{\varepsilon}\omega\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega\overset{\rm
def}{=}\mathcal{K}_{11}+\mathcal{K}_{12}.$

In particular, observe that $\mathcal{K}_{11}$ already matches the first term
in $\mathcal{I}_{1}$.

Regarding $\mathcal{K}_{12}$, we perform an integration by parts followed by
(1.21) to deduce that

$\displaystyle\mathcal{K}_{12}$
$\displaystyle=-\int_{\mathbb{R}^{2}}u\cdot\nabla\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega\Delta_{0}^{\varepsilon}\Delta_{1}^{\varepsilon}\omega$
$\displaystyle=-\int_{\mathbb{R}^{2}}u\cdot\nabla\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega\Delta_{0}^{\varepsilon}\Delta_{1}^{\varepsilon}\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega,$

which, employing commutators, can be rewritten as

$\displaystyle\mathcal{K}_{12}$
$\displaystyle=-\int_{\mathbb{R}^{2}}\left[\Delta_{0}^{\varepsilon}\Delta_{1}^{\varepsilon},u\cdot\nabla\right]\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega$
$\displaystyle\quad-\int_{\mathbb{R}^{2}}u\cdot\nabla\Delta_{0}^{\varepsilon}\Delta_{1}^{\varepsilon}\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega.$

Then, we use (1.21), again, to find that

$\displaystyle\mathcal{K}_{12}$
$\displaystyle=-\int_{\mathbb{R}^{2}}\left[\Delta_{0}^{\varepsilon}\Delta_{1}^{\varepsilon},u\cdot\nabla\right]\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega-\int_{\mathbb{R}^{2}}u\cdot\nabla\Delta_{0}^{\varepsilon}\Delta_{1}^{\varepsilon}\omega\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega$
$\displaystyle=-\int_{\mathbb{R}^{2}}\left[\Delta_{0}^{\varepsilon}\Delta_{1}^{\varepsilon},u\cdot\nabla\right]\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega-\mathcal{K}_{12}.$

Consequently, we arrive at the conclusion that

$\mathcal{K}_{12}=-\frac{1}{2}\int_{\mathbb{R}^{2}}\left[\Delta_{0}^{\varepsilon}\Delta_{1}^{\varepsilon},u\cdot\nabla\right]\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega\left(\Delta_{0}^{\varepsilon}+\Delta_{1}^{\varepsilon}\right)\omega,$

thereby completing the proof of the lemma by successfully identifying all the
terms in $\mathcal{I}_{1}$. ∎

######  Lemma 1.7.

For any $u\in\dot{H}^{1}(\mathbb{R}^{2})$ and $\omega\in
L^{4}(\mathbb{R}^{2})$, it holds that

$\displaystyle\left|\int_{\mathbb{R}^{2}}u\cdot\nabla\Delta_{j}^{\varepsilon}\omega\Delta_{j+1}^{\varepsilon}\omega\right|\lesssim\left\|\nabla
u\right\|_{L^{2}}\left(\left\|\Delta_{j}^{\varepsilon}\omega\right\|_{L^{4}}^{2}+\left\|\Delta_{j+1}^{\varepsilon}\omega\right\|_{L^{4}}^{2}\right),$

for all $j\in\mathbb{Z}$.

More precisely, there is a decomposition

$\int_{\mathbb{R}^{2}}u\cdot\nabla\Delta_{j}^{\varepsilon}\omega\Delta_{j+1}^{\varepsilon}\omega=\mathcal{I}_{1}+\mathcal{I}_{2},$

where

$|\mathcal{I}_{1}|\lesssim\left\|\nabla
u\right\|_{L^{q}}\left(\left\|\Delta_{j}^{\varepsilon}\omega\right\|_{L^{\frac{2q}{q-1}}}^{2}+\left\|\Delta_{j+1}^{\varepsilon}\omega\right\|_{L^{\frac{2q}{q-1}}}^{2}\right)$

and

$|\mathcal{I}_{2}|\lesssim\left\|\mathds{1}_{|D|>\frac{\Theta_{\varepsilon}2^{j}}{3}}\nabla
u\right\|_{L^{2}}\left\|\Delta_{j}^{\varepsilon}\omega\right\|_{L^{2p}}\left\|\Delta_{j+1}^{\varepsilon}\omega\right\|_{L^{\frac{2p}{p-1}}},$

for any $p,q\in[1,\infty]$ and $j\in\mathbb{Z}$.

###### Proof.

The decomposition into $\mathcal{I}_{1}+\mathcal{I}_{2}$ follow from an
application of Lemma 1.6. Then, a direct application of Hölder’s inequality
gives that

$\displaystyle|\mathcal{I}_{2}|$
$\displaystyle\lesssim\left\|\mathds{1}_{|D|>\frac{\Theta_{\varepsilon}2^{j}}{3}}u\right\|_{L^{2}}\left\|\nabla\Delta_{j}^{\varepsilon}\omega\right\|_{L^{2p}}\left\|\Delta_{j+1}^{\varepsilon}\omega\right\|_{L^{\frac{2p}{p-1}}}$
$\displaystyle\lesssim
2^{j}\Theta_{\varepsilon}\left\|\mathds{1}_{|D|>\frac{\Theta_{\varepsilon}2^{j}}{3}}u\right\|_{L^{2}}\left\|\Delta_{j}^{\varepsilon}\omega\right\|_{L^{2p}}\left\|\Delta_{j+1}^{\varepsilon}\omega\right\|_{L^{\frac{2p}{p-1}}}$
$\displaystyle\lesssim\left\|\mathds{1}_{|D|>\frac{\Theta_{\varepsilon}2^{j}}{3}}\nabla
u\right\|_{L^{2}}\left\|\Delta_{j}^{\varepsilon}\omega\right\|_{L^{2p}}\left\|\Delta_{j+1}^{\varepsilon}\omega\right\|_{L^{\frac{2p}{p-1}}},$

which is the desired bound on $\mathcal{I}_{2}$.

The control of $\mathcal{I}_{1}$ is slightly more involved. It requires the
use of a classical commutator estimate, which can be found in [2, Lemma 2.97]
and gives that

$\left\|\left[\Delta_{j+1}^{\varepsilon},u\cdot\nabla\right]\Delta_{j}^{\varepsilon}\omega\right\|_{L^{\frac{2q}{q+1}}}\lesssim(2^{j}\Theta_{\varepsilon})^{-1}\left\|\nabla
u\right\|_{L^{q}}\left\|\nabla\Delta_{j}^{\varepsilon}\omega\right\|_{L^{\frac{2q}{q-1}}}\lesssim\left\|\nabla
u\right\|_{L^{q}}\left\|\Delta_{j}^{\varepsilon}\omega\right\|_{L^{\frac{2q}{q-1}}}.$

Combining this bound with another use of Hölder’s inequality provides a
suitable estimate on the first term in $\mathcal{I}_{1}$. The control of the
second term in $\mathcal{I}_{1}$ is similar, which completes the proof of the
Lemma. ∎

##  2\. Rate of convergence of the inviscid limit in Yudovich’s class

###  2.1. Second main result

Here, we show that our new method of proof of Theorem 1.2 leads to a refined
understanding of the inviscid limit in the two-dimensional Navier–Stokes
system

(2.1) $\begin{cases}\begin{aligned}
\partial_{t}u^{\varepsilon}+u^{\varepsilon}\cdot\nabla
u^{\varepsilon}-\varepsilon\Delta u^{\varepsilon}+\nabla
p^{\varepsilon}&=0,\\\ \operatorname{div}u^{\varepsilon}&=0,\\\
{u^{\varepsilon}}|_{t=0}&=u_{0}.\end{aligned}\end{cases}$

Specifically, we prove the following result.

