# Vectorial variational problems in $L^{\infty}$ constrained by the Navier-
Stokes equations

Ed Clark  E. C., Department of Mathematics and Statistics, University of
Reading, Whiteknights Campus, Pepper Lane, Reading RG6 6AX, United Kingdom
<EMAIL_ADDRESS>,  Nikos Katzourakis  N. K., Department of
Mathematics and Statistics, University of Reading, Whiteknights Campus, Pepper
Lane, Reading RG6 6AX, United Kingdom<EMAIL_ADDRESS>and  Boris
Muha  B. M., Department of Mathematics, Faculty of Science, University of
Zagreb, Zagreb, Trg Republike Hrvatske 14, 10000, Croatia<EMAIL_ADDRESS>
###### Abstract.

We study a minimisation problem in $L^{p}$ and $L^{\infty}$ for certain cost
functionals, where the class of admissible mappings is constrained by the
Navier-Stokes equations. Problems of this type are motivated by variational
data assimilation for atmospheric flows arising in weather forecasting. Herein
we establish the existence of PDE-constrained minimisers for all $p$, and also
that $L^{p}$ minimisers converge to $L^{\infty}$ minimisers as $p\to\infty$.
We further show that $L^{p}$ minimisers solve an Euler-Lagrange system.
Finally, all special $L^{\infty}$ minimisers constructed via approximation by
$L^{p}$ minimisers are shown to solve a divergence PDE system involving
measure coefficients, which is a divergence-form counterpart of the
corresponding non-divergence Aronsson-Euler system.

###### Key words and phrases:

Navier-Stokes equations; Calculus of Variations in $L^{\infty}$; PDE-
Constrained Optimisation; Euler-Lagrange equations; Absolute minimisers;
Aronsson-Euler systems, Generalised Lagrange Multipliers; Data Assimilation;
Weather forecasting

###### 2010 Mathematics Subject Classification:

35Q30, 35D35, 35A15, 49J40, 49K20, 49K35.

E.C. has been financially supported through the UK EPSRC scholarship GS19-055

B.M. has been partially financially supported through the Croatian Science
Foundation project IP-2019-04-1140

###### Contents

  1. 1 Introduction and main results
  2. 2 Preliminaries and the main estimates
  3. 3 Minimisers of $L^{p}$ problems and convergence as $p\to\infty$
  4. 4 The equations for $L^{p}$ PDE-constrained minimisers
  5. 5 The equations for $L^{\infty}$ PDE-constrained minimisers

##  1\. Introduction and main results

Let $\Omega\subseteq\mathbb{R}^{n}$ be an open bounded set and let also $n\geq
2$ and $\nu,T>0$. Consider the Navier-Stokes equations

(1.1) $\left\\{\ \begin{array}[]{lll}&\ \ \partial_{t}u\,-\nu\Delta
u\,+(u\cdot\mathrm{D})u\,+\mathrm{D}\mathrm{p}\,-f=\,y,&\ \ \text{ in
}\Omega\times(0,T),\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ &\ \
\mathrm{div}\,u\,=0,&\ \ \text{ in }\Omega\times(0,T),\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ &\ \ u(\cdot,0)=u_{0},&\ \ \text{ on }\Omega,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ &\ \ u=\,0,&\ \ \text{ on
}\partial\Omega\times(0,T),\end{array}\right.$

and for brevity let us henceforth symbolise $\nabla
u:=(\mathrm{D}u,\partial_{t}u)$ and $\Omega_{T}:=\Omega\times(0,T)$, where
$\mathrm{D}u=(\partial_{x_{1}}u,...,\partial_{x_{n}}u)\in\mathbb{R}^{n\times
n}$ symbolises the spatial gradient. The system of PDEs (1.1) describes the
velocity $u:\Omega_{T}\longrightarrow\mathbb{R}^{n}$ and the pressure
$\mathrm{p}:\Omega_{T}\longrightarrow\mathbb{R}$ of a flow, for some given
initial data $u_{0}:\Omega\longrightarrow\mathbb{R}^{n}$ with source
$f:\Omega_{T}\longrightarrow\mathbb{R}^{n}$. Here the map
$y:\Omega_{T}\longrightarrow\mathbb{R}^{n}$ is a parameter and should be
understood as a (deterministic) noise or error. Let also $N\in\mathbb{N}$ and
suppose we are given a mapping

(1.2) $\mathrm{Q}\ :\ \
\Omega_{T}\times\big{(}\mathbb{R}^{n}\times{\mathbb{R}}^{(n+1)\times
n}\times\mathbb{R}\big{)}\longrightarrow{\mathbb{R}}^{N}.$

A problem of interest in the geosciences, in particular in data assimilation
for atmospheric flows in relation to weather forecasting (see e.g. [16, 17,
18]), can be formulated as follows: find solutions $(u,\mathrm{p})$ to (1.1)
such that, in an appropriate sense,

(1.3) $\left\\{\ \ \begin{split}y\,\approx\,0,&\\\
\mathrm{Q}(\cdot,\cdot,u,\nabla u,\mathrm{p})-q\,\approx\,0,&\ \ \ \ \ \ \
\end{split}\right.$

where $q:\Omega_{T}\longrightarrow\mathbb{R}^{N}$ is a vector of given
measurable “data” arising from some specific measurements, taken through the
“observation operator” $\mathrm{Q}$ of (1.2). In (1.1) and (1.3), $y$
represents an error in the measurements which forces the Navier-Stokes
equations to be satisfied only approximately for solenoidal (divergence-free)
vector fields. Namely, we are looking for solutions to (1.1) such that
simultaneously the error $y$ vanishes, and also the measurements $q$ match the
prediction of the solution $(u,\mathrm{p})$ through the observation operator
(1.2). In application, $\mathrm{Q}$ is typically some component (e.g. linear
projection or nonlinear submersion) of the atmospheric flow that we can
observe. Unfortunately, the data fitting problem (1.3) is severely ill-posed;
an exact solution may well not exist, and even if it does, it may not be
unique.

In this paper, inspired by the methodology of data assimilation, especially
variational data assimilation in continuous time (for relevant works we refer
e.g. to [13, 19, 26, 29, 30, 44, 46, 52]), we seek to minimise the misfit
functional

$(u,\mathrm{p},y)\,\mapsto\,(1-\lambda)\big{\|}\mathrm{Q}(\cdot,\cdot,u,\nabla
u,\mathrm{p})-q\big{\|}\,+\,\lambda\big{\|}y\big{\|}$

over all admissible triplets $(u,\mathrm{p},y)$ which satisfy (1.1), for a
fixed weight $\lambda\in(0,1)$. The role of this weight is to obtain
essentially a Pareto family of extremals, one for each value $\lambda$, even
though in this paper we do not pursue further this viewpoint of vector-valued
minimisation (the interested reader may e.g. consult [21]). The standard
approach to data assimilation is to use Hilbert space methods (or least
squares in the discrete case), hence minimising in $L^{2}$. The novelty of our
approach, which is also justified from the viewpoint of applications, is to
consider instead _minimisation in $L^{\infty}$_, namely by interpreting the
norms above as $L^{\infty}$ ones (or maxima in the discrete case). There is a
significant advantage of considering a min-max problem instead of minimising
the squared averages: the misfit becomes uniformly small throughout the space-
time domain $\Omega_{T}$ and not just on average, hence large “spikes” of
deviations from zero are at the outset excluded.

When one passes from a variational problem for an integral norm to one for the
supremum norm, even though this is justified from the viewpoint of desired
outputs, it poses some serious challenges. The $L^{\infty}$ norm is neither
differentiable nor strictly convex, and the space $L^{\infty}$ is neither
reflexive nor separable. Additionally, with respect to the domain argument,
the $L^{\infty}$ norm is not additive but only sub-additive. Further, one
would also need estimates for (1.1) in appropriate subspaces of $L^{\infty}$
for weakly differentiable functions, which, to the best of our knowledge, do
not exist even for linear strongly elliptic systems (see e.g. [33]). Even
then, if one somehow solves the $L^{\infty}$ minimisation problem (by using,
for instance, the direct method of the Calculus of Variations as in [24],
under the appropriate quasiconvexity assumptions for $|\mathrm{Q}-q|+|y|$ as
in [12]), the analogue of the Euler-Lagrange equations for the $L^{\infty}$
problem cannot be derived directly by perturbation/sensitivity methods due to
the lack of smoothness of the $L^{\infty}$ norm.

In this paper, to overcome the difficulties described above, we follow the
methodology of the relatively new field of Calculus of Variations in
$L^{\infty}$ (see e.g. [22, 36] for a general introduction to the scalar-
valued theory), and in particular the ideas from [38, 39, 40, 41] involving
higher order and vectorial problems, as well as problems involving PDE-
constraints, which have only recently started being investigated. To this end,
we follow the approach of solving the desired $L^{\infty}$ variational problem
by solving respective approximating $L^{p}$ variational problems for all $p$,
and obtain appropriate compactness estimates which allow to pass to the limit
as $p\to\infty$. The case of finite $p>2$ studied herein is also of
independent interest, especially for numerical discretisation schemes in
$L^{\infty}$ (see e.g. [42, 43]), but in this paper we treat it mostly as an
approximation device to solve efficiently the $L^{\infty}$ problem. The idea
of this approach is based on the observation that, for a fixed essentially
bounded function on a domain of finite measure, the $L^{p}$ norm tends to the
$L^{\infty}$ norm of the function as $p\to\infty$.

In order to state our hypotheses and main results, let us set

(1.4)
$\mathrm{K}\big{(}x,t,\eta,A,a,r\big{)}\,:=\,\mathrm{Q}\big{(}x,t,\eta,A,a,r\big{)}\,-\,q(x,t)$

(note that in (1.4) $(x,t)\in\Omega_{T}$ is treated as two arguments and the
two arguments $(A,a)$ are for $\nabla u=(\mathrm{D}u,\partial_{t}u)$, which we
conveniently display abbreviated as one) and, by considering the (strong)
divergence operator
$\mathrm{div}:W^{1,1}(\Omega;\mathbb{R}^{n})\longrightarrow L^{1}(\Omega)$, we
henceforth assume that

(1.5) $\left\\{\ \ \begin{array}[]{ll}(a)&\Omega\text{ is bounded and has
$C^{2}$ boundary }\partial\Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
(b)&u_{0}\in\big{(}W^{2,{\infty}}\cap
W^{1,\infty}_{0}\big{)}({\Omega};{\mathbb{R}}^{n})\cap\mathrm{ker}(\mathrm{div}),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ (c)&f\in
L^{\infty}({\Omega}_{T};{\mathbb{R}}^{n})\ \ \ \&\ \ \ q\in
L^{\infty}({\Omega}_{T};{\mathbb{R}}^{N}),\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ (d)&\mathrm{K}(x,t,\cdot,\cdot,\cdot,\cdot)\text{ is $C^{1}$ for
almost every }(x,t),\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
(e)&\mathrm{K}(\cdot,\cdot,\eta,A,a,r)\text{ is $L^{\infty}$ for all
}(\eta,A,a,r),\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
(f)&|\mathrm{K}(x,t,\eta,A,\cdot,\cdot)|^{2}\text{ is convex for all
}(x,t,\eta,A).\end{array}\right.$

Then, for any $p\in(1,\infty)$, we define the $L^{p}$ misfit
$\mathrm{E}_{p}:\mathfrak{X}^{p}({\Omega}_{T})\longrightarrow\mathbb{R}$ by
setting

(1.6)
${\mathrm{E}}_{p}\big{(}u,\mathrm{p},y\big{)}\,:=\,(1-\lambda){\big{\|}\mathrm{K}(\cdot,u,\nabla
u,\mathrm{p})\big{\|}}_{\dot{L}^{p}({\Omega}_{T})}+\,\lambda\|y\|_{\dot{L}^{p}({\Omega}_{T})}.$

We note that in (1.6) and subsequently, the dotted $\dot{L}^{p}$ quantities
are regularisations of the respective norms at the origin, obtained by
regularising the Euclidean norm in the respective target space:

