# An extrapolation result in the variational setting: improved regularity,
compactness, and applications to quasilinear systems

Sebastian Bechtel  Delft Institute of Applied Mathematics, Delft University of
Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
<EMAIL_ADDRESS>and  Mark Veraar<EMAIL_ADDRESS>
###### Abstract.

In this paper we consider the variational setting for SPDE on a Gelfand triple
$(V,H,V^{*})$. Under the standard conditions on a linear coercive pair
$(A,B)$, and a symmetry condition on $A$ we manage to extrapolate the
classical $\mathrm{L}^{2}$-estimates in time to $\mathrm{L}^{p}$-estimates for
some $p>2$ without any further conditions on $(A,B)$. As a consequence we
obtain several other a priori regularity results of the paths of the solution.

Under the assumption that $V$ embeds compactly into $H$, we derive a universal
compactness result quantifying over all $(A,B)$. As an application of the
compactness result we prove global existence of weak solutions to a system of
second order quasi-linear equations.

###### Key words and phrases:

Stochastic evolution equations, variational methods, compactness, tightness
stochastic partial differential equations, quasi- and semi-linear, coercivity,
extrapolation, Sneiberg’s lemma

###### 2020 Mathematics Subject Classification:

Primary: 60H15. Secondary: 35A01, 35B65, 35K59, 35K90, 47H05, 47J35.

The first author is supported by the Alexander von Humboldt foundation by a
Feodor Lynen grant. The second author is supported by the VICI subsidy
VI.C.212.027 of the Netherlands Organisation for Scientific Research (NWO)

##  1\. Introduction

In this paper we consider stochastic evolution equations in the variational
setting. This is the stochastic version of Lions classical setting [22] and
goes back to the work of [5, 20, 27]. For details we refer the reader to the
monographs [23, 30].

This paper consists of two main parts. In the first part we prove an
extrapolation result, which improves the usual a priori regularity estimates
for linear equations with operators that satisfy the usual coercivity
conditions. As a consequence we derive a new type of compactness result for
linear equations, which is universal in the sense that for given constants in
the estimates it varies over all admissible operators and data. In the second
part we apply the compactness result to obtain global existence for a class of
stochastic parabolic systems which is not well-understood yet.

The triple $(V,H,V^{*})$ are Hilbert spaces such that $V\hookrightarrow
H\hookrightarrow V^{*}$ densely, where we identify the Hilbert space $H$ with
its dual, and where $V^{*}$ is the dual with respect to the inner product of
$H$. The stochastic equations we consider can be written as:

(LP) $\left\\{\begin{aligned}
\,\mathrm{d}u(t)+A(t)u(t)\,\,\mathrm{d}t&=f(t)\,\,\mathrm{d}t+(B(t)u(t)+g(t))\,\,\mathrm{d}W(t),\\\
u(0)&=u_{0}.\end{aligned}\right.$

Here $A:(0,T)\times\Omega\to\mathcal{L}(V,V^{*})$ and
$B:(0,T)\times\Omega\to\mathcal{L}_{2}(U,H)$, and $W$ is a $U$-cylindrical
Brownian motion, where $U$ is a real separable Hilbert space. The time $T>0$
is finite. For the inhomogeneities $(f,g,u_{0})$ we assume
$f:(0,T)\times\Omega\to V^{*}$, $g:(0,T)\times\Omega\to\mathcal{L}_{2}(U,H)$
and $u_{0}:\Omega\to H$, and the standard measurability conditions are
supposed to be satisfied.

Consider the following condition on the pair $(A,B)$.

######  Assumption 1.1.

There exist constants $\Lambda,\lambda>0$ and $M\geq 0$ such that pointwise in
$[0,T]\times\Omega$ for all $v\in V$,

$\displaystyle\operatorname{Re}\,\langle
Av,v\rangle-\frac{1}{2}\|Bv\|_{\mathcal{L}_{2}(U,H)}^{2}$
$\displaystyle\geq\lambda\|v\|_{V}^{2}-M\|v\|_{H}^{2}$ (coercivity),
$\displaystyle\|Av\|_{V^{*}}\leq\Lambda\|v\|_{V}\quad$
$\displaystyle\&\quad\|Bv\|_{\mathcal{L}_{2}(U,H)}^{2}\leq\Lambda\|v\|_{V}^{2}$
$\displaystyle\text{(boundedness)}.$

Under Assumption 1.1 it is standard that (LP) has a unique strong solution
$u\in\mathrm{L}^{2}(0,T;V)\cap\mathrm{C}([0,T];H)$ a.s. and the following a
priori estimate holds for a constant $C$ only depending on $T\vee 1$,
$\lambda$, $\Lambda$, and $M$:

(1) $\displaystyle\begin{aligned}
\|u\|_{\mathrm{L}^{2}(\Omega\times(0,T);V)}+\|u\|_{\mathrm{L}^{2}(\Omega;\mathrm{C}([0,T];H)}\leq
C\big{[}\|u_{0}\|_{\mathrm{L}^{2}(\Omega;H)}&+\|f\|_{\mathrm{L}^{2}(\Omega\times(0,T);V^{*})}\\\
&+\|g\|_{\mathrm{L}^{2}(\Omega\times(0,T);\mathcal{L}_{2}(U,H))}\big{]}.\end{aligned}$

In the deterministic setting it follows from [13] that one can find some
$p_{0}>2$ depending on $T\vee 1$, $\lambda$, $\Lambda$, and $M$ such that for
every $p\in[2,p_{0}]$ there exists a constant $C_{p}$ such that

(2)
$\displaystyle\|u\|_{\mathrm{L}^{p}(0,T;V)}+\|u^{\prime}\|_{\mathrm{L}^{p}(0,T;V)}\leq
C_{p}\big{[}\|u_{0}\|_{(H,V)_{1-\frac{2}{p}}}$
$\displaystyle+\|f\|_{\mathrm{L}^{p}(0,T;V^{*})}\big{]}.$

Actually this even holds true outside the setting of Hilbert space and
variational problems. On the other hand, (2) does not hold for all
$p\in(2,\infty)$. In [13] the estimate (2) is used to obtain a compactness
result, which in turn is used to prove global existence of solutions to a
quasi-linear parabolic PDE. On the other hand, if $A$ does not depend on time,
then (2) holds for all $p\in(1,\infty)$ (see [18, Theorem 17.2.26 (4)]). A
stochastic version of an extrapolation result holds if $B=0$ (or small), and
was proved in [24]. In the general case $B\neq 0$ the result is false since
the $\mathrm{L}^{p}(\Omega)$-integrability can fail to hold (see [10]) and a
more restrictive coercivity condition is needed (see [15]). However, it would
be interesting to obtain some result with mixed integrability
$\mathrm{L}^{2}(\Omega;\mathrm{L}^{p}(0,T;V))$ in case of general $(A,B)$
satisfying Assumption 1.1 and any $p\in[2,\infty)$.

To prove (2) one can use Sneiberg’s lemma for complex interpolation. In the
latter argument one needs an operator $T\in\mathcal{L}(X_{i},Y_{i})$, where
$(X_{0},X_{1})$ and $(Y_{0},Y_{1})$ are Banach couples. By complex
interpolation also $T\in\mathcal{L}(X_{\theta},Y_{\theta})$ for all
$\theta\in(0,1)$. Sneiberg’s result states that the interval of
$\theta\in(0,1)$ for which $T$ is invertible is open. In the deterministic
setting one can take
$X_{i}={}_{0}\mathrm{W}^{1,p_{i}}(0,T;V^{*})\cap\mathrm{L}^{p_{i}}(0,T;V)$ and
$Y_{i}=\mathrm{L}^{p_{i}}(0,T;V^{*})$ with $1<p_{0}<2<p_{1}<\infty$ and
$Tu=u^{\prime}+Au$. The boundedness of $T$ is trivial. Now if one takes
$\theta\in(0,1)$ such that
$\frac{1-\theta}{p_{0}}=\frac{\theta}{p_{1}}=\frac{1}{2}$, then
$T:X_{\theta}\to Y_{\theta}$ is invertible by (2) for $p=2$ and with
$u_{0}=0$. Thus, Sneiberg’s lemma provides us with an open interval around
this value of $\theta$, which yields the required result. As far as we could
see it seems impossible to define an operator $T$ in the stochastic framework
so that the above steps can be completed.

### New estimates for the linear problem

In this paper we circumvent the above problem at the prize of a symmetry
condition on $A$, and prove the following result, where we emphasise that we
do not need any regularity conditions on $(A,B)$.