(In the statement below, we use the classical notation $B^{s}_{p,q}$ to denote
the usual Besov space of regularity $s\in\mathbb{R}$, integrability
$p\in[1,\infty]$ and frequency summability $q\in[1,\infty]$. We will also make
use of the homogeneous version of Besov spaces defined in the usual way.)

######  Theorem 2.1.

Let $T>0$, $s\in(0,1)$ and $u_{0}\in
H^{1}\cap\dot{B}^{1+s}_{2,\infty}\cap\dot{B}^{1+s}_{p,\infty}$, for some
$p\in(2,\infty)$. Assume also that $\omega_{0}=\operatorname{curl}u_{0}\in
L^{\infty}$. Let $u^{\varepsilon}$ be the unique solution of (2.1) and $u$ be
the unique Yudovich solution of the Euler equation (1.2) with the same initial
data $u_{0}$ and no source term, i.e., $g=0$.

Further assume that

(2.2)
$\left\|u^{\varepsilon}-u\right\|_{L^{\infty}([0,T];L^{2})}=O\left(\varepsilon^{\alpha}\right),$

for some $\alpha>0$, as $\varepsilon\to 0$, and that

(2.3)
$t\mapsto\left\|(\omega^{\varepsilon},\omega)(t)\right\|_{B^{s(t)}_{2,\infty}\cap
B^{s(t)}_{p,\infty}(\mathbb{R}^{2})}\in L^{\infty}([0,T]),$

uniformly in $\varepsilon$, for some nonnegative function $s(t)\in
C^{1}([0,T])$, with $s(0)=s$ and $s^{\prime}(t)<0$, for all $t\in[0,T]$.

Then, it holds that

$\left\|u^{\varepsilon}-u\right\|_{L^{\infty}([0,T];\dot{H}^{1})}=O\left(\left(\frac{\varepsilon^{2\alpha
s(T)}}{|\log\varepsilon|}\right)^{\frac{1}{2(1+s(T))}}\right),$

as $\varepsilon\to 0.$

Observe that (2.3) yields that $\omega^{\varepsilon}$ and $\omega$ are bounded
in $L^{\infty}([0,T];\dot{B}^{s(T)}_{2,\infty})$, uniformly in $\varepsilon$.
Therefore, employing the classical convexity inequality

$\displaystyle\left\|u^{\varepsilon}-u\right\|_{L^{\infty}([0,T];\dot{H}^{1})}$
$\displaystyle\lesssim\left\|u^{\varepsilon}-u\right\|_{L^{\infty}([0,T];\dot{B}^{1}_{2,1})}$
$\displaystyle\lesssim\left\|u^{\varepsilon}-u\right\|_{L^{\infty}([0,T];\dot{B}^{0}_{2,\infty})}^{\frac{s(T)}{1+s(T)}}\left\|u^{\varepsilon}-u\right\|_{L^{\infty}([0,T];\dot{B}^{1+s(T)}_{2,\infty})}^{\frac{1}{1+s(T)}}$
$\displaystyle\lesssim\left\|u^{\varepsilon}-u\right\|_{L^{\infty}([0,T];L^{2})}^{\frac{s(T)}{1+s(T)}}\left\|u^{\varepsilon}-u\right\|_{L^{\infty}([0,T];\dot{B}^{1+s(T)}_{2,\infty})}^{\frac{1}{1+s(T)}}$

along with the bounds (2.2) and (2.3) yields the interpolated rate of
convergence

(2.4)
$\left\|u^{\varepsilon}-u\right\|_{L^{\infty}([0,T];\dot{H}^{1})}=O\left(\varepsilon^{\frac{\alpha
s(T)}{1+s(T)}}\right).$

This rate of convergence has already been discussed by other authors in [5,
Corollary 2]. The conclusion of Theorem 2.1 establishes a subtle logarithmic
refinement of the above rate of convergence as a consequence of the fact that
the regularity of $\omega^{\varepsilon}$ and $\omega$ is strictly larger than
$s(T)$ for times $t<T$. This is achieved by building on our new method of
proof of Theorem 1.2.

Thus, the assumption that $s^{\prime}(t)$ be negative is essential and
describes a “memory effect” of the higher regularity at previous times.
Moreover, it seems plausible that this phenomenon is not exclusive to
hydrodynamical models. It therefore deserves further investigation.

######  Remark 2.1.

A straightforward adaptation of the proof of convergence of velocities in
$L^{\infty}_{t}L^{2}$ given in Section 1.5.1 shows that (2.2) holds with the
value $\alpha=\frac{1}{2}\exp(-C_{*}T)$, where
$C_{*}=C\left\|\omega_{0}\right\|_{L^{2}\cap L^{\infty}}$, for some universal
constant $C>0$. A precise justification for this result can be found in [2,
Theorem 7.37]. Furthermore, under the assumptions of Theorem 2.1, classical
regularity results on transport flows (see [2, Theorem 3.28]) show that the
unique solution $u^{\varepsilon}$ to (2.1) enjoys the decaying regularity
estimate (2.3) with $s(t)=s\exp(-C_{*}t)$. Hence, an application of Theorem
2.1 establishes the rate of convergence

$\left\|u^{\varepsilon}-u\right\|_{L^{\infty}([0,T];\dot{H}^{1})}=O\left(\left(\frac{\varepsilon^{s\exp(-2C_{*}T)}}{|\log\varepsilon|}\right)^{\frac{1}{2(1+s\exp(-C_{*}T))}}\right),$

as $\varepsilon\to 0$. Note that there are other results (see [3], for
instance) showing that (2.3) holds with even slower regularity decays, thereby
further improving the ensuing rates of convergence stemming from the
application of Theorem 2.1.

######  Remark 2.2.

The preceding theorem covers vortex patches with non-smooth boundary, i.e,
solutions which correspond to the initial data

$\omega_{0}=\mathds{1}_{\Omega},$

where $\Omega$ is a bounded domain in $\mathbb{R}^{2}$ whose boundary has a
Minkowski dimension $D$ which is strictly less than $2$. Indeed, in that case,
one can show (see [6, Lemma 3.2]) that

$\omega_{0}\in B^{\frac{2-D}{p}}_{p,\infty}(\mathbb{R}^{2}),$

for any $p\in[1,\infty)$. Accordingly, it follows that

$\omega_{0}\in B^{s}_{2,\infty}\cap B^{s}_{p,\infty}(\mathbb{R}^{2}),$

for some $s\in(0,1)$.