(1.7)
$\|h\|_{\dot{L}^{p}({\Omega}_{T})}:=\,\big{\|}|h|_{(p)}\big{\|}_{L^{p}({\Omega}_{T})},\
\ \ \ |\cdot|_{(p)}\,:=\,\sqrt{|\cdot|^{2}+p^{-2}}.$

Further, since we will only be dealing with finite measures, we will always be
using the normalised $L^{p}$ norms in which we replace the integral over the
domain with the respective average, for example for $L^{p}(\Omega_{T})$ with
the $(n+1)$-Lebesgue measure, the norm will be

$\|h\|_{L^{p}(\Omega_{T})}:=\left(\,-\displaystyle\int_{\Omega_{T}}|h|^{p}\,\mathrm{d}\mathcal{L}^{n+1}\\!\right)^{\\!\\!1/p}.$

The admissible minimisation class $\mathfrak{X}^{p}({\Omega}_{T})$ over which
$\mathrm{E}_{p}$ is considered, is defined as follows:

(1.8)
$\mathfrak{X}^{p}(\Omega_{T})\,:=\,\Big{\\{}(u,\mathrm{p},y)\in\mathcal{W}^{p}(\Omega_{T})\
:\ (u,\mathrm{p},y)\text{ satisfies weakly }\eqref{1.1}\Big{\\}},$

where

(1.9)
$\mathcal{W}^{p}(\Omega_{T})\,:=\,W^{2,1;p}_{\mathrm{L},\sigma}(\Omega_{T};\mathbb{R}^{n})\times
W^{1,0;p}_{\sharp}(\Omega_{T})\times L^{p}(\Omega_{T};\mathbb{R}^{n}).$

The rather complicated functional spaces appearing in (1.9) are defined as
follows. The space $W^{2,1;p}_{\mathrm{L},\sigma}(\Omega_{T};\mathbb{R}^{n})$
consists of solenoidal maps which are $W^{2,p}$ in space and $W^{1,p}$ in
time, and also laterally vanishing on $\partial\Omega\times(0,T)$:

(1.10)
$\left\\{\begin{split}W^{2,1;p}_{\mathrm{L},\sigma}(\Omega_{T};\mathbb{R}^{n})\,&:=\,L^{p}\big{(}(0,T);W^{2,p}_{0,\sigma}(\Omega;\mathbb{R}^{n})\big{)}\bigcap
W^{1,p}\big{(}(0,T);L^{p}(\Omega;\mathbb{R}^{n})\big{)},\\\
W^{2,p}_{0,\sigma}(\Omega;\mathbb{R}^{n})\,&:=\,\big{(}W^{2,p}\cap
W^{1,p}_{0}\big{)}(\Omega;\mathbb{R}^{n})\cap\mathrm{ker}(\mathrm{div}).\end{split}\right.\\!\\!$

The space $W^{1,0;p}_{\sharp}(\Omega_{T})$ consists of scalar-valued functions
which are $W^{1,p}$ in space with zero average, and $L^{p}$ in time:

(1.11) $\left\\{\ \
\begin{split}W^{1,0;p}_{\sharp}(\Omega_{T})\,&:=\,L^{p}\big{(}(0,T);W^{1,p}_{\sharp}(\Omega)\big{)},\\\
W^{1,p}_{\sharp}(\Omega)\,&:=\,\left\\{g\in W^{1,p}(\Omega)\ :\
\int_{\Omega}g\,\mathrm{d}\mathcal{L}^{n}=0\right\\}.\end{split}\right.$

The associated norms in these spaces are the expected ones, namely

(1.12) $\left\\{\ \
\begin{split}\|v\|_{W^{2,1;p}_{\mathrm{L},\sigma}(\Omega_{T})}\,&:=\,\|v\|_{L^{p}(\Omega_{T})}\,+\,\|\nabla
v\|_{L^{p}(\Omega_{T})}\,+\,\|\mathrm{D}^{2}v\|_{L^{p}(\Omega_{T})},\\\
\|g\|_{W^{1,0;p}_{\sharp}(\Omega_{T})}\,&:=\,\|g\|_{L^{p}(\Omega_{T})}\,+\,\|\mathrm{D}g\|_{L^{p}(\Omega_{T})}.\end{split}\right.$

Note also that the divergence-free condition for $u$ in (1.1) has now been
incorporated in the functional space
$W^{2,1;p}_{\mathrm{L},\sigma}(\Omega_{T})$. Finally, the $L^{\infty}$ misfit
$\mathrm{E}_{\infty}:\mathfrak{X}^{\infty}({\Omega}_{T})\longrightarrow\mathbb{R}$
is defined by setting

(1.13)
$\mathrm{E}_{\infty}\big{(}u,\mathrm{p},y\big{)}\,:=\,(1-\lambda){\big{\|}\mathrm{K}(\cdot,\cdot,u,\nabla
u,\mathrm{p})\big{\|}}_{L^{\infty}({\Omega}_{T})}+\,\lambda\|y\|_{L^{\infty}({\Omega}_{T})},$

where the admissible class $\mathfrak{X}^{\infty}(\Omega_{T})$ is given by

(1.14)
$\mathfrak{X}^{\infty}(\Omega_{T})\,:=\,\bigcap_{1<p<\infty}\mathfrak{X}^{p}(\Omega_{T}).$

Note that the natural topology of $\mathfrak{X}^{\infty}(\Omega_{T})$ is not
induced by a complete norm in a Banach space, but instead its topology is
defined as the locally convex topology induced from the ambient Frechét space
$\smash{\bigcap_{1<p<\infty}\mathcal{W}^{p}(\Omega_{T})}$. Notwithstanding, no
difficulties will arise from this fact, which is a necessity that stems from
the lack of $W^{2,\infty}$-$W^{1,\infty}$ estimates for (1.1). In particular,
$\mathfrak{X}^{\infty}(\Omega_{T})$ is _not_ obtained by considering the
strictly smaller Cartesian product space

$\mathcal{W}^{\infty}(\Omega_{T})\,=\,W^{2,1;\infty}_{\mathrm{L},\sigma}(\Omega_{T})\times\smash{W^{1,0;\infty}_{\sharp}(\Omega_{T})}\times
L^{\infty}(\Omega_{T};\mathbb{R}^{n}).$

However, we will assume that the solution $(u,\mathrm{p})$ to (1.1) is strong
and satisfies $W^{2,p}$-$W^{1,p}$ estimates for any finite $p$. This is
deducible under assumption (1.5) in the case of $n=2$ (see e.g. [32, 53]), and
also under smallness conditions on $u_{0}$ in any dimension $n\geq 3$ (see
e.g. [2, 34, 54]). Hence, our additional hypothesis is

(1.15) $\left\\{\ \ \begin{split}&\text{Either}\\\ &\hskip
130.0ptn=2,\phantom{{}_{\big{]}}}\\\ &\text{or, $n\geq 3$ but for any
$p\in(1,\infty)$, exists $C>0$ depending only on $p$ and}\\\ &\text{on
$\partial\Omega$, $T$, $\|u_{0}\|_{L^{2}(\Omega)},\|f\|_{L^{2}(\Omega_{T})}$,
such that}\\\ &\ \
\|u\|_{W^{2,1;p}_{\mathrm{L},\sigma}(\Omega_{T})}+\,\|\mathrm{p}\|_{W^{1,0;p}_{\sharp}(\Omega_{T})}\,\leq\,C\big{(}\|u_{0}\|_{W^{2-\frac{2}{p},p}(\Omega)}+\,\|f\|_{L^{p}(\Omega_{T})}\big{)},\phantom{\bigg{]}}\\\
&\text{when $(u,\mathrm{p})$ solves weakly \eqref{1.1} with $y\equiv
0$.}\end{split}\right.\\!\\!\\!$

Assumption (1.15), albeit restrictive, is compatible with situations of
interest in weather forecasting (see e.g. [16, 17, 18]). Our first main result
concerns the existence of $\mathrm{E}_{p}$-minimisers in
$\mathfrak{X}^{p}(\Omega_{T})$, the existence of
$\mathrm{E}_{\infty}$-minimisers in $\mathfrak{X}^{\infty}(\Omega_{T})$ and
the approximability of the latter by the former as $p\to\infty$.

######  Theorem 1 ($\mathrm{E}_{\infty}$-minimisers,
$\mathrm{E}_{p}$-minimisers & convergence as $p\to\infty$).

Suppose that (1.5) and (1.15) hold true. Then, for any $p\in(n+2,\infty]$, the
functional $\mathrm{E}_{p}$ (given by (1.6) for $p<\infty$ and by (1.13) for
$p=\infty$) has a constrained minimiser $(u_{p},\mathrm{p}_{p},y_{p})$ in the
admissible class $\mathfrak{X}^{p}(\Omega_{T})$:

(1.16)
$\mathrm{E}_{p}\big{(}u_{p},\mathrm{p}_{p},y_{p}\big{)}\,=\,\inf\Big{\\{}\mathrm{E}_{p}\big{(}u,\mathrm{p},y\big{)}\,:\
\big{(}u,\mathrm{p},y\big{)}\in\mathfrak{X}^{p}(\Omega_{T})\Big{\\}}.$

Additionally, there exists a subsequence of indices $(p_{j})_{1}^{\infty}$
such that the sequence of respective $\mathrm{E}_{p_{j}}$-minimisers
$(u_{p_{j}},\mathrm{p}_{p_{j}},y_{p_{j}})$ satisfies
$(u_{p},\mathrm{p}_{p},y_{p})\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup(u_{\infty},\mathrm{p}_{\infty},y_{\infty})$
in $\mathcal{W}^{q}(\Omega_{T})$ for any $q\in(1,\infty)$, as
$p_{j}\to\infty$. Additionally,

(1.17) $\left\\{\ \
\begin{array}[]{ll}u_{p}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup
u_{\infty},&\ \ \text{in
}W^{2,1;q}_{\mathrm{L},\sigma}(\Omega_{T};\mathbb{R}^{n}),\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ u_{p}\longrightarrow u_{\infty},&\ \ \text{in
}C\big{(}\overline{\Omega_{T}};\mathbb{R}^{n}\big{)},\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\ \mathrm{D}u_{p}\longrightarrow\mathrm{D}u_{\infty},&\ \
\text{in }C\big{(}\overline{\Omega_{T}};\mathbb{R}^{n\times n}\big{)},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\mathrm{p}_{p}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup\mathrm{p}_{\infty},&\
\ \text{in }W^{1,0;q}_{\\#}(\Omega_{T};\mathbb{R}^{n}),\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\ y_{p}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup
y_{\infty},&\ \ \text{in }L^{q}(\Omega_{T}),\end{array}\right.$

for any $q\in(1,\infty)$, and also

(1.18)
$\mathrm{E}_{p}(u_{p},\mathrm{p}_{p},y_{p})\longrightarrow\mathrm{E}_{\infty}(u_{\infty},\mathrm{p}_{\infty},y_{\infty})$

as $p_{j}\to\infty$.

Given the existence of constrained minimisers established by Theorem 1 above,
the next natural question concerns the existence of necessary conditions in
the form of PDEs governing the constrained minimisers. We first consider the
case of $p<\infty$. Unsurprisingly, the PDE constraint of (1.1) used in
defining (1.8) gives rise to a generalised Lagrange multiplier in the Euler-
Lagrange equations, obtained by utilising well-known results on the Kuhn-
Tucker theory from [57]. Interestingly, however, the incorporation of the
solenoidality constraint into the functional space (recall (1.10)), allows us
to have only one generalised multiplier corresponding only to the parabolic
system in (1.1), instead of two.