######  Theorem 1.2 (Extrapolation of integrability and regularity).

Suppose that Assumption 1.1 holds and that on $[0,T]\times\Omega$ and for
$u,v\in V$ the symmetry condition $\langle Au,v\rangle=\langle Av,u\rangle$
holds. Then there exists $p_{0}>2$ only depending on $\Lambda,\lambda,T\vee 1$
and $M$ such that for all $p\in[2,p_{0}]$,
$f\in\mathrm{L}^{p}_{\mathcal{F}}(\Omega\times(0,T);V^{*})$,
$g\in\mathrm{L}^{p}_{\mathcal{F}}(\Omega\times(0,T);\mathcal{L}_{2}(U,H))$ and
$u_{0}\in\mathrm{L}^{p}_{\mathcal{F}_{0}}(\Omega;(H,V)_{1-\nicefrac{{2}}{{p}},p})$,
problem (LP) admits a unique solution
$u\in\mathrm{L}^{p}(\Omega\times(0,T);V)$. Moreover, for $\delta\in(1/p,1/2)$
and $\theta\in[0,1/2)$ we have

$u\in\mathrm{L}^{p}(\Omega;\mathrm{C}^{\delta-\frac{1}{p}}([0,T];[H,V]_{1-2\delta})),\
\ \text{and}\ \
u\in\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];(H,V)_{1-\frac{2}{p},p})).$

In the above we used the notation $[\cdot,\cdot]_{\theta}$ and
$(\cdot,\cdot)_{\theta,p}$ for complex and real interpolation, respectively.
The main novelties are $\mathrm{L}^{p}$-integrability in $\Omega\times(0,T)$
and the improved smoothness of the paths, which is much better than the
classical path space $\mathrm{C}([0,T];H)$ used in (1).

Theorem 1.2 is a special case of Theorem 2.2, where additional linear
estimates are stated as well. As mentioned, we cannot use Sneiberg’s lemma to
prove Theorem 1.2 . Instead we will use abstract Stein interpolation. The
proof was inspired by the recent work [7] on $\mathrm{L}^{p}$-bounds for heat
semigroups.

The symmetry condition can be written equivalently as $A^{*}=A$. We do not
know if the symmetry condition is needed in Theorem 1.2. It is certainly not
necessary. For many operators $A$ satisfying Assumption 1.1, one can always
apply Theorem 1.2 to its symmetric part $\frac{A+A^{*}}{2}$ and use a
perturbation argument $A=\frac{A+A^{*}}{2}+R$ with $R=\frac{A-A^{*}}{2}$ (see
[3, Theorem 3.2]). The only thing which needs to be checked is the following
relative boundedness condition: for every $\varepsilon>0$ there exists a
$C_{\varepsilon}$ such that

$\|Rv\|_{V^{*}}\leq\varepsilon\|v\|_{V}+C_{\varepsilon}\|v\|_{H}.$

For instance for $2m$-th elliptic differential operators with smooth
coefficients in space the highest order differentiation cancels out and $R$
turns out to be of order $2m-1$, and thus it satisfies the above relative
boundedness condition.

Theorem 1.2 implies that the $\mathrm{L}^{2}$-theory in [1] can be extended to
$\mathrm{L}^{p}$-theory for some range $p\in(2,p_{0}]$. Although this leads to
minimal changes in the proof, it has many consequences for applications since
for several problems it turns out that any $p>2$ is sufficient. For instance
this occurs for SPDEs in dimension $d=2$. Due to the fact that taking for
instance $H=\mathrm{H}^{1}$, $V=\mathrm{H}^{2}$ and $V^{*}=\mathrm{L}^{2}$,
the space $H$ does not embed into $\mathrm{L}^{\infty}$ as this instance of
the Sobolev embedding fails. Using $\mathrm{L}^{p}$-theory instead one only
needs that $[H,V]_{1-\frac{2}{p}}=\mathrm{H}^{2-\frac{1}{p}}$ embeds into
$\mathrm{L}^{\infty}$, which is true in dimension $d=2$. We will investigate
these applications in a subsequent paper.

### Universal compactness

Thanks to Theorem 1.2 we obtain a compactness/tightnesss result for the
solution mapping $(f,g,u_{0})\mapsto u$ for (LP) for the path space
$\mathrm{C}([0,T];H)$. We expect that this has many consequences. In earlier
works tightness of laws is obtained with $H$ replaced by $V^{*}$ or even
larger spaces for different but related settings (cf. [11] and [29]).
Tightness plays a key role in the stochastic compactness method (see [8]) for
obtaining weak solutions. Using a smaller path space can give more information
on the weak solution and its approximation.

######  Theorem 1.3 (Universal compactness for variational problems).

Suppose that the embedding $V\hookrightarrow H$ is compact. Let
$\lambda,\Lambda,T>0$, $M\geq 0$ and $K>0$ be fixed. Then there exists a
$p_{0}>2$ only depending on $\lambda,\Lambda,T\vee 1,M$ such that for all
$p\in(2,p_{0}]$, the laws $\\{\mathscr{L}(u):u\\}$ on $C([0,T];H)$ are tight,
where $u$ runs over all strong solutions to (LP) and all $(A,B)$ satisfying
Assumption 1.1 with $A=A^{*}$, and all
$f\in\mathrm{L}^{p}_{\mathcal{F}}(\Omega\times(0,T);V^{*})$,
$g\in\mathrm{L}^{p}_{\mathcal{F}}(\Omega\times(0,T);\mathcal{L}_{2}(U,H))$ and
$u_{0}\in\mathrm{L}^{p}_{\mathcal{F}_{0}}(\Omega;(H,V)_{1-\nicefrac{{2}}{{p}},p})$
satisfying $\|f\|\leq K$, $\|g\|\leq K$ and $\|u_{0}\|\leq K$ in the
respective norms.

As before, the condition $A=A^{*}$ can be weakened by a perturbation argument.

To illustrate the power of Theorem 1.3 we will show that it can be effectively
used in the stochastic compactness method to prove global existence of a
system of quasi-linear SPDEs (see Theorem 6.6 below). A technical issue in
applying the stochastic compactness method is that the a.s. limit, obtained by
tightness and the Skorohod embedding theorem, needs to be identified as a
solution. The following advantages turn up as a consequence of the uniform
estimate we obtain in the proof of Theorem 1.2:

  1. (i)

Tightness in $\mathrm{C}([0,T];H)$ (if $V\hookrightarrow H$ is compact);

  2. (ii)

uniform estimates in $\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];H))$;

  3. (iii)

uniform estimates in $\mathrm{L}^{p}(\Omega\times(0,T);V)$.

Here we let $u$ run over all solution with data $(A,B,f,g,u_{0})$ as in
Theorem 1.3. As a consequence of (1) we obtain subsequences which converge in
$\mathrm{C}([0,T];H)$. As a consequence of (2) and (3) we obtain weak
compactness in $\mathrm{L}^{p}(\Omega\times(0,T);V)$ and uniform integrability
of $\\{\|u\|_{\mathrm{C}([0,T];H)}^{2}:u\\}$ and
$\\{\|u\|_{\mathrm{L}^{2}(\Omega\times(0,T);V)}^{2}:u\\}$, where $u$ runs over
all strong solutions for the given data $(A,B,f,g,u_{0})$ The uniform
integrability can be effectively combined with Vitali’s convergence theorem.

### Application to a quasi-linear system of SPDEs

On an open and bounded set $D\subseteq\mathbb{R}^{d}$ consider the quasilinear
system

(QLP) $\displaystyle\left\\{\quad\begin{aligned}
\,\mathrm{d}u^{\alpha}&=\big{[}\partial_{i}(a^{\alpha\beta}_{ij}(u)\partial_{j}u^{\beta})+\partial_{i}\Phi^{\alpha}_{i}(u)+\phi^{\alpha}(u)\big{]}\,\mathrm{d}t\\\
&\qquad\qquad+\sum_{n\geq
1}\big{[}b^{\alpha\beta}_{n,j}(u^{\beta})\partial_{j}u^{\beta}+g^{\alpha}_{n}(u)\big{]}\,\mathrm{d}w_{n},\
\ &\text{ on }D,\\\ u^{\alpha}&=0,\ \ &\text{ on }\partial D,\\\
u^{\alpha}(0)&=u^{\alpha}_{0},\ \ &\text{ on }D.\end{aligned}\right.$

We state a special case of Theorem 6.6.

######  Theorem 1.4 (Weak existence for quasi-linear systems).

Let $h>1$. Suppose Assumption 6.1. Then there is $p_{0}>2$, depending on
$T\vee 1$, $\lambda$, $\Lambda$, $q$ and $C$ from the assumption such that, if
$p\in(2,p_{0}]$ and $q\in(ph,\infty)$, then, given
$u_{0}\in\mathrm{L}^{p}_{\mathcal{F}_{0}}(\Omega;\mathrm{B}^{1-\frac{2}{p}}_{2,p,0}(D))\cap\mathrm{L}^{q}_{\mathcal{F}_{0}}(\Omega\times
D)$, there exists a weak solution
$(\widetilde{u},\widetilde{W},\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{\mathbb{P}},(\widetilde{\mathcal{F}}_{t})_{t\geq
0})$ of (QLP).