###  2.2. Proof of Theorem 2.1

This proof consists in refining the ideas leading to Theorem 1.2 by exploiting
the available additional regularity and optimizing the choice of parameters.
We proceed in three steps:

  1. (1)

In Section 2.2.1, we revisit the arguments laid out in Section 1.5.2.

  2. (2)

In Section 2.2.2, we exploit the additional regularity of solutions to improve
the ideas behind the Compactness Extrapolation Lemma (Lemma 1.4).

  3. (3)

In Section 2.2.3, we optimize the choice of $\Theta_{\varepsilon}$, which is
an essential parameter describing the convergence of solutions.

####  2.2.1. Bounds on low frequencies

We see that

$\left\|\mathds{1}_{|D|\leq\Theta_{\varepsilon}}(\omega^{\varepsilon}-\omega)\right\|_{L^{\infty}([0,T];L^{2})}\lesssim\Theta_{\varepsilon}\varepsilon^{\alpha},$

for any parameter $\Theta_{\varepsilon}$, which will be chosen such that

(2.5) $\lim_{\varepsilon\rightarrow 0}\Theta_{\varepsilon}=\infty\quad\text{
and }\quad\lim_{\varepsilon\rightarrow
0}\Theta_{\varepsilon}\varepsilon^{\alpha}=0$

to ensure the convergence of low frequencies.

####  2.2.2. Bounds on high frequencies

We focus now on the evanescence of high frequencies. To that end, we write
that

$\left\|\mathds{1}_{|D|\geq\Theta_{\varepsilon}}(\omega^{\varepsilon}-\omega)\right\|_{L^{\infty}([0,T];L^{2})}\leq\left\|\mathds{1}_{|D|\geq\Theta_{\varepsilon}}\omega^{\varepsilon}\right\|_{L^{\infty}([0,T];L^{2})}+\left\|\mathds{1}_{|D|\geq\Theta_{\varepsilon}}\omega\right\|_{L^{\infty}([0,T];L^{2})}$

and we estimate each term in the right-hand side, above, by refining the
techniques laid out in Section 1.5.3 and exploiting the available additional
regularity of velocities to obtain a sharp rate of convergence. We only
outline the control of the high frequencies of $\omega^{\varepsilon}$, for the
control of $\omega$ is identical.

From (1.15), we find that

(2.6)
$\displaystyle\frac{1}{2}\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega^{\varepsilon}(t)\right\|_{L^{2}}^{2}$
$\displaystyle=\frac{1}{2}\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega_{0}\right\|_{L^{2}}^{2}-\varepsilon\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega^{\varepsilon}\right\|_{L^{2}_{t}\dot{H}^{1}}^{2}-\mathcal{J}(t)$
$\displaystyle\leq\frac{1}{2}\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega_{0}\right\|_{L^{2}}^{2}+\sum_{i=1}^{3}|\mathcal{J}_{i}(t)|,$

where $\\{\mathcal{J}_{i}\\}_{i\in\\{1,2,3\\}}$ are given in (1.16).

Then, by applying Lemma 1.7, we find that

$\displaystyle|\mathcal{J}_{1}(t)|$
$\displaystyle\lesssim\int_{0}^{t}\left\|\nabla
u^{\varepsilon}(\tau)\right\|_{L^{q}}\left(\left\|\Delta_{-1}^{\varepsilon}\omega^{\varepsilon}(\tau)\right\|_{L^{\frac{2q}{q-1}}}^{2}+\left\|\Delta_{0}^{\varepsilon}\omega^{\varepsilon}(\tau)\right\|_{L^{\frac{2q}{q-1}}}^{2}\right)d\tau$
$\displaystyle\quad+\int_{0}^{t}\left\|\mathds{1}_{|D|>\frac{\Theta_{\varepsilon}}{6}}\nabla
u^{\varepsilon}(\tau)\right\|_{L^{2}}\left\|\Delta_{-1}^{\varepsilon}\omega^{\varepsilon}(\tau)\right\|_{L^{\infty}}\left\|\Delta_{0}^{\varepsilon}\omega^{\varepsilon}(\tau)\right\|_{L^{2}}d\tau,$

where $q\in(2,\infty)$ is fixed. It follows that

$|\mathcal{J}_{1}(t)|\lesssim\left\|\omega^{\varepsilon}\right\|_{L^{\infty}([0,t];L^{2}\cap
L^{\infty})}\int_{0}^{t}\left\|(\mathrm{Id}-S_{-3}^{\varepsilon})\omega^{\varepsilon}(\tau)\right\|_{L^{\frac{2q}{q-1}}\cap
L^{2}}^{2}d\tau,$

where we denoted

$S_{k}^{\varepsilon}=\sum_{j\leq k-1}\Delta_{j}^{\varepsilon},$

for any $k\in\mathbb{Z}$. This takes care of $\mathcal{J}_{1}$.

As for $\mathcal{J}_{2}$ and $\mathcal{J}_{3}$, it is readily seen from (1.18)
and (1.19) that

$|\mathcal{J}_{2}(t)|+|\mathcal{J}_{3}(t)|\lesssim\left\|\omega^{\varepsilon}\right\|_{L^{\infty}([0,t];L^{\infty})}\int_{0}^{t}\left\|(\mathrm{Id}-S_{-4}^{\varepsilon})\omega^{\varepsilon}(\tau)\right\|_{L^{2}}^{2}d\tau.$

All in all, by plugging the foregoing bounds into (2.6) and utilizing (1.3)
with $p\in\\{2,\infty\\}$, we obtain that

$\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega^{\varepsilon}(t)\right\|_{L^{2}}^{2}\lesssim_{q}\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega_{0}\right\|_{L^{2}}^{2}+\int_{0}^{t}\left\|(\mathrm{Id}-S_{-4}^{\varepsilon})\omega^{\varepsilon}(\tau)\right\|_{L^{2}\cap
L^{\frac{2q}{q-1}}}^{2}d\tau.$

At last, observing that the same estimate holds for $\omega$ in place of
$\omega^{\varepsilon}$, we deduce that

$\displaystyle\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\left(\omega^{\varepsilon}-\omega\right)(t)\right\|_{L^{2}}^{2}$
$\displaystyle\lesssim_{q}\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\omega_{0}\right\|_{L^{2}}^{2}+\int_{0}^{t}\left\|(\mathrm{Id}-S_{-4}^{\varepsilon})(\omega^{\varepsilon},\omega)(\tau)\right\|_{L^{2}\cap
L^{\frac{2q}{q-1}}}^{2}d\tau$
$\displaystyle\lesssim_{q}\Theta_{\varepsilon}^{-2s}\left\|\omega_{0}\right\|_{\dot{B}_{2,\infty}^{s}}^{2}$
$\displaystyle\quad+\left(\int_{0}^{t}\Theta_{\varepsilon}^{-2s(\tau)}d\tau\right)\sup_{\tau\in[0,t]}\left\|(\omega^{\varepsilon},\omega)(\tau)\right\|_{\dot{B}^{s(\tau)}_{2,\infty}\cap\dot{B}^{s(\tau)}_{\frac{2q}{q-1},\infty}}^{2},$

for any $t\in[0,T]$. In particular, employing (2.3), we conclude that

$\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\left(\omega^{\varepsilon}-\omega\right)(t)\right\|_{L^{2}}^{2}\lesssim\Theta_{\varepsilon}^{-2s}+\int_{0}^{t}\Theta_{\varepsilon}^{-2s(\tau)}d\tau,$

where we chose $q$ so that $p=\frac{2q}{q-1}$ (without any loss of generality,
we may assume here that $p\in(2,4)$).