To state our second main result, we first need to introduce some notation. For
any $M\in\mathbb{N}$ and $p\in(1,\infty)$, we define the operator

$\mathfrak{M}_{p}\ :\ \ L^{p}(\Omega_{T};\mathbb{R}^{M})\longrightarrow
L^{p^{\prime}}(\Omega_{T};\mathbb{R}^{M}),$

where $p^{\prime}:=p/(p-1)$, by setting

(1.19)
$\mathfrak{M}_{p}(V)\,:=\,\frac{|V|_{(p)}^{p-2}\,V}{\big{(}\|V\|_{\dot{L}^{p}(\Omega_{T})}\big{)}^{p-1}}.$

Here $|\cdot|_{(p)}$ is the regularisation of the Euclidean norm of
$\mathbb{R}^{M}$, as defined in (1.7). By Hölder’s inequality it is immediate
to verify that (for the normalised $\smash{L^{p^{\prime}}}$ norm) we actually
have

$\big{\|}\mathfrak{M}_{p}(V)\big{\|}_{L^{p^{\prime}}(\Omega_{T})}\,\leq\,1,$

and therefore $\mathfrak{M}_{p}$ is valued in the unit ball of
$\smash{L^{p^{\prime}}(\Omega_{T};\mathbb{R}^{M})}$. Further, for brevity we
will use the notation

(1.20) $\left\\{\ \
\begin{split}\mathrm{K}[u,\mathrm{p}]\,:=&\,\mathrm{K}\big{(}\cdot,\cdot,u,\nabla
u,\mathrm{p}\big{)},\\\
\mathrm{K}_{\eta}[u,\mathrm{p}]\,:=&\,\mathrm{K}_{\eta}\big{(}\cdot,\cdot,u,\nabla
u,\mathrm{p}\big{)},\\\
\mathrm{K}_{(A,a)}[u,\mathrm{p}]\,:=&\,\mathrm{K}_{(A,a)}\big{(}\cdot,\cdot,u,\nabla
u,\mathrm{p}\big{)},\\\
\mathrm{K}_{r}[u,\mathrm{p}]\,:=&\,\mathrm{K}_{r}\big{(}\cdot,\cdot,u,\nabla
u,\mathrm{p}\big{)},\end{split}\right.$

for $\mathrm{K}$ and its partial derivatives
$\mathrm{K}_{\eta},\mathrm{K}_{(A,a)},\mathrm{K}_{r}$ with respect to the
arguments for $u,\nabla u$ and $\mathrm{p}$ respectively.

######  Theorem 2 (Variational Equations in $L^{p}$).

Suppose that (1.5) and (1.15) hold true. Then, for any $p\in(n+2,\infty)$,
there exists a Lagrange multiplier

(1.21)
$\Psi_{p}\,\in\,\Big{(}W^{2-\frac{2}{p},p}_{0,\sigma}(\Omega;\mathbb{R}^{n})\Big{)}^{*}$

associated with the constrained minimisation problem (1.16), such that the
minimising triplet
$(u_{p},\mathrm{p}_{p},y_{p})\in\mathfrak{X}^{p}(\Omega_{T})$ satisfies the
relations

(1.22) $\left\\{\
\begin{split}&(1-\lambda)\int_{\Omega_{T}}\Big{(}\mathrm{K}_{\eta}[u_{p},\mathrm{p}_{p}]\cdot
u\,+\,\mathrm{K}_{(A,a)}[u_{p},\mathrm{p}_{p}]:\nabla
u\Big{)}\cdot\mathfrak{M}_{p}\big{(}\mathrm{K}[u_{p},\mathrm{p}_{p}]\big{)}\,\mathrm{d}\mathcal{L}^{n+1}\\\
&=-\lambda\int_{\Omega_{T}}\Big{(}\partial_{t}u-\nu\Delta
u+(u\cdot\mathrm{D})u_{p}+(u_{p}\cdot\mathrm{D})u\Big{)}\cdot\mathfrak{M}_{p}(y_{p})\,\mathrm{d}{\mathcal{L}}^{n+1}\,+\,\big{\langle}\Psi_{p},u(\cdot,0)\big{\rangle},\end{split}\right.$
(1.23)
$\begin{split}&(1-\lambda)\int_{\Omega_{T}}\mathrm{K}_{r}[u_{p},\mathrm{p}_{p}]\,\mathrm{p}\cdot\mathfrak{M}_{p}\big{(}\mathrm{K}[u_{p},\mathrm{p}_{p}]\big{)}\,\mathrm{d}\mathcal{L}^{n+1}\,=-\lambda\int_{\Omega_{T}}\mathrm{D}\mathrm{p}\cdot\mathfrak{M}_{p}(y_{p})\,\mathrm{d}\mathcal{L}^{n+1}\\\
\end{split}$

for all test mappings

$(u,\mathrm{p})\,\in\,W^{2,1;p}_{\mathrm{L},\sigma}(\Omega_{T};\mathbb{R}^{n})\times
W^{1,0;p}_{\sharp}(\Omega_{T}),$

where the operators
$\mathrm{K},\mathrm{K}_{\eta},\mathrm{K}_{(A,a)},\mathrm{K}_{r}$ are given by
(1.20).

Now we consider the case of $p=\infty$. For this extreme case, which is
obtained by an appropriate passage to limits as $p\to\infty$ in Theorem 2, we
need to assume additionally that the operator $\mathrm{K}[u,\mathrm{p}]$ does
not depend on $(\partial_{t}u,\mathrm{p})$, hence in this case we will
symbolise

(1.24) $\left\\{\ \
\begin{split}\mathrm{K}[u]\,:=&\,\mathrm{K}\big{(}\cdot,\cdot,u,\mathrm{D}u\big{)},\\\
\mathrm{K}_{\eta}[u]\,:=&\,\mathrm{K}_{\eta}\big{(}\cdot,\cdot,u,\mathrm{D}u\big{)},\\\
\mathrm{K}_{A}[u]\,:=&\,\mathrm{K}_{A}\big{(}\cdot,\cdot,u,\mathrm{D}u\big{)},\end{split}\right.$

for $\mathrm{K}$ and its partial derivatives
$\mathrm{K}_{\eta},\mathrm{K}_{A}$ with respect to the arguments for
$u,\mathrm{D}u$ respectively, all of which will also need to be assumed to be
continuous. We note that, when $p=\infty$, there is no direct analogue of the
divergence structure Euler Lagrange equations. Instead, one of the central
points of Calculus of Variations in $L^{\infty}$ is that Aronsson-Euler PDE
systems may be derived, under appropriate (stringent) assumptions. Even in the
unconstrained case, these PDE systems are always non-divergence and even fully
nonlinear and with discontinuous coefficients (see e.g. [7, 8, 23, 37, 43]).
The case of $L^{\infty}$ problems involving only first order derivative of
scalar-valued functions is nowadays a well established field which originated
from the work of Aronsson in the 1960 [4, 5], today largely interconnected to
the theory of Viscosity Solution to nonlinear elliptic PDE (for a general
pedagogical introduction see e.g. [22, 36]). However, vectorial and higher
$L^{\infty}$ variational problems involving constraints, have only recently
began being explored (see [39, 40], but also the relevant earlier
contributions [3, 6, 10]). For several interesting developments on
$L^{\infty}$ variational problems we refer the interested reader to [9, 11,
14, 15, 20, 26, 31, 45, 49, 50, 51].

In this paper, motivated by recent progress on higher order and on constrained
$L^{\infty}$ variational problems made in [41] by the author jointly with
Moser and by the author in [39, 40] (inspired by earlier contributions by
Moser and Schwetlick deployed in a geometric setting in [48]), we follow a
slightly different approach which does not lead an Aronsson-Euler type system;
instead, it leads to a _divergence structure_ PDE system. However, there is a
toll to be paid, as the divergence PDEs arising as necessary conditions
involve measures as auxiliary parameters whose determination becomes part of
the problem. Notwithstanding, the central point of this idea is to use a
scaling in the Euler-Lagrange equations before letting $p\to\infty$, which is
different from the scaling used to (formally) derive the Aronsson-Euler
equations as $p\to\infty$.

In the light of the above comments, our final main result concerns the
satisfaction of necessary PDE conditions for the PDE-constrained minimisers in
$L^{\infty}$ constructed in Theorem 1, and reads as follows.

######  Theorem 3 (Variational Equations in $L^{\infty}$).

Suppose that (1.5) and (1.15) hold true, and that additionally $\mathrm{K}$
does not depend on $(\partial_{t}u,\mathrm{p})$ with
$\mathrm{K},\mathrm{K}_{\eta},\mathrm{K}_{A}$ in (1.24) being continuous on
$\overline{\Omega_{T}}\times\mathbb{R}^{n}\times\mathbb{R}^{n\times n}$. Then,
there exists a linear functional

(1.25)
$\Psi_{\infty}\in\bigcap_{r>n+2}\Big{(}W^{2-\frac{2}{r},r}_{0,\sigma}(\Omega;\mathbb{R}^{n})\Big{)}^{*}$

which is a Lagrange multiplier associated with the constrained minimisation
problem (1.16) for $p=\infty$. There also exist vector measures

(1.26)
$\Sigma_{\infty}\in\mathcal{M}\big{(}\overline{\Omega_{T}};\mathbb{R}^{N}\big{)},\
\ \
\sigma_{\infty}\in\mathcal{M}\big{(}\overline{\Omega_{T}};\mathbb{R}^{n}\big{)}$

such that the minimising triplet
$(u_{\infty},\mathrm{p}_{\infty},y_{\infty})\in\mathfrak{X}^{\infty}(\Omega_{T})$
satisfies the relations

(1.27) $\left\\{\
\begin{split}&(1-\lambda)\int_{\overline{\Omega_{T}}}\Big{(}\mathrm{K}_{\eta}[u_{\infty}]\cdot
u\,+\,\mathrm{K}_{A}[u_{\infty}]:\mathrm{D}u\Big{)}\cdot\mathrm{d}\Sigma_{\infty}\\\
&=-\lambda\int_{\overline{\Omega_{T}}}\Big{(}\partial_{t}u-\nu\Delta
u+(u\cdot\mathrm{D})u_{\infty}+(u_{\infty}\cdot\mathrm{D})u\Big{)}\cdot\mathrm{d}\sigma_{\infty}\,+\,\big{\langle}\Psi_{\infty},u(\cdot,0)\big{\rangle},\end{split}\right.$
(1.28)
$\int_{\overline{\Omega_{T}}}\,\mathrm{D}\mathrm{p}\cdot\mathrm{d}\sigma_{\infty}\,=\,0,$

for all test mappings

$(u,\mathrm{p})\,\in\,\Big{(}W^{2,1;\infty}_{\mathrm{L},\sigma}(\Omega_{T};\mathbb{R}^{n})\cap
C^{2}\big{(}\overline{\Omega_{T}};\mathbb{R}^{n}\big{)}\Big{)}\times\Big{(}W^{1,0;\infty}_{\sharp}(\Omega_{T})\cap
C^{1}\big{(}\overline{\Omega_{T}}\big{)}\Big{)}.$

Further, the multiplier $\Psi_{\infty}$ and the measures
$\Sigma_{\infty},\sigma_{\infty}$ can be approximated as follows:

(1.29) $\left\\{\ \
\begin{split}&\Psi_{p}\,\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\,\Psi_{\infty},\
\,\text{ in
}\smash{\big{(}W^{2-{2}/{r},r}_{0,\sigma}(\Omega;\mathbb{R}^{n})\big{)}^{*}},\text{
for all }r>n+2,\\\
&\Sigma_{p}\,\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\,\Sigma_{\infty},\
\ \text{ in }\mathcal{M}\big{(}\overline{\Omega_{T}};\mathbb{R}^{N}\big{)},\\\
&\sigma_{p}\,\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\,\sigma_{\infty},\
\ \ \text{ in
}\mathcal{M}\big{(}\overline{\Omega_{T}};\mathbb{R}^{n}\big{)},\end{split}\right.$

along a subsequence $p_{j}\to\infty$, where

(1.30) $\left\\{\ \
\begin{split}\Sigma_{p}\,&:=\,\mathfrak{M}_{p}\big{(}\mathrm{K}[u_{p}]\big{)}\mathcal{L}^{n+1}\text{\LARGE$\llcorner$}_{\Omega_{T}},\\\
\sigma_{p}\,&:=\,\mathfrak{M}_{p}(y_{p})\mathcal{L}^{n+1}\text{\LARGE$\llcorner$}_{\Omega_{T}}.\end{split}\right.$