Besides that we are able to treat highly coupled systems, even in the case of
scalar quasilinear equations the result of Theorem 1.4 contains new features:

  1. (i)

less regularity on the initial data is required;

  2. (ii)

equations on domains with boundary condition can be considered;

  3. (iii)

gradient/transport noise terms are included.

Often when applying the stochastic compactness method only the torus is
considered for simplicity. Actually, equations on domains with for instance
Dirichlet boundary condition can lead to complications:

  * •

Bootstrapping regularity can be problematic due the presence of boundary
values;

  * •

It is often unclear in what extrapolation space to formulate
compactness/tightness.

### Open problems

Given the well-posedness theory and the deterministic extrapolation results it
is natural to state the following open problem.

######  Problem 1.5.

Do Theorems 1.2 and 1.3 hold without the symmetry condition $A^{*}=A$ ?

The well-posedness theory of Lions is usually stated in the setting of
nonlinear operators $(A,B)$ satisfying a continuity, monotonicity and
coercivity condition. We covered only the linear setting in Theorems 1.2 and
1.3. This leads to the following natural question.

######  Problem 1.6.

Do Theorems 1.2 and 1.3 hold in the setting of nonlinear monotone operators
$(A,B)$ ?

This problem seems to be open even in the deterministic setting.

### Organization of the paper

In Section 2 we present the precise assumptions and state a more general
version of Theorem 1.2, and we derive Theorem 1.3 from it. In Section 3 we
present a result on analytic dependency of solutions on the equation. This
will be needed later on in order to apply Stein interpolation. In Section 4 we
discuss a case in which we have
$\mathrm{L}^{p}(\Omega\times(0,T);V)$-estimates for all $p\in[2,\infty)$,
which is used as one of the endpoints in the Stein interpolation; the other
endpoint is $\mathrm{L}^{2}(\Omega\times(0,T);V)$. Finally, in Section 5 we
prove the main result Theorem 2.2, a more general version of Theorem 1.2.

The main weak existence result on quasi-linear systems is stated in Section 6.
First, a proof in the case of Lipschitz nonlinearities will be given in
Section 7 via Theorem 1.3 and the stochastic compactness method. This
intermediate result has its own value as it imposes even less structural
assumptions. The general case is covered in Section 8, again via an a priori
estimate in $\mathrm{L}^{q}$ and the stochastic compactness method.

### Notation

  * •

The notation $a\lesssim_{P}b$ means that there is a constant $C$ only
depending on the parameter $P$ such that $a\leq Cb$.

  * •

$(V,H,V^{*})$ is the notation for the Gelfand triple of Hilbert spaces.

  * •

$(\cdot,\cdot)_{\theta,p}$ stands for real interpolation.

  * •

$[\cdot,\cdot]_{\theta}$ stands for complex interpolation.

  * •

$W$ denotes a cylindrical Brownian motion on the real separable Hilbert space
$U$.

  * •

$\mathcal{L}_{2}$ denotes the Hilbert–Schmidt operators.

For further unexplained notation the reader is referred to [1, 16, 23].

##  2\. Main extrapolation result for linear equations

We consider a linear stochastic partial differential equation of the form

(P)
$\displaystyle\,\mathrm{d}u(t)+A(t)u(t)\,\mathrm{d}t=f(t)\,\mathrm{d}t+(B(t)u(t)+g(t))\,\mathrm{d}W(t),\quad
u(0)=u_{0}.$

Let us make precise the setting and assumptions for this part.

######  Assumption 2.1.

Fix $0<T<\infty$. Let $U$ be a real separable Hilbert space, and $H$ and $V$
be complex separable Hilbert spaces, where $V\subseteq H$ continuously and
densely. Fix a probability space $\Omega$ with filtration
$\mathcal{F}=(\mathcal{F}_{t})_{0\leq t\leq T}$ and let $W$ be a real-valued
$U$-cylindrical Brownian motion adapted to $\mathcal{F}$. Consider
$A\colon\Omega\times(0,T)\to\mathcal{L}(V,V^{*})$ and
$B\colon\Omega\times(0,T)\to\mathcal{L}(V,\mathcal{L}_{2}(U,H))$, where
$\mathcal{L}_{2}(U,H)$ denotes the space of Hilbert Schmidt operators from $U$
to $H$, both progressively measurable. Suppose that there exist
$\Lambda(A),\Lambda(B),\lambda>0$ and $M\geq 0$ such that pointwise on
$[0,T]\times\Omega$ and $v\in V$ one has

(3) $\displaystyle\operatorname{Re}\,\langle
Av,v\rangle-\frac{1}{2}\|Bv\|_{\mathcal{L}_{2}(U,H)}^{2}\geq\lambda\|v\|_{V}^{2}-M\|v\|_{H}^{2},$

and

(4)
$\displaystyle\|Av\|_{V^{*}}\leq\Lambda(A)\|v\|_{V}\quad\&\quad\|Bv\|_{\mathcal{L}_{2}(U,H)}^{2}\leq\Lambda(B)\|v\|_{V}^{2}.$

Put $\Lambda=\Lambda(A)+\Lambda(B)$.

The main result of this part is the following.

######  Theorem 2.2.

Suppose that Assumption 2.1 holds and $\mapsto\langle Au,v\rangle=\langle
Av,u\rangle$ pointwise in $[0,T]\times\Omega$. Then there exists a $p_{0}>2$
such that, for all $p\in[2,p_{0}]$,
$f\in\mathrm{L}^{p}_{\mathcal{F}}(\Omega\times(0,T);V^{*})$,
$g\in\mathrm{L}^{p}_{\mathcal{F}}(\Omega\times(0,T);\mathcal{L}_{2}(U,H))$ and
$u_{0}\in\mathrm{L}^{p}_{\mathcal{F}_{0}}(\Omega;(H,V)_{1-\nicefrac{{2}}{{p}},p})$,
problem (P) admits a unique solution
$u\in\mathrm{L}^{p}(\Omega\times(0,T);V)$, and for any
$\theta\in[0,\nicefrac{{1}}{{2}})$ there is a constant $K_{\theta}$ depending
only on $\Lambda(A)$, $\Lambda(B)$, $\lambda$, $M$, $T\vee 1$, $\theta$ such
that

(5)
$\displaystyle\|u\|_{\mathrm{L}^{p}(\Omega;\mathrm{H}^{\theta,p}(0,T;[H,V]_{1-2\theta}))}\leq
K_{\theta}C_{u_{0},f,g},$

where

(6) $\displaystyle
C_{u_{0},f,g}=\|u_{0}\|_{\mathrm{L}^{p}(\Omega;(H,V)_{1-\nicefrac{{2}}{{p}},p})}+\|f\|_{\mathrm{L}^{p}(\Omega\times(0,T);V^{*})}+\|g\|_{\mathrm{L}^{p}(\Omega\times(0,T);\mathcal{L}_{2}(U,H))}.$

Also, there is a constant $K_{0}$ depending only on $\Lambda(A)$,
$\Lambda(B)$, $\lambda$, $M$, $T\vee 1$, and $p$ such that

(7)
$\displaystyle\|u\|_{\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];(H,V)_{1-\nicefrac{{2}}{{p}},p}))}\leq
K_{0}C_{u_{0},f,g}.$

Moreover, for $\theta\in(\nicefrac{{1}}{{p}},\nicefrac{{1}}{{2}})$, there is a
constant $K_{\theta}$ depending only on $\Lambda(A)$, $\Lambda(B)$, $\lambda$,
$T\vee 1$, and $\theta$ such that

(8)
$\displaystyle\|u\|_{\mathrm{L}^{p}(\Omega;\mathrm{C}^{\theta-\nicefrac{{1}}{{p}}}([0,T];[H,V]_{1-2\theta}))}$
$\displaystyle\leq K_{\theta}C_{u_{0},f,g}.$

The symbol $\mathrm{H}^{\theta,p}$ denotes the (vector-valued) Bessel
potential space (see [16, 18]), and the subscripts $\mathcal{F}$ and
$\mathcal{F}_{0}$ refer to the subspaces of $\mathrm{L}^{p}$ of
$\mathcal{F}$-progressively measurable and $\mathcal{F}_{0}$-adapted
functions.

######  Remark 2.3 (real problems).

In virtue of complexification, the result does also apply to problems over
real Hilbert spaces. We are going to use this fact later on.