Finally, evaluating that

$\displaystyle\int_{0}^{t}\Theta_{\varepsilon}^{-2s(\tau)}d\tau$
$\displaystyle=\int_{0}^{t}\frac{-1}{2s^{\prime}(\tau)\log\Theta_{\varepsilon}}\frac{d}{d\tau}\left(\Theta_{\varepsilon}^{-2s(\tau)}\right)d\tau$
$\displaystyle\leq\frac{-1}{2s^{\prime}(t)\log\Theta_{\varepsilon}}\int_{0}^{t}\frac{d}{d\tau}\left(\Theta_{\varepsilon}^{-2s(\tau)}\right)d\tau$
$\displaystyle=\frac{-1}{2s^{\prime}(t)\log\Theta_{\varepsilon}}\left(\Theta_{\varepsilon}^{-2s(t)}-\Theta_{\varepsilon}^{-2s}\right)$
$\displaystyle\lesssim\frac{\Theta_{\varepsilon}^{-2s(t)}-\Theta_{\varepsilon}^{-2s}}{\log\Theta_{\varepsilon}},$

we arrive at

$\left\|\sqrt{\mathrm{Id}-S_{0}^{\varepsilon}}\left(\omega^{\varepsilon}-\omega\right)(t)\right\|_{L^{2}}^{2}\lesssim\Theta_{\varepsilon}^{-2s}+\frac{\Theta_{\varepsilon}^{-2s(t)}}{\log\Theta_{\varepsilon}}\lesssim\frac{\Theta_{\varepsilon}^{-2s(t)}}{\log\Theta_{\varepsilon}}.$

####  2.2.3. Optimization of $\Theta_{\varepsilon}$

By combining the low-frequency and high-frequency estimates established in the
previous two steps, we obtain that

(2.7)
$\left\|\omega^{\varepsilon}-\omega\right\|_{L^{\infty}([0,T];L^{2})}\lesssim\Theta_{\varepsilon}\varepsilon^{\alpha}+\frac{\Theta_{\varepsilon}^{-s(T)}}{\log^{\frac{1}{2}}\Theta_{\varepsilon}}.$

We are now going to choose a specific value for $\Theta_{\varepsilon}$
satisfying (2.5) which will optimize the above estimate.

To that end, notice first that the value

$\Theta_{\varepsilon}=\left(\frac{s(T)}{\varepsilon^{\alpha}}\right)^{\frac{1}{1+s(T)}}$

is the global minimum of the function

$\Theta_{\varepsilon}\mapsto\Theta_{\varepsilon}\varepsilon^{\alpha}+\Theta_{\varepsilon}^{-s(T)}.$

Accordingly, if we ignore the logarithmic correction in (2.7), this choice of
$\Theta_{\varepsilon}$ entails the rate of convergence displayed in (2.4),
which is not optimal.

In order to improve this rate of convergence, we need now to exploit the
logarithmic decay of the last term in (2.7). Thus, we define

$\Theta_{\varepsilon}\overset{\rm
def}{=}\varepsilon^{-\frac{\alpha}{1+s(T)}}|\log\varepsilon|^{-\beta},$

where $\beta\in\mathbb{R}$ will be determined shortly. In particular, observe
that

$\log\Theta_{\varepsilon}\gtrsim|\log\varepsilon|,$

as $\varepsilon\to 0$. Therefore, incorporating this value into (2.7) yields
that

$\displaystyle\left\|\omega^{\varepsilon}-\omega\right\|_{L^{\infty}([0,T];L^{2})}$
$\displaystyle\lesssim\varepsilon^{\frac{\alpha
s(T)}{1+s(T)}}\left(|\log\varepsilon|^{-\beta}+|\log\varepsilon|^{\beta
s(T)}\log^{-\frac{1}{2}}\Theta_{\varepsilon}\right)$
$\displaystyle\lesssim\varepsilon^{\frac{\alpha
s(T)}{1+s(T)}}\left(|\log\varepsilon|^{-\beta}+|\log\varepsilon|^{\beta
s(T)-\frac{1}{2}}\right).$

At last, optimizing the value of $\beta$ by setting

$\beta\overset{\rm def}{=}\frac{1}{2(1+s(T))}$

leads to the rate of convergence claimed in the statement of Theorem 2.1,
which completes its proof. ∎

##  3\. Two-dimensional incompressible plasmas

###  3.1. Third main result

We are now interested in two-dimensional incompressible plasmas described by
the Euler–Maxwell equations

(3.1) $\begin{cases}\begin{aligned} \text{\tiny(Euler's
equation)}&&&\partial_{t}u+u\cdot\nabla u=-\nabla p+j\times
B,&\operatorname{div}u=0,&\\\ \text{\tiny(Amp\\`{e}re's
equation)}&&&\frac{1}{c}\partial_{t}E-\nabla\times
B=-j,&\operatorname{div}E=0,&\\\ \text{\tiny(Faraday's
equation)}&&&\frac{1}{c}\partial_{t}B+\nabla\times
E=0,&\operatorname{div}B=0,&\\\ \text{\tiny(Ohm's
law)}&&&j=\sigma\big{(}cE+P(u\times
B)\big{)},&\operatorname{div}j=0,&\end{aligned}\end{cases}$

where $(t,x)\in[0,\infty)\times\mathbb{R}^{2}$ and
$P=\mathrm{Id}-\Delta^{-1}\nabla\mathrm{div}$ denotes Leray’s projector onto
divergence-free vector fields, supplemented with initial data
$(u,E,B)_{|t=0}=(u_{0},E_{0},B_{0})$ such that
$\operatorname{div}u_{0}=\operatorname{div}E_{0}=\operatorname{div}B_{0}=0$.

The positive parameters $\sigma$ and $c$ refer to the electric conductivity
and the speed of light, respectively, whereas the physical variables $u(t,x)$,
$E(t,x)$ and $B(t,x)$ are vector fields with values in $\mathbb{R}^{3}$ which
are interpreted as the fluid velocity, the electric field and the magnetic
field, respectively. Finally, the field $j(t,x)$ denotes the electric current
density and is determined by Ohm’s law.