Finally, $\Sigma_{\infty}$ concentrates on the set whereon
$|\mathrm{K}[u_{\infty}]|$ is maximised over $\overline{\Omega_{T}}$

(1.31)
$\Sigma_{\infty}\Big{(}\big{\\{}\big{|}\mathrm{K}[u_{\infty}]\big{|}<\big{\|}\mathrm{K}[u_{\infty}]\big{\|}_{L^{\infty}(\Omega_{T})}\big{\\}}\Big{)}=\,0,$

and $\sigma_{\infty}$ asymptotically concentrates on the set whereon
$|y_{\infty}|$ is approximately maximised over $\overline{\Omega_{T}}$, in the
sense that for any $\varepsilon>0$ small,

(1.32)
$\lim_{p\to\infty}\,\sigma_{p}\Big{(}\Big{\\{}|y_{p}|<\|y_{\infty}\|_{L^{\infty}(\Omega_{T})}-\varepsilon\Big{\\}}\Big{)}=\,0.$

Even though the weak interpretation of the equations (1.22)-(1.23) is
relatively obvious, this is not the case for (1.27)-(1.28) despite having a
simpler form. The reason is that the limiting measures
$(\Sigma_{\infty},\sigma_{\infty})$ are not product measures on
$\overline{\Omega_{T}}=\overline{\Omega}\times[0,T]$ in order to use the
Fubini theorem, therefore due to the temporal dependence, (1.28) cannot be
simply interpreted as “$\mathrm{div}(\sigma_{\infty})=0$”. Similar arguments
can be made for (1.27) as well. Since this point is not utilised any further
in this paper, we only provide a brief discussion in the next section.

We conclude this introduction with some remarks regarding the organisation of
this paper. This introduction is followed by Section 2, in which we discuss
some preliminaries and also establish some basic estimates which are utilised
subsequently to establish our main results. In Section 3 we prove Theorem 1 by
establishing the existence of constrained minimisers for all $p$ including
$p=\infty$, as well as the convergence of minimiser of the former problems to
those of the latter. In Section 4 we prove Theorem 2, deriving the necessary
PDE conditions which constrained minimisers in $L^{p}$ satisfy. Finally, in
Section 5 prove Theorem 3, deriving the necessary PDE conditions that
constrained minimisers in $L^{\infty}$ satisfy, as well as the additional
properties that the measures arising in these PDEs satisfy. A key ingredient
here is that we establish appropriate weak* compactness for the Lagrange
multipliers arising in the $L^{p}$ problems in order to pass to the limit as
$p\to\infty$.

##  2\. Preliminaries and the main estimates

We begin by recording for later use the following modified Hölder inequality
for the dotted $\dot{L}^{p}$ regularised “norms” defined in (1.7): for any
$1\leq q\leq p<\infty$ and $h\in L^{p}(\Omega_{T})$, we have the inequality

$\|h\|_{\dot{L}^{q}(\Omega_{T})}\,\leq\,\|h\|_{\dot{L}^{p}(\Omega_{T})}\,+\,\sqrt{q^{-2}-p^{-2}},$

which can be very easily confirmed by a direct computation. Next, we continue
with a brief discussion regarding the weak interpretation of the equations
(1.27)-(1.28). As already noted in the introduction, since
$(\Sigma_{\infty},\sigma_{\infty})$ are not in general neither product
measures nor absolutely continuous with respect to the $(n+1)$-Lebesgue
measure on $\overline{\Omega_{T}}=\overline{\Omega}\times[0,T]$, one needs to
use the disintegration “slicing” theorem for Young measures in order to
express them appropriately, as follows. Since $\sigma_{\infty}$ is a vector
measure in $\mathcal{M}(\overline{\Omega_{T}};\mathbb{R}^{n})$, by the Radon-
Nikodym theorem, we may decompose

$\sigma_{\infty}\,=\,\frac{\mathrm{d}\sigma_{\infty}}{\mathrm{d}\|\sigma_{\infty}\|}\,\|\sigma_{\infty}\|,$

where $\|\sigma_{\infty}\|\in\mathcal{M}(\overline{\Omega_{T}})$ is the scalar
total variation measure and
${\mathrm{d}\sigma_{\infty}}/{\mathrm{d}\|\sigma_{\infty}\|}$ is the vector-
valued Radon-Nikodym derivative of $\sigma_{\infty}$ with respect to
$\|\sigma_{\infty}\|$. Fix now any $h\in
L^{1}(\overline{\Omega_{T}},\|\sigma_{\infty}\|)$. By the disintegration
“slicing” theorem for Young measure (see se.g. [27, Theorem 3.2, p. 179]), we
have the representation formula

$\int_{\overline{\Omega_{T}}}h\,\mathrm{d}\|\sigma_{\infty}\|\,=\,\int_{[0,T]}\\!\bigg{(}\int_{\overline{\Omega}}h(x,t)\,\mathrm{d}\|\sigma_{\infty}\|_{t}(x)\bigg{)}\mathrm{d}\|\sigma_{\infty}\|^{o}(t)$

where the measure $\|\sigma_{\infty}\|^{o}\in\mathcal{M}([0,T])$ and the
family of measures
$(\|\sigma_{\infty}\|_{t})_{t\in[0,T]}\subseteq\mathcal{M}(\overline{\Omega})$
are defined as follows:

$\|\sigma_{\infty}\|^{o}\,:=\,\|\sigma_{\infty}\|\big{(}\overline{\Omega}\times\cdot\,\big{)},\
\ \ \ \
\|\sigma_{\infty}\|_{t}(A)\,:=\,\frac{\mathrm{d}\|\sigma_{\infty}\|\big{(}A\times\cdot\,\big{)}}{\mathrm{d}\|\sigma_{\infty}\|\big{(}\overline{\Omega}\times\cdot\,\big{)}}(t),\text{
for }A\subseteq\overline{\Omega}\text{ Borel}.$

Namely, $\|\sigma_{\infty}\|^{o}$ is one of the marginals of $\sigma_{\infty}$
and for $\|\sigma_{\infty}\|^{o}$-a.e. $t\in[0,T]$, the measure
$\|\sigma_{\infty}\|_{t}$ evaluated at $A$ is defined as the Radon-Nikodym
derivative of the measure $\|\sigma_{\infty}\|\big{(}A\times\cdot\,\big{)}$
with respect to
$\|\sigma_{\infty}\|\big{(}\overline{\Omega}\times\cdot\,\big{)}$ at the point
$t\in[0,T]$. Then, in view of (1.28), by choosing $\mathrm{p}$ in the form
$\mathrm{p}(x,t)=\pi(x)\tau(t)$, we have

$\begin{split}0\,&=\,\int_{\overline{\Omega_{T}}}\mathrm{D}\mathrm{p}\cdot\mathrm{d}\sigma_{\infty}\\\
&=\,\int_{\overline{\Omega_{T}}}\bigg{(}\mathrm{D}\mathrm{p}\cdot\frac{\mathrm{d}\sigma_{\infty}}{\mathrm{d}\|\sigma_{\infty}\|}\bigg{)}\,\mathrm{d}\|\sigma_{\infty}\|\\\
&=\,\int_{[0,T]}\\!\bigg{(}\int_{\overline{\Omega}}\bigg{(}\mathrm{D}\mathrm{p}\cdot\frac{\mathrm{d}\sigma_{\infty}}{\mathrm{d}\|\sigma_{\infty}\|}\bigg{)}(x,t)\,\mathrm{d}\|\sigma_{\infty}\|_{t}(x)\bigg{)}\mathrm{d}\|\sigma_{\infty}\|^{o}(t)\\\
&=\,\int_{[0,T]}\\!\bigg{(}\int_{\overline{\Omega}}\bigg{(}\mathrm{D}\pi(x)\cdot\frac{\mathrm{d}\sigma_{\infty}}{\mathrm{d}\|\sigma_{\infty}\|}(x,t)\bigg{)}\,\mathrm{d}\|\sigma_{\infty}\|_{t}(x)\bigg{)}\tau(t)\,\mathrm{d}\|\sigma_{\infty}\|^{o}(t).\end{split}$

The arbitrariness of $\tau$ implies that for $\|\sigma_{\infty}\|^{o}$-a.e.
$t\in[0,T]$, we have

$\int_{\overline{\Omega}}\bigg{(}\mathrm{D}\pi(x)\cdot\frac{\mathrm{d}\sigma_{\infty}}{\mathrm{d}\|\sigma_{\infty}\|}(x,t)\bigg{)}\,\mathrm{d}\|\sigma_{\infty}\|_{t}(x)\,=\,0.$

When restricting our attention to those test function for which
$\pi|_{\partial\Omega}\equiv 0$, we obtain the next weak interpretation of
(1.28):

$\mathrm{div}\bigg{(}\frac{\mathrm{d}\sigma_{\infty}}{\mathrm{d}\|\sigma_{\infty}\|}(\cdot,t)\,\|\sigma_{\infty}\|_{t}\bigg{)}\,=\,0,\
\ \ \text{ in }\Omega,$

for $\|\sigma_{\infty}\|^{o}$-a.e. $t\in[0,T]$. Similar considerations apply
also to equation (1.27), but the arguments are considerably more complicated.

Next we prove a general compact embedding lemma by means of interpolation
theory.

######  Lemma 4.

Suppose that $p>n+2$. Then, there exists $\alpha\in(0,1)$ such that the space

$W^{2,1;p}(\Omega_{T})\,:=\,L^{p}\big{(}(0,T);W^{2,p}(\Omega)\big{)}\bigcap
W^{1,p}\big{(}(0,T);L^{p}(\Omega)\big{)}$

is compactly embedded in the space
$C^{0,\alpha}\big{(}[0,T];C^{1,\alpha}(\overline{\Omega})\big{)}$.

Proof of Lemma 4. Let us use the abbreviated space notation

$\mathrm{X}_{1}\,:=\,W^{2,p}(\Omega),\ \ \mathrm{X}_{0}\,:=\,L^{p}(\Omega)$

and select $\theta$ such that

$\frac{p+n}{2p}\,<\,\theta\,<\,\frac{p-1}{p},$

which is possible since

$\frac{p-1}{p}-\frac{p+n}{2p}\,=\,\frac{p-(n+2)}{2p}\,>\,0.$

Since $1-\theta>1/p$, direct application of the interpolation result in [1,
Theorem 5.2] for the exponents $s_{0}:=1$, $s_{1}:=0$ and $p_{0}\equiv
p_{1}:=p$ yields that space $W^{2,1;p}(\Omega_{T})$ is compactly embedded in
the space $C^{0,\alpha}\big{(}[0,T];\mathrm{X}\big{)}$, where
$0<\alpha<1-\theta-1/p$ and
$\mathrm{X}=(\mathrm{X}_{0},\mathrm{X}_{1})_{\theta,p}$ symbolises the real
interpolation between the Banach spaces $\mathrm{X}_{0}$ and $\mathrm{X}_{1}$.
Now it remains to identify the space $\mathrm{X}$. By using standard results
in interpolation theory (see e.g. [55, Theorem 4.3.1.1 and formula (2.4.2/9)]
or [56] for Lipschitz domains) we get:

$\big{(}L^{p}(\Omega),W^{2,p}(\Omega)\big{)}_{\theta,p}\,=\,\mathrm{B}^{2\theta}_{pp}(\Omega)\,=\,W^{2\theta,p}(\Omega).$

Since $2\theta>1+n/p$, by the standard Sobolev embedding theorem for
fractional spaces (e.g. [25, Theorem 8.2], we have that
$W^{2\theta,p}(\Omega)$ is continuously embedded in the space
$C^{1,\alpha}(\overline{\Omega})$, where $0<\alpha\leq 2\theta-1-n/p$. The
conclusion ensues. ∎

######  Remark 5.

Let us now record for later use the following simple inclusion of space (which
is in fact a continuous embedding):

$C^{0,\alpha}\big{(}[0,T];C^{0,\alpha}(\overline{\Omega})\big{)}\,\subseteq\,C^{0,\alpha}\big{(}\overline{\Omega_{T}}\big{)}.$

Indeed, for any $h\in
C^{0,\alpha}\big{(}[0,T];C^{0,\alpha}(\overline{\Omega})\big{)}$, we compute

$\begin{split}\big{|}h(t_{1},x_{1})-h(t_{2},x_{2})\big{|}\,&\leq\,|h(t_{1},x_{1})-h(t_{2},x_{1})|+|h(t_{2},x_{1})-h(t_{2},x_{2})|\\\
&\leq\,\|h(t_{1},\cdot)-h(t_{2},\cdot)\|_{C(\overline{\Omega})}\,+\,\|h(t_{2},\cdot)\|_{C^{0,\alpha}(\overline{\Omega})}|x_{1}-x_{2}|^{\alpha}\\\
&\leq\,\big{(}|t_{1}-t_{2}|^{\alpha}+|x_{1}-x_{2}|^{\alpha}\big{)}\|h\|_{C^{0,\alpha}([0,T];C^{0,\alpha}(\overline{\Omega}))}\end{split}$

which establishes the claim.

