# On a general variational framework for existence and uniqueness in
Differential Equations

Pablo Pedregal

###### Abstract.

Starting from the classic contraction mapping principle, we establish a
general, flexible, variational setting that turns out to be applicable to many
situations of existence in Differential Equations. We show its potentiality
with some selected examples including initial-value, Cauchy problems for ODEs;
non-linear, monotone PDEs; linear and non-linear hyperbolic problems; and
steady Navier-Stokes systems.

Supported by the Spanish Ministerio de Ciencia, Innovación y Universidades
through project MTM2017-83740-P

##  1\. Introduction

Possibly the most fundamental result yielding existence and uniqueness of
solutions of an equation is the classic Banach contraction mapping principle.

######  Theorem 1.1.

Let $\mathbf{T}:\mathbb{H}\to\mathbb{H}$ be a mapping from a Banach space
$\mathbb{H}$ into itself that is contractive in the sense

$\|\mathbf{T}\mathbf{x}-\mathbf{T}\mathbf{y}\|\leq
k\|\mathbf{x}-\mathbf{y}\|,\quad
k\in[0,1),\mathbf{x},\mathbf{y}\in\mathbb{H}.$

Then $\mathbf{T}$ admits a unique fixed point
$\overline{\mathbf{x}}\in\mathbb{H}$,

$\mathbf{T}\overline{\mathbf{x}}=\overline{\mathbf{x}}.$

The proof is well-known, elementary, and independent of dimension. The most
fascinating issue is that this basic principle is at the heart of many
uniqueness results in Applied Analysis and Differential Equations. Our aim is
to stress this fact from a variational stand-point. This means that we would
like to rephrase the previous principle into a variational form that could be
directly and flexibly used in many of the situations where uniqueness of
solutions is known or expected. Our basic principle is the following.

######  Proposition 1.2.

Let $E:\mathbb{H}\to\mathbb{R}^{+}$ be a non-negative, lower semi-continuous
functional in a Banach space $\mathbb{H}$, such that

(1.1) $\|\mathbf{x}-\mathbf{y}\|\leq C(E(\mathbf{x})+E(\mathbf{y})),\quad
C>0,\mathbf{x},\mathbf{y}\in\mathbb{H}.$

Suppose, in addition, that

(1.2) $\inf_{\mathbf{z}\in\mathbb{H}}E(\mathbf{z})=0.$

Then there is a unique minimizer, i.e. a unique
$\overline{\mathbf{x}}\in\mathbb{H}$ such that $E(\overline{\mathbf{x}})=0$,
and

(1.3) $\|\mathbf{x}-\overline{\mathbf{x}}\|\leq
CE(\mathbf{x}),\quad\mathbf{x}\in\mathbb{H}.$

The proof again is elementary, because every minimizing sequence
$\\{\mathbf{x}_{j}\\}$ with $E(\mathbf{x}_{j})\searrow 0$ must be a Cauchy
sequence in $\mathbb{H}$, according to (1.1), and so it converges to some
$\overline{\mathbf{x}}\in\mathbb{H}$. The lower semicontinuity implies that

$0\leq E(\overline{\mathbf{x}})\leq\liminf_{j\to\infty}E(\mathbf{x}_{j})=0,$

and $\overline{\mathbf{x}}$ is a minimizer. Condition (1.1) implies
automatically that such minimizer is unique, and leads to (1.3).

Condition (1.3) is a very clear statement that functional $E$ in Proposition
1.2 is a measure of how far we are from $\overline{\mathbf{x}}$, the unique
point where $E$ vanishes. Indeed, this consequence already points in the
direction in which to look for functionals $E$ in specific situations: they
should be setup as a way to measure departure from solutions sought. This will
be taken as a guiding principle in concrete examples. The usual least-square
method (see [1], [4], for example), suitably adapted to each situation, stands
as a main, natural possibility for $E$.

It is not surprising that Proposition 1.2 is more general than Theorem 1.1, in
the sense that the latter is a consequence of the former by considering the
natural functional

(1.4) $E(\mathbf{x})=\|\mathbf{T}\mathbf{x}-\mathbf{x}\|.$

Indeed, for an arbitrary pair $\mathbf{x},\mathbf{y}\in\mathbb{H}$,

$\|\mathbf{x}-\mathbf{y}\|\leq\|\mathbf{x}-\mathbf{T}\mathbf{x}\|+\|\mathbf{T}\mathbf{x}-\mathbf{T}\mathbf{y}\|+\|\mathbf{T}\mathbf{y}-\mathbf{y}\|,$

and

$\|\mathbf{x}-\mathbf{y}\|\leq
E(\mathbf{x})+E(\mathbf{y})+k\|\mathbf{x}-\mathbf{y}\|.$

From here, we immediately find (1.1)

$\|\mathbf{x}-\mathbf{y}\|\leq\frac{1}{1-k}(E(\mathbf{x})+E(\mathbf{y})).$

Along every sequence of iterates, we have (1.2) if $\mathbf{T}$ is contactive.
Of course, minimizers for $E$ in (1.4) are exactly fixed points for
$\mathbf{T}$.

Our objective is to argue that the basic variational principle in Proposition
1.2 is quite flexible, and can be implemented in many of the situations in
Differential Equations where uniqueness of solutions is known.

There are two main requisites in Proposition 1.2. The first one (1.1) has to
be shown directly in each particular scenario where uniqueness is sought. Note
that it is some kind of enhanced coercivity, and, as such, stronger than plain
coercivity. Concerning (1.2), there is, however, a general strategy based on
smoothness that can be applied to most of the interesting situations in
practice. For the sake of simplicity, we restrict attention to a Hilbert space
situation, and regard $\mathbb{H}$ as a Hilbert space henceforth. If a non-
negative functional $E:\mathbb{H}\to\mathbb{R}^{+}$ is $\mathcal{C}^{1}$-,
then

$\inf_{\mathbf{x}\in\mathbb{H}}\|E^{\prime}(\mathbf{x})\|=0.$

Therefore, it suffices to demand that

$\lim_{E^{\prime}(\mathbf{x})\to\mathbf{0}}E(\mathbf{x})=0$

to enforce (1.2). Proposition 1.2 becomes then:

######  Proposition 1.3.

Let $E:\mathbb{H}\to\mathbb{R}^{+}$ be a non-negative, $\mathcal{C}^{1}$\-
functional in a Hilbert space $\mathbb{H}$, such that

(1.5) $\|\mathbf{x}-\mathbf{y}\|\leq C(E(\mathbf{x})+E(\mathbf{y})),\quad
C>0,\mathbf{x},\mathbf{y}\in\mathbb{H}.$

Suppose, in addition, that

(1.6) $\lim_{E^{\prime}(\mathbf{x})\to\mathbf{0}}E(\mathbf{x})=0.$

Then there is a unique $\overline{\mathbf{x}}\in\mathbb{H}$ such that
$E(\overline{\mathbf{x}})=0$, and

$\|\mathbf{x}-\overline{\mathbf{x}}\|\leq CE(\mathbf{x})$

for every $\mathbf{x}\in\mathbb{H}$.

Though the following is a simple observation, it is worth to note it
explicitly.

######  Proposition 1.4.

Under the same conditions as in Proposition 1.3, the functional $E$ enjoys the
Palais-Smale condition.

We remind readers that the fundamental Palais-Smale condition reads:

> If the sequence $\\{\mathbf{x}_{j}\\}$ is bounded in $\mathbb{H}$, and
> $E^{\prime}(\mathbf{x}_{j})\to\mathbf{0}$ in $\mathbb{H}$, then, at least
> for some subsequence, $\\{\mathbf{x}_{j}\\}$ converges in $\mathbb{H}$.

Again, it is not difficult to suspect the proof. Condition (1.6) informs us
that Palais-Smale sequences ($\\{\mathbf{x}_{j}\\}$, bounded and
$E^{\prime}(\mathbf{x}_{j})\to\mathbf{0}$) are always minimizing sequences for
$E$ ($E(\mathbf{x}_{j})\to 0$), while the estimate (1.5) ensures that (the
full) such sequence is a Cauchy sequence in $\mathbb{H}$. Notice, however,
that, due to (1.6), $0$ is the only possible critical value of $E$, and so
critical points become automatically global minimizers regardless of convexity
considerations.

In view of the relevance of conditions (1.5) and (1.6), we adopt the following
definition in which we introduce some simple, helpful changes to broaden its
applicability. We also change the notation to stress that vectors in
$\mathbb{H}$ will be functions for us.

######  Definition 1.1.

A non-negative, $\mathcal{C}^{1}$-functional

$E(\mathbf{u}):\mathbb{H}\to\mathbb{R}^{+}$

defined over a Hilbert space $\mathbb{H}$ is called an error functional if

  1. (1)

behavior as $E^{\prime}\to\mathbf{0}$:

(1.7) $\lim_{E^{\prime}(\mathbf{u})\to\mathbf{0}}E(\mathbf{u})=0$

over bounded subsets of $\mathbb{H}$; and

  2. (2)

enhanced coercivity: there is a positive constant $C$, such that for every
pair $\mathbf{u},\mathbf{v}\in\mathbb{H}$ we have

(1.8) $\|\mathbf{u}-\mathbf{v}\|^{2}\leq C(E(\mathbf{u})+E(\mathbf{v})).$

Our basic result Proposition 1.3 remains the same.

######  Proposition 1.5.

Let $E:\mathbb{H}\to\mathbb{R}^{+}$ be an error functional according to
Definition 1.1. Then there is a unique minimizer
$\mathbf{u}_{\infty}\in\mathbb{H}$ such that $E(\mathbf{u}_{\infty})=0$, and

(1.9) $\|\mathbf{u}-\mathbf{u}_{\infty}\|^{2}\leq CE(\mathbf{u}),$

for every $\mathbf{u}\in\mathbb{H}$.

It is usually said that the contraction mapping principle Theorem 1.1, though
quite helpful in ODEs, is almost inoperative for PDEs. We will try to make an
attempt at convincing readers that, on the contrary, Proposition 1.3 is
equally helpful for ODEs and PDEs. To this end, we will examine several
selected examples as a sample of the potentiality of these ideas.
Specifically, we will look at the following situations, though none of our
existence results is new at this stage:

  1. (1)

Cauchy, initial-value problems for ODEs;

  2. (2)

linear hyperbolic examples;

  3. (3)

non-linear, monotone PDEs;

  4. (4)

non-linear wave models;

  5. (5)

steady Navier-Stokes system.

We systematically will have to check the two basic properties (1.8) and (1.7)
in each situation treated. We can be dispensed with condition (1.7), and
replace it by (1.2) if more general results not requiring smoothness are
sought. On the other hand, in many regular situations linearization may lead
in a systematic way to the following.

######  Proposition 1.6.

Let

$E(\mathbf{u}):\mathbb{H}\to\mathbb{R}^{+}$

be a $\mathcal{C}^{1}$-functional verifying the enhanced coercivity condition
(1.8). Suppose there is $\mathbf{T}:\mathbb{H}\to\mathbb{H}$, a locally
Lipschitz operator, such that

(1.10) $\langle
E^{\prime}(\mathbf{u}),\mathbf{T}\mathbf{u}\rangle=-dE(\mathbf{u})$

for every $\mathbf{u}\in\mathbb{H}$, and some constant $d>0$. Then $E$ is an
error functional (according to Definition 1.1), and, consequently, there is a
unique global minimizer $\mathbf{u}_{\infty}$ with $E(\mathbf{u}_{\infty})=0$,
and (1.9) holds

(1.11) $\|\mathbf{u}-\mathbf{u}_{\infty}\|^{2}\leq CE(\mathbf{u}),$

for some constant $C$, and every $\mathbf{u}\in\mathbb{H}$.

Note how condition (1.10) leads immediately to (1.7). In this contribution, we
will assume smoothness in all of our examples.

Typically our Hilbert spaces $\mathbb{H}$ will be usual Sobolev spaces in
different situations, so standard facts about these spaces will be taken for
granted. In particular, the following Hilbert spaces will play a basic role
for us in those various situations mentioned above

$H^{1}(0,T;\mathbb{R}^{N}),\quad H^{1}_{0}(\mathbb{R}^{N}_{+}),\quad
H^{1}(\mathbb{R}^{N}_{+}),\quad H^{1}_{0}(\Omega),\quad
H^{1}_{0}(\Omega;\mathbb{R}^{N}),$

for a domain $\Omega\subset\mathbb{R}^{N}$ as regular as we may need it to be.