A straightforward calculation shows that solutions to the above system
satisfy, at least formally, the energy inequality

(3.2)
$\left\|u(t)\right\|_{L^{2}}^{2}+\left\|E(t)\right\|_{L^{2}}^{2}+\left\|B(t)\right\|_{L^{2}}^{2}+\frac{2}{\sigma}\int_{0}^{t}\left\|j(\tau)\right\|_{L^{2}}^{2}d\tau\leq\mathcal{E}_{0}^{2},$

for all $t\geq 0$, where we denote

$\mathcal{E}_{0}\overset{\rm
def}{=}\left(\left\|u_{0}\right\|_{L^{2}}^{2}+\left\|E_{0}\right\|_{L^{2}}^{2}+\left\|B_{0}\right\|_{L^{2}}^{2}\right)^{\frac{1}{2}}.$

Current methods fail to produce the existence of solutions to (3.1) for
initial data merely satisfying that $\mathcal{E}_{0}$ is finite. In general,
when studying solutions of (3.1), it is therefore natural to suppose that
$(u_{0},E_{0},B_{0})$ enjoys some suitable enhanced regularity.

Henceforth, we further assume that the fields $u$, $E$ and $B$ satisfy the
two-dimensional normal structure

(3.3) $u(t,x)=\begin{pmatrix}u_{1}(t,x)\\\ u_{2}(t,x)\\\ 0\end{pmatrix},\qquad
E(t,x)=\begin{pmatrix}E_{1}(t,x)\\\ E_{2}(t,x)\\\
0\end{pmatrix}\qquad\text{and}\qquad B(t,x)=\begin{pmatrix}0\\\ 0\\\
b(t,x)\end{pmatrix}.$

This structure played a significant role in our recent work [1], where we
established the global existence and uniqueness of solutions to (3.1), with
the structure (3.3), provided the speed of light $c$ is sufficiently large
when compared to the size of the initial data in suitable norms. The condition
therein relating $c$ to the initial data can be interpreted as a strengthening
of the physical principle that the velocity of the fluid cannot exceed the
speed of light. A precise reformulation of the well-posedness result from [1]
is contained in the following theorem.

######  Theorem 3.1 ([1]).

Let $s$ be any real number in $(\frac{7}{4},2)$ and consider a bounded family
of initial data

$\big{\\{}(u_{0}^{c},E_{0}^{c},B_{0}^{c})\big{\\}}_{c>0}\subset\left(H^{1}\times
H^{s}\times H^{s}\right)(\mathbb{R}^{2}),$

with
$\operatorname{div}u_{0}^{c}=\operatorname{div}E_{0}^{c}=\operatorname{div}B_{0}^{c}$
and the two-dimensional normal structure (3.3), such that the initial
vorticities $\omega_{0}^{c}\overset{\rm def}{=}\nabla\times u_{0}^{c}$ form a
bounded family of $L^{\infty}(\mathbb{R}^{2})$.

There is a constant $c_{0}>0$, such that, for any speed of light
$c\in(c_{0},\infty)$, there is a global weak solution $(u^{c},E^{c},B^{c})$ to
the two-dimensional Euler–Maxwell system (3.1), with the normal structure
(3.3) and initial data $(u_{0}^{c},E_{0}^{c},B_{0}^{c})$, satisfying the
energy inequality (3.2) and enjoying the additional regularity

(3.4) $\begin{gathered}u^{c}\in L^{\infty}(\mathbb{R}^{+};L^{\infty}\cap
H^{1}),\quad\omega^{c}\overset{\rm def}{=}\nabla\times u^{c}\in
L^{\infty}(\mathbb{R}^{+};L^{\infty}),\quad(E^{c},B^{c})\in
L^{\infty}(\mathbb{R}^{+};H^{s}),\\\ (cE^{c},B^{c})\in
L^{2}(\mathbb{R}^{+};\dot{H}^{1}\cap\dot{H}^{s}),\quad(E^{c},B^{c})\in
L^{2}(\mathbb{R}^{+};\dot{W}^{1,\infty}),\quad j^{c}\in
L^{2}(\mathbb{R}^{+};L^{2}\cap L^{\infty}),\end{gathered}$

where $j^{c}\overset{\rm def}{=}\sigma\big{(}cE^{c}+P(u^{c}\times
B^{c})\big{)}$. It is to be emphasized that the bounds in (3.4) are uniform in
$c\in(c_{0},\infty)$, for any bounded family of initial data.

Furthermore, for each $c\in(c_{0},\infty)$, the solution $(u^{c},E^{c},B^{c})$
is unique in the space of all solutions $(\bar{u},\bar{E},\bar{B})$ to the
Euler–Maxwell system (3.1) satisfying the bounds, locally in time,

$(\bar{u},\bar{E},\bar{B})\in L^{\infty}_{t}L^{2}_{x},\qquad\bar{u}\in
L^{2}_{t}L^{\infty}_{x},\qquad\bar{j}\in L^{2}_{t,x},$

and having the same initial data.

######  Remark 3.1.

We should note that the uniform bound $j^{c}\in
L^{2}(\mathbb{R}^{+};L^{\infty})$ in (3.4) cannot be found in the statements
of the theorems from [1]. However, it is established explicitely in Section
3.8 therein, as an intermediate step of the proof of the uniform bound
$\omega^{c}\in L^{\infty}(\mathbb{R}^{+};L^{\infty})$.

In view of the uniform bounds (3.4), it is possible to establish the
convergence

$(u^{c},E^{c},B^{c})\stackrel{{\scriptstyle
c\to\infty}}{{\longrightarrow}}(u,0,B),$

in the strong topology of
$L^{2}_{\mathrm{loc}}(\mathbb{R}^{+}\times\mathbb{R}^{2})$, of the solution
from Theorem 3.1, where $(u,B)$ is the unique solution to the
magnetohydrodynamic system

(3.5) $\left\\{\begin{aligned} &\partial_{t}u+u\cdot\nabla u=-\nabla
p,&\operatorname{div}u=0,&\\\ &\partial_{t}B+u\cdot\nabla
B-\frac{1}{\sigma}\Delta B=0,&&\end{aligned}\right.$

corresponding to the initial data $(u_{0},B_{0})$ and satisfying the bounds

$\begin{gathered}u\in L^{\infty}(\mathbb{R}^{+};L^{\infty}\cap
H^{1}),\quad\omega\overset{\rm def}{=}\nabla\times u\in
L^{\infty}(\mathbb{R}^{+};L^{\infty}),\\\ B\in
L^{\infty}(\mathbb{R}^{+};H^{1})\cap
L^{2}(\mathbb{R}^{+};\dot{H}^{1}\cap\dot{H}^{2}\cap\dot{W}^{1,\infty}).\end{gathered}$

This convergence result and the ensuing bounds follow from a standard
compactness argument and is formulated in Corollary 1.2 from [1].

Our aim is now to improve on this asymptotic characterization of (3.1) by
showing that the convergence holds globally in space, uniformly in time, in
adequate functional spaces. Our main global convergence result for inviscid
two-dimensional plasmas is contained in the following theorem.