######  Lemma 6.

Suppose that assumptions (1.5) and (1.15) are satisfied. We have that

$\mathfrak{X}^{\infty}(\Omega_{T})\,\neq\,\emptyset$

(and consequently we have $\mathfrak{X}^{p}(\Omega_{T})\neq\emptyset$ for all
$p>1$). Further, if $(u,\mathrm{p},y)\in\mathfrak{X}^{p}(\Omega_{T})$ for some
$p>1$ which satisfies

$\mathrm{E}_{p}(u,\mathrm{p},y)\,\leq\,M$

for some $M>0$, then for any $q\leq p$ there exists $C(q,M)>0$ such that

$\|u\|_{W^{2,1;q}_{\mathrm{L},\sigma}(\Omega_{T})}+\,\|\mathrm{p}\|_{W^{1,0;q}_{\sharp}(\Omega_{T})}+\,\|y\|_{L^{q}(\Omega_{T})}\,\leq\,C(q,M).$

Further, if $p>n+2$ and $q\in(n+2,p]$, then there exists $\alpha\in(0,1)$ and
a constant $C(M,q)>0$ such that additionally

$\|u\|_{C^{0,\alpha}(\Omega_{T})}+\,\|\mathrm{D}u\|_{C^{0,\alpha}(\Omega_{T})}\,\leq\,C(q,M).$

We note that the constants above also depend on
$n,\partial\Omega,T,f,u_{0},\lambda$, but as all these are fixed throughout
this paper, we suppress denoting the explicit dependence on them.

Proof of Lemma 6. By assumptions (1.5)(b)-(1.5)(c), we have that the triplet
$(u_{0},0,y_{0})$, where

$y_{0}\,:=\,-\nu\Delta u_{0}\,+\,(u_{0}\cdot\mathrm{D})u_{0}\,-\,f$

satisfies that $(u_{0},0,y_{0})\in\mathfrak{X}^{\infty}(\Omega_{T})$, and in
fact lies also in the smaller space

$W^{2,1;\infty}_{\mathrm{L},\sigma}(\Omega_{T})\times\smash{W^{1,0;\infty}_{\sharp}(\Omega_{T})}\times
L^{\infty}(\Omega_{T};\mathbb{R}^{n}).$

Next, if $(u,\mathrm{p},y)\in\mathfrak{X}^{p}(\Omega_{T})$ with
$\mathrm{E}_{p}(u,\mathrm{p},y)\leq M$, then we readily have that

$\|y\|_{L^{q}(\Omega_{T})}\,\leq\,\|y\|_{\dot{L}^{p}(\Omega_{T})}\,\leq\,\frac{M}{\lambda},$

whilst by assumptions (1.15) and (1.5)(c) we have that

$\|u\|_{W^{2,1;q}_{\mathrm{L},\sigma}(\Omega_{T})}+\,\|\mathrm{p}\|_{W^{1,0;q}_{\sharp}(\Omega_{T})}\,\leq\,C(q)\Big{(}1+\frac{M}{\lambda}\Big{)}.$

for some $q$-dependent constant $C(q)$, for any $q\in(n,p]$. Further, suppose
$p>n+2$ and $n+2<q\leq p$. Then, the above estimate in particular implies

$\|u\|_{L^{q}(\Omega_{T})}+\,\|\nabla u\|_{L^{q}(\Omega_{T})}\,\leq\,C(q,M),$

whereat application of the Morrey imbedding theorem yields

$\|u\|_{C^{0,\alpha^{\prime}}(\Omega_{T})}\,\leq\,C(q,M),$

for a new constant $C(q,M)$ and some $\alpha^{\prime}\in(0,1)$. Next, by Lemma
4, Remark 5 and the established estimate for $q>n+2$, we have

$C(q)\|\mathrm{D}u\|_{C^{0,\alpha^{\prime\prime}}(\Omega_{T})}\,\leq\,\|u\|_{W^{2,1;q}(\Omega_{T})}\,\leq\,C(q,M),$

for some $\alpha^{\prime\prime}\in(0,1)$ and some constant $C(q)>0$. By
choosing $\alpha:=\min\\{\alpha^{\prime},\alpha^{\prime\prime}\\}$, the
conclusion ensues. ∎

##  3\. Minimisers of $L^{p}$ problems and convergence as $p\to\infty$

In this section we establish Theorem 1, by utilising the results of Section 2.

Proof of Theorem 1. Fix $p\in(n+2,\infty)$. By Lemma 4,
$\mathfrak{X}^{p}(\Omega_{T})\neq\emptyset$, therefore
$0\leq\inf_{\mathfrak{X}^{p}(\Omega_{T})}\mathrm{E}_{p}<\infty$. By Lemma 6,
it follows that $\mathfrak{X}^{p}(\Omega_{T})$ is sequentially weakly compact.
Note now that $y\mapsto\|y\|^{p}_{\dot{L}^{p}(\Omega_{T})}$ is trivially
convex, and by the identity

$\big{\|}\mathrm{K}\big{(}\cdot,u,\mathrm{D}u,\partial_{t}u,\mathrm{p}\big{)}\big{\|}^{p}_{\dot{L}^{p}(\Omega_{T})}\,=\,\int_{\Omega_{T}}\Big{(}\big{|}\mathrm{K}\big{(}\cdot,u,\mathrm{D}u,\partial_{t}u,\mathrm{p}\big{)}|^{2}+p^{-2}\Big{)}^{\\!\frac{p}{2}}\mathrm{d}\mathcal{L}^{n+1},$

assumption (1.5)(f) yields that

$(\partial_{t}u,\mathrm{p})\,\mapsto\,\big{\|}\mathrm{K}\big{(}\cdot,u,\mathrm{D}u,\partial_{t}u,\mathrm{p}\big{)}\big{\|}^{p}_{\dot{L}^{p}(\Omega_{T})}$

is also convex. Again by Lemma 6 and by standard results in the Calculus of
Variations (see e.g. [24]), it follows that $\mathrm{E}_{p}$ is weakly lower
semicontinuous in $\mathfrak{X}^{p}(\Omega_{T})$ as the convex combination of
the $p$-th roots of two weakly lower semicontinuous functionals. Hence,
$\mathrm{E}_{p}$ attains its infimum at some
$(u_{p},\mathrm{p}_{p},y_{p})\in\mathfrak{X}^{p}(\Omega_{T})$.

Consider now the family of minimisers $(u_{p},\mathrm{p}_{p},y_{p})_{p>n+2}$.
For any $(u,\mathrm{p},y)\in\mathfrak{X}^{\infty}(\Omega_{T})$ and any $q\leq
p$, minimality and the Hölder inequality for the dotted $\dot{L}^{p}$
functionals yield

$\mathrm{E}_{p}(u_{p},\mathrm{p}_{p},y_{p})\,\leq\,\mathrm{E}_{p}(u,\mathrm{p},y)\,\leq\,\mathrm{E}_{\infty}(u,\mathrm{p},y)\,+\,p^{-1}.$

By choosing $(u,\mathrm{p},y)=(u_{0},0,y_{0})$, by Lemma 6 and a standard
diagonal argument, we have that the family of minimisers is weakly precompact
in $\mathcal{W}^{q}(\Omega_{T})$ for all $q\in(n+2,\infty)$. Further, by Lemma
4 and Remark 5, $W^{2,1;q}_{\mathrm{L},\sigma}(\Omega_{T};\mathbb{R}^{n})$ is
compactly embedded in
$\smash{C^{0,\alpha}\big{(}[0,T];C^{1,\alpha}(\overline{\Omega};\mathbb{R}^{n})\big{)}}$.
Hence, for any sequence of indices $p_{j}\to\infty$, there exists
$(u_{\infty},\mathrm{p}_{\infty},y_{\infty})\in\cap_{q\in(n+2,\infty)}\mathcal{W}^{q}(\Omega_{T})$
and a subsequence denoted again as $(p_{j})_{1}^{\infty}$ such that (1.17)
holds true. Additionally, due to these modes of convergence, it follows that
$(u_{\infty},\mathrm{p}_{\infty},y_{\infty})$ solves (1.1), therefore in fact
$(u_{\infty},\mathrm{p}_{\infty},y_{\infty})\in\mathfrak{X}^{\infty}(\Omega_{T})$.
Again now by minimality and the Hölder inequality for the dotted $\dot{L}^{p}$
functionals, for any $(u,\mathrm{p},y)\in\mathfrak{X}^{\infty}(\Omega_{T})$ we
have

$\mathrm{E}_{q}(u_{p},\mathrm{p}_{p},y_{p})-\sqrt{q^{-2}-p^{-2}}\,\leq\,\mathrm{E}_{p}(u_{p},\mathrm{p}_{p},y_{p})\,\leq\,\mathrm{E}_{\infty}(u,\mathrm{p},y)\,+\,p^{-1}.$

Since as we have already shown, $\mathrm{E}_{q}$ is weakly lower
semicontinuous in $\mathfrak{X}^{q}(\Omega_{T})$, by letting $p\to\infty$
along the subsequence in the above inequality yields

$\begin{split}\mathrm{E}_{q}(u_{\infty},\mathrm{p}_{\infty},y_{\infty})-\sqrt{q^{-2}}\,&\leq\,\liminf_{p_{j}\to\infty}\mathrm{E}_{p}(u_{p},\mathrm{p}_{p},y_{p})\\\
&\leq\,\limsup_{p_{j}\to\infty}\mathrm{E}_{p}(u_{p},\mathrm{p}_{p},y_{p})\\\
&\leq\,\mathrm{E}_{\infty}(u,\mathrm{p},y).\end{split}$

By further letting $q\to\infty$, we obtain

$\begin{split}\mathrm{E}_{\infty}(u_{\infty},\mathrm{p}_{\infty},y_{\infty})\,&\leq\,\liminf_{p_{j}\to\infty}\mathrm{E}_{p}(u_{p},\mathrm{p}_{p},y_{p})\\\
&\leq\,\limsup_{p_{j}\to\infty}\mathrm{E}_{p}(u_{p},\mathrm{p}_{p},y_{p})\\\
&\leq\,\mathrm{E}_{\infty}(u,\mathrm{p},y),\end{split}$

for any $(u,\mathrm{p},y)\in\mathfrak{X}^{\infty}(\Omega_{T})$. The above
inequality establishes on the one hand that
$(u_{\infty},\mathrm{p}_{\infty},y_{\infty})$ minimises $\mathrm{E}_{\infty}$
over $\mathfrak{X}^{\infty}(\Omega_{T})$, and on the other hand by choosing
$(u,\mathrm{p},y):=(u_{\infty},\mathrm{p}_{\infty},y_{\infty})$ that (1.18)
holds true. Hence, Theorem 1 has been established. ∎

##  4\. The equations for $L^{p}$ PDE-constrained minimisers

In this section we establish the proof of Theorem 2. We begin with some
preparation. Firstly, it will be convenient to consider the functional
$\mathrm{E}_{p}$ of (1.6) as being defined in the wider Banach space
$\mathcal{W}^{p}(\Omega_{T})$ defined in (1.9):

$\mathrm{E}_{p}\ :\ \ \mathcal{W}^{p}(\Omega_{T})\longrightarrow\mathbb{R}.$

Next, we introduce a mapping on $\mathcal{W}^{p}(\Omega_{T})$ which
incorporates the PDE constraint (1.1) appearing in (1.8) as follows. We define

$\mathrm{G}=\left[\begin{array}[]{c}\mathrm{G}_{1}\\\
\mathrm{G}_{2}\end{array}\right]\ :\ \
\mathcal{W}^{p}(\Omega_{T})\longrightarrow
L^{p}(\Omega_{T};\mathbb{R}^{n})\times
W^{2-\frac{2}{p},p}_{0,\sigma}(\Omega;\mathbb{R}^{n})$

by setting

$\left\\{\ \
\begin{split}\mathrm{G}_{1}(u,\mathrm{p},y)\,&:=\,\partial_{t}u\,-\nu\Delta
u\,+(u\cdot\mathrm{D})u\,+\mathrm{D}\mathrm{p}\,-(y+f),\\\
\mathrm{G}_{2}(u,\mathrm{p},y)\,&:=\,u(\cdot,0)-u_{0}.\end{split}\right.$

Then, we may express (1.8) as

$\mathfrak{X}^{p}(\Omega_{T})\,=\,\mathcal{W}^{p}(\Omega_{T})\cap\\{\mathrm{G}=0\\}.$

We are now ready to prove our second main result.