If one is interested in numerical or practical approximation of solutions
$\mathbf{u}_{\infty}$, note how (1.11) is a clear invitation to seek
approximations to $\mathbf{u}_{\infty}$ by minimizing $E(\mathbf{u})$. The
standard way to take a functional to its minimum value is to use a steepest
descent algorithm or some suitable variant of it. It is true that such
procedure is designed, in fact, to lead the derivative
$E^{\prime}(\mathbf{u})$ to zero; but precisely, condition (1.7) is
guaranteeing that in doing so we are also converging to $\mathbf{u}_{\infty}$
always. We are not pursuing this direction here, though it has been
implemented in some scenarios ([7], [8]).

Definition 1.1 is global. A local parallel concept may turn out necessary for
some situations. We will show this in our final example dealing with the
steady Navier-Stokes system. The application to parabolic problems, though
still feasible, is, in general, more delicate.

##  2\. Cauchy problems for ODEs

As a preliminary step, we start testing our ideas with a typical initial-
value, Cauchy problem for the non-linear system

(2.1) $\mathbf{x}^{\prime}(t)=\mathbf{f}(\mathbf{x}(t))\hbox{ in
}(0,+\infty),\quad\mathbf{x}(0)=\mathbf{x}_{0}$

where the map

$\mathbf{f}(\mathbf{y}):\mathbb{R}^{N}\to\mathbb{R}^{N}$

is smooth and globally Lipschitz, and $\mathbf{x}_{0}\in\mathbb{R}^{N}$. Under
these circumstances, it is well-known that (2.1) possesses a unique solution.
Let us pretend not to know anything about problem (2.1), and see if our
formalism could be applied in this initial situation to prove the following
classical theorem.

######  Theorem 2.1.

If the mapping $\mathbf{f}(\mathbf{y})$ is globally Lipschitz, there is unique
absolutely continuous solution

$\mathbf{x}(t):[0,\infty)\to\mathbb{R}^{N}$

for (2.1).

According to our previous discussion, we need a functional
$E:\mathbb{H}\to\mathbb{R}^{+}$ defined on an appropriate Hilbert space
$\mathbb{H}$ complying with the necessary properties.

For a fixed, but otherwise arbitrary, positive time $T$, we will take

$\displaystyle\mathbb{H}=\\{\mathbf{z}(t):[0,T]\to\mathbb{R}^{N}:\mathbf{z}\in
H^{1}(0,T;\mathbb{R}^{N}),\mathbf{z}(0)=\mathbf{0}\\},$ (2.2) $\displaystyle
E(\mathbf{z})=\frac{1}{2}\int_{0}^{T}|\mathbf{z}^{\prime}(s)-\mathbf{f}(\mathbf{x}_{0}+\mathbf{z}(s))|^{2}\,ds.$

$\mathbb{H}$ is a subspace of the standard Sobolev space
$H^{1}(0,T;\mathbb{R}^{N})$, under the norm (recall that
$\mathbf{z}(0)=\mathbf{0}$ for paths in $\mathbb{H}$)

$\|\mathbf{z}\|^{2}=\int_{0}^{T}|\mathbf{z}^{\prime}(s)|^{2}\,ds.$

Note that paths $\mathbf{x}\in\mathbb{H}$ are absolutely continuous, and hence
$E(\mathbf{z})$ is well-defined over $\mathbb{H}$. We first focus on (1.8).

######  Lemma 2.2.

For paths $\mathbf{y}$, $\mathbf{z}$ in $\mathbb{H}$, we have

$\|\mathbf{y}-\mathbf{z}\|^{2}\leq C(E(\mathbf{y})+E(\mathbf{z})),\quad C>0.$

###### Proof.

The proof is, in fact, pretty elementary. Suppose that

$\mathbf{z}(0)=\mathbf{y}(0)=\mathbf{x}_{0},$

so that $\mathbf{y}-\mathbf{z}\in\mathbb{H}$. Then

$\displaystyle\mathbf{y}(t)-\mathbf{z}(t)=$
$\displaystyle\int_{0}^{t}(\mathbf{y}^{\prime}(s)-\mathbf{z}^{\prime}(s))\,ds$
$\displaystyle=$
$\displaystyle\int_{0}^{t}(\mathbf{y}^{\prime}(s)-\mathbf{f}(\mathbf{y}(s)))\,ds+\int_{0}^{t}(\mathbf{f}(\mathbf{y}(s))-\mathbf{f}(\mathbf{z}(s)))\,ds$
$\displaystyle+\int_{0}^{t}(\mathbf{f}(\mathbf{z}(s))-\mathbf{z}^{\prime}(s))\,ds.$

From here, we immediately find

$|\mathbf{y}(t)-\mathbf{z}(t)|^{2}\leq
C(E(\mathbf{y})+E(\mathbf{z}))+CM^{2}\int_{0}^{t}|\mathbf{y}(s)-\mathbf{z}(s)|^{2}\,ds,$

if $M$ is the Lipschitz constant for the map $\mathbf{f}$, and $C$ is a
generic, universal constant we will not care to change. From Gronwall’s lemma,
we can have

(2.3) $|\mathbf{y}(t)-\mathbf{z}(t)|^{2}\leq
C(E(\mathbf{y})+E(\mathbf{z}))e^{CM^{2}T}$

for all $t\in[0,T]$. This means

$\displaystyle\|\mathbf{y}-\mathbf{z}\|_{L^{\infty}(0,T;\mathbb{R}^{N})}\leq
e^{CM^{2}T/2}\sqrt{C(E(\mathbf{y})+E(\mathbf{z}))},$ (2.4)
$\displaystyle\|\mathbf{y}-\mathbf{z}\|_{L^{2}(0,T;\mathbb{R}^{N})}^{2}\leq
CTe^{CM^{2}T}(E(\mathbf{y})+E(\mathbf{z})).$

But once we can rely on this information, the above decomposition allows us to
write in a similar manner

$\int_{0}^{t}|\mathbf{y}^{\prime}(s)-\mathbf{z}^{\prime}(s)|^{2}\,ds\leq
C(E(\mathbf{y})+E(\mathbf{z}))+CM^{2}\int_{0}^{t}|\mathbf{y}(s)-\mathbf{z}(s)|^{2}\,ds$

and

$\|\mathbf{y}^{\prime}-\mathbf{z}^{\prime}\|^{2}_{L^{2}(0,T;\mathbb{R}^{N})}\leq
C(E(\mathbf{y})+E(\mathbf{z}))+CM^{2}\|\mathbf{y}-\mathbf{z}\|^{2}_{L^{2}(0,T;\mathbb{R}^{N})},$

and thus, taking into account (2.4),

$\|\mathbf{y}^{\prime}-\mathbf{z}^{\prime}\|^{2}_{L^{2}(0,T;\mathbb{R}^{N})}\leq(C+C^{2}M^{2}Te^{CM^{2}T})(E(\mathbf{y})+E(\mathbf{z})).$

Our estimate (1.8) is then a consequence that the norm in $\mathbb{H}$ can be
taken to be the $L^{2}$-norm of the derivative. ∎

The second basic ingredient is (1.7). We will be using Proposition 1.6. We
assume further that the mapping $\mathbf{f}$ is smooth with a derivative
uniformly bounded to guarantee the uniform Lipschitz condition. For the
operator $\mathbf{T}$, we will put $\mathbf{Z}=\mathbf{T}\mathbf{z}$ for
$\mathbf{z}\in\mathbb{H}$, and linearize (2.1) at the path
$\mathbf{x}_{0}+\mathbf{z}(t)$ to write

(2.5)
$\displaystyle\mathbf{Z}^{\prime}(t)=\mathbf{f}(\mathbf{x}_{0}+\mathbf{z}(t))+\nabla\mathbf{f}(\mathbf{x}_{0}+\mathbf{z}(t))\mathbf{Z}(t)-\mathbf{z}^{\prime}(t)\hbox{
in }[0,T],$ $\displaystyle\mathbf{Z}(0)=\mathbf{0}.$

This is a linear, differential, non-constant coefficient system for
$\mathbf{Z}$ with coefficients depending on $\mathbf{z}$. Under smoothness
assumptions, which we take for granted, such operator $\mathbf{T}$ is locally
Lipschitz because the image $\mathbf{Z}=\mathbf{T}\mathbf{z}$ is defined
through a linear initial-value, Cauchy problem with coefficients depending
continuously on $\mathbf{z}$.

The important property to be checked, concerning $\mathbf{T}$, is (1.10). It
is elementary to see, under smoothness assumptions which, as indicated, we
have taken for granted, that

$\langle
E^{\prime}(\mathbf{z}),\mathbf{Z}\rangle=\int_{0}^{T}(\mathbf{z}^{\prime}(s)-\mathbf{f}(\mathbf{x}_{0}+\mathbf{z}(s)))(\mathbf{Z}^{\prime}(s)-\nabla\mathbf{f}(\mathbf{x}_{0}+\mathbf{z}(s))\mathbf{Z}(s))\,ds$

Hence, for $\mathbf{Z}=\mathbf{T}\mathbf{z}$ coming from (2.5), we immediately
deduce that

$\langle E^{\prime}(\mathbf{z}),\mathbf{Z}\rangle=-2E(\mathbf{z}).$

We are, then, entitled to apply Proposition 1.6 to conclude that functional
$E$ in (2.2) is an error functional after Definition 1.1, and we are entitled
to utilize Proposition 1.5 to conclude the following.

######  Theorem 2.3.

If the mapping $\mathbf{f}(\mathbf{y}):\mathbb{R}^{N}\to\mathbb{R}^{N}$ is
$\mathcal{C}^{1}$\- with a globally bounded gradient, then, for arbitrary
$\mathbf{x}_{0}\in\mathbb{R}^{N}$ and $T>0$, problem (2.1) admits a unique
$\mathcal{C}^{1}$-solution

$\overline{\mathbf{x}}(t):[0,T)\to\mathbb{R}^{N},$

and there is a positive constant $C$ such that

$\|\mathbf{x}-\overline{\mathbf{x}}\|_{\mathbb{H}}^{2}\leq
C\int_{0}^{T}|\mathbf{x}^{\prime}(s)-\mathbf{f}(\mathbf{x}_{0}+\mathbf{x}(s))|^{2}\,ds$

for every $\mathbf{x}\in\mathbb{H}$.

There is no difficulty in showing a local version of this result by using the
same ideas.

##  3\. Linear hyperbolic example

Since most likely readers will not be used to think about hyperbolic problems
in these terms, we will treat the most transparent example of a linear,
hyperbolic problem from this perspective, and later apply the method to a non-
linear wave equation.

We seek a (weak) solution $u(t,\mathbf{x})$ of some sort of the problem

(3.1) $\displaystyle u_{tt}(t,\mathbf{x})-\Delta
u(t,\mathbf{x})-u(t,\mathbf{x})=f(t,\mathbf{x})\hbox{ in }\mathbb{R}^{N}_{+},$
$\displaystyle u(0,\mathbf{x})=0,u_{t}(0,\mathbf{x})=0\hbox{ on }t=0,$

for $f\in L^{2}(\mathbb{R}^{N}_{+})$. Here $\mathbb{R}^{N}_{+}$ is the upper
half hyperspace $[0,+\infty)\times\mathbb{R}^{N}$. We look for

$u(t,\mathbf{x})\in H^{1}_{0}(\mathbb{R}^{N}_{+})$

(jointly in time and space) such that

(3.2)
$\displaystyle\int_{\mathbb{R}^{N}_{+}}[u_{t}(t,\mathbf{x})w_{t}(t,\mathbf{x})-\nabla
u(t,\mathbf{x})\cdot\nabla w(t,\mathbf{x})$
$\displaystyle+(f(t,\mathbf{x})+u(t,\mathbf{x}))w(t,\mathbf{x})]\,d\mathbf{x}\,dt=0$

for every test function

$w(t,\mathbf{x})\in H^{1}(\mathbb{R}^{N}_{+}).$

Note how the arbitrary values of the test function $w$ for $t=0$ imposes the
vanishing initial velocity $u_{t}(0,\mathbf{x})=0$.