As a consequence of Theorem 2.2 we can already prove the compactness/tightness
result of Theorem 1.3.

###### Proof of Theorem 1.3.

Let $S$ denote the set of solutions described in Theorem 1.3. Fix
$\theta\in(1/p,1/2)$. By Theorem 2.2, $S$ is bounded in
$\mathrm{L}^{p}(\Omega;C^{\theta-\frac{1}{p}}([0,T];[H,V]_{1-2\theta}))$ by
$K_{\theta}C_{u_{0},f,g}$. Let $\varepsilon>0$. Choose $R>0$ such that
$K_{\theta}^{p}C_{u_{0},f,g}^{p}R^{-p}\leq\varepsilon$. Then for all $u\in S$,

$\mathbb{P}(\|u\|_{\mathrm{C}^{\theta-\frac{1}{p}}([0,T];[H,V]_{1-2\theta})}\geq
R)\leq
R^{-p}\mathbb{E}\|u\|_{\mathrm{C}^{\theta-\frac{1}{p}}([0,T];[H,V]_{1-2\theta})}\leq\varepsilon.$

By the vector-valued Arzéla-Ascoli theorem [21, Theorem III.3.1], the
embedding

$\mathrm{C}^{\theta-\frac{1}{p}}([0,T];[H,V]_{1-2\theta})\hookrightarrow\mathrm{C}([0,T];H)$

is compact, and therefore $\\{\mathcal{L}(u):u\in S\\}$ is tight. ∎

##  3\. $\mathrm{L}^{2}$-estimates and analytic dependency

The central finding of this section is that solutions to a complex family of
variational problems have analytic dependence provided this was the case for
the complex family, see Proposition 3.3 and Corollary 3.4. Before, we will
briefly discuss well-posedness of (P) in a complex setting.

To ease notation, we introduce data and solution spaces for the variational
setting.

######  Definition 3.1.

Define the data spaces
$\mathrm{E}_{0}\coloneqq\mathrm{L}^{2}_{\mathcal{F}}(\Omega\times(0,T);V^{*})$
and
${\mathbb{E}_{\frac{1}{2}}}\coloneqq\mathrm{L}^{2}_{\mathcal{F}}(\Omega\times(0,T);\mathcal{L}_{2}(U,H))$,
and the solution space
$\mathrm{E}_{1}\coloneqq\mathrm{L}^{2}_{\mathcal{F}}(\Omega\times(0,T);V)\cap\mathrm{L}^{2}_{\mathcal{F}}(\Omega;\mathrm{C}([0,T];H))$,
where the subscript $\mathcal{F}$ indicates the subspace of progressively
measurable functions.

###  3.1. A (complex) variational setting

The following variational well-posedness result is well-known in its real
formulation [23]. The complex version follows by “forgetting” the complex
structure. For instance, we can interpret a complex vector space $H$ as a real
vector space if we equip it with the inner product
$(\cdot|\cdot)_{H_{\mathbb{R}}}=\operatorname{Re}(\cdot|\cdot)_{H}$ and so on.

######  Proposition 3.2.

Let $f\in\mathrm{E}_{0}$, $g\in{\mathbb{E}_{\frac{1}{2}}}$. Suppose Assumption
2.1 holds. Then (P) with $u_{0}\in\mathrm{L}^{2}_{\mathcal{F}_{0}}(H)$ has a
unique solution $u\in\mathrm{E}_{1}$, and there exists a constant $C_{L}$
depending on $\Lambda$, $\lambda$, $M$, and $T\vee 1$, such that

(9) $\displaystyle\|u\|_{\mathrm{E}_{1}}\leq
C_{L}\bigl{(}\|u_{0}\|_{\mathrm{L}^{2}(H)}+\|f\|_{\mathrm{E}_{0}}+\|g\|_{{\mathbb{E}_{\frac{1}{2}}}}\bigr{)}.$

###  3.2. Analytic dependence of solutions

In the following result we show analytic dependence of solutions.

######  Proposition 3.3.

Let $O\subseteq\mathbb{C}$ be open and for $z\in O$ let $A_{z}$ and $B_{z}$ be
operator functions that satisfy Assumption 2.1 uniformly in $z$ and that
depend analytically on $z$ in the uniform operator topology. Furthermore, let
$f\in\mathrm{E}_{0}$, $g\in{\mathbb{E}_{\frac{1}{2}}}$ and $u_{0}\in H$. Then,
for fixed $z$, the unique solution $u_{z}$ of the problem

(10)
$\displaystyle\,\mathrm{d}u(t)+A_{z}(t)u(t)\,\mathrm{d}t=f(t)\,\mathrm{d}t+(B_{z}(t)u(t)+g(t))\,\mathrm{d}W(t),\quad
u(0)=u_{0},$

gives rise to an analytic function $O\ni z\mapsto u_{z}\in\mathrm{E}_{1}$.

###### Proof.

Since we consider differences below, we have zero initial data in the
equations in this proof. For fixed $z$, Proposition 3.2 yields a unique
solution $u_{z}$ of (10).

Step 1: $O\ni z\mapsto u_{z}\in\mathrm{E}_{1}$ is continuous.

Let $z\in O$ and $h$ small enough so that $z+h\in O$. The function $v\coloneqq
u_{z+h}-u_{z}$ is the unique solution of

(11)
$\displaystyle\,\mathrm{d}v(t)+A_{z+h}v(t)\,\mathrm{d}t=-(A_{z+h}-A_{z})u_{z}\,\mathrm{d}t+(B_{z+h}v(t)+(B_{z+h}-B_{z})u_{z})\,\mathrm{d}W(t).$

Therefore, by Proposition 3.2,

(12) $\displaystyle\|v\|_{\mathrm{E}_{1}}$ $\displaystyle\leq
C_{L}\bigl{(}\|(A_{z+h}-A_{z})u_{z}\|_{\mathrm{E}_{0}}+\|(B_{z+h}-B_{z})u_{z}\|_{{\mathbb{E}_{\frac{1}{2}}}}\bigr{)}.$

Hence, continuity of $A_{z}$ and $B_{z}$ yield the claim.

Step 2: $O\ni z\mapsto u_{z}\in\mathrm{E}_{1}$ is analytic.

Write $D^{h}$ for difference quotients of $u$, $A$, and $B$ with respect to
$z$, for example $D^{h}u=\frac{u(z+h)-u(z)}{h}$. One has that $D^{h}u$ is the
unique solution of

(13)
$\displaystyle\,\mathrm{d}v(t)+A_{z}v\,\mathrm{d}t=-(D^{h}A_{z})u_{z+h}\,\mathrm{d}t+(B_{z}v+(D^{h}B_{z})u_{z+h})\,\mathrm{d}W(t).$

We show that $D^{h}u$ is a Cauchy sequence in $\mathrm{E}_{1}$. Then, by the
very definition of complex differentiability, $z\mapsto u_{z}$ is analytic. To
this end, consider $h_{1}\neq h_{2}$. The difference $w\coloneqq
D^{h_{1}}u-D^{h_{2}}u$ solves

(14)
$\displaystyle\,\mathrm{d}w(t)+A_{z}w\,\mathrm{d}t=-\Bigl{(}\bigl{(}D^{h_{1}}A-D^{h_{2}}A\bigr{)}u_{z+h_{1}}+(D^{h_{2}}A)(u_{z+h_{1}}-u_{z+h_{2}})\Bigr{)}\,\mathrm{d}t$
(15)
$\displaystyle+\Bigl{(}B_{z}w+\bigl{(}D^{h_{1}}B-D^{h_{2}}B\bigr{)}u_{z+h_{1}}+(D^{h_{2}}B)(u_{z+h_{1}}-u_{z+h_{2}})\Bigr{)}\,\mathrm{d}W(t).$

A calculation using Proposition 3.2 shows

(16) $\displaystyle\|D^{h_{1}}u-D^{h_{2}}u\|_{\mathrm{E}_{1}}$ (17)
$\displaystyle\leq{}$ $\displaystyle
C_{L}^{2}\Bigl{(}\|D^{h_{1}}A-D^{h_{2}}A\|_{\mathcal{L}(V,V^{*})}+\|D^{h_{1}}B-D^{h_{2}}B\|_{\mathcal{L}(V,\mathcal{L}_{2}(U,H))}\Bigr{)}(\|f\|_{\mathrm{E}_{0}}+\|g\|_{{\mathbb{E}_{\frac{1}{2}}}})$
(18)
$\displaystyle\quad+C_{L}\Bigl{(}\|D^{h_{2}}A\|_{\mathcal{L}(V,V^{*})}+\|D^{h_{2}}B\|_{\mathcal{L}(V,\mathcal{L}_{2}(U,H))}\Bigr{)}\|u_{z+h_{1}}-u_{z+h_{2}}\|_{\mathrm{E}_{1}}.$