Proof of Theorem 2. By assumption (1.5), for any $p\in(n+2,\infty)$ the
functional
$\mathrm{E}_{p}:\mathcal{W}^{p}(\Omega_{T})\longrightarrow\mathbb{R}$ is
Frechét differentiable and its derivative

$\begin{split}\mathrm{d}\mathrm{E}_{p}\ &:\ \
\mathcal{W}^{p}(\Omega_{T})\longrightarrow\big{(}\mathcal{W}^{p}(\Omega_{T})\big{)}^{*},\\\
(\mathrm{d}\mathrm{E}_{p})_{(\bar{u},\bar{\mathrm{p}},\bar{y})}(u,\mathrm{p},y)\,&=\,\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\Big{|}_{\varepsilon=0}\mathrm{E}_{p}\Big{(}\bar{u}+\varepsilon
u,\bar{\mathrm{p}}+\varepsilon\mathrm{p},\bar{y}+\varepsilon
y\Big{)}\end{split}$

can be easily computed and is given by the formula

$\begin{split}(\mathrm{d}\mathrm{E}_{p})_{(\bar{u},\bar{\mathrm{p}},\bar{y})}(u,\mathrm{p},y)\,=\,p(1-\lambda)\,&{-\displaystyle\int_{\Omega_{T}}}\\!\Big{(}\mathrm{K}_{\eta}[\bar{u},\bar{\mathrm{p}}]\cdot
u\,+\,\mathrm{K}_{(A,a)}[\bar{u},\bar{\mathrm{p}}]:\nabla
u+\,\mathrm{K}_{r}[\bar{u},\bar{\mathrm{p}}]\,\mathrm{p}\Big{)}\cdot\\\
&\cdot\mathfrak{M}_{p}\big{(}\mathrm{K}[\bar{u},\bar{\mathrm{p}}]\big{)}\,\mathrm{d}\mathcal{L}^{n+1}\,+\,p\lambda\,{-\displaystyle\int_{\Omega_{T}}}\mathfrak{M}_{p}(\bar{y})\cdot
y\,\mathrm{d}\mathcal{L}^{n+1},\end{split}$

where the operator
$\mathfrak{M}_{p}:L^{p}(\Omega_{T};\mathbb{R}^{M})\longrightarrow
L^{p^{\prime}}(\Omega_{T};\mathbb{R}^{M})$ (for $M\in\\{N,n\\}$) is given by
(1.19) and we have used the notation introduced in (1.20). Next, we note that
the mapping $\mathrm{G}$ which incorporates the PDE constraint is also Fréchet
differentiable and it can be easily confirmed that its derivative

$\begin{split}\mathrm{d}\mathrm{G}\ &:\ \
\mathcal{W}^{p}(\Omega_{T})\longrightarrow\mathcal{B}\bigg{(}\mathcal{W}^{p}(\Omega_{T}),L^{p}(\Omega_{T};\mathbb{R}^{n})\times
W^{2-\frac{2}{p},p}_{0,\sigma}(\Omega;\mathbb{R}^{n})\bigg{)},\\\
&(\mathrm{d}\mathrm{G})_{(\bar{u},\bar{\mathrm{p}},\bar{y})}(u,\mathrm{p},y)\,=\,\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\Big{|}_{\varepsilon=0}\mathrm{G}\Big{(}\bar{u}+\varepsilon
u,\bar{\mathrm{p}}+\varepsilon\mathrm{p},\bar{y}+\varepsilon
y\Big{)}\end{split}$

is given by the formula

$\begin{split}\big{(}\mathrm{d}\mathrm{G}\big{)}_{(\bar{u},\bar{\mathrm{p}},\bar{y})}(u,\mathrm{p},y)\,=\,\left[\begin{array}[]{c}\partial_{t}u-\nu\Delta
u+(u\cdot\mathrm{D})\bar{u}+(\bar{u}\cdot\mathrm{D})u+\mathrm{D}\mathrm{p}-y\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ u(\cdot,0)\end{array}\right].\end{split}$

We now claim that the differential

$(\mathrm{d}\mathrm{G})_{(\bar{u},\bar{\mathrm{p}},\bar{y})}\ :\ \
\mathcal{W}^{p}(\Omega_{T})\longrightarrow
L^{p}(\Omega_{T};\mathbb{R}^{n})\times
W^{2-\frac{2}{p},p}_{0,\sigma}(\Omega;\mathbb{R}^{n})$

is a surjective map, for any
$(\bar{u},\bar{\mathrm{p}},\bar{y})\in\mathcal{W}^{p}(\Omega)$. This is
equivalent to the statement that for any $p>n+2$, the linearised Navier-Stokes
problem

$\left\\{\ \ \begin{array}[]{rr}\partial_{t}u-\nu\Delta
u+(u\cdot\mathrm{D})\bar{u}+(\bar{u}\cdot\mathrm{D})u+\mathrm{D}\mathrm{p}\,=\,F,&\text{
in }\Omega_{T},\\\ \mathrm{div}\,u\,=\,0,&\text{ in }\Omega_{T},\\\
u(\cdot,0)\,=\,v,&\text{ on }\Omega,\\\ u\,=\,0,&\text{ on
}\partial\Omega\times(0,T),\end{array}\right.$

has a solution $(u,\mathrm{p})\in
W^{2,1;p}_{\mathrm{L},\sigma}(\Omega_{T};\mathbb{R}^{n})\times
W^{1,0;p}_{\sharp}(\Omega_{T})$, for any $\bar{u}\in
W^{2,1;p}_{\mathrm{L},\sigma}(\Omega_{T};\mathbb{R}^{n})$ and any data

$(F,v)\,\in\,L^{p}(\Omega_{T};\mathbb{R}^{n})\times
W^{2-\frac{2}{p},p}_{0,\sigma}(\Omega;\mathbb{R}^{n}).$

This is indeed the case, and it is a consequence of a classical result of
Solonnikov [53, Th. 4.2] for $n=3$ and of Giga-Sohr [35, Th. 2.8] for $n>3$,
as a perturbation of the Stokes problem. As a consequence, the assumptions of
the generalised Kuhn-Tucker theorem hold true (see e.g. Zeidler [57, Cor.
48.10 & Th. 48B]). Hence, there exists a Lagrange multiplier

$\begin{split}\Lambda_{p}\ \in\ &\Big{(}L^{p}(\Omega_{T};\mathbb{R}^{n})\times
W^{2-\frac{2}{p},p}_{0,\sigma}(\Omega_{T};\mathbb{R}^{n})\Big{)}^{*}\end{split}$

such that

$\begin{split}\big{(}\mathrm{d}\mathrm{E}_{p}\big{)}_{(u_{p},\mathrm{p}_{p},y_{p})}(u,\mathrm{p},y)\,=\,\Big{\langle}\big{(}\mathrm{d}\mathrm{G}\big{)}_{(u_{p},\mathrm{p}_{p},y_{p})}(u,\mathrm{p},y),\,\Lambda_{p}\Big{\rangle},\\\
\end{split}$

for any $(u,{\mathrm{p}},y)\in\mathcal{W}^{p}(\Omega)$. By standard duality
arguments, the Riesz representation theorem and by taking into account the
form of the differentials $\mathrm{d}\mathrm{E}_{p}$ and
$\mathrm{d}\mathrm{G}$, we may identify $\Lambda_{p}$ with a pair of Lagrange
multipliers

$(\phi_{p},\Psi_{p})\,\in\,L^{p^{\prime}}(\Omega_{T};\mathbb{R}^{n})\times\Big{(}W^{2-\frac{2}{p},p}_{0,\sigma}(\Omega_{T};\mathbb{R}^{n})\Big{)}^{*}$

such that, the constrained minimiser
$\big{(}u_{p},p_{p},y_{p}\big{)}\in\mathfrak{X}^{p}(\Omega_{T})$ satisfies the
equation

$\begin{split}&(1-\lambda)\int_{\Omega_{T}}\\!\Big{(}\mathrm{K}_{\eta}[u_{p},{\mathrm{p}}_{p}]\cdot
u\,+\,\mathrm{K}_{(A,a)}[u_{p},{\mathrm{p}}_{p}]:\nabla
u+\,\mathrm{K}_{r}[u_{p},{\mathrm{p}}_{p}]\,\mathrm{p}\Big{)}\cdot\\\ &\hskip
50.0pt\cdot\mathfrak{M}_{p}\big{(}\mathrm{K}[u_{p},{\mathrm{p}}_{p}]\big{)}\,\mathrm{d}\mathcal{L}^{n+1}\
+\ \lambda\int_{\Omega_{T}}\mathfrak{M}_{p}(y_{p})\cdot
y\,\mathrm{d}\mathcal{L}^{n+1}\\\
&=\int_{\Omega_{T}}\Big{(}\partial_{t}u-\nu\Delta
u+(u\cdot\mathrm{D})u_{p}+(u_{p}\cdot\mathrm{D})u+\mathrm{D}\mathrm{p}-y\Big{)}\cdot\phi_{p}\,\mathrm{d}\mathcal{L}^{n+1}\,+\,\langle\Psi_{p},u(\cdot,0)\rangle,\end{split}$

for any $(u,\mathrm{p},y)\in\mathcal{W}^{p}(\Omega_{T})$. We note that here we
have tacitly rescaled $(\phi_{p},\Psi_{p})$ by multiplying them with the
factor $p(\mathcal{L}^{n+1}(\Omega_{T}))^{-1}$, in order to remove the
averages arising from $\mathrm{\mathrm{E}}_{p}$ on the left hand side and to
be able to obtain non-trivial limits as $p\to\infty$ of the multipliers
themselves later on. By using linear independence, the above equation actually
decouples to the triplet of relations