To setup a suitable error functional

$E(u):H^{1}_{0}(\mathbb{R}^{N}_{+})\to\mathbb{R}^{+}$

for every $u(t,\mathbf{x})$, and not just for the solution we seek, we utilize
a natural least-square concept as indicated in the Introduction. Define an
appropriate defect or residual function

$U(t,\mathbf{x})\in H^{1}(\mathbb{R}^{N}_{+}),$

for each such $u\in H^{1}_{0}(\mathbb{R}^{N}_{+})$, as the unique variational
solution of

(3.3) $\int_{\mathbb{R}^{N}_{+}}[(u_{t}+U_{t})w_{t}-(\nabla u-\nabla
U)\cdot\nabla w+(f+u+U)w]\,d\mathbf{x}\,dt=0$

valid for every $w\in H^{1}(\mathbb{R}^{N}_{+})$. This function $U$ is indeed
the unique minimizer over $H^{1}(\mathbb{R}^{N}_{+})$ of the strictly convex,
quadratic functional

$I(w)=\int_{\mathbb{R}^{N}_{+}}\left(\frac{1}{2}[(w_{t}+u_{t})^{2}+|\nabla
w-\nabla u|^{2}+(u+w)^{2}]+fw\right)\,d\mathbf{x}\,dt$

for each fixed $u$. The size of $U$ is regarded as a measure of the departure
of $u$ from being a solution of our problem

$\displaystyle E:H^{1}_{0}(\mathbb{R}^{N}_{+})\to\mathbb{R}^{+},$ (3.4)
$\displaystyle
E(u)=\int_{\mathbb{R}^{N}_{+}}\frac{1}{2}[U^{2}_{t}(t,\mathbf{x})+|\nabla
U(t,\mathbf{x})|^{2}+U^{2}(t,\mathbf{x})]\,d\mathbf{x}\,dt.$

We can also put, in a short form,

(3.5) $E(u)=\frac{1}{2}\|U\|_{H^{1}(\mathbb{R}^{N}_{+})}^{2};$

or even

$E(u)=\frac{1}{2}\|u_{tt}-\Delta u-u-f\|^{2}_{H^{-1}(\mathbb{R}^{N}_{+})},$

though we will stick to (3.5) to better manipulate $E$.

We would like to apply Proposition 1.5 in this situation, and hence, we set to
ourselves the task of checking the two main assumptions in Definition 1.1. Our
functional $E$ is definitely smooth and non-negative to begin with.

It is not surprising that in order to work with the wave equation the
following two linear operators

$\displaystyle\mathbb{S}:H^{1}(\mathbb{R}^{N}_{+})\mapsto
H^{1}(\mathbb{R}^{N}_{+}),\quad\mathbb{S}w(t,\mathbf{x})=w(t,-\mathbf{x}),$
$\displaystyle\mathcal{S}:H^{1}(\mathbb{R}^{N}_{+})\mapsto
H^{1}(\mathbb{R}^{N}_{+})^{*},$
$\displaystyle\mathcal{S}u(t,\mathbf{x})=(u(t,-\mathbf{x}),u_{t}(t,-\mathbf{x}),\nabla
u(t,-\mathbf{x})),$

will play a role. $H^{1}(\mathbb{R}^{N}_{+})^{*}$ is here the dual space of
$H^{1}(\mathbb{R}^{N}_{+})$, not to be mistaken with
$H^{-1}(\mathbb{R}^{N}_{+})$. Put
$\mathbb{H}=\mathcal{S}(H^{1}_{0}(\mathbb{R}^{N}_{+}))$. The following fact is
elementary. Check for instance [2].

######  Lemma 3.1.

  1. (1)

The map $\mathbb{S}$ is an isometry.

  2. (2)

$\mathbb{H}$ is a closed subspace of $H^{1}(\mathbb{R}^{N}_{+})^{*}$, and

$\mathcal{S}:H^{1}_{0}(\mathbb{R}^{N}_{+})\to\mathbb{H}$

is a bijective, continuous mapping. In fact, we clearly have

(3.6)
$\|u\|_{H^{1}_{0}(\mathbb{R}^{N}_{+})}\leq\|\mathcal{S}u\|_{H^{1}(\mathbb{R}^{N}_{+})^{*}}.$

We can now proceed to prove inequality (1.8) in this new context.

######  Proposition 3.2.

There is a constant $K>0$ such that

$\|u-v\|_{H^{1}_{0}(\mathbb{R}^{N}_{+})}^{2}\leq K(E(u)+E(v)),$

for every pair $u,v\in H^{1}_{0}(\mathbb{R}^{N}_{+})$.

###### Proof.

Let $U,V\in H^{1}(\mathbb{R}^{N})$ be the respective residual functions
associated with $u$ and $v$. Because we are in a linear situation, if we
replace

$u-v\mapsto u,\quad U-V\mapsto U,$

we would have

(3.7) $\int_{\mathbb{R}^{N}_{+}}[(u_{t}+U_{t})w_{t}-(\nabla u-\nabla
U)\cdot\nabla w+(u+U)w]\,d\mathbf{x}\,dt=0,$

for every $w\in H^{1}(\mathbb{R}^{N}_{+})$. If we use $\mathbb{S}w$ in (3.7)
instead of $w$, we immediately find

$\displaystyle\int_{\mathbb{R}^{N}_{+}}[(u_{t}+U_{t})w_{t}(t,-\mathbf{x})+(\nabla
u-\nabla U)\cdot\nabla w(t,-\mathbf{x})$
$\displaystyle+(u+U)w(t,-\mathbf{x})]\,d\mathbf{x}\,dt=0.$

The terms involving $U$ can be written in compact form as

$\langle U,\mathbb{S}w\rangle_{\langle
H^{1}(\mathbb{R}^{N}_{+}),H^{1}(\mathbb{R}^{N}_{+})\rangle}$

while a natural change of variables in the terms involving $u$ leads to
writing these in the form

$\langle w,\mathcal{S}u\rangle_{\langle
H^{1}(\mathbb{R}^{N}_{+}),H^{1}(\mathbb{R}^{N}_{+})^{*}\rangle}.$

Hence, for every $w\in H^{1}(\mathbb{R}^{N}_{+})$, we find

$\langle U,\mathbb{S}w\rangle_{\langle
H^{1}(\mathbb{R}^{N}_{+}),H^{1}(\mathbb{R}^{N}_{+})\rangle}+\langle
w,\mathcal{S}u\rangle_{\langle
H^{1}(\mathbb{R}^{N}_{+}),H^{1}(\mathbb{R}^{N}_{+})^{*}\rangle}=0.$

Bearing in mind this identity, we have, through the Lemma 3.1,

$\displaystyle\|u\|_{H^{1}_{0}(\mathbb{R}^{N}_{+})}\leq$
$\displaystyle\|\mathcal{S}u\|_{H^{1}(\mathbb{R}^{N}_{+})^{*}}$
$\displaystyle=$ $\displaystyle\sup_{\|w\|_{H^{1}(\mathbb{R}^{N}_{+})}\leq
1}\langle w,\mathcal{S}u\rangle_{\langle
H^{1}(\mathbb{R}^{N}_{+}),H^{1}(\mathbb{R}^{N}_{+})^{*}\rangle}$
$\displaystyle\leq$
$\displaystyle\|U\|_{H^{1}(\mathbb{R}^{N}_{+})}\,\|w\|_{H^{1}(\mathbb{R}^{N}_{+})}$
$\displaystyle\leq$ $\displaystyle\|U\|_{H^{1}(\mathbb{R}^{N}_{+})}.$

If we go back to

$u\mapsto u-v,\quad U\mapsto U-V,$

we are led to

$\displaystyle\|u-v\|^{2}_{H^{1}_{0}(\mathbb{R}^{N}_{+})}\leq$
$\displaystyle\|U-V\|^{2}_{H^{1}(\mathbb{R}^{N}_{+})}$ $\displaystyle\leq$
$\displaystyle
C\left(\|U\|^{2}_{H^{1}(\mathbb{R}^{N}_{+})}+\|V\|^{2}_{H^{1}(\mathbb{R}^{N}_{+})}\right)$
$\displaystyle\leq$ $\displaystyle C\left(E(u)+E(v)\right),$

for some constant $C>0$. ∎

The second main ingredient to apply Proposition 1.5 is to show that $E$
defined in (3.4) complies with (1.7) too. To this end, we need to compute the
derivative $E^{\prime}(u)$, and so we perform a perturbation

$u+\epsilon v\mapsto U+\epsilon V,$

in (3.3) to write

$\displaystyle\int_{\mathbb{R}^{N}_{+}}[(u_{t}+\epsilon v_{t}+U_{t}+\epsilon
V_{t})w_{t}-(\nabla u+\epsilon\nabla v-\nabla U-\epsilon\nabla V)\cdot\nabla
w$ $\displaystyle+(f+u+\epsilon v+U+\epsilon V)w]\,d\mathbf{x}\,dt=0.$

The term to order 1 in $\epsilon$ should vanish

(3.8) $\int_{\mathbb{R}^{N}_{+}}[(v_{t}+V_{t})w_{t}-(\nabla v-\nabla
V)\cdot\nabla w+(v+V)w]\,d\mathbf{x}\,dt=0$

for every $w\in H^{1}(\mathbb{R}^{N}_{+})$. On the other hand, by
differentiating

$E(u+\epsilon v)=\int_{\mathbb{R}^{N}_{+}}\frac{1}{2}((U+\epsilon
V)^{2}_{t}+|\nabla U+\epsilon\nabla V|^{2}+(U+\epsilon
V)^{2})\,d\mathbf{x}\,dt,$

with respect to $\epsilon$, and setting $\epsilon=0$, we arrive at

$\langle E^{\prime}(u),v\rangle=\int_{\mathbb{R}^{N}_{+}}(U_{t}V_{t}+\nabla
U\cdot\nabla V+UV)\,d\mathbf{x}\,dt.$

By taking $w=U$ in (3.8), we can also write

$\displaystyle\langle E^{\prime}(u),v\rangle=$
$\displaystyle\int_{\mathbb{R}^{N}_{+}}(\nabla v\cdot\nabla
U-v_{t}U_{t}-vU)\,d\mathbf{x}\,dt$ $\displaystyle=$
$\displaystyle-\langle\mathbb{S}v,\mathcal{S}U\rangle_{\langle
H^{1}(\mathbb{R}^{N}_{+}),H^{1}(\mathbb{R}^{N}_{+})^{*}\rangle}.$

From this identity, which ought to be valid for every $v\in
H^{1}_{0}(\mathbb{R}^{N}_{+})$, we clearly conclude that if
$E^{\prime}(u)\to\mathbf{0}$ then $\mathcal{S}U\to\mathbf{0}$ as well, because
$\mathbb{S}$ preserves the norm. Realizing that

$E(u)=\frac{1}{2}\|U\|_{H^{1}(\mathbb{R}^{N}_{+})}^{2}\leq\frac{1}{2}\|\mathcal{S}U\|_{H^{1}(\mathbb{R}^{N}_{+})^{*}}^{2},$

by estimate (3.6), we conclude the following.

######  Proposition 3.3.

The functional $E$ in (3.4) is an error functional in the sense of Definition
1.1.

Our main abstract result Proposition 1.5 applies in this situation too, and we
can conclude

######  Theorem 3.4.

Problem (3.1) admits a unique weak solution $u\in
H^{1}_{0}(\mathbb{R}^{N}_{+})$ in the sense (3.2), and for every other $v\in
H^{1}_{0}(\mathbb{R}^{N}_{+})$, we have

$\|u-v\|_{H^{1}_{0}(\mathbb{R}^{N}_{+})}^{2}\leq KE(v),$

for some positive constant $K$.

##  4\. Non-linear monotone problems

Suppose we would like to solve, or approximate the solution of, a certain non-
linear elliptic system of PDEs of the form

$\operatorname{div}[\Phi(\nabla u)]=0\hbox{ in }\Omega,\quad u=u_{0}\hbox{ on
}\partial\Omega,$

for a non-linear, smooth map

$\Phi(\mathbf{a}):\mathbb{R}^{N}\to\mathbb{R}^{N}.$

$\Omega\subset\mathbb{R}^{N}$ is assumed to be a regular, bounded domain. One
can set up a natural, suitable, non-negative, smooth functional based on the
least-squares idea, as already introduced,

(4.1) $E(v):H^{1}_{0}(\Omega)\to\mathbb{R}$

by putting

$E(v)=\frac{1}{2}\int_{\Omega}|\nabla U(\mathbf{x})|^{2}\,d\mathbf{x}$

where

$\operatorname{div}[\Phi(\nabla v+\nabla u_{0})+\nabla U]=0\hbox{ in }\Omega$

and $U\in H^{1}_{0}(\Omega)$. We can also put

$E(v)=\frac{1}{2}\|\operatorname{div}[\Phi(\nabla v+\nabla
u_{0})]\|^{2}_{H^{-1}(\Omega)}.$

Our goal is to apply again Proposition 1.5, or, since we are now in a non-
linear situation, Proposition 1.6. Anyhow, (1.8) is necessary.

######  Lemma 4.1.

Let $\Phi(\mathbf{a}):\mathbb{R}^{N}\to\mathbb{R}^{N}$ be a smooth-map with
linear growth at infinity, i.e.