First, the difference quotients of $A$ and $B$ are Cauchy sequences for $A$
and $B$ are holomorphic. Second, the difference quotients of $A$ and $B$ are,
again by analyticity, uniformly bounded. Third,
$\|u_{z+h_{1}}-u_{z+h_{2}}\|_{\mathrm{E}_{1}}$ tends to zero by Step 1. We
conclude that $D^{h}u$ is a Cauchy sequence as claimed. ∎

######  Corollary 3.4.

Let $O\subseteq\mathbb{C}$ be open and for $z\in O$ let $A_{z}$ and $B_{z}$ be
operator functions that satisfy Assumption 2.1 uniformly in $z$ and that
depend analytically on $z$ in the uniform operator topology. Furthermore, let
$f:O\to\mathrm{E}_{0}$, $g:O\to{\mathbb{E}_{\frac{1}{2}}}$ and $u_{0}:O\to H$
be analytic. Then, for fixed $z$, the unique solution $u_{z}$ of the problem

(19)
$\displaystyle\,\mathrm{d}u(t)+A_{z}(t)u(t)\,\mathrm{d}t=f_{z}(t)\,\mathrm{d}t+(B_{z}(t)u(t)+g_{z}(t))\,\mathrm{d}W(t),\quad
u(0)=u_{0},$

gives rise to an analytic function $O\ni z\mapsto u_{z}\in\mathrm{E}_{1}$.

###### Proof.

From Proposition 3.3 it follows that
$S_{z}:O\to\mathcal{L}(\mathrm{E}_{0}\times{\mathbb{E}_{\frac{1}{2}}}\times
H,\mathrm{E}_{1})$ given by $S_{z}=u_{z}$, where $u_{z}$ is the solution to
(10) is analytic in the strong operator topology. Therefore, $S$ is analytic
in the uniform operator topology (see [4, Proposition A.3]). Now the
analyticity follows as the solution to (19) is given by
$S_{z}(f_{z},g_{z},u_{0,z})$ and is the composition of analytic functions. ∎

##  4\. $\mathrm{L}^{p}$-estimates for small perturbations of the autonomous
case

In this section, we establish $\mathrm{L}^{p}$-theory by means of a
perturbation result with the autonomous case. The quantities in the assumption
of Theorem 4.6 look cumbersome on the first glance, but we will see in Section
5 that they are, in fact, very closely tied to the concept of coercivity.

Throughout this section, fix some $p>2$. We extend Definition 3.1.

######  Definition 4.1.

Put
$\mathrm{E}^{p}_{0}\coloneqq\mathrm{L}^{p}_{\mathcal{F}}(\Omega\times(0,T);V^{*})$
and
${\mathbb{E}^{p}_{\frac{1}{2}}}\coloneqq\mathrm{L}^{p}_{\mathcal{F}}(\Omega\times(0,T);\mathcal{L}_{2}(U,H))$
for the data spaces and
$\mathrm{E}^{p}_{1}\coloneqq\mathrm{L}^{p}_{\mathcal{F}}(\Omega\times(0,T);V)\cap\mathrm{L}^{p}_{\mathcal{F}}(\Omega;\mathrm{C}([0,T];(H,V)_{1-\nicefrac{{2}}{{p}},p}))$
for the solution space.

Let us introduce a reference operator for the perturbation argument.

######  Definition 4.2.

Consider the operator $A_{0}\colon V\to V^{*}$ given by $\langle
A_{0}(u),v\rangle\coloneqq(u|v)_{V}$.

######  Remark 4.3.

The operator $A_{0}$ is invertible, positive and self-adjoint.

The following deterministic maximal $L^{p}$-regularity result holds.

######  Lemma 4.4.

There is a constant $C_{p}$ such that, for all $\mu>0$,
$f\in\mathrm{L}^{p}(0,T;V^{*})$, and $u$ a strong solution to $u^{\prime}+(\mu
A_{0})u=f$ with $u(0)=0$, one has the a-priori estimate

(20) $\displaystyle\mu\|u\|_{\mathrm{L}^{p}(0,T;V)}\leq
C_{p}\|f\|_{\mathrm{L}^{p}(0,T;V^{*})}.$

###### Proof.

The fact that $A_{0}$ has maximal $L^{p}$-regularity on $\mathbb{R}_{+}$
follows from de Simon’s result (see [18, Corollary 17.3.8]). From the proof of
[18, Theorem 17.2.26] it follows that $\mu A_{0}$ has maximal
$L^{p}$-regularity on $\mathbb{R}_{+}$ with the same constant $C_{p}$.
Therefore, by [18, Lemma 17.2.16]), $\mu A_{0}$ has maximal $L^{p}$-regularity
on $(0,T)$ with the same constant $C_{p}$. This gives the desired bound with
constant independent of $\mu$ ∎

Next we will present a similar result in the stochastic setting, but without
the optimal scaling in $\mu$ as this will not be needed later.

######  Lemma 4.5.

For any $\mu>0$, the operator $\mu A_{0}$ admits stochastic maximal
$\mathrm{L}^{p}$-regularity, that is to say, there is $c(p,\mu)>0$ such that
for all $f\in\mathrm{E}^{p}_{0}$ and $g\in{\mathbb{E}^{p}_{\frac{1}{2}}}$
there is a unique strong solution $u\in\mathrm{E}_{1}$ to

(21) $\displaystyle\,\mathrm{d}u(t)+\mu
A_{0}u\,\mathrm{d}t=f(t)\,\mathrm{d}t+g(t)\,\mathrm{d}W(t),\qquad u(0)=0,$

satisfying the estimate

(22) $\displaystyle\|u\|_{\mathrm{E}^{p}_{1}}\leq
c(p,\mu)\bigl{(}\|f\|_{\mathrm{E}^{p}_{0}}+\|g\|_{{\mathbb{E}^{p}_{\frac{1}{2}}}}\bigr{)}.$

###### Proof.

Let $\mu>0$. By linearity and Lemma 4.4 it suffices to consider $f=0$. As $\mu
A_{0}$ is positive and self-adjoint on $V^{*}$, it is also positive and self-
adjoint on $H$ (see [26, Proposition 1.24]). Therefore, it has a bounded
$\mathrm{H}^{\infty}$-calculus with constant $1$ (see [17, Proposition
10.2.23]). Moreover, $H$ is, as a separable Hilbert space, isomorphic to
$\mathrm{L}^{2}(\mathbb{R})$. Therefore, from the definition of $A_{0}$ and
from [25] we obtain

$\mu^{1/2}\|u\|_{\mathrm{E}^{p}_{1}}=\|(\mu
A_{0})^{1/2}u\|_{L^{p}(\mathbb{R}_{+};H)}\leq
c(p)\|g\|_{L^{p}(\mathbb{R}_{+};\mathcal{L}_{2}(U,H))}=c(p)\|g\|_{{\mathbb{E}^{p}_{\frac{1}{2}}}}.$

∎

######  Theorem 4.6.

Let $f\in\mathrm{E}^{p}_{0}$ and $g\in{\mathbb{E}^{p}_{\frac{1}{2}}}$. Employ
Assumptions 2.1 with $M=0$ and recall the constant $\Lambda(B)$. Suppose that
there are $\varepsilon,\delta>0$ such that

  1. (i)

for some $\mu>0$ one has $C_{p}\mu^{-1}\|A-\mu
A_{0}\|_{\mathcal{L}(V,V^{*})}\leq 1-\varepsilon$, where $C_{p}$ is the
constant from Lemma 4.4, and

  2. (ii)

$c\Lambda(B)\leq 1-\delta$, where $c=c(\Lambda,\mu,T,\varepsilon,p)$.

Then there exists a unique strong solution $u\in\mathrm{E}^{p}_{1}$ to (P)
with $u_{0}=0$ satisfying the estimate

(23)
$\displaystyle\|u\|_{\mathrm{E}^{p}_{1}}\leq\frac{c}{\delta}\bigl{(}\|f\|_{\mathrm{E}^{p}_{0}}+\|g\|_{{\mathbb{E}^{p}_{\frac{1}{2}}}}\bigr{)}.$

###### Proof.

We tacitly impose the initial condition $u(0)=0$ whenever we speak about
equations in this proof. First, we investigate the problem with $B=0$, that is
to say, we want to show that there exists a unique strong solution
$u\in\mathrm{E}^{p}_{1}$ to the stochastic problem

(24)
$\displaystyle\,\mathrm{d}u(t)+A(t)u(t)\,\mathrm{d}t=f(t)\,\mathrm{d}t+g(t)\,\mathrm{d}W(t)$

satisfying the estimate

(25) $\displaystyle\|u\|_{\mathrm{E}^{p}_{1}}\leq
c(\|f\|_{\mathrm{E}^{p}_{0}}+\|g\|_{\mathrm{E}^{p}_{0}})$

with a constant $c$ depending only on $\varepsilon$, $\Lambda$, $\mu$, $T\vee
1$, and $p$. This constant is the constant postulated in hypothesis (ii) of
the theorem.