######  Theorem 3.2.

Let $\big{\\{}(u_{0}^{c},E_{0}^{c},B_{0}^{c})\big{\\}}_{c>0}$ be a bounded
family of initial data as required in Theorem 3.1, for some fixed
$s\in(\frac{7}{4},2)$, and consider the unique solution $(u^{c},E^{c},B^{c})$
to (3.1) provided by the same theorem, for large $c$. If $u_{0}^{c}$ converges
to $u_{0}$ in ${H}^{1}(\mathbb{R}^{2})$ and $(E_{0}^{c},B_{0}^{c})$ converges
to $(0,B_{0})$ in $L^{2}(\mathbb{R}^{2})$, then $(u^{c},B^{c})$ converges
strongly to $(u,B)$, the unique global solution of (3.5), on any finite time
interval $[0,T]$, in the sense that

$\lim_{c\rightarrow\infty}\left\|u^{c}-u\right\|_{L^{\infty}([0,T];H^{1}(\mathbb{R}^{2}))}=0,$

and

$\lim_{c\rightarrow\infty}\left(\left\|B^{c}-B\right\|_{L^{\infty}([0,T];L^{2}(\mathbb{R}^{2}))}+\left\|B^{c}-B\right\|_{L^{2}([0,T];\dot{H}^{1}(\mathbb{R}^{2}))}\right)=0.$

Furthermore, the electric current density
$j^{c}=\sigma\big{(}cE^{c}+P(u^{c}\times B^{c})\big{)}$ converges strongly to
$\nabla\times B$ in the sense that

$\lim_{c\to\infty}\left\|j^{c}-\nabla\times
B\right\|_{L^{2}([0,T];L^{2}(\mathbb{R}^{2}))}=0.$

######  Remark 3.2.

It is possible to quantify the convergence of $u^{c}$ and $B^{c}$ in the
preceding theorem with a rate $O(c^{-\alpha})$, for some $\alpha>0$. This will
be clear in the proof of the theorem, below. Moreover, the electric field
$E^{c}$ vanishes in $L^{2}(\mathbb{R}^{+};\dot{H}^{1})$ with a rate
$O(c^{-1})$ due to the uniform bounds given in (3.4).

######  Remark 3.3.

In the statements of the above results, the restriction on the range of the
regularity parameter $s\in(\frac{7}{4},2)$ is only necessary in the
construction of global solutions given in Theorem 3.1. However, Theorem 3.2
would hold for the wider range $s\in[1,2]$ provided solutions to (3.1)
satisfying the uniform bounds (3.4) could be constructed for that range.

The proof of Theorem 3.2 is given in Section 3.3, below. It relies on a self-
contained asymptotic analysis of Ampère’s equation, which is detailed in the
next section.

###  3.2. Asymptotic analysis of Ampère’s equation

A formal asymptotic analysis of Ampère’s equation readily shows that $j^{c}$
converges to $\nabla\times B$, in the regime $c\to\infty$. The proof of our
main result Theorem 3.2 hinges upon a refined asymptotic analysis of the same
equation. More precisely, the following proposition provides a key estimate on
the distance between $\nabla\times B^{c}$ and $j^{c}$, as $c\to\infty$.

######  Proposition 3.3.

Let $\big{\\{}(u_{0}^{c},E_{0}^{c},B_{0}^{c})\big{\\}}_{c>0}$ be a bounded
family of initial data as required in Theorem 3.1, for some fixed
$s\in(\frac{7}{4},2)$, and consider the unique solution $(u^{c},E^{c},B^{c})$
provided by the same theorem, for $c>c_{0}$. Then, one has the estimate

$\left\|\nabla\times
B^{c}-j^{c}\right\|_{L^{2}(\mathbb{R}^{+};\dot{H}^{\eta-1})}\lesssim\frac{1}{c}\left\|\nabla\times
B^{c}_{0}-j^{c}_{0}\right\|_{\dot{H}^{\eta-1}}+\frac{1}{c}\left\|E^{c}_{0}\right\|_{\dot{H}^{\eta}}+\frac{C_{*}}{c^{2}},$

for every $\eta\in[1,s]$ and $c>c_{0}$, where the initial electric current
density is given by

$j_{0}^{c}\overset{\rm def}{=}\sigma\left(cE_{0}^{c}+P(u_{0}^{c}\times
B_{0}^{c})\right)$

and $C_{*}>0$ depends on the family of initial data.

######  Remark 3.4.

Notice that the family of initial data considered in the preceding proposition
is bounded uniformly in the spaces

$u_{0}^{c}\in L^{2},\qquad E_{0}^{c}\in H^{1},\qquad
B_{0}^{c}\in\dot{H}^{1}\cap L^{\infty}.$

Therefore, setting $\eta=1$, the above result provides us with the estimate

(3.6) $\left\|\nabla\times
B^{c}-j^{c}\right\|_{L^{2}(\mathbb{R}^{+};L^{2})}\lesssim\left\|E^{c}_{0}\right\|_{L^{2}}+\frac{1}{c},$

which shows that $\nabla\times B^{c}-j^{c}$ vanishes in the topology of
$L^{2}$ if the initial electric field converges strongly to zero.

###### Proof.

First applying a time derivative to Ampère’s equation and combining it with
Faraday’s equation and Ohm’s law yields the damped wave equation

$\frac{1}{c}\partial_{t}^{2}E^{c}-c\Delta E^{c}+\sigma
c\partial_{t}E^{c}=-\sigma\partial_{t}P(u^{c}\times B^{c}).$

Then, performing a classical energy estimate by multiplying the above damped
wave equation by $(-\Delta)^{\eta-1}\partial_{t}E^{c}$ and integrating in
space (this can be done rigorously by employing a Littlewood–Paley
decomposition, for instance), we find that

$\displaystyle\frac{1}{2}\frac{d}{dt}\left(\frac{1}{c}\left\|\partial_{t}E^{c}(t)\right\|_{\dot{H}^{\eta-1}}^{2}+c\left\|E^{c}(t)\right\|_{\dot{H}^{\eta}}^{2}\right)$
$\displaystyle+\sigma
c\left\|\partial_{t}E^{c}(t)\right\|_{\dot{H}^{\eta-1}}^{2}$
$\displaystyle\leq\sigma\left\|\partial_{t}P(u^{c}\times
B^{c})(t)\right\|_{\dot{H}^{\eta-1}}\left\|\partial_{t}E^{c}(t)\right\|_{\dot{H}^{\eta-1}}$
$\displaystyle\leq\frac{\sigma}{2c}\left\|\partial_{t}P(u^{c}\times
B^{c})(t)\right\|_{\dot{H}^{\eta-1}}^{2}+\frac{\sigma
c}{2}\left\|\partial_{t}E^{c}(t)\right\|_{\dot{H}^{\eta-1}}^{2},$

whereby

$\frac{d}{dt}\left(\frac{1}{c}\left\|\partial_{t}E^{c}(t)\right\|_{\dot{H}^{\eta-1}}^{2}+c\left\|E^{c}(t)\right\|_{\dot{H}^{\eta}}^{2}\right)+\sigma
c\left\|\partial_{t}E^{c}(t)\right\|_{\dot{H}^{\eta-1}}^{2}\leq\frac{\sigma}{c}\left\|\partial_{t}P(u^{c}\times
B^{c})(t)\right\|_{\dot{H}^{\eta-1}}^{2}.$