(4.2) $|\Phi(\mathbf{a})|\leq C_{1}|\mathbf{a}|+C_{0},$

with $C_{1}>0$, and strictly monotone in the sense

(4.3)
$(\Phi(\mathbf{a}_{1})-\Phi(\mathbf{a}_{0}))\cdot(\mathbf{a}_{1}-\mathbf{a}_{0})\geq
c|\mathbf{a}_{1}-\mathbf{a}_{0}|^{2},\quad c>0,$

for every pair of vectors $\mathbf{a}_{i}$, $i=0,1$. Then there is a positive
constant $C$ such that

$\|u-v\|_{H^{1}_{0}(\Omega)}^{2}\leq C(E(u)+E(v)),$

for every pair $u,v\in H^{1}_{0}(\Omega)$.

###### Proof.

Let $u,v\in H^{1}_{0}(\Omega)$, and let $U,V\in H^{1}_{0}(\Omega)$ be their
respective residuals in the sense

(4.4) $\operatorname{div}[\Phi(\nabla u+\nabla u_{0})+\nabla
U]=0,\quad\operatorname{div}[\Phi(\nabla v+\nabla u_{0})+\nabla V]=0$

in $\Omega$, and

$E(u)=\frac{1}{2}\|\nabla U\|^{2}_{L^{2}(\Omega;\mathbb{R}^{N})},\quad
E(v)=\frac{1}{2}\|\nabla V\|^{2}_{L^{2}(\Omega;\mathbb{R}^{N})}.$

If we use $u-v$ as test field in (4.4), we find

$\displaystyle\int_{\Omega}\Phi(\nabla u+\nabla u_{0})\cdot(\nabla u-\nabla
v)\,d\mathbf{x}=-\int_{\Omega}\nabla U\cdot(\nabla u-\nabla v)\,d\mathbf{x},$
$\displaystyle\int_{\Omega}\Phi(\nabla v+\nabla u_{0})\cdot(\nabla u-\nabla
v)\,d\mathbf{x}=-\int_{\Omega}\nabla V\cdot(\nabla u-\nabla v)\,d\mathbf{x}.$

The monotonicity condition, together with these identities, takes us, by
subtracting one from the other, to

$c\int_{\Omega}|\nabla u-\nabla v|^{2}\,d\mathbf{x}\leq\int_{\Omega}(\nabla
V-\nabla U)\cdot(\nabla u-\nabla v)\,d\mathbf{x}.$

The standard Cauchy-Schwarz inequality implies that

$c\|\nabla u-\nabla v\|_{L^{2}(\Omega;\mathbb{R}^{N})}\leq\|\nabla U-\nabla
V\|_{L^{2}(\Omega;\mathbb{R}^{N})},$

and thus, taking into account the triangular inequality, we have

$c^{2}\|\nabla u-\nabla v\|_{L^{2}(\Omega;\mathbb{R}^{N})}^{2}\leq
4E(u)+4E(v).$

The use of Poincaré’s inequality yields our statement. ∎

The second ingredient, to apply Theorem 1.6, is the operator $\mathbf{T}$
which comes directly from linearization or from Newton’s method. Given an
approximation of the solution $v+u_{0}$, $v\in H^{1}_{0}(\Omega)$, we seek a
better approximation $V\in H^{1}_{0}(\Omega)$ in the form

(4.5) $\operatorname{div}[\Phi(\nabla v+\nabla u_{0})+\nabla\Phi(\nabla
v+\nabla u_{0})\nabla V]=0\hbox{ in }\Omega,$

as a linear approximation of

$\operatorname{div}[\Phi(\nabla v+\nabla u_{0}+\nabla V)]=0\hbox{ in }\Omega.$

We therefore define

(4.6) $\mathbf{T}:H^{1}_{0}(\Omega)\to H^{1}_{0}(\Omega),\quad\mathbf{T}v=V,$

where $V$ is the solution of (4.5). The fact that $\mathbf{T}$ is well-defined
is a direct consequence of the standard Lax-Milgram lemma and the
identification

$\mathbf{A}=\nabla\Phi(\nabla v+\nabla u_{0}),\quad\mathbf{a}=\Phi(\nabla
v+\nabla u_{0}),$

provided

$|\nabla\Phi(\mathbf{v})|\leq
M,\quad\mathbf{u}\cdot\nabla\Phi(\mathbf{v})\mathbf{u}\geq
c|\mathbf{u}|^{2},\quad M,c>0.$

The first bound is compatible with linear growth at infinity, (4.2), while the
second one is a consequence of monotonicity (4.3). On the other hand, the
smoothness of $\mathbf{T}$ depends directly on the smoothness of $\Phi$,
specifically, we assume $\Phi$ to be $\mathcal{C}^{2}$. Since $\mathbf{T}$
comes from Newton’s method, condition (1.10) is guaranteed. We are hence
entitled to apply Proposition 1.6 and conclude that

######  Theorem 4.2.

Let $\Phi(\mathbf{a}):\mathbb{R}^{N}\to\mathbb{R}^{N}$ be a
$\mathcal{C}^{2}$-mapping such that

$\displaystyle|\nabla\Phi(\mathbf{a})|\leq M,$
$\displaystyle(\Phi(\mathbf{a}_{1})-\Phi(\mathbf{a}_{0}))\cdot(\mathbf{a}_{1}-\mathbf{a}_{0})\geq
c|\mathbf{a}_{1}-\mathbf{a}_{0}|^{2},$

for constants $M,c>0$, and every $\mathbf{a}$, $\mathbf{a}_{1}$,
$\mathbf{a}_{0}$ in $\mathbb{R}^{N}$. There is a unique weak solution $u\in
u_{0}+H^{1}_{0}(\Omega)$, for arbitrary $u_{0}\in H^{1}(\Omega)$, of the
equation

$\operatorname{div}[\Phi(\nabla u)]=0\hbox{ in }\Omega,\quad u-u_{0}\in
H^{1}_{0}(\Omega).$

Moreover

$\|u-v\|^{2}_{H^{1}_{0}(\Omega)}\leq CE(v)$

for every other $v\in u_{0}+H^{1}_{0}(\Omega)$.

It is not hard to design appropriate sets of assumptions to deal with more
general equations of the form

$\operatorname{div}[\Phi(\nabla v(\mathbf{x}),v(\mathbf{x}),\mathbf{x})]=0.$

##  5\. Non-linear waves

We would like to explore non-linear equations of the form

$u_{tt}(t,\mathbf{x})-\Delta u(t,\mathbf{x})-f(\nabla
u(t,\mathbf{x}),u_{t}(t,\mathbf{x}),u(t,\mathbf{x}))=0\hbox{ in
}(t,\mathbf{x})\in\mathbb{R}^{N}_{+},$

subjected to initial conditions

$u(0,\mathbf{x})=u_{0}(\mathbf{x}),\quad
u_{t}(0,\mathbf{x})=u_{1}(\mathbf{x})$

for appropriate data $u_{0}$ and $u_{1}$ belonging to suitable spaces to be
determined. Dimension $N$ is taken to be at least two. Though more complicated
situations could be considered allowing for a monotone main part in the
equation, as in the previous section, to better understand the effect of the
term incorporating lower-order terms, we will restrict ourselves to the
equation above. Conditions on the non-linear term

$f(\mathbf{z},z,u):\mathbb{R}^{N}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$

will be specified along the way as needed.

Our ambient space will be $H^{1}(\mathbb{R}^{N}_{+})$ so that weak solutions
$u$ are sought in $H^{1}(\mathbb{R}^{N}_{+})$. If we assume

$u_{0}\in H^{1/2}(\mathbb{R}^{N}),\quad u_{1}\in L^{2}(\mathbb{R}^{N}),$

we can take for granted, without loss of generality, that both $u_{0}$ and
$u_{1}$ identically vanish and $u\in H^{1}_{0}(\mathbb{R}^{N}_{+})$, at the
expense of permitting

$f(\mathbf{z},z,u,t,\mathbf{x}):\mathbb{R}^{N}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{R}.$

We will therefore stick to the problem

(5.1) $u_{tt}-\Delta u-f(\nabla u,u_{t},u,t,\mathbf{x})=0\hbox{ in
}(t,\mathbf{x})\in\mathbb{R}^{N}_{+},$

subjected to initial conditions

(5.2) $u(0,\mathbf{x})=0,\quad u_{t}(0,\mathbf{x})=0.$

A weak solution $u\in H^{1}_{0}(\mathbb{R}^{N}_{+})$ of (5.1) is such that

(5.3) $\int_{\mathbb{R}^{N}_{+}}[-u_{t}w_{t}+\nabla u\cdot\nabla w-f(\nabla
u,u_{t},u,t,\mathbf{x})w]\,d\mathbf{x}\,dt=0$

for every $w\in H^{1}(\mathbb{R}^{N}_{+})$. This weak formulation asks for the
non-linear term recorded in the function $f$ to comply with

(5.4) $|f(\mathbf{z},z,u,t,\mathbf{x})|\leq
C(|\mathbf{z}|+|z|+|u|^{(N+1)/(N-1)})+f_{0}(t,\mathbf{x})$

for a function $f_{0}\in L^{2}(\mathbb{R}^{N}_{+})$, in such a way that the
composition

$f(\nabla u,u_{t},u,t,\mathbf{x})\in L^{2}(\mathbb{R}^{N}_{+})$

for every $u\in H^{1}(\mathbb{R}^{N}_{+})$. As expected, for every $u\in
H^{1}_{0}(\mathbb{R}^{N}_{+})$ we define its residual $U\in
H^{1}(\mathbb{R}^{N}_{+})$ through

(5.5) $\int_{\mathbb{R}^{N}_{+}}[(U_{t}+u_{t})w_{t}-(\nabla u-\nabla
U)\cdot\nabla w+(U+f(\nabla u,u_{t},u,t,\mathbf{x}))w]\,d\mathbf{x}\,dt=0$

which ought to be correct for every test $w\in H^{1}(\mathbb{R}^{N}_{+})$; and
the functional

(5.6) $E(u):H^{1}_{0}(\mathbb{R}^{N}_{+})\to\mathbb{R}^{+},\quad
E(u)=\frac{1}{2}\|U\|^{2}_{H^{1}(\mathbb{R}^{N}_{+})},$

as a measure of departure of $u$ from being a weak solution of (5.1). Note how
(5.5) determines $U$ in a unique way. Indeed, such $U$ is the unique minimizer
of the strictly convex, quadratic functional

$\frac{1}{2}\int_{\mathbb{R}^{N}_{+}}\left[(U_{t}+u_{t})^{2}+|\nabla U-\nabla
u|^{2}+(U+f(\nabla u,u_{t},u,t,\mathbf{x}))^{2}\right]\,d\mathbf{x}\,dt$

define for $U\in H^{1}(\mathbb{R}^{N}_{+})$. We claim that under appropriate
additional hypotheses, we can apply Proposition 1.6 to this situation. To
explain things in the most affordable way, however, we will show that
Proposition 1.5 can also be applied directly. This requires to check that $E$
in (5.6) is indeed an error functional in the sense of Definition 1.1.

We will be using the operators and the formalism right before Lemma 3.1, as
well as bound (3.6) in this lemma.

######  Lemma 5.1.

Suppose the function $f(\mathbf{z},z,u,t,\mathbf{x})$ is such that

  1. (1)

$f(\mathbf{0},0,0,t,\mathbf{x})\in L^{2}(\mathbb{R}^{N}_{+})$;

  2. (2)

the difference $f(\mathbf{z},z,u,t,\mathbf{x})-u$ is globally Lipschitz with
respect to triplets $(\mathbf{z},z,u)$ in the sense

$\displaystyle|f(\mathbf{z},z,u,t,\mathbf{x})-u-f(\mathbf{y},y,v,t,\mathbf{x})+v|\leq$
$\displaystyle
M\left(|\mathbf{z}-\mathbf{y}|+|z-y|+\frac{1}{D}|u-v|^{(N+1)/(N-1)}\right),$

where $D$ is the constant of the corresponding embedding

$H^{1}(\mathbb{R}^{N}_{+})\subset L^{2(N+1)/(N-1)}(\mathbb{R}^{N}_{+}),$

and $M<1$.

Then there is a positive constant $K$ with

$\|u-v\|_{H^{1}_{0}(\mathbb{R}^{N}_{+})}^{2}\leq K(E(u)+E(v)),$

for every pair $u,v\in H^{1}_{0}(\mathbb{R}^{N}_{+})$, where $E$ is given in
(5.6).