$\left\\{\ \
\begin{split}&(1-\lambda)\int_{\Omega_{T}}\\!\Big{(}\mathrm{K}_{\eta}[u_{p},{\mathrm{p}}_{p}]\cdot
u\,+\,\mathrm{K}_{(A,a)}[u_{p},{\mathrm{p}}_{p}]:\nabla
u\Big{)}\cdot\mathfrak{M}_{p}\big{(}\mathrm{K}[u_{p},{\mathrm{p}}_{p}]\big{)}\,\mathrm{d}\mathcal{L}^{n+1}\
\\\ &=\int_{\Omega_{T}}\Big{(}\partial_{t}u-\nu\Delta
u+(u\cdot\mathrm{D})u_{p}+(u_{p}\cdot\mathrm{D})u\Big{)}\cdot\phi_{p}\,\mathrm{d}\mathcal{L}^{n+1}\,+\,\langle\Psi_{p},u(\cdot,0)\rangle,\end{split}\right.$
$\begin{split}(1-\lambda)\int_{\Omega_{T}}\\!\big{(}\mathrm{K}_{r}[u_{p},{\mathrm{p}}_{p}]\,\mathrm{p}\big{)}\cdot\mathfrak{M}_{p}\big{(}\mathrm{K}[u_{p},{\mathrm{p}}_{p}]\big{)}\,\mathrm{d}\mathcal{L}^{n+1}\,=\,\int_{\Omega_{T}}\mathrm{D}\mathrm{p}\cdot\phi_{p}\,\mathrm{d}\mathcal{L}^{n+1},\end{split}$
$\begin{split}\lambda\int_{\Omega_{T}}\mathfrak{M}_{p}(y_{p})\cdot
y\,\mathrm{d}\mathcal{L}^{n+1}\,=\,-\int_{\Omega_{T}}y\cdot\phi_{p}\,\mathrm{d}\mathcal{L}^{n+1}.\end{split}$

The arbitrariness of $y\in L^{p}(\Omega_{T};\mathbb{R}^{n})$ in the third
equation readily yields that the multiplier $\phi_{p}$ equals

$\phi_{p}\,=\,-\lambda\,\mathfrak{M}_{p}(y_{p}).$

By substituting this into the first two equations, we see that the theorem has
been established. ∎

##  5\. The equations for $L^{\infty}$ PDE-constrained minimisers

In this section we establish our final main result. In this case we need to
assume that $\mathrm{K}$ does not depend on $(\partial_{t}u,\mathrm{p})$ and
we will invoke the symbolisations (1.24) for $\mathrm{K}$ and its derivatives
$\mathrm{K}_{\eta},\mathrm{K}_{A}$, all of which are additionally assumed to
be continuous.

Proof of Theorem 3. By Theorem 2 it follows that for any $p\in(n+2,\infty)$,
the minimising triplet
$(u_{p},\mathrm{p}_{p},y_{p})\in\mathfrak{X}^{p}(\Omega_{T})$ satisfies

$\left\\{\
\begin{split}&(1-\lambda)\int_{\Omega_{T}}\Big{(}\mathrm{K}_{\eta}[u_{p}]\cdot
u\,+\,\mathrm{K}_{A}[u_{p}]:\mathrm{D}u\Big{)}\cdot\mathfrak{M}_{p}\big{(}\mathrm{K}[u_{p}]\big{)}\,\mathrm{d}\mathcal{L}^{n+1}\\\
&=-\lambda\int_{\Omega_{T}}\Big{(}\partial_{t}u-\nu\Delta
u+(u\cdot\mathrm{D})u_{p}+(u_{p}\cdot\mathrm{D})u\Big{)}\cdot\mathfrak{M}_{p}(y_{p})\,\mathrm{d}{\mathcal{L}}^{n+1}\,+\,\big{\langle}\Psi_{p},u(\cdot,0)\big{\rangle},\end{split}\right.$

and also

$\begin{split}&\int_{\Omega_{T}}\mathrm{D}\mathrm{p}\cdot\mathfrak{M}_{p}(y_{p})\,\mathrm{d}\mathcal{L}^{n+1}\,=\,0,\\\
\end{split}$

for all test mappings $(u,\mathrm{p})\in
W^{2,1;p}_{\mathrm{L},\sigma}(\Omega_{T};\mathbb{R}^{n})\times
W^{1,0;p}_{\sharp}(\Omega_{T})$. The first goal is to pass to the limit as
$p\to\infty$ in these equations in order to obtain (1.27)-(1.28). Since by
(1.19) we readily have that $\mathfrak{M}_{p}(y_{p})$ and
$\mathfrak{M}_{p}\big{(}\mathrm{K}[u_{p}]\big{)}$ are valued in the unit balls
of $\smash{L^{p^{\prime}}}(\Omega_{T};\mathbb{R}^{n})$ and of
$\smash{L^{p^{\prime}}}(\Omega_{T};\mathbb{R}^{N})$ respectively, by defining
$\Sigma_{p}$ and $\sigma_{p}$ as in (1.30), the existence of limiting measures
$\Sigma_{\infty},\sigma_{\infty}$ is guaranteed along perhaps a further
subsequence such that

$\Sigma_{p}\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\Sigma_{\infty}$
in $\mathcal{M}\big{(}\overline{\Omega_{T}};\mathbb{R}^{N}\big{)}$ and
$\sigma_{p}\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\sigma_{\infty}$
in $\mathcal{M}\big{(}\overline{\Omega_{T}};\mathbb{R}^{n}\big{)}$,

as $p_{j}\to\infty$. Further, by Lemma 6 we have that $u_{p}\longrightarrow
u_{\infty}$ and $\mathrm{D}u_{p}\longrightarrow\mathrm{D}u_{\infty}$, both
uniformly on $\overline{\Omega_{T}}$ as $p_{j}\to\infty$. Also, by the
continuity assumption on $\mathrm{K},\mathrm{K}_{\eta},\mathrm{K}_{A}$ on
$\overline{\Omega_{T}}\times\mathbb{R}^{n}\times\mathbb{R}^{n\times n}$ and
again Lemma 6, it follows that

$\mathrm{K}[u_{p}]\longrightarrow\mathrm{K}[u_{\infty}]$,
$\mathrm{K}_{\eta}[u_{p}]\longrightarrow\mathrm{K}_{\eta}[u_{\infty}]$ and
$\mathrm{K}_{A}[u_{p}]\longrightarrow\mathrm{K}_{A}[u_{\infty}]$,

all uniformly on $\overline{\Omega_{T}}$ as $p_{j}\to\infty$. Putting all this
together, we see that the remaining main point is to obtain a uniform estimate
on the family of Lagrange multipliers $(\Psi_{p})_{p>n+2}$ in order to deduce
that

$\Psi_{p}\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\Psi_{\infty}$
in $\smash{\big{(}W^{2-2/r,r}_{0,\sigma}(\Omega;\mathbb{R}^{n})\big{)}^{*}}$,
for all $r>n+2$,

which would allow to pass to the limit as $p_{j}\to\infty$. Once this has been
achieved, passing to the limit in the equations follows by standard duality
pairing arguments, which are made possible by restricting the class of test
functions $(u,\mathrm{p})$ to those which are continuous together with those
derivatives appearing in the relations.

In order to derive the desired estimate on $(\Psi_{p})_{p>n+2}$, we argue as
follows. Consider (1.22) for $\mathrm{K}_{a}\equiv 0$ (the first equation
appearing in this proof) and let us fix the initial value on
$\Omega\times\\{0\\}$

$u(\cdot,0)\,\equiv\,\hat{u}\,\in
W^{2,\infty}_{0,\sigma}(\Omega;\mathbb{R}^{n})$

of the arbitrary test function $u$, but we will select $u$ on $\Omega_{T}$
such that the term in the bracket in the integral on the right-hand-side
becomes a gradient. Then, this term will vanish identically as a consequence
of (1.23) when $\mathrm{K}_{r}\equiv 0$ (the second equation appearing in this
proof). Indeed, let $p>n+2$ and let also $(\tilde{u},\tilde{\mathrm{p}})$ be
the (unique) solution to

$\left\\{\ \
\begin{array}[]{rr}\partial_{t}\tilde{u}-\nu\Delta\tilde{u}+(\tilde{u}\cdot\mathrm{D})u_{p}+(u_{p}\cdot\mathrm{D})\tilde{u}+\mathrm{D}\tilde{\mathrm{p}}\,=\,0,&\text{
in }\Omega_{T},\\\ \mathrm{div}\,\tilde{u}\,=\,0,&\text{ in }\Omega_{T},\\\
\tilde{u}(\cdot,0)\,=\,\hat{u},&\text{ on }\Omega,\\\ \tilde{u}\,=\,0,&\text{
on }\partial\Omega\times(0,T),\end{array}\right.$

The solvability of the above problem is a consequence of the classical result
of Solonnikov [53, Th. 4.2] for $n=3$ and of Giga-Sohr [35, Th. 2.8] for
$n>3$, as a perturbation of the Stokes problem: by choosing $q>n+2$ in
Solonnikov’s assumption (4.14), a solution as claimed does exist. Further,
since $\hat{u}$ is in $W^{2,\infty}_{0,\sigma}(\Omega;\mathbb{R}^{n})$, by
[53, Cor. 2, p. 489] we have the uniform estimate

$\|\tilde{u}\|_{W^{2,1;r}_{\mathrm{L},\sigma}(\Omega_{T})}+\,\|\tilde{\mathrm{p}}\|_{W^{1,0;r}_{\sharp}(\Omega_{T})}\,\leq\,C(r)\|\hat{u}\|_{W^{2-\frac{2}{r},r}_{0,\sigma}(\Omega)},$

for any $r\in(1,\infty)$. By Lemmas 4 and 6 and Remark 5, if we restrict our
attention to $r\in(n+2,\infty)$, we additionally have the bound

$\|\tilde{u}\|_{L^{\infty}(\Omega_{T})}+\,\|\mathrm{D}\tilde{u}\|_{L^{\infty}(\Omega_{T})}\leq\,C(r)\|\hat{u}\|_{W^{2-\frac{2}{r},r}_{0,\sigma}(\Omega)},$

for some new constant $C(r)$ (which is unbounded as $r\searrow n+2$). By
setting

$\left\\{\ \
\begin{split}K_{\infty}\,:=&\,\sup\Big{\\{}|\mathrm{K}_{\eta}|+|\mathrm{K}_{A}|:\
\Omega_{T}\times\mathbb{B}^{n}_{R_{\infty}}(0)\times\mathbb{B}^{n\times
n}_{R_{\infty}}(0)\Big{\\}},\\\
R_{\infty}\,:=&\,\sup_{p>n+2}\Big{(}\|u_{p}\|_{L^{\infty}(\Omega_{T})}+\,\|\mathrm{D}u_{p}\|_{L^{\infty}(\Omega_{T})}\Big{)},\end{split}\right.$

where $\mathbb{B}^{n}_{R_{\infty}}(0)$ and $\mathbb{B}^{n\times
n}_{R_{\infty}}(0)$ denote the balls of radius $R_{\infty}$ centred at the
origin of $\mathbb{R}^{n}$ and of $\mathbb{R}^{n\times n}$ respectively, we
estimate by using (1.22)-(1.23) (for $\mathrm{K}_{a}\equiv 0$,
$\mathrm{K}_{r}\equiv 0$) and that by (1.19) we have
$\smash{\big{\|}\mathfrak{M}_{p}\big{(}\mathrm{K}[u_{p}]\big{)}\big{\|}_{L^{1}(\Omega_{T})}\leq
1}$ (for the normalised $L^{1}$ norm):