Further integrating in time, we deduce that

(3.7)
$\displaystyle\frac{1}{c}\left\|\partial_{t}E^{c}\right\|_{L^{\infty}(\mathbb{R}^{+};\dot{H}^{\eta-1})}+$
$\displaystyle\left\|E^{c}\right\|_{L^{\infty}(\mathbb{R}^{+};\dot{H}^{\eta})}+\left\|\partial_{t}E^{c}\right\|_{L^{2}(\mathbb{R}^{+};\dot{H}^{\eta-1})}$
$\displaystyle\lesssim\frac{1}{c}\left\|\partial_{t}E^{c}_{0}\right\|_{\dot{H}^{\eta-1}}+\left\|E^{c}_{0}\right\|_{\dot{H}^{\eta}}+\frac{1}{c}\left\|\partial_{t}P(u^{c}\times
B^{c})\right\|_{L^{2}(\mathbb{R}^{+};\dot{H}^{\eta-1})}.$

We are now left with estimating $\partial_{t}P(u^{c}\times B^{c})$, above. To
this end, we exploit Euler and Faraday’s equations from (3.1) to write

$\displaystyle\partial_{t}P(u^{c}\times B^{c})$
$\displaystyle=P\big{(}\partial_{t}u^{c}\times
B^{c}+u^{c}\times\partial_{t}B^{c}\big{)}$
$\displaystyle=P\Big{(}\big{(}P(j^{c}\times B^{c})-P(u^{c}\cdot\nabla
u^{c})\big{)}\times B^{c}-cu^{c}\times(\nabla\times
E^{c})\Big{)}=\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3},$

where

$\displaystyle\mathcal{I}_{1}$ $\displaystyle=P\Big{(}\big{(}P(j^{c}\times
B^{c})\big{)}\times B^{c}\Big{)},$ $\displaystyle\mathcal{I}_{2}$
$\displaystyle=-P\Big{(}\big{(}P(u^{c}\cdot\nabla u^{c})\big{)}\times
B^{c}\Big{)},$ $\displaystyle\mathcal{I}_{3}$
$\displaystyle=-cP\Big{(}u^{c}\times(\nabla\times E^{c})\Big{)}.$

Next, we utilize the paradifferential product laws for normal vector fields
established in [1] to control each term individually. This is a step where the
normal structure (3.3) plays a crucial role. For convenience, we have gathered
the relevant product estimates from [1] in the appendix.

Thus, by Lemma A.1 and the energy inequality (3.2), we obtain that

$\displaystyle\left\|\mathcal{I}_{1}\right\|_{L^{2}_{t}\dot{H}^{\eta-1}}$
$\displaystyle\lesssim\left\|P(j^{c}\times
B^{c})\right\|_{L^{2}_{t}\dot{H}^{\frac{\eta-1}{2}}}\left\|B^{c}\right\|_{L^{\infty}_{t}\dot{H}^{\frac{\eta+1}{2}}}$
$\displaystyle\lesssim\left\|j^{c}\right\|_{L^{2}_{t}L^{2}}\left\|B^{c}\right\|_{L^{\infty}_{t}\dot{H}^{\frac{\eta+1}{2}}}^{2}\lesssim\mathcal{E}_{0}\left\|B^{c}\right\|_{L^{\infty}_{t}{H}^{\eta}}^{2}.$

As for the second term $\mathcal{I}_{2}$, by applying Lemma A.1, we find that

$\displaystyle\left\|\mathcal{I}_{2}\right\|_{L^{2}_{t}\dot{H}^{\eta-1}}$
$\displaystyle\lesssim\left\|u^{c}\cdot\nabla
u^{c}\right\|_{L^{\infty}_{t}L^{2}}\left\|B^{c}\right\|_{L^{2}_{t}\dot{H}^{\eta}}$
$\displaystyle\lesssim\left\|u^{c}\right\|_{L^{\infty}_{t}L^{\infty}}\left\|u^{c}\right\|_{L^{\infty}_{t}\dot{H}^{1}}\left\|B^{c}\right\|_{L^{2}_{t}\dot{H}^{\eta}}.$

Finally, for the last term $\mathcal{I}_{3}$, using Lemma A.1, again, yields
that

$\displaystyle\left\|\mathcal{I}_{3}\right\|_{L^{2}_{t}\dot{H}^{\eta-1}}$
$\displaystyle\lesssim\left\|u^{c}\right\|_{L^{\infty}_{t}(L^{\infty}\cap\dot{H}^{1})}\left\|c\nabla\times
E^{c}\right\|_{L^{2}_{t}\dot{H}^{\eta-1}}$
$\displaystyle\lesssim\left\|u^{c}\right\|_{L^{\infty}_{t}(L^{\infty}\cap\dot{H}^{1})}\left\|cE^{c}\right\|_{L^{2}_{t}\dot{H}^{\eta}}.$

All in all, in view of the uniform bounds (3.4), we deduce that the control

$\partial_{t}P(u^{c}\times B^{c})\in
L^{2}(\mathbb{R}^{+};\dot{H}^{\eta-1}),\quad\forall\eta\in[1,s],$

holds uniformly in $c$.

Now, by incorporating the control of $\partial_{t}P(u^{c}\times B^{c})$ into
(3.7), we find that

$\frac{1}{c}\left\|\partial_{t}E^{c}\right\|_{L^{\infty}(\mathbb{R}^{+};\dot{H}^{\eta-1})}+\left\|E^{c}\right\|_{L^{\infty}(\mathbb{R}^{+};\dot{H}^{\eta})}+\left\|\partial_{t}E^{c}\right\|_{L^{2}(\mathbb{R}^{+};\dot{H}^{\eta-1})}\lesssim\frac{1}{c}\left\|\partial_{t}E^{c}_{0}\right\|_{\dot{H}^{\eta-1}}+\left\|E^{c}_{0}\right\|_{\dot{H}^{\eta}}+\frac{C_{*}}{c}.$

At last, employing Ampère’s equation to substitute
$\frac{1}{c}\partial_{t}E^{c}$ with $\nabla\times B^{c}-j^{c}$ completes the
proof. ∎

###  3.3. Proof of Theorem 3.2

We proceed in two steps. First, we show the strong global convergence of the
velocity field $u^{c}$ through an application of Theorem 1.2 combined with
Proposition 3.3. Then, we show the stability of the magnetic field $B^{c}$ by
performing a suitable energy estimate on a transport–diffusion equation with a
vanishing source term.

####  3.3.1. Stability of the velocity field

Employing the normal structure (3.3), we write the momentum equation from
(3.1) as

$\partial_{t}u^{c}+u^{c}\cdot\nabla
u^{c}+\nabla\left(p^{c}+\frac{|B^{c}|^{2}}{2}-(\mathrm{Id}-P)g^{c}\right)=Pg^{c},$

where

$g^{c}\overset{\rm def}{=}\left(j^{c}-\nabla\times B^{c}\right)\times B^{c}.$

We want now to apply Theorem 1.2 to the above system to deduce the convergence
of $u^{c}$, as $c\rightarrow\infty$. To that end, we need to establish, first,
that $Pg^{c}$ vanishes in $L^{1}([0,T];H^{1}(\mathbb{R}^{2}))$ and, second,
that $\operatorname{curl}g^{c}$ remains uniformly bounded in
$L^{1}([0,T];L^{\infty}(\mathbb{R}^{2}))$.