###### Proof.

Note how our hypotheses on the nonlinearity $f$ imply the bound (5.4) by
taking

$\mathbf{y}=\mathbf{0},\quad y=v=0,\quad
f_{0}(t,\mathbf{x})=f(\mathbf{0},0,0,t,\mathbf{x}).$

If $u$, $v$ belong to $H^{1}_{0}(\mathbb{R}^{N}_{+})$, and $U$, $V$ in
$H^{1}(\mathbb{R}^{N}_{+})$ are their respective residuals, then

$\displaystyle\int_{\mathbb{R}^{N}_{+}}[(U_{t}+u_{t})w_{t}-(\nabla u-\nabla
U)\cdot\nabla w+(U+f(\nabla u,u_{t},u,t,\mathbf{x}))w]\,d\mathbf{x}\,dt=0,$
$\displaystyle\int_{\mathbb{R}^{N}_{+}}[(V_{t}+v_{t})w_{t}-(\nabla v-\nabla
V)\cdot\nabla w+(V+f(\nabla v,v_{t},v,t,\mathbf{x}))w]\,d\mathbf{x}\,dt=0,$

for every $w\in H^{1}(\mathbb{R}^{N}_{+})$. By subtracting one from the other,
and letting

$s=u-v,\quad S=U-V,$

we find

$\displaystyle\int_{\mathbb{R}^{N}_{+}}[(S_{t}+s_{t})w_{t}-(\nabla s-\nabla
S)\cdot\nabla w+$ $\displaystyle(S+f(\nabla u,u_{t},u,t,\mathbf{x})-f(\nabla
v,v_{t},v,t,\mathbf{x}))w]\,d\mathbf{x}\,dt=0,$

for every $w\in H^{1}(\mathbb{R}^{N}_{+})$. We can recast this identity, by
using the formalism in the corresponding linear situation around Lemma 3.1, as

$\displaystyle\langle S,\mathbb{S}w\rangle_{\langle
H^{1}(\mathbb{R}^{N}_{+}),H^{1}(\mathbb{R}^{N}_{+})\rangle}+\langle
w,\mathcal{S}s\rangle_{\langle
H^{1}(\mathbb{R}^{N}_{+}),H^{1}(\mathbb{R}^{N}_{+})^{*}\rangle}$
$\displaystyle=-\int_{\mathbb{R}^{N}_{+}}[f(\nabla
u,u_{t},u,t,\mathbf{x})-f(\nabla
v,v_{t},v,t,\mathbf{x})-s)w]\,d\mathbf{x}\,dt.$

The same manipulations as in the proof of Proposition 3.2, together with the
assumed Lipschitz property on $f$, lead immediately to

$\displaystyle\|s\|_{H^{1}_{0}(\mathbb{R}^{N}_{+})}\leq$
$\displaystyle\|\mathcal{S}s\|_{H^{1}(\mathbb{R}^{N}_{+})^{*}}$
$\displaystyle=$ $\displaystyle\sup_{\|w\|_{H^{1}(\mathbb{R}^{N}_{+})}\leq
1}|\langle w,\mathcal{S}s\rangle_{\langle
H^{1}(\mathbb{R}^{N}_{+}),H^{1}(\mathbb{R}^{N}_{+})^{*}\rangle}|$
$\displaystyle=$ $\displaystyle\sup_{\|w\|_{H^{1}(\mathbb{R}^{N}_{+})}\leq
1}\left|\langle
S,\mathbb{S}w\rangle+\int_{\mathbb{R}^{N}_{+}}(f_{u}-f_{v}-s)w\,d\mathbf{x}\,dt\right|$
$\displaystyle\leq$
$\displaystyle\|S\|_{H^{1}(\mathbb{R}^{N}_{+})}\,\|w\|_{H^{1}(\mathbb{R}^{N}_{+})}+M\|s\|_{H^{1}_{0}(\mathbb{R}^{N}_{0})}\|w\|_{H^{1}(\mathbb{R}^{N}_{+})}$
$\displaystyle\leq$
$\displaystyle\|S\|_{H^{1}(\mathbb{R}^{N}_{+})}+M\|s\|_{H^{1}_{0}(\mathbb{R}^{N}_{+})}.$

We are putting

$f_{u}=f(\nabla u,u_{t},u,t,\mathbf{x}),\quad f_{v}=f(\nabla
v,v_{t},v,t,\mathbf{x}),$

for the sake of notation. Note also the use of the embedding constant. The
resulting final inequality, and the relative sizes of these constants, show
our claim. ∎

We turn to the second important property for $E$ to become an error
functional, namely,

$\lim_{E^{\prime}(\mathbf{u})\to\mathbf{0}}E(\mathbf{u})=0$

over bounded subsets of $H^{1}_{0}(\mathbb{R}^{N}_{+})$. We assume that the
non-linearity $f$ is $C^{1}$\- with respect to $(\mathbf{z},z,u)$, and its
partial derivatives are uniformly bounded. To compute the derivative
$E^{\prime}(u)$ at an arbitrary $u\in H^{1}_{0}(\mathbb{R}^{N}_{+})$, we
perform, as usual, the perturbation to first-order

$u\mapsto u+\epsilon v,\quad U\mapsto U+\epsilon V,$

and introduce them in (5.5). After differentiation with respect to $\epsilon$,
and setting $\epsilon=0$, we find

(5.7) $\int_{\mathbb{R}^{N}_{+}}[(V_{t}+v_{t})w_{t}-(\nabla v-\nabla
V)\cdot\nabla w+(V+\overline{f}_{\mathbf{z}}\cdot\nabla
v+\overline{f}_{z}v_{t}+\overline{f}_{u}v)w]\,d\mathbf{x}\,dt=0,$

for all $w\in H^{1}(\mathbb{R}^{N}_{+})$, where

$\overline{f}_{\mathbf{z}}(t,\mathbf{x})=f_{\mathbf{z}}(\nabla
u(t,\mathbf{x}),u_{t}(t,\mathbf{x}),u(t,\mathbf{x}),t,\mathbf{x}),$

and the same for $\overline{f}_{z}(t,\mathbf{x})$ and
$\overline{f}_{u}(t,\mathbf{x})$. On the other hand,

$\langle E^{\prime}(u),v\rangle=\lim_{\epsilon\to
0}\frac{1}{\epsilon}(E(u+\epsilon v)-E(u))$

is clearly given by

$\langle E^{\prime}(u),v\rangle=\int_{\mathbb{R}^{N}_{+}}\left(\nabla
U\cdot\nabla V+U_{t}\,V_{t}+U\,V\right)\,d\mathbf{x}\,dt.$

If we use $w=U$ in (5.7), we can write

$\langle
E^{\prime}(u),v\rangle=\int_{\mathbb{R}^{N}_{+}}\left[-v_{t}\,U_{t}+\nabla
v\cdot\nabla U-(\overline{f}_{\mathbf{z}}\cdot\nabla
v+\overline{f}_{z}v_{t}+\overline{f}_{u}v)U\right]\,d\mathbf{x}\,dt.$

The validity of this representation for every $v\in
H^{1}_{0}(\mathbb{R}^{N}_{+})$ enables us to identify $E^{\prime}(u)$ with the
triplet

$(-U_{t}-\overline{f}_{z}U,\nabla
U-U\overline{f}_{\mathbf{z}},-\overline{f}_{u}U),$

in the sense $E^{\prime}(u)=\mathcal{S}U$ where the linear operator

$\mathcal{S}:H^{1}(\mathbb{R}^{N}_{+})\mapsto H^{-1}(\mathbb{R}^{N}_{+})$

is precisely determined by

$\langle\mathcal{S}U,v\rangle=\int_{\mathbb{R}^{N}_{+}}\left[-v_{t}\,U_{t}+\nabla
v\cdot\nabla U-(\overline{f}_{\mathbf{z}}\cdot\nabla
v+\overline{f}_{z}v_{t}+\overline{f}_{u}v)U\right]\,d\mathbf{x}\,dt$

for every $v\in H^{1}_{0}(\mathbb{R}^{N}_{+})$. Notice how this operator
$\mathcal{S}$ is well-defined because the non-linearity $f$ has been assumed
to be globally Lipschitz with partial derivatives uniformly bounded. To
conclude that $E^{\prime}(u)\to\mathbf{0}$ implies $U\to 0$ and, hence
$E(u)=0$, we need to ensure that this operator $\mathcal{S}$ is injective. We
conjecture that this is so, without further requirements; but to simplify the
argument here, we add the assumption that

$|f_{u}(\mathbf{z},z,u,t,z)|\geq\epsilon>0$

for every $(\mathbf{z},z,u,t,z)$. Under this additional hypothesis, the
condition

$(-U_{t}-\overline{f}_{z}U,\nabla
U-U\overline{f}_{\mathbf{z}},-\overline{f}_{u}U)=\mathbf{0}$

automatically implies

$U=\nabla U=U_{t}=0$

and hence $E(u)=0$.

######  Theorem 5.2.

Suppose the non-linearity $f(\mathbf{z},z,u,t,\mathbf{x})$ is
$\mathcal{C}^{1}$\- with respect to variables $(\mathbf{z},z,u)$, and:

  1. (1)

$f(\mathbf{0},0,0,t,\mathbf{x})\in L^{2}(\mathbb{R}^{N}_{+})$;

  2. (2)

the difference $f(\mathbf{z},z,u,t,\mathbf{x})-u$ is Lipschitz with respect to
triplets $(\mathbf{z},z,u)$ in the sense

$\displaystyle|f(\mathbf{z},z,u,t,\mathbf{x})-u-f(\mathbf{y},y,v,t,\mathbf{x})+v|\leq$
$\displaystyle
M\left(|\mathbf{z}-\mathbf{y}|+|z-y|+\frac{1}{D}|u-v|^{(N+1)/(N-1)}\right),$

where $D$ is the constant of the corresponding embedding

$H^{1}(\mathbb{R}^{N}_{+})\subset L^{2(N+1)/(N-1)}(\mathbb{R}^{N}_{+}),$

and $M<1$;

  3. (3)

non-vanishing of $f_{u}$: there is some $\epsilon>0$ with

$|f_{u}(\mathbf{z},z,u,t,z)|\geq\epsilon>0.$

Then the problem

$u_{tt}-\Delta u+f(\nabla u,u_{t},u,t,\mathbf{x})=0\hbox{ in
}(t,\mathbf{x})\in\mathbb{R}^{N}_{+}$

under vanishing initial conditions

$u(0,\mathbf{x})=u_{t}(0,\mathbf{x})=0,\quad\mathbf{x}\in\mathbb{R}^{N},$

admits a unique weak solution $u\in H^{1}_{0}(\mathbb{R}^{N}_{+})$ in the
sense (5.3), and

$\|v-u\|^{2}_{H^{1}_{0}(\mathbb{R}^{N}_{+})}\leq KE(v),$

for any other $v\in H^{1}_{0}(\mathbb{R}^{N}_{+})$.

Without the global Lipschitzianity condition on $f$ in the previous statement,
but only the smoothness with respect to triplets $(\mathbf{z},z,u)$, only a
local existence result is possible. This is standard.

##  6\. The steady Navier-Stokes system

For a bounded, Lipschitz, connected domain $\Omega\subset\mathbb{R}^{N}$,
$N=2,3$, we are concerned with the steady Navier-Stokes system

(6.1) $-\nu\Delta\mathbf{u}+\nabla\mathbf{u}\,\mathbf{u}+\nabla
u=\mathbf{f},\quad\operatorname{div}\mathbf{u}=0\hbox{ in }\Omega,$

for a vector field $\mathbf{u}\in H^{1}_{0}(\Omega;\mathbb{R}^{N})$, and a
scalar, pressure field $u\in L^{2}(\Omega)$. The external force field
$\mathbf{f}$ is assumed to belong to the dual space
$H^{-1}(\Omega;\mathbb{R}^{N})$. The parameter $\nu>0$ is viscosity. Because
of the incompressibility condition, the system can also be written in the form

$-\nu\Delta\mathbf{u}+\operatorname{div}(\mathbf{u}\otimes\mathbf{u})+\nabla
u=\mathbf{f},\quad\operatorname{div}\mathbf{u}=0\hbox{ in }\Omega.$

A weak solution is a divergence-free vector field $\mathbf{u}\in
H^{1}_{0}(\Omega;\mathbb{R}^{N})$, and a scalar field $u\in L^{2}(\Omega)$,
normalized by demanding vanishing average in $\Omega$, such that

$\int_{\Omega}[\nu\nabla\mathbf{u}(\mathbf{x}):\nabla\mathbf{v}(\mathbf{x})-\mathbf{u}(\mathbf{x})\nabla\mathbf{v}(\mathbf{x})\mathbf{u}(\mathbf{x})-u(\mathbf{x})\operatorname{div}\mathbf{v}(\mathbf{x})]\,d\mathbf{x}=\langle\mathbf{f},\mathbf{v}\rangle$

where the right-hand side stands for the duality pairing

$H^{-1}(\Omega;\mathbb{R}^{N})-H^{1}_{0}(\Omega;\mathbb{R}^{N}).$

We propose to look at this problem incorporating the incompressibility
constraint into the space as part of feasibility as is usually done; there is
also the alternative to treat the same situation incorporating a penalization
on the divergence into the functional, instead of including it into the class
of admissible fields (see [5]). The pressure field rises as the multiplier
corresponding to the divergence-free constraint.