Step 1: Reduction to an a-priori estimate.

For $\theta\in[0,1]$ consider $A_{\theta}\coloneqq(1-\theta)\mu A_{0}+\theta
A$. Since $A_{\theta}-\mu A_{0}=\theta(A-\mu A_{0})$, one has

$C_{p}\mu^{-1}\|A_{\theta}-\mu
A_{0}\|_{\mathcal{L}(V,V^{*})}=C_{p}\mu^{-1}\theta\|A-\mu
A_{0}\|_{\mathcal{L}(V,V^{*})}\leq 1-\varepsilon.$

Therefore, in the light of a stochastic version of the method of continuity
(see [28, Prop. 3.10]), it suffices to show the a priori estimate (25) for any
strong solution $u$ of (24).

Step 2: Show the a-priori estimate.

Let $u$ a strong solution of (24). By Lemma 4.5 there is a strong solution
$v\in\mathrm{E}^{p}_{1}$ of

$\,\mathrm{d}v(t)+(\mu
A_{0})v(t)\,\mathrm{d}t=f(t)\,\mathrm{d}t+g(t)\,\mathrm{d}W(t)$

satisfying the estimate

(26) $\displaystyle\|v\|_{\mathrm{E}^{p}_{1}}\leq
c(p,\mu)(\|f\|_{\mathrm{E}^{p}_{0}}+\|g\|_{{\mathbb{E}^{p}_{\frac{1}{2}}}}).$

Define $w\coloneqq u-v\in\mathrm{E}^{p}_{1}$. Fix $\omega\in\Omega$. Use the
shorthand notation $w_{\omega}(t)=w(\omega,t)$, and so on. By construction,
$w_{\omega}$ is almost surely a strong solution to the deterministic problem

$\partial_{t}w_{\omega}+(\mu A_{0})w_{\omega}=-(A_{\omega}-\mu
A_{0})u_{\omega}.$

Lemma 4.4 lets us compute

(27) $\displaystyle\mu\|w_{\omega}\|_{\mathrm{L}^{p}(0,T;V)}$
$\displaystyle\leq C_{p}\|(A_{\omega}-\mu
A_{0})u_{\omega}\|_{\mathrm{L}^{p}(0,T;V^{*})}$ (28) $\displaystyle\leq
C_{p}\|A_{\omega}-\mu
A_{0}\|_{\mathcal{L}(V,V^{*})}\Bigl{(}\|v_{\omega}\|_{\mathrm{L}^{p}(0,T;V)}+\|w_{\omega}\|_{\mathrm{L}^{p}(0,T;V)}\Bigr{)}.$

Divide this by $\mu$ and use the assumption to give

(29) $\displaystyle\|w_{\omega}\|_{\mathrm{L}^{p}(0,T;V)}$ $\displaystyle\leq
C_{p}\mu^{-1}\|(A_{\omega}-\mu
A_{0})\|_{\mathcal{L}(V,V^{*})}\Bigl{(}\|v_{\omega}\|_{\mathrm{L}^{p}(0,T;V)}+\|w_{\omega}\|_{\mathrm{L}^{p}(0,T;V)}\Bigr{)}$
(30)
$\displaystyle\leq(1-\varepsilon)\|v_{\omega}\|_{\mathrm{L}^{p}(0,T;V)}+(1-\varepsilon)\|w_{\omega}\|_{\mathrm{L}^{p}(0,T;V)}.$

Absorb the second term on the right-hand side into the left-hand side to
deduce

(31)
$\displaystyle\|w_{\omega}\|_{\mathrm{L}^{p}(0,T;V)}\leq\frac{1-\varepsilon}{\varepsilon}\|v_{\omega}\|_{\mathrm{L}^{p}(0,T;V)}.$

According to [28, Step 1 in Thm. 3.9], $u$ is progressively measurable. Hence,
averaging the last bound over $\Omega$ yields

(32)
$\displaystyle\|w\|_{\mathrm{L}^{p}(\Omega\times(0,T);V)}\leq\frac{1-\varepsilon}{\varepsilon}\|v\|_{\mathrm{L}^{p}(\Omega\times(0,T);V)}.$

In conjunction with $u=v+w$ and (26) we obtain in summary

(33) $\displaystyle\|u\|_{\mathrm{L}^{p}(\Omega\times(0,T);V)}$
$\displaystyle\leq\|v\|_{\mathrm{L}^{p}(\Omega\times(0,T);V)}+\|w\|_{\mathrm{L}^{p}(\Omega\times(0,T);V)}\leq\frac{c(p,\mu)}{\varepsilon}\bigl{(}\|f\|_{\mathrm{E}^{p}_{0}}+\|g\|_{{\mathbb{E}^{p}_{\frac{1}{2}}}}\bigr{)}.$

Step 3: Including $B$.

As is standard (see [28, Prop. 3.10], for instance), we solve the problem (P)
with $u_{0}=0$,

(34)
$\displaystyle\,\mathrm{d}u(t)+A(t)u(t)\,\mathrm{d}t=f(t)\,\mathrm{d}t+(B(t)u(t)+g(t))\,\mathrm{d}W(t),$

using a fixed-point argument. For $\Phi\in\mathrm{E}^{p}_{1}$, replace $Bu$ by
$B\Phi$ in (34). This problem possesses a unique solution in
$\mathrm{E}^{p}_{1}$ in virtue of the first part of the proof. Write $R(\Phi)$
for it. To show unique solvability of (34), it suffices to show that $R$ is a
strict contraction.

Let $\Phi_{1},\Phi_{2}\in\mathrm{E}^{p}_{1}$. By linearity,
$R(\Phi_{1})-R(\Phi_{2})$ is the unique strong solution to the problem

$\,\mathrm{d}u(t)+A(t)u(t)\,\mathrm{d}t=B(t)(\Phi_{1}(t)-\Phi_{2}(t))\,\mathrm{d}W(t).$

Hence, (25) with $f=0$ and $g=B(\Phi_{1}-\Phi_{2})$ gives

(35) $\displaystyle\|R(\Phi_{1})-R(\Phi_{2})\|_{\mathrm{E}^{p}_{1}}$
$\displaystyle\leq
C\|B(\Phi_{1}-\Phi_{2})\|_{{\mathbb{E}^{p}_{\frac{1}{2}}}}\leq
C\Lambda(B)\|\Phi_{1}-\Phi_{2}\|_{\mathrm{E}^{p}_{1}}.$

Thus, $R$ is a strict contraction by the assumption $C\Lambda(B)\leq
1-\delta$.

For the estimate, apply the a-priori estimate (25) once more to find

(36) $\displaystyle\|u\|_{\mathrm{E}^{p}_{1}}$ $\displaystyle\leq
C(\|f\|_{\mathrm{E}^{p}_{0}}+\|g\|_{{\mathbb{E}^{p}_{\frac{1}{2}}}}+\|Bu\|_{{\mathbb{E}^{p}_{\frac{1}{2}}}})\leq
C(\|f\|_{\mathrm{E}^{p}_{0}}+\|g\|_{{\mathbb{E}^{p}_{\frac{1}{2}}}})+(1-\delta)\|u\|_{\mathrm{E}^{p}_{1}}.$

We can absorb the second term of the right-hand side into the left-hand side
to conclude. ∎

##  5\. Extrapolation of integrability in time

Our main result follows from an interpolation argument using abstract Stein
interpolation [32]. For details on interpolation the reader is referred to [6,
16, 31]. The last two sections constitute the endpoint cases of this
interpolation. Before we can conclude in Section 5.2, we have to construct an
analytic family of auxiliary problems first.

######  Assumption 5.1.

For fixed $t$, the form $V\times V\ni(u,v)\mapsto\langle
A(\omega,t)u,v\rangle$ is hermitian almost surely. To ease notation, write
$a(u,v)\coloneqq a_{\omega,t}(u,v)\coloneqq\langle A(\omega,t)u,v\rangle$.

This symmetry assumption is used in a crucial way in Lemmas 5.4 and 5.7.

######  Convention 5.2.

We ignore the dependence on $(\omega,t)$ in our notation. For instance, we let
$A$ denote the operator $V\to V^{*}$ given by $\langle
Au,v\rangle\coloneqq\langle A(\omega,t)u,v\rangle$. Also, we put
$a(u,v)\coloneqq\langle Au,v\rangle$ and $a[u]\coloneqq a(u,u)$.