In order to prove the vanishing of $Pg^{c}$, we employ the paradifferential
calculus estimate from Lemma A.1 to find that

$\left\|Pg^{c}\right\|_{L^{1}_{t}L^{2}}\lesssim\left\|\nabla\times
B^{c}-j^{c}\right\|_{L^{2}_{t}L^{2}}\left\|B^{c}\right\|_{L^{2}_{t}\dot{H}^{1}}.$

Furthermore, noticing that

$\operatorname{curl}g^{c}=\left(\nabla\times B^{c}-j^{c}\right)\cdot\nabla
B^{c}=-j^{c}\cdot\nabla B^{c},$

we see, by Hölder’s inequality, that

$\left\|Pg^{c}\right\|_{L^{1}_{t}\dot{H}^{1}}\lesssim\left\|\operatorname{curl}g^{c}\right\|_{L^{1}_{t}L^{2}}\lesssim\left\|\nabla\times
B^{c}-j^{c}\right\|_{L^{2}_{t}L^{2}}\left\|B^{c}\right\|_{L^{2}_{t}\dot{W}^{1,\infty}}.$

Hence, since the assumptions of Theorem 3.2 guarantee that $E^{c}_{0}$
vanishes in $L^{2}$ and that $B^{c}$ remains bounded in
$L^{2}_{t}(\dot{H}^{1}\cap\dot{W}^{1,\infty})$, we deduce from (3.6) that
$Pg^{c}\to 0$ in $L^{1}([0,T];H^{1}(\mathbb{R}^{2}))$.

As for the uniform bound of $\operatorname{curl}g^{c}$ in
$L^{1}([0,T];L^{\infty}(\mathbb{R}^{2}))$, it follows immediately upon
noticing that both $j^{c}$ and $\nabla B^{c}$ are uniformly bounded in
$L^{2}([0,T];L^{\infty}(\mathbb{R}^{2}))$, as emphasized in (3.4).

We can now apply Theorem 1.2 to conclude that

(3.8)
$\lim_{c\rightarrow\infty}\sup_{t\in[0,T]}\left\|u^{c}(t)-u(t)\right\|_{H^{1}(\mathbb{R}^{2})}=0,$

thereby completing the proof of stability of the velocity field.

####  3.3.2. Stability of the magnetic field.

By suitably combining Faraday’s equation with Ohm’s law from (3.1), we observe
that

$\partial_{t}B^{c}+u^{c}\cdot\nabla B^{c}-\frac{1}{\sigma}\Delta
B^{c}=\frac{1}{\sigma}\nabla\times(\nabla\times B^{c}-j^{c}).$

Then, subtracting this equation with (3.5), we see that

$\partial_{t}\widetilde{B}^{c}+u^{c}\cdot\nabla\widetilde{B}^{c}-\frac{1}{\sigma}\Delta\widetilde{B}^{c}=-(u^{c}-u)\cdot\nabla
B+\frac{1}{\sigma}\nabla\times(\nabla\times B^{c}-j^{c}),$

where $\widetilde{B}^{c}\overset{\rm def}{=}B^{c}-B$, with the initial data
$\widetilde{B}^{c}_{0}\overset{\rm def}{=}B^{c}_{0}-B_{0}$.

By a classical $L^{2}$-energy estimate, we find that

$\displaystyle\|\widetilde{B}^{c}\|_{L^{\infty}([0,T];L^{2})}$
$\displaystyle+\|\widetilde{B}^{c}\|_{L^{2}([0,T];\dot{H}^{1})}$
$\displaystyle\lesssim\|\widetilde{B}^{c}_{0}\|_{L^{2}}+\left\|\nabla\times\left((u^{c}-u)\times
B+\frac{1}{\sigma}(\nabla\times
B^{c}-j^{c})\right)\right\|_{L^{2}([0,T];\dot{H}^{-1})}$
$\displaystyle\lesssim\|\widetilde{B}^{c}_{0}\|_{L^{2}}+\left\|u^{c}-u\right\|_{L^{\infty}([0,T];L^{2})}\left\|B\right\|_{{L^{2}([0,T];\dot{H}^{1})}}+\left\|\nabla\times
B^{c}-j^{c}\right\|_{{L^{2}([0,T];L^{2})}},$

where we applied Lemma A.1 to deduce the obtain estimate

$\displaystyle\left\|\nabla\times\left((u^{c}-u)\times
B\right)\right\|_{L^{2}([0,T];\dot{H}^{-1})}$
$\displaystyle\lesssim\left\|P\left((u^{c}-u)\times
B\right)\right\|_{L^{2}([0,T];L^{2})}$
$\displaystyle\lesssim\left\|u^{c}-u\right\|_{L^{\infty}([0,T];L^{2})}\left\|B\right\|_{{L^{2}([0,T];\dot{H}^{1})}}.$

It then follows from (3.6) and (3.8) that

$\lim_{c\rightarrow\infty}\left(\|\widetilde{B}^{c}\|_{L^{\infty}([0,T];L^{2})}+\|\widetilde{B}^{c}\|_{L^{2}([0,T];\dot{H}^{1})}\right)=0,$

which completes the proof of stability of the magnetic field.

At last, writing

$\left\|j^{c}-\nabla\times
B\right\|_{L^{2}([0,T];L^{2})}\leq\left\|B^{c}-B\right\|_{L^{2}([0,T];\dot{H}^{1})}+\left\|j^{c}-\nabla\times
B^{c}\right\|_{L^{2}([0,T];L^{2})},$

it is readily seen that $j^{c}$ converges to $\nabla\times B$ in
$L^{2}([0,T];L^{2})$, which completes the proof of the theorem. ∎

##  Appendix A Paradifferential calculus and the normal structure

It was noticed in [1] that the normal structure (3.3) of solutions to the
Euler–Maxwell system could be exploited to extend the classical product laws
of paradifferential calculus to a wider range of parameter. We recall here a
simplified statement of Lemma 3.4 from [1] which provides the product laws
which are relevant to the present work. A complete justification of this
result can be found in [1].

######  Lemma A.1.

Let $s\in(-1,2)$ and let $F,G:\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}$ be
divergence-free vector fields having the normal structure

$F(x)=\begin{pmatrix}F_{1}(x)\\\ F_{2}(x)\\\
0\end{pmatrix}\qquad\text{and}\qquad G(x)=\begin{pmatrix}0\\\ 0\\\
G_{3}(x)\end{pmatrix}.$

Then, for any $\eta\in(-\infty,1)$ and $s\in(-\infty,2)$, with $\eta+s>0$, one
has the product law

$\left\|P(F\times
G)\right\|_{\dot{H}^{\eta+s-1}}\lesssim\left\|F\right\|_{\dot{H}^{\eta}}\left\|G\right\|_{\dot{H}^{s}}.$

Moreover, in the endpoint case $\eta=1$, one has that

$\left\|P(F\times
G)\right\|_{\dot{H}^{s}}\lesssim\left\|F\right\|_{L^{\infty}\cap\dot{H}^{1}}\left\|G\right\|_{\dot{H}^{s}},$

for any $s\in(-1,2)$.

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