Let

$\mathbb{D}\equiv H^{1}_{0,div}(\Omega;\mathbb{R}^{N})=\\{\mathbf{u}\in
H^{1}_{0}(\Omega;\mathbb{R}^{N}):\operatorname{div}\mathbf{u}=0\hbox{ in
}\Omega\\}.$

For every such $\mathbf{u}$, we determine its residual $\mathbf{U}$, in a
unique way, as the solution of the restricted variational problem

$\hbox{Minimize in
}\mathbf{V}\in\mathbb{D}:\quad\int_{\Omega}\left[\frac{1}{2}|\nabla\mathbf{V}|^{2}+(\nu\nabla\mathbf{u}-\mathbf{u}\otimes\mathbf{u}):\nabla\mathbf{V}\right]\,d\mathbf{x}-\langle\mathbf{f},\mathbf{V}\rangle.$

The pressure $v$ comes as the corresponding multiplier for the divergence-free
constraint, in such a way that the unique minimizer $\mathbf{U}$ is determined
through the variational equality

$\int_{\Omega}(\nabla\mathbf{U}:\nabla\mathbf{V}+(\nu\nabla\mathbf{u}-\mathbf{u}\otimes\mathbf{u}):\nabla\mathbf{V}+v\operatorname{div}\mathbf{V})\,d\mathbf{x}-\langle\mathbf{f},\mathbf{V}\rangle$

valid for every test field $\mathbf{V}\in H^{1}_{0}(\Omega;\mathbb{R}^{N})$.
This is the weak form of the optimality condition associated with the previous
variational problem

(6.2)
$-\Delta\mathbf{U}-\nu\Delta\mathbf{u}+\operatorname{div}(\mathbf{u}\otimes\mathbf{u})-\mathbf{f}+\nabla
v=\mathbf{0}\hbox{ in }\Omega,$

for $\mathbf{U}\in\mathbb{D}$. The multiplier $v\in L^{2}_{0}(\Omega)$
(square-integrable fields with a vanishing average) is the pressure. We define

(6.3) $E(\mathbf{u}):\mathbb{D}\to\mathbb{R}^{+},\quad
E(\mathbf{u})=\frac{1}{2}\int_{\Omega}|\nabla\mathbf{U}(\mathbf{x})|^{2}\,d\mathbf{x}.$

For $\mathbf{u},\mathbf{v}\in\mathbb{D}$, let
$\mathbf{U},\mathbf{V}\in\mathbb{D}$ be their respective residuals, and $u,v$
their respective pressure fields. Put

$\mathbf{w}=\mathbf{u}-\mathbf{v}\in\mathbb{D},\quad\mathbf{W}=\mathbf{U}-\mathbf{V}\in\mathbb{D},\quad
w=u-v\in L^{2}(\Omega).$

It is elementary to find, by subtraction of the corresponding system (6.2) for
$\mathbf{u}$ and $\mathbf{v}$, that

(6.4)
$-\Delta\mathbf{W}-\nu\Delta\mathbf{w}+\operatorname{div}(\mathbf{u}\otimes\mathbf{u}-\mathbf{v}\otimes\mathbf{v})+\nabla
w=\mathbf{0}\hbox{ in }\Omega.$

It is the presence of the non-linear term
$\operatorname{div}(\mathbf{u}\otimes\mathbf{u})$, so fundamental to the
Navier-Stokes system, what makes the situation different compared to a linear
setting.

We write

$\operatorname{div}(\mathbf{u}\otimes\mathbf{u}-\mathbf{v}\otimes\mathbf{v})=\operatorname{div}(\mathbf{w}\otimes\mathbf{u})+\operatorname{div}(\mathbf{v}\otimes\mathbf{w}),$

and bear in mind the well-know fact

(6.5)
$\int_{\Omega}(\mathbf{v}\otimes\mathbf{v}:\nabla\mathbf{u}+\mathbf{u}\otimes\mathbf{v}:\nabla\mathbf{v})\,d\mathbf{x}=\int_{\Omega}\mathbf{u}\otimes\mathbf{v}:\nabla\mathbf{u}\,d\mathbf{x}=0$

for every $\mathbf{u},\mathbf{v}\in\mathbb{D}$. If we use $\mathbf{w}$ as a
test function in (6.4), we find that

(6.6)
$\int_{\Omega}[\nabla\mathbf{W}:\nabla\mathbf{w}+\nu|\nabla\mathbf{w}|^{2}-(\mathbf{w}\otimes\mathbf{u}):\nabla\mathbf{w}-(\mathbf{v}\otimes\mathbf{w}):\nabla\mathbf{w}]\,d\mathbf{x}=0.$

Note that the integral involving $w$ vanishes because $\mathbf{w}$ is
divergence-free. By (6.5),

$\displaystyle\int_{\Omega}(\mathbf{w}\otimes\mathbf{u}):\nabla\mathbf{w}\,d\mathbf{x}=0,$
$\displaystyle\int_{\Omega}(\mathbf{v}\otimes\mathbf{w}):\nabla\mathbf{w}\,d\mathbf{x}=-\int_{\Omega}(\mathbf{w}\otimes\mathbf{w}):\nabla\mathbf{v}\,d\mathbf{x}.$

Identity (6.6) becomes

(6.7)
$\int_{\Omega}[\nabla\mathbf{W}:\nabla\mathbf{w}+\nu|\nabla\mathbf{w}|^{2}+(\mathbf{w}\otimes\mathbf{w}):\nabla\mathbf{v}]\,d\mathbf{x}=0.$

We can use $\mathbf{v}$ as a test function in the corresponding system (6.2)
for $\mathbf{v}$ to have

$\int_{\Omega}(\nabla\mathbf{V}:\nabla\mathbf{v}+\nu|\nabla\mathbf{v}|^{2}-\mathbf{f}\cdot\mathbf{v})\,d\mathbf{x}=0.$

Again we have utilized that fields in $\mathbb{D}$ are divergence-free, and
the second identity in (6.5). This last identity implies, in an elementary
way, that

(6.8)
$\nu\|\mathbf{v}\|_{H^{1}_{0}(\Omega;\mathbb{R}^{N})}\leq\|\mathbf{f}\|_{H^{-1}(\Omega;\mathbb{R}^{N})}+\sqrt{2E(\mathbf{v})}.$

Recall that

$2E(\mathbf{v})=\|\mathbf{V}\|^{2}_{H^{1}_{0}(\Omega;\mathbb{R}^{N})}.$

We now have all the suitable elements to exploit (6.7). If $C=C(n)$ is the
constant of the Sobolev embedding of $H^{1}(\Omega)$ into $L^{4}(\Omega)$ for
$N\leq 4$, then (6.7) leads to

$\nu\|\mathbf{w}\|^{2}\leq\|\mathbf{W}\|\,\|\mathbf{w}\|+C^{2}\|\mathbf{w}\|^{2}\|\mathbf{v}\|$

where all norms here are in $H^{1}_{0}(\Omega;\mathbb{R}^{N})$. On the other
hand, if we replace the size of $\mathbf{v}$ by the estimate (6.8), we are
carried to

$\nu\|\mathbf{w}\|^{2}\leq\|\mathbf{W}\|\,\|\mathbf{w}\|+\frac{C^{2}}{\nu}\|\mathbf{w}\|^{2}\left(\|\mathbf{f}\|_{H^{-1}(\Omega;\mathbb{R}^{N})}+\sqrt{2E(\mathbf{v})}\right),$

or

$\left(\nu-\frac{C^{2}}{\nu}\left(\|\mathbf{f}\|+\sqrt{2E(\mathbf{v})}\right)\right)\|\mathbf{w}\|^{2}\leq\|\mathbf{W}\|\,\|\mathbf{w}\|.$

Since

$\|\mathbf{W}\|=\|\mathbf{U}-\mathbf{V}\|\leq\|\mathbf{U}\|+\|\mathbf{V}\|,$

we would have

$\left(\nu-\frac{C^{2}}{\nu}\left(\|\mathbf{f}\|+\sqrt{2E(\mathbf{v})}\right)\right)^{2}\|\mathbf{u}-\mathbf{v}\|^{2}\leq
4(E(\mathbf{u})+E(\mathbf{v})).$

The form of this inequality leads us to the following interesting
generalization of Definition 1.1.

######  Definition 6.1.

A non-negative, $\mathcal{C}^{1}$-functional

$E(\mathbf{u}):\mathbb{H}\to\mathbb{R}^{+}$

defined over a Hilbert space $\mathbb{H}$ is called an error functional if
there is some positive constant $c$ (including $c=+\infty$) such that:

  1. (1)

behavior as $E^{\prime}\to\mathbf{0}$:

$\lim_{E^{\prime}(\mathbf{u})\to\mathbf{0}}E(\mathbf{u})=0$

over bounded subsets of $\mathbb{H}$; and

  2. (2)

enhanced coercivity: there is a positive constant $C$ (that might depend on
$c$), such that for every pair $\mathbf{u},\mathbf{v}$ belonging to the sub-
level set $\\{E\leq c\\}$, we have

$\|\mathbf{u}-\mathbf{v}\|^{2}\leq C(E(\mathbf{u})+E(\mathbf{v})).$

It is interesting to note that the sub-level sets $\\{E\leq d\\}$ for $d<c$,
for a functional $E$ verifying Definition 6.1, cannot maintain several
connected components.

Because our basic result Proposition 1.5 is concerned with zeros of $E$, it is
still valid under Definition 6.1.

######  Proposition 6.1.

Let $E:\mathbb{H}\to\mathbb{R}^{+}$ be an error functional according to
Definition 6.1. Then there is a unique minimizer
$\mathbf{u}_{\infty}\in\mathbb{H}$ such that $E(\mathbf{u}_{\infty})=0$, and

$\|\mathbf{u}-\mathbf{u}_{\infty}\|^{2}\leq CE(\mathbf{u}),$

for every $\mathbf{u}\in\mathbb{H}$ provided $E(\mathbf{u})$ is sufficiently
small ($E(\mathbf{u})\leq c$, the constant in Definition 6.1).

The calculations that motivated this generalization yield the following.

######  Proposition 6.2.

Let $N\leq 4$, and $\Omega\subset\mathbb{R}^{N}$, a bounded, Lipschitz,
connected domain. If $\nu>0$ and $\mathbf{f}\in H^{-1}(\Omega;\mathbb{R}^{N})$
are such that the quotient $\|\mathbf{f}\|/\nu^{2}$ is sufficiently small,
then the functional $E$ in (6.3) complies with Definition 6.1.

We now turn to examining the interconnection between $E$ and $E^{\prime}$. To
this end, we gather here (6.2) and (6.3)

$\displaystyle
E(\mathbf{u})=\frac{1}{2}\int_{\Omega}|\nabla\mathbf{U}(\mathbf{x})|^{2}\,d\mathbf{x},$
$\displaystyle-\Delta\mathbf{U}-\nu\Delta\mathbf{u}+\operatorname{div}(\mathbf{u}\otimes\mathbf{u})-\mathbf{f}+\nabla
u=\mathbf{0}\hbox{ in }\Omega,$

for $\mathbf{u},\mathbf{U}\in\mathbb{D}$ and $u\in L^{2}_{0}(\Omega)$. If we
replace

$\mathbf{u}\mapsto\mathbf{u}+\epsilon\mathbf{v},\quad\mathbf{U}\mapsto\mathbf{U}+\epsilon\mathbf{V},\quad
u\mapsto u+\epsilon v,$

to first-order in $\epsilon$, we would have

$\displaystyle
E(\mathbf{u}+\epsilon\mathbf{v})=\frac{1}{2}\int_{\Omega}|\nabla\mathbf{U}+\epsilon\nabla\mathbf{V}|^{2}\,d\mathbf{x},$
$\displaystyle-\Delta(\mathbf{U}+\epsilon\mathbf{V})-\nu\Delta(\mathbf{u}+\epsilon\mathbf{v})+\operatorname{div}((\mathbf{u}+\epsilon\mathbf{v})\otimes(\mathbf{u}+\epsilon\mathbf{v}))$
$\displaystyle-\mathbf{f}+\nabla(u+\epsilon v)=\mathbf{0}.$

By differentiating with respect to $\epsilon$, and setting $\epsilon=0$, we
arrive at

$\displaystyle\langle
E^{\prime}(\mathbf{u}),\mathbf{v}\rangle=\int_{\Omega}\nabla\mathbf{U}\cdot\nabla\mathbf{V}\,d\mathbf{x},$
$\displaystyle-\Delta\mathbf{V}-\nu\Delta\mathbf{v}+\operatorname{div}(\mathbf{u}\otimes\mathbf{v}+\mathbf{v}\otimes\mathbf{u})+\nabla
v=\mathbf{0}.$

If we use $\mathbf{U}$ as a test function in this last system, we realize that

$\langle
E^{\prime}(\mathbf{u}),\mathbf{v}\rangle=\int_{\Omega}[-\nu\nabla\mathbf{v}\cdot\nabla\mathbf{U}+\mathbf{u}\otimes\mathbf{v}:\nabla\mathbf{U}+\mathbf{v}\otimes\mathbf{u}:\nabla\mathbf{U}]\,d\mathbf{x}$

for every $\mathbf{v}\in\mathbb{D}$. If we set
$\mathbf{w}=E^{\prime}(\mathbf{u})\in\mathbb{D}$, then

$\int_{\Omega}[\nabla\mathbf{w}\cdot\nabla\mathbf{v}+\nu\nabla\mathbf{U}\cdot\nabla\mathbf{v}+(\nabla\mathbf{U}\mathbf{u}+\mathbf{u}\nabla\mathbf{U})\cdot\mathbf{v}]\,d\mathbf{x},$

for every $\mathbf{v}\in\mathbb{D}$. In particular, if we plug
$\mathbf{v}=\mathbf{U}$ in, bearing in mind that due to (6.5) the last two
terms drop out, we are left with

$\nu\|\nabla\mathbf{U}\|^{2}=-\langle\nabla\mathbf{w},\nabla\mathbf{U}\rangle,$

or

$\nu\|\nabla\mathbf{U}\|\leq\|\nabla\mathbf{w}\|.$

If the term on the right-hand side, which is $\|E^{\prime}(\mathbf{u})\|$,
tends to zero, so does the one on the left-hand side, which is
$\nu\sqrt{2E(\mathbf{u})}$. This shows the second basic property of an error
functional. As a result, Proposition 6.1 can be applied.