The following lemma translates upper bounds on the diagonal of $V\times V$ to
the whole form. That the optimal constant is $1$ is a consequence of
Assumption 5.1, but is actually not needed later. A simpler polarization
argument without Assumption 5.1 would lead to an extra factor $2$.

######  Lemma 5.3.

Suppose that Assumption 5.1 holds. If for some $c>0$ one has $|a[u]|\leq
c\|u\|_{V}^{2}$, then also $|a(u,v)|\leq c\|u\|_{V}\|v\|_{V}$ for all $u,v\in
V$.

###### Proof.

Let $u,v\in V$. We can of course assume $u\neq 0\neq v$. There is a complex
number $z$ with $|z|=1$ such that $za(u,v)$ is real and positive. By the
polarization identity one has

(37) $\displaystyle
za(u,v)=a(zu,v)=\frac{1}{4}\bigl{(}a[zu+v]-a[zu-v]+\mathrm{i}a[zu+\mathrm{i}v]-\mathrm{i}a[zu-\mathrm{i}v]\bigr{)}.$

Since $a$ is hermitian, $a[u]$ is real for any $u\in V$. Hence, taking the
real part of (37) gives

$|a(u,v)|=za(u,v)=\frac{1}{4}\bigl{(}a[zu+v]-a[zu-v]\bigr{)}.$

Expanding the quadratic forms in conjunction with the bound on the diagonal
leads to

(38)
$\displaystyle|a(u,v)|=\frac{1}{4}\bigl{(}2a[zu]+2a[v]\bigr{)}\leq\frac{c}{2}\bigl{(}\|u\|_{V}+\|v\|_{V}\bigr{)}.$

It remains to use $|a(u,v)|=|a(\lambda u,\lambda^{-1}v)|$, apply (38), and
minimize over $\lambda>0$. ∎

###  5.1. Complex family of auxiliary problems

######  Lemma 5.4.

Suppose that Assumption 2.1 holds with $M=0$ and that Assumption 5.1 holds.
Put $\mu=\Lambda(A)$ and $\rho=\frac{\lambda}{\Lambda(A)}$. Then one has

$\Bigl{|}\mu^{-1}\Bigl{(}a[v]-\frac{1}{2}\|Bv\|_{\mathcal{L}_{2}(U,H)}^{2}\Bigr{)}-\|v\|_{V}^{2}\Bigr{|}\leq(1-\rho)\|v\|_{V}^{2}.$

###### Proof.

To ease notation, put $c[v]\coloneqq
a[v]-\frac{1}{2}\|Bv\|_{\mathcal{L}_{2}(U,H)}^{2}$. By Assumptions 2.1 (with
$M=0$) and 5.1 one has $\lambda\|v\|_{V}^{2}\leq
c[v]\leq\Lambda(A)\|v\|_{V}^{2}$. Let $s\geq 0$. Assumption 5.1 implies that
$sc[v]-\|v\|_{V}^{2}$ is real, hence

(39)
$\displaystyle|sc[v]-\|v\|_{V}^{2}|=\max(sc[v]-\|v\|_{V}^{2},\|v\|_{V}^{2}-sc[v]).$

Consider the first term. Since

$sc[v]-\|v\|_{V}^{2}\leq(s\Lambda(A)-1)\|v\|_{V}^{2},$

the maximum in (39) coincides with the second term for any choice
$s\leq\frac{1}{\Lambda(A)}$. On the other hand,

$\|v\|_{V}^{2}-sc[v]\leq(1-s\lambda)\|v\|_{V}^{2},$

so the right-hand side of the last estimate is minimal for the maximal
admissible choice $s=(\Lambda(A))^{-1}$. Put $\mu=s^{-1}=\Lambda(A)$ and
$\rho=s\lambda=\frac{\lambda}{\Lambda(A)}$. In summary, we obtain

(40) $\displaystyle|\mu^{-1}c[v]-\|v\|_{V}^{2}|\leq(1-\rho)\|v\|_{V}^{2}.$

The same calculation but applied to $a[v]$ instead of
$a[v]-\frac{1}{2}\|Bv\|_{\mathcal{L}_{2}(U,H)}^{2}$ in conjunction with Lemma
5.3 gives the following.

######  Corollary 5.5.

Suppose that Assumption 2.1 holds with $M=0$ and that Assumption 5.1 holds.
For the same $\mu$ and $\rho$ as in Lemma 5.4 one has
$\|\mu^{-1}A-A_{0}\|_{\mathcal{L}(V,V^{*})}\leq 1-\rho$.

We use $\mu$ and $\rho$ from the lemma to define complex perturbations of $A$
and $B$.

######  Definition 5.6.

With $\mu$ and $\rho$ from Lemma 5.4, fix numbers $0<r<1<R$ satisfying

  1. (i)

$R(1-\rho)<1$, and

  2. (ii)

$r\min(C_{p}(1-\rho),c\Lambda(B))<1$, where $C_{p}$ is the constant from Lemma
4.4 and $c$ is the constant from Theorem 4.6.

Moreover, with the holomorphic function $F(z)=r\mathrm{e}^{z\log(R/r)}$
defined on the open unit strip $S\subseteq\mathbb{C}$, define

(41) $\displaystyle
A_{z}\coloneqq\mu\bigl{(}F(z)(\mu^{-1}A-A_{0})+A_{0}\bigr{)}\quad\&\quad
B_{z}\coloneqq F(z)^{\frac{1}{2}}B.$

######  Lemma 5.7.

Suppose that Assumption 2.1 holds with $M=0$ and that Assumption 5.1 holds.
Then the coercivity condition (3) in Assumption 2.1 is satisfied for $A_{z}$
and $B_{z}$, uniformly in $z\in\overline{S}$. To be more precise, the implied
constant depends only on $\lambda$, $\Lambda(A)$, and $R$.

###### Proof.

Write $\|\cdot\|_{\mathcal{L}_{2}}\coloneqq\|\cdot\|_{\mathcal{L}_{2}(U,H)}$.
Recall $\mu=\Lambda(A)$ from Lemma 5.4. Consequently,
$\|\mu^{-1}A\|_{\mathcal{L}(V,V^{*})}\leq 1$. Since $a$ is hermitian by
Assumption 5.1, it follows that $\mu^{-1}a[v]-\|v\|_{V}^{2}$ is real and non-
positive. Recall

$\mu^{-1}A_{z}=F(z)(\mu^{-1}A-A_{0})+A_{0}.$

Calculate for $\|v\|_{V}=1$ that

(42)
$\displaystyle\operatorname{Re}\mu^{-1}\bigl{(}a_{z}[v]-\frac{1}{2}\|B_{z}(v)\|_{\mathcal{L}_{2}}^{2}\bigr{)}$
$\displaystyle={}1+\bigl{(}\operatorname{Re}F(z)\bigr{)}(\mu^{-1}a[v]-\|v\|_{V}^{2})-\frac{|F(z)|}{2\mu}\|B(v)\|_{\mathcal{L}_{2}}^{2}$
(43)
$\displaystyle\geq{}1+|F(z)|(\mu^{-1}a[v]-\|v\|_{V}^{2})-\frac{|F(z)|}{2\mu}\|B(v)\|_{\mathcal{L}_{2}}^{2}$
(44)
$\displaystyle={}1+|F(z)|\Bigl{(}\mu^{-1}\bigl{(}a[v]-\frac{1}{2}\|B(v)\|_{\mathcal{L}_{2}}^{2}\bigr{)}-\|v\|_{V}^{2}\Bigr{)}$
(45)
$\displaystyle\geq{}1-|F(z)|\Bigl{|}\mu^{-1}\bigl{(}a[v]-\frac{1}{2}\|B(v)\|_{\mathcal{L}_{2}}^{2}\bigr{)}-\|v\|_{V}^{2}\Bigr{|},$

where we exploited in the first line that $\mu^{-1}a[v]-\|v\|_{V}^{2}$ is real
and the second line that it is non-positive. Therefore, Lemma 5.4 reveals

(46)
$\displaystyle\operatorname{Re}\mu^{-1}\bigl{(}a_{z}[v]-\frac{1}{2}\|B_{z}(v)\|_{\mathcal{L}_{2}}^{2}\bigr{)}\geq
1-|F(z)|(1-\rho),$

so that the right-hand side is strictly positive in virtue of Definition 5.6
(i). Eventually, multiplying by $\mu$ completes the proof. ∎

Now consider on the unit strip $S$ the complex family of problems

($P_{z}$)
$\displaystyle\,\mathrm{d}u(t)+A_{z}(t)u(t)\,\mathrm{d}t=f(t)\,\mathrm{d}t+(B_{z}(t)u(t)+g(t))\,\mathrm{d}W(t)$

with initial condition $u(0)=0$.