$\begin{split}\big{|}\big{\langle}\Psi_{p},\hat{u}\big{\rangle}\big{|}\,\leq&\
\lambda\,\bigg{|}\\!\int_{\Omega_{T}}\mathrm{D}\tilde{\mathrm{p}}\cdot\mathfrak{M}_{p}(y_{p})\,\mathrm{d}{\mathcal{L}}^{n+1}\bigg{|}\\\
&+(1-\lambda)\bigg{|}\\!\int_{\Omega_{T}}\\!\Big{(}\mathrm{K}_{\eta}[u_{p}]\cdot\tilde{u}\,+\,\mathrm{K}_{A}[u_{p}]:\mathrm{D}\tilde{u}\Big{)}\cdot\mathfrak{M}_{p}\big{(}\mathrm{K}[u_{p}]\big{)}\,\mathrm{d}\mathcal{L}^{n+1}\bigg{|}\\\
=&\
(1-\lambda)T\mathcal{L}^{n}(\Omega)\bigg{|}\,{-\displaystyle\int_{\Omega_{T}}}\\!\Big{(}\mathrm{K}_{\eta}[u_{p}]\cdot\tilde{u}\,+\,\mathrm{K}_{A}[u_{p}]:\mathrm{D}\tilde{u}\Big{)}\cdot\mathfrak{M}_{p}\big{(}\mathrm{K}[u_{p}]\big{)}\,\mathrm{d}\mathcal{L}^{n+1}\bigg{|}\\\
\leq&\
(1-\lambda)T\mathcal{L}^{n}(\Omega)K_{\infty}\Big{(}\|\tilde{u}\|_{L^{\infty}(\Omega_{T})}+\,\|\mathrm{D}\tilde{u}\|_{L^{\infty}(\Omega_{T})}\Big{)}\\\
\leq&\
(1-\lambda)T\mathcal{L}^{n}(\Omega)K_{\infty}C(r)\|\hat{u}\|_{W^{2-\frac{2}{r},r}_{0,\sigma}(\Omega)},\end{split}$

for any $r$ fixed. Therefore, for any $r\in(n+2,\infty)$ and any $\hat{u}\in
W^{2,\infty}_{0,\sigma}(\Omega;\mathbb{R}^{n})$, we have the estimate

$\begin{split}\big{|}\big{\langle}\Psi_{p},\hat{u}\big{\rangle}\big{|}\,\leq\big{(}T\mathcal{L}^{n}(\Omega)K_{\infty}\big{)}C(r)\|\hat{u}\|_{W^{2-\frac{2}{r},r}_{0,\sigma}(\Omega)}.\end{split}$

Since $W^{2,\infty}_{0,\sigma}(\Omega;\mathbb{R}^{n})$ is dense in
$W^{2-2/r,r}_{0,\sigma}(\Omega;\mathbb{R}^{n})$, by the Hahn-Banach theorem,
the above estimate implies that for any fixed $p>n+2$, the bounded linear
functional

$\Psi_{p}\ :\ \
W^{2-\frac{2}{p},p}_{0,\sigma}(\Omega;\mathbb{R}^{n})\longrightarrow\mathbb{R}$

can be (uniquely) extended to a functional
$\Psi_{p}:W^{2-2/r,r}_{0,\sigma}(\Omega;\mathbb{R}^{n})\longrightarrow\mathbb{R}$
for all $r\in(n+2,p]$, whose extension we denote again by $\Psi_{p}$.
Therefore, $\Psi_{p}$ can be extended to a unique continuous linear functional

$\Psi_{p}\ :\ \
\bigcup_{r>n+2}W^{2-\frac{2}{r},r}_{0,\sigma}(\Omega;\mathbb{R}^{n})\longrightarrow\mathbb{R}$

on the above Fréchet space, whose topology can be defined in the standard
locally convex sense by the family of seminorms

$\Big{\\{}\|\cdot\,\|_{W^{2-{2}/{r},r}(\Omega)}:\ r>n+2\Big{\\}}.$

Additionally, the uniformity of the estimate with respect to $p$ implies that

$(\Psi_{p})_{p>n+2}$ is bounded in
$\bigg{(}\bigcup_{r>n+2}W^{2-\frac{2}{r},r}_{0,\sigma}(\Omega;\mathbb{R}^{n})\bigg{)}^{*}$

(in the locally convex sense). Hence, as it can be seen by a customary
diagonal argument in the scale of Banach spaces
$\smash{\big{\\{}W^{2-{2}/{r},r}_{0,\sigma}(\Omega_{T};\mathbb{R}^{n}):r>n+2\big{\\}}}$
comprising the Fréchet space, there exists a continuous linear functional

$\Psi_{\infty}\ :\ \
\bigcup_{r>n+2}W^{2-\frac{2}{r},r}_{0,\sigma}(\Omega;\mathbb{R}^{n})\longrightarrow\mathbb{R}$

and a further subsequence as $p\to\infty$ such that along which we have
$\Psi_{p}\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\Psi_{\infty}$
in the locally convex sense. Additionally, since

$\Psi_{\infty}\in\bigcap_{r>n+2}\Big{(}W^{2-\frac{2}{r},r}_{0,\sigma}(\Omega;\mathbb{R}^{n})\Big{)}^{*}$

the convergence
$\Psi_{p}\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\Psi_{\infty}$
is equivalent to weak* convergence in the Banach space
$\smash{W^{2-{2}/{r},r}_{0,\sigma}(\Omega;\mathbb{R}^{n})}$ for any fixed
$r>n+2$. In conclusion, we see that (1.27)-(1.28) have now been established.

Now we complete the proof of Theorem 3 by establishing (1.31)-(1.32). Since
$\mathrm{K}[u_{p}]\longrightarrow\mathrm{K}[u_{\infty}]$ in
$C\big{(}\overline{\Omega_{T}};\mathbb{R}^{N}\big{)}$, by applying [38, Prop.
10], we immediately obtain that $\Sigma_{\infty}$ concentrates on the set
whereon $|\mathrm{K}[u_{\infty}]|$ is maximised over $\overline{\Omega_{T}}$:

$\Sigma_{\infty}\bigg{(}\Big{\\{}\big{|}\mathrm{K}[u_{\infty}]\big{|}<\max_{\overline{\Omega_{T}}}\big{|}\mathrm{K}[u_{\infty}]\big{|}\Big{\\}}\bigg{)}=\,0.$

This proves (1.31). For (1.32), we argue as follows. We first note that

$\|y_{p}\|_{\dot{L}^{p}(\Omega_{T})}\longrightarrow\,\|y_{\infty}\|_{L^{\infty}(\Omega_{T})}$

as $p\to\infty$, along a subsequence. In view of (1.6) and (1.13), this is a
consequence of (1.18) and the fact that
$\mathrm{K}[u_{p}]\longrightarrow\mathrm{K}[u_{\infty}]$ uniformly on
$\overline{\Omega_{T}}$, which implies

$\big{\|}\mathrm{K}[u_{p}]\big{\|}_{\dot{L}^{p}(\Omega_{T})}\longrightarrow\,\big{\|}\mathrm{K}[u_{\infty}]\big{\|}_{L^{\infty}(\Omega_{T})}.$

As a consequence of the convergence of $\|y_{p}\|_{\dot{L}^{p}(\Omega_{T})}$
to $\|y_{\infty}\|_{L^{\infty}(\Omega_{T})}$, for any $\varepsilon>0$ we may
choose $p$ large so that

$\|y_{p}\|_{\dot{L}^{p}(\Omega_{T})}\geq\,\|y_{\infty}\|_{L^{\infty}(\Omega_{T})}-\frac{\varepsilon}{2}.$

Let us define now the following subset of $\Omega_{T}$, which without loss of
generality we may assume it has positive $\mathcal{L}^{n+1}$-measure:

$A_{p,\varepsilon}\,:=\,\Big{\\{}|y_{p}|\leq\|y_{\infty}\|_{L^{\infty}(\Omega_{T})}-\varepsilon\Big{\\}}.$

In particular, if $\mathcal{L}^{n+1}(A_{p,\varepsilon})>0$, then necessarily
$\|y_{\infty}\|_{L^{\infty}(\Omega_{T})}>0$. For any Borel set
$B\subseteq\Omega_{T}$ such that $\mathcal{L}^{n+1}(\Omega_{T}\cap B)>0$, we
estimate by using (1.30), (1.19), (1.7) and the above:

$\begin{split}\sigma_{p}(A_{p,\varepsilon}\cap
B)\,&\leq\,\frac{\mathcal{L}^{n+1}(A_{p,\varepsilon}\cap
B)}{\|y_{p}\|^{p-1}_{\dot{L}^{p}(\Omega_{T})}}\,{-\displaystyle\int_{A_{p,\varepsilon}\cap
B}}\big{(}|y_{p}|_{(p)}\big{)}^{p-1}\,\mathrm{d}\mathcal{L}^{n+1}\\\
&\leq\,\frac{\mathcal{L}^{n+1}(A_{p,\varepsilon}\cap
B)}{\|y_{p}\|^{p-1}_{\dot{L}^{p}(\Omega_{T})}}\,{-\displaystyle\int_{A_{p,\varepsilon}\cap
B}}\Big{(}\|y_{\infty}\|_{L^{\infty}(\Omega_{T})}-\varepsilon\Big{)}^{\\!p-1}\,\mathrm{d}\mathcal{L}^{n+1}\\\
&\leq\,\frac{\mathcal{L}^{n+1}(A_{p,\varepsilon}\cap
B)}{\|y_{p}\|^{p-1}_{\dot{L}^{p}(\Omega_{T})}}\Big{(}\|y_{\infty}\|_{L^{\infty}(\Omega_{T})}-\varepsilon\Big{)}^{\\!p-1}\\\
&\leq\,\mathcal{L}^{n+1}(A_{p,\varepsilon}\cap
B)\bigg{(}\frac{\|y_{\infty}\|_{L^{\infty}(\Omega_{T})}-\varepsilon}{\|y_{\infty}\|_{L^{\infty}(\Omega_{T})}-\frac{\varepsilon}{2}}\bigg{)}^{\\!p-1}.\end{split}$

As a result, for any $\varepsilon>0$ small, any $p$ large enough and any Borel
set $B\subseteq\Omega_{T}$ with $\mathcal{L}^{n+1}(\Omega_{T}\cap B)>0$, we
have obtained the density estimate

$\begin{split}\frac{\sigma_{p}(A_{p,\varepsilon}\cap
B)}{\mathcal{L}^{n+1}(A_{p,\varepsilon}\cap
B)}\,\leq\,\bigg{(}1-\frac{\varepsilon}{2\|y_{\infty}\|_{L^{\infty}(\Omega_{T})}-\varepsilon}\bigg{)}^{\\!p-1}.\end{split}$

The above estimate in particular implies that
$\sigma_{p}(A_{p,\varepsilon})\longrightarrow 0$ as $p\to\infty$ for any
$\varepsilon>0$ fixed, therefore establishing (1.32). The proof of Theorem 3
is now complete. ∎

######  Remark 7.

It is perhaps worth noting (in relation to the preceding arguments in the
proof of (1.32)) that the modes of convergence

$\|y_{p}\|_{L^{p}(\Omega_{T})}\longrightarrow\,\|y_{\infty}\|_{L^{\infty}(\Omega_{T})}\
\ \text{ and }\ \
y_{p}\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}y_{\infty}\
\text{ in }L^{\infty}(\Omega_{T};\mathbb{R}^{n})$

as $p\to\infty$, in general by themselves do not suffice to obtain
$y_{p}\longrightarrow y_{\infty}$ in any strong sense, hence precluding the
derivation of a stronger property than (1.32), along the lines of (1.31). A
simple counter-example, even in one dimension, is the following: let $p\in
2\mathbb{N}$ and set

$y_{p}\,:=\,\sum_{j=0}^{(p-2)/2}\bigg{[}\chi_{\big{(}\frac{2j}{p},\frac{2j+1}{p}\big{)}}-\chi_{\big{(}\frac{2j+1}{p},\frac{2j+2}{p}\big{)}}\bigg{]}\,+\,\chi_{(1,2)},$

and also $y_{\infty}:=\chi_{(1,2)}$. Then, we have $|y_{p}|=1$
$\mathcal{L}^{1}$-a.e. on $(0,2)$ for all $p$, hence we deduce that
$\|y_{p}\|_{L^{p}(0,2)}\longrightarrow\,\|y_{\infty}\|_{L^{\infty}(0,2)}$,
whilst we also have $\
y_{p}\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}y_{\infty}$
in $L^{\infty}(0,2)$ as $p\to\infty$, but $y_{p}\,\,\not\\!\\!\longrightarrow
y_{\infty}$ neither a.e., nor in $L^{1}$ or in measure.

### Acknowledgement

N.K. is indebted to Jochen Bröcker for his expert insights involving
scientific discussions about variational data assimilation in continuous time
in relation to meteorology.

## References

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