######  Theorem 6.3.

If $\Omega\subset\mathbb{R}^{N}$, $N\leq 4$, is a bounded, Lipschitz,
connected domain, and $\nu>0$ and $\mathbf{f}\in
H^{-1}(\Omega;\mathbb{R}^{N})$ in the steady Navier-Stokes system (6.1) are
such that the quotient $\|\mathbf{f}\|/\nu^{2}$ is sufficiently small, then
there is a unique weak solution $\mathbf{u}$ in $\mathbb{D}$, and

$\|\mathbf{u}-\mathbf{v}\|^{2}_{H^{1}_{0}(\Omega;\mathbb{R}^{N})}\leq
CE(\mathbf{v})$

provided $E(\mathbf{v})$ is sufficiently small.

## References

  * [1] B. Bochev, M. Gunzburger, Least-Squares Finite Element Methods, Applied Mathematical Sciences, vol. 166, Springer, New York, 2009.
  * [2] Brezis, Haim Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
  * [3] Evans, Lawrence C. Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.
  * [4] Glowinski, R. (1983) Numerical Methods for Nonlinear Variational Problems, Springer series in Computational Physics, Springer, New York, NY.
  * [5] Lemoine, J., Münch, A., Pedregal, P., Analysis of continuous H1-least-squares methods for the steady Navier-Stokes system. Appl. Math. Optim. 83 (2021), no. 1, 461–488.
  * [6] Münch, Arnaud; Pedregal, Pablo A least-squares formulation for the approximation of null controls for the Stokes system. C. R. Math. Acad. Sci. Paris 351 (2013), no. 13-14, 545–550.
  * [7] Münch, Arnaud; Pedregal, Pablo About least-squares type approach to address direct and controllability problems. Evolution equations: long time behavior and control, 118–136, London Math. Soc. Lecture Note Ser., 439, Cambridge Univ. Press, Cambridge, 2018.
  * [8] Pedregal, P., A variational approach to dynamical systems and its numerical simulation. Numer. Funct. Anal. Optim. 31 (2010), no. 7-9, 814-830.


# MDsrv - visual sharing and analysis of molecular dynamics simulations

Michelle Kampfrath§
Leipzig University
Image and Signal Processing Group
Leipzig, Germany
René Staritzbichler§
Leipzig University
Institute of Biophysics and Medical Physics
Leipzig, Germany
Guillermo Pérez Hernández
Charité Universitätsmedizin Berlin
Institute of Medical Physics and Biophysics
Berlin, Germany
Alexander S. Rose
Mol* Consortium

Johanna K.S. Tiemann
University of Copenhagen
Linderstrøm-Lang Centre for Protein Science
Copenhagen, Denmark
Gerik Scheuermann
Leipzig University
Image and Signal Processing Group
Leipzig, Germany

Daniel Wiegreffe§✉
Leipzig University
Image and Signal Processing Group
Leipzig, Germany
Peter W. Hildebrand§✉
Leipzig University
Institute of Biophysics and Medical Physics
Leipzig, Germany

###### Abstract

Molecular dynamics simulation is a proven technique for computing and
visualizing the time-resolved motion of macromolecules at atomic resolution.
The MDsrv is a tool that streams MD trajectories and displays them
interactively in web browsers without requiring advanced skills, facilitating
interactive exploration and collaborative visual analysis. We have now
enhanced the MDsrv to further simplify the upload and sharing of MD
trajectories and improve their online viewing and analysis. With the new
instance, the MDsrv simplifies the creation of sessions, which allows the
exchange of MD trajectories with preset representations and perspectives. An
important innovation is that the MDsrv can now access and visualize
trajectories from remote datasets, which greatly expands its applicability and
use, as the data no longer needs to be accessible on a local server. In
addition, initial analyses such as sequence or structure alignments, distance
measurements, or RMSD calculations have been implemented, which optionally
support visual analysis. Finally,  based on Mol*, MDsrv now provides faster
and more efficient visualization of even large trajectories  compared to its
predecessor tool NGL.

Correspondence<EMAIL_ADDRESS>and<EMAIL_ADDRESS>leipzig.de
§ Authors contributed equally

## Graphical Abstract

Figure 1:  MDsrv helps experts prepare interactive MD sessions and share them
by simply sending a URL link to students, staff, or by including the URL in
publications.

## Introduction

Molecular dynamics (MD) simulations are a well-established technique to
investigate time-resolved motions of biological macromolecules at atomic
resolution [1, 2]. MD simulations are routinely applied to answer many
biologically relevant questions and have proven their value in deciphering
functional mechanisms of proteins such as of G protein-coupled receptors
(GPCRs), DNA, or RNA and thereby aiding also in the design of novel drugs
(e.g. [3]). Recent approaches using machine learning that push the limitations
of classical MD simulation [4] give a glimpse of future applications that may
even exceed the time scales of coarse-grained dynamics [5, 6]. MD simulations
can provide unprecedented insight into temporal and spatial details by
calculating the dynamic behavior of proteins in their natural cellular
environment in silico.

In recent years, researchers using MD simulations have helped make the field
of MD more FAIR (findability, accessibility, interoperability, reproducibility
of data [7]) by depositing their data in general-purpose scientific data
repositories, making them accessible for further study and research [8]. Even
if access to MD trajectories is granted, some degree of expert knowledge is
required to even load and visually examine a simulation. Visualization itself
is a great way to bridge technical knowledge gaps, for example between
experimental and computational researchers, as it facilitates and guides data
analysis and the complementary strengths of human and machine analysis are
potentiated when led by interactive visualization [9]. Web-enabled
visualization solutions help reach a wider audience. Web visualization tools
are now increasingly being developed and made available for visualizing MD
trajectories.  The popularity of these applications is also due to the free
and easy accessibility of the tools for visualization of MD trajectories and
their user-friendliness.

The _MDsrv_ was the first tool that made interactive streaming and
visualization of MD trajectories in web browsers more accessible [10]; others
followed including HTMoL [11], UnityMol [12], PCAViz [13], and 3dRS [14].  The
applications of these tools are broad, and they all cover the same
functionalities of in-browser visualization and sharing of remote data. Each
one, however, differentiates itself by adding different features, e.g. the
programmatic manipulation of the trajectory via text-console [11], the
visualization of principal components [13] or of arbitrary collective
variables [14]. As of to our knowledge, with its simplicity, the _MDsrv_ and
in particular the NGL Viewer allowed an easy and streamline embedding into
other websites. An example is the GPCRmd platform, which hosts MD simulations
of more than 200 GPCRs and visualizes the intrinsic flexibility of the
3D-GPCRome that in turn provides valuable insights into signaling processes
that are also relevant to drug development [15].  Furthermore, the NGL Viewer
and _MDsrv_ were the basis for 3dRS that improved even more the user
experience and facilitate session generation with the goal to make such MD
streaming viewers more applicable by the community. 3dRS is well adapted to
sharing structures and trajectories, but only up to a certain size. Due to the
ever increasing amount of data, MDsrv addresses this aspect by being optimized
for arbitrarily large data. This approach led in the development also to
smaller restrictions (e.g., possible file formats) that will be addressed in
future releases.

_MDsrv_ already allowed MD experts to set up a session to share an intuitive
visualization of a trajectory with collaborators. Although it has been adopted
by many computational biophysicists, usability was still somewhat limited.
First, creating a session required knowledge of JavaScript, and extracting
accurate representations, colors, or alignments was tedious. Simple and
helpful analysis functions such as distance measurements required even more
expert knowledge of the underlying functions of the _MDsrv_ which in turn
relied on the NGL Viewer visualization [16]. This complicated the application
and limited the visualization setup to experts. Here we introduce a new
instance of _MDsrv_ that allows non-experts to load and share MD trajectory
sessions more easily ( see as overview Fig 1, available at
https://proteinformatics.informatik.uni-leipzig.de/mdsrv).

The underlying viewer, NGL, has been replaced by its successor Mol* [17],
which combines the strengths of the NGL Viewer and the LiteMol Suite [18].
Mol* offers a number of benefits for _MDsrv_ , including superior visual
quality, improved secondary structure assignment, trajectory animation video
export, and fully GUI-driven session support. We created a custom viewer built
on top of Mol*.  This approach not only allowed the _MDsrv_ to directly
benefit from the above-mentioned features of Mol*, but also opened up the
possibility of integrating necessary changes directly into Mol*, in addition
to creating further specifications at the level of the MDsrv. We contributed
an importer for Zenodo (https://zenodo.org/) datasets to Mol* that greatly
simplifies access and visualization of trajectories from Zenodo and alleviates
the need for one’s own _MDsrv_ instance.  In addition, interactive trajectory
analysis features for distance, RMSD, angle, or dihedral measurements have
been added. Finally, we modularized the software for MDsrv in containers to
simplify the installation of MDsrv. With our update of the _MDsrv_ , we take
another step towards our goal of making MD simulations even more accessible to
a broader user base in line with the FAIR principles.

## Methods and Description

### Server

General architecture: The _MDsrv_ web service is designed for sharing and
interactively viewing of trajectories from MD simulations. It can be used, for
example, to facilitate collaboration between researchers working in
experimental and computational groups on different continents. To this end, we
provide _MDsrv_ , a platform (see Fig. 2) where MD trajectories can be
uploaded and their visualization configured. _MDsrv_ is divided into two
components. The  static server provides access to the _MDsrv_ website and
delivers all the functionality of _MDsrv_ to the client. In addition, the
streaming server delivers the MD trajectories frame by frame. The client’s
local web browser then processes this data locally in the  frontend provided
by the  static server. This separation allows _MDsrv_ to scale easily since
only the streaming server needs to be expanded in case of heavy usage. The MD
expert, who delivers the trajectory data can in the next step design so-called
sessions that are saved in the _MDsrv_ to make them publicly available. Such
sessions are preconfigured visualizations e.g. a zoom on a ligand-binding
pocket or a specific representation. For this purpose, the MD expert or any
user can either directly upload their local trajectories or load publicly
available data, such as those deposited at Zenodo directly into _MDsrv_ ,
since _MDsrv_ can obtain them via their record number. By accessing the
frontend of _MDsrv_ , a user can then open these sessions and get the
preconfigured visualization presented, where the view, zoom, and also other
settings can still be manipulated interactively.

Streaming server: With the increase in the size of MD simulation trajectories,
sharing and visualization becomes a challenge, especially when the trajectory
must be loaded over the World Wide Web and then read in entirely by the
visualization program. Meanwhile, the size of an MD trajectory can quickly
exceed the size of a standard desktop computer’s memory and in order to not
crash the application, the transferred data of the trajectory must be reduced.
We, therefore, introduced a streaming server (see Fig. 2) for storing and
processing before streaming the trajectory to the viewing client. Sharing MD
trajectories via a web application has now been further enhanced for _MDsrv_.
For this purpose, a session is stored on the streaming server after being
saved by the MD expert. This streaming server then sends only the necessary
simulation steps to the user on request, which makes the memory consumption on
the user side negligible. This allows, for example, to open a several gigabyte
large MD simulation on a mobile device.

Docker container virtualization: We have modularized the software for _MDsrv_
in containers, as this allows us to simplify the installation of the web
service, to offer the software in a more sustainable way, and additionally to
provide an easy local installation. In order to make the sessions reusable
according to the FAIR principles, _MDsrv_ should also be sustainable in terms
of installation and reusability. To archive a session permanently, you need
not only the data but also an instance of _MDsrv_ that was used to process the
data, since future updates might change the results, e.g. the representation
or the distance plots, depending on their default criteria. The software must
also be stored in a platform-independent manner to simplify future execution.
A promising approach here is the container virtualization by Docker. We have
wrapped the frontend and the streaming server of _MDsrv_(see Fig 2) in
containers, so that _MDsrv_ can be operated independently of the underlying
operating system. This makes it possible to deploy _MDsrv_ in the future on
systems that would no longer be able to do so due to software updates. This
principle is also successfully implemented by, for example, the Galaxy Browser
[19]. For general usage without the necessity to install _MDsrv_ locally, as
it was previously the case, we provide a publicly available instance of
_MDsrv_ where all sessions are publicly accessible
(https://proteininformatics.informatik.uni-leipzig.de/). To allow MD experts
to share private MD trajectories e.g. with a reviewer during review process,
_MDsrv_ can be easily installed on dedicated hardware by using the Docker
images of the frontend and the streaming server.

Web viewer: We updated the underlying viewer application in _MDsrv_ from NGL
to Mol*. A crucial feature of Mol* is its support for saving the application
state at any time to reload it later. This allows _MDsrv_ to offer the
creation of sessions without the need for programming knowledge.  MDsrv
extends this functionality so that the previously local sessions can now be
persistently stored using MDsrv in a publicly accessible manner along with the
associated data. To fulfill all the needs of _MDsrv_ we made a number of
contributions to Mol*. We added support for more coordinates and topology file
formats  that can be accessed from the client locally, namely, top, prmtop,
trr, and nctraj, in addition to the existing support for psf, dcd, and xtc.
Further, we added a Zenodo import extension that allows easy loading of
structures, trajectories, and zip archives from any Zenodo dataset. Finally,
we leveraged the modular GUI of Mol* to create a custom viewer for _MDsrv_
that adds GUI components for MD analyses including linked time-traces of
measurements and precomputed sequence alignments.

Limitations: The software architecture we have chosen has limitations in
addition to the aforementioned advantages. Although _MDsrv_ has only low
requirements on the client’s hardware, the streaming server must have
sufficient computing and storage capacity to process the trajectories.
Furthermore, for a public usage scenario with multiple users, the server
should be connected to the internet with sufficient network bandwidth. In
addition, the streaming server could be hosted in a content delivery network,
so that it is optimally accessible worldwide. For users to be able to stream
the trajectories, they need a stable internet connection.

Figure 2: The system overview of _MDsrv_. MD experts can configure a new
session via the frontend  provided by the static server of _MDsrv_ and store
the data on the Streaming Server of _MDsrv_. Afterward, other users can access
this session and stream the provided MD simulation. The streaming works
automatically without any additional user interaction.

###  MD analysis

The analysis of MD trajectories can range from simple visual inspection of
salient features, like (un)binding events, domain motion, interface formation
etc, to more data-driven approaches that uncover relationships and events that
might otherwise go unnoticed, like subtle bond-breaking, allosteric effects,
and transient effects. In order to integrate the visual with the data-driven
analysis, some molecular visualization tools (e.g., VMD [20] or
PyMOL111Schrödinger, LLC (November, 2015) The PyMOL Molecular Graphics System,
Version 1.8.) offer the on-the-fly computation of a variety of (geometric)
measurements on the displayed geometry. This way, time-traces of distances,
angles, dihedrals and RMSDs can be visualized in-program, without resorting to
a growing number specialized standalone libraries (MDanalysis [21, 22], MDtraj
[23], MDciao222Pérez Hernández, G. and Tiwari, S. gph82/mdciao:
https://zenodo.org/record/5643177. (November, 2021), BioPython [24]).

On par with the visualization programs mentioned above, _MDsrv_ can compute
time-traces of the geometric parameters. However, _MDsrv_ goes one step
further in user-friendliness, by making these time-traces interactive and
navigable per mouse-click. Key events that might go unnoticed in the visual
inspection, become sharp transitions in the navigable time-trace (see Fig. 3).
Just by clicking on these transitions (or anywhere in the time-trace), the
matching 3D frame will also appear in the main panel, making it easier to
correlate events in the time-trace with events in the 3D trajectory. When
combined with the sharing capabilities of _MDsrv_ , these navigable time-
traces make the analysis considerably more accessible to the un-trained user,
who might otherwise not distinguish these transitions from the continuous
oscillations inherent to MD. Additionally, the values of the plot can be
sorted or filtered to a given numerical range (e.g. when a distance between
two atoms is between 3 and 4 $\AA$), making geometries/frames of interest even
easier to locate.

Sequence Alignment is one the most widely used tools in bioinformatics for a
broad range of purposes [25, 26]. Alignments provide insight into both
similarities as well as differences between pairs or multiple sequences. The
full interpretation of an alignment, however, requires mapping the alignment
onto 3D structures. Only in a 3D context can the spatial relationships between
distant sequence positions become apparent. _MDsrv_ now offers as a novel
feature the upload of an alignment together with two or more structures. By
uploading multiple structures and an alignment, one can overlay or superimpose
the structures based on the alignment.

### Usage

Creating, preparing and sharing sessions can be done via our provided server,
or you can set up your own using Docker.

Sharing a prepared session using a URL: A new session can be created by
selecting Start new Session on our server
(https://proteininformatics.informatik.uni-leipzig.de/mdsrv), or by setting up
your own. Using the Home menu on the left, structures and trajectories can be
uploaded directly into the client from remote data repositories or from local
sources. Afterward, the session can be prepared as desired and shared via a
URL. The Remote Session feature in the lower panel is used for this purpose.
To further describe the session, it can be given a name , description and
source to document the origin of the displayed data and establish authorship
for the session. This information is then displayed when such a session is
used. After selecting the Upload button, the session will be stored on our
server. If the session is to be stored on another _MDsrv_ instance, the
address of the target server can be changed  in the webinterface. The uploaded
session will now appear in the list of sessions in the Remote Session list on
the left side in the home menu. Right-clicking on the session opens it in a
new tab with its own URL. This URL can then be shared. An already saved
session can also be further modified and uploaded again as a new session.

Upload and stream a trajectory: The Add Trajectory to Stream Server feature
enables the user to upload a trajectory to our streaming server instead of
directly viewing the simulations in our frontend which would normally be too
large to be held in the memory provided for the visualization client on the
users’ machine. Once the URL to the file, a name, a more detailed description,
and source is provided, the file will be saved on the server  and is made
available to users when viewing this data. The streaming server always
downloads the data to be streamed using a URL, since the data to be streamed
is usually not stored on client devices due to its size. Since the streaming
server can also be deployed in local environments, users can use their local
instance to host and stream non-public data in their access-restricted usage
scenario (e.g., within a lab). The trajectory will now be available to stream
in the Match Stream Trajectory feature. A corresponding structure that matches
the trajectory must be imported into the client to be able to view the
trajectory frame by frame. The trajectory can be played once it has been
matched to the structure using the Match Stream Trajectory feature.  Currently
it is only possible to stream xtc formats, but we plan to expand the supported
formats in the future.

Time-traces for trajectories: We extended the measurement function for
structures and trajectories by adding the possibility to display a time-trace
of the measurement over all frames for a trajectory. After importing a
trajectory and adding a measurement by selection, e.g. 2 atoms/residues for
the distance or 3 for an angle, its time-trace can be displayed in the Time-
trace plot feature. Different interactions with the plot are possible, e.g.
sorting by frame, ascending, or descending, filters for values, selection of a
specific value in the plot to skip to the corresponding frame, and displaying
the RMSD value for the whole structure where the comparison frame can be
defined.

Zenodo import: Files from a Zenodo record can be imported using the Zenodo
Import feature. The user only has to specify the Record number to get the
associated files displayed for import. The Type parameter allows to select
what kind of data should be imported: structure, trajectory, volume, or
compressed. All the files to the corresponding type will be displayed in a
drop-down menu below. If there are no supported files in the record, the user
will be notified.

Alignment: Based on the functionality of Mol* to superpose structures, _MDsrv_
extends this functionality by additionally importing externally generated
sequence alignments. The quality of a structural alignment depends crucially
on the quality of the underlying sequence alignment. Obtaining high quality
sequence alignments is far from trivial and often out of the scope of the
default methods also implemented in Mol*.  It is possible to superpose two or
more structures based on an imported alignment. After a ClustalW file and
additionally the structures from the alignment have been provided, the
sequences in the alignment have to be matched with the corresponding sequences
in the structures. The matching is done using the Match Sequence Alignment
feature. Once the alignment is complete, the superposition of the structures
is calculated based on the alignment. The alignment can be displayed in the
View Sequence Alignment feature.

More detailed descriptions including video materials on how to use the
features are available online (https://proteininformatics.informatik.uni-
leipzig.de/mdsrv/).

### Example Output

Figure 3 shows an example of the time-trace for a selected atomic distance,
which makes it possible to identify key events during the course of a
trajectory.

Figure 3: Composite of screenshots showing the navigable time-trace (bottom-
left) functionality. We have chosen replica one of reference [27], where a
membrane cholesterol molecule spontaneously enters the adenosine A2A receptor
(A2AR) from the lipid phase through transmembrane 5 and 6 into the binding
pocket. The time-trace shows the distance between the terminal Oxygen atom of
the cholesterol and the backbone Hydrogen atom of a TM Alanine (ALA57). With
this metric, several events can be appreciated and navigated to per mouse-
click, throughout one $\mathrm{-long}$ trajectory: the starting point (frame
4, first arrow, top-right), the entering event (sharp drop in distance-plot at
around frame 15, second arrow, top-right) and the forming of a Hydrogen bond
(small drop in distance plot down to 2.91 $\AA$, frame 88, third arrow,
bottom-right). The _MDsrv_ session for this figure can be found at
http://proteinformatics.uni-leipzig.de/mdsrv_ms/fig2.html.

## Conclusion

The _MDsrv_ web service is a simple and fast solution for sharing and
interactively visualizing MD simulations in the web browser. Users can inspect
MD simulations without having to install special software. The streaming
approach thus eliminates the need for sophisticated hardware for use on the
user side. In addition, _MDsrv_ offers various tools for basic analysis of MD
simulations that can be used directly in the web service.  With the simplified
installation using Docker, additional public or private instances can be
easily created.

The further development of visualization tools and the requirements for a tool
to easily exchange MD trajectories made further adaptations of thee _MDsrv_
necessary. Despite the widespread use of NGL by many web services, the switch
to Mol* enabled the immediate use of some features, including superior visual
quality, improved secondary structure assignment, trajectory animation video
export, and fully GUI-driven session support of this visualization tool, which
can be understood as an evolution of NGL and LiteMol Suite. As part of the
update, we have added new features to Mol* such as support for additional
coordinate and topology file formats and a Zenodo importer extension that
allows easy loading of structures, trajectories, and zip archives from any
Zenodo dataset. Other new features of _MDsrv_ include easy session creation
and interactive RMSD and distance angle analysis. Finally, we have modularized
the software for _MDsrv_ in containers to simplify installation and ensure
sustainability. This improvement makes the _MDsrv_ user experience even more
accessible and easier to use for a growing community.

## Data Availability

We made _MDsrv_ accessible as open source software at
https://github.com/dwiegreffe/mdsrv.
A release of the docker images is available at
https://hub.docker.com/r/dwiegreffe/mdsrv-viewer for the viewer and
https://hub.docker.com/r/dwiegreffe/mdsrv-remote for the remote server.
A hosted instance of the webservice can be accessed at
https://proteininformatics.informatik.uni-leipzig.de.
A companion website with examples, FAQ and manual can be found at:
https://proteininformatics.informatik.uni-leipzig.de/mdsrv/.

## ACKNOWLEDGEMENTS

We thank all the users of our service for providing us with valuable feedback
and specifically Adam Zech and Nikola Ristic for testing the latest update.

## Funding

This research was funded by the Development Bank of Saxony (SAB) under project
number 100335729 (to G.S.) and the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) through CRC 1423, project number 421152132,
subproject Z04 (to P.W.H.). JKST acknowledges funding by the Novo Nordisk
Foundation (NNF18OC0033950, to Kresten Lindorff-Larsen).

#### Conflict of interest statement.

None declared.

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