By construction, the mappings $z\mapsto A_{z}$ and $z\mapsto B_{z}$ are
analytic, and Lemma 5.7 assures that Assumption 2.1 is satisfied uniformly in
$z$. We conclude with Proposition 3.3 that $z\mapsto u_{z}\in\mathrm{E}_{1}$,
where $u_{z}$ is the unique solution of ($P_{z}$), is analytic.

###  5.2. Conclusion by Stein interpolation

The following proposition is the basis for our main result.

######  Proposition 5.8.

Under Assumptions 2.1 with $M=0$ and 5.1 there exists a $p_{0}>2$ such that,
for all $p\in[2,p_{0}]$, $f\in\mathrm{E}_{0}^{p}$ and
$g\in{\mathbb{E}^{p}_{\frac{1}{2}}}$, problem (P) with $u_{0}=0$ admits a
unique solution $u\in\mathrm{E}_{1}^{p}$, and there is a constant $C$
depending only on $\Lambda(A)$, $\Lambda(B)$, $\lambda$, $T\vee 1$ such that

(47) $\displaystyle\|u\|_{\mathrm{E}_{1}^{p}}\leq
C\bigl{(}\|f\|_{\mathrm{E}_{0}^{p}}+\|g\|_{{\mathbb{E}^{p}_{\frac{1}{2}}}}\bigr{)}.$

###### Proof.

Step 1: Applying abstract Stein interpolation to ($P_{z}$).

For $z\in\overline{S}$ define an operator
$T_{z}\colon\mathrm{E}_{0}\times{\mathbb{E}_{\frac{1}{2}}}\to\mathrm{E}_{1}$
which maps a data pair $(f,g)$ to the unique solution $u$ of ($P_{z}$). By
linearity of the equation, $T_{z}$ is linear and the family of operator
$(T_{z})_{z\in\overline{S}}$ is uniformly bounded according to Proposition
3.2.

Fix some $q>2$. We apply the results of the last section with $p$ replaced by
$q$. Let
$(f,g)\in\mathrm{E}^{q}_{0}\times{\mathbb{E}^{q}_{\frac{1}{2}}}\subseteq\mathrm{E}_{0}\times{\mathbb{E}_{\frac{1}{2}}}$.
We claim that $T_{\mathrm{i}t}(f,g)\in\mathrm{E}^{q}_{1}$ uniformly. Indeed,
this follows from Theorem 4.6 if we check its smallness assumptions.
Hypothesis (ii) follows immediately from Definition 5.6. Moreover, since
$A_{0}-\mu^{-1}A_{z}=F(z)(A_{0}-\mu^{-1}A)$, it follows from Corollary 5.5
that

(48) $\displaystyle
C_{q}\|A_{0}-\mu^{-1}A_{\mathrm{i}t}\|_{\mathcal{L}(V,V^{*})}=C_{q}|F(\mathrm{i}t)|\|A_{0}-\mu^{-1}A\|_{\mathcal{L}(V,V^{*})}\leq
rC_{q}(1-\rho).$

Therefore, by choice of $r$ in Definition 5.6,
$C_{q}|A_{0}-\mu^{-1}A_{\mathrm{i}t}\|_{\mathcal{L}(V,V^{*})}<1$ uniformly in
$z=\mathrm{i}t$.

Finally, we have argued in Section 3.2 that for fixed $(f,g)$ the map $S\ni
z\mapsto T_{z}(f,g)\in\mathrm{E}_{1}$ is holomorphic. In summary, this allows
us to invoke abstract Stein interpolation [32] to deduce

(49) $\displaystyle\|u_{\theta}\|_{\mathrm{E}_{1}^{q_{\theta}}}\leq
C_{\theta}\bigl{(}\|f\|_{\mathrm{E}_{0}^{q_{\theta}}}+\|g\|_{\mathbb{E}^{q_{\theta}}_{\frac{1}{2}}}\bigr{)},\quad\theta\in(0,1),f\in\mathrm{E}^{q}_{0},g\in{\mathbb{E}^{q}_{\frac{1}{2}}},$

where $\frac{1}{q_{\theta}}=\frac{1-\theta}{2}+\frac{\theta}{q}$ and
$C_{\theta}=C_{L}^{1-\theta}(c/\delta)^{\theta}$, $u_{\theta}$ is the unique
solution to ($P_{z}$) with $z=\theta$, and where we use the constants from
Proposition 3.2 and Theorem 4.6.

Step 2: Specializing to (P).

Choose $\theta=-\frac{\log(r)}{\log(R/r)}\in(0,1)$. Then $F(\theta)=1$ and we
find $A_{\theta}=A$ and $B_{\theta}=B$. Write $p_{0}\coloneqq q_{\theta}$,
then Step 1 shows

(50) $\displaystyle\|u\|_{\mathrm{E}_{1}^{p_{0}}}\leq
C\bigl{(}\|f\|_{\mathrm{E}_{0}^{p_{0}}}+\|g\|_{\mathbb{E}^{p_{0}}_{\frac{1}{2}}}\bigr{)},\quad
f\in\mathrm{E}^{q}_{0},g\in{\mathbb{E}^{q}_{\frac{1}{2}}},$

where $2<p_{0}<q$ and $u$ is the unique solution to (P) with data $(f,g)$. By
an approximation argument, we extend well-posedness to
$(f,g)\in\mathrm{E}_{0}^{p_{0}}\times\mathbb{E}^{p_{0}}_{\frac{1}{2}}$.

Now to obtain the statement for $p\in(2,p_{0}]$, one can either use complex
interpolation with the case $p=2$, or lower the value of $r$ in the above
proof. ∎

Now we can prove the main regularity result for linear equations stated in
Theorem 2.2.

###### Proof of Theorem 2.2.

By considering $e^{\lambda t}u(t)$ for suitable $\lambda\in\mathbb{R}$ we can
reduce to the case $M=0$ in Assumption 2.1. Then Proposition 5.8 yields a
unique strong solution to (P) with $u_{0}=0$ that satisfies the
$\mathrm{L}^{p}$-estimate corresponding to $\theta=0$ in the theorem. Still
with $u_{0}=0$, we can first deduce higher regularity in time from [2, Prop.
3.8], using the reference operators $\widetilde{B}=0$ and
$\widetilde{A}=A_{0}$. In a second step, we include initial values
$u_{0}\in\mathrm{L}^{p}_{\mathcal{F}_{0}}(\Omega;(H,V)_{1-\nicefrac{{2}}{{p}},p})$
in virtue of [2, Prop. 3.10]. At both stages, [2, Prop. 2.10] can be used to
get the maximal estimate missing in the case $\theta=0$. ∎

##  6\. Quasilinear problem and well-posedness result

On an open and bounded set $D\subseteq\mathbb{R}^{d}$ consider the quasilinear
system

(QL) $\displaystyle\left\\{\quad\begin{aligned}
\,\mathrm{d}u^{\alpha}&=\big{[}\partial_{i}(a^{\alpha\beta}_{ij}(u)\partial_{j}u^{\beta})+\partial_{i}\Phi^{\alpha}_{i}(u)+\phi^{\alpha}(u)\big{]}\,\mathrm{d}t\\\
&\qquad\qquad+\sum_{n\geq
1}\big{[}b^{\alpha\beta}_{n,j}(u^{\beta})\partial_{j}u^{\beta}+g^{\alpha}_{n}(u)\big{]}\,\mathrm{d}w_{n},\
\ &\text{ on }D,\\\ u^{\alpha}&=0,\ \ &\text{ on }\partial D,\\\
u^{\alpha}(0)&=u^{\alpha}_{0},\ \ &\text{ on }D,\end{aligned}\right.$

where $W=(w_{n}(t)\,:\,t\geq 0)_{n\geq 1}$ are independent standard Brownian
motions on a probability space $\Omega$, $i,j=1,\dots,d$, and
$\alpha,\beta=1,\dots,N$ for some system size $N\geq 1$. For simplicity, we
employed Einstein’s summation convention above and in the sequel. We will
ignore the system size $N$ in the notation for function spaces if this cannot
cause any confusion. Let $\mathcal{F}$ be the filtration on $\Omega$ induced
by $W$.

Let $h>1$. For our main result with growing non-linearities, Theorem 6.6, we
introduce the following assumption. As an intermediate step, we also treat
“the case $h=1$” in Theorem 7.2, with reduced structural assumptions, and thus
of independent interest.

######  Assumption 6.1 ($h$).

  1. (i)

The system is diagonal, that is $a_{ij}^{\alpha\beta}$ and
$b_{j}^{\alpha\beta}$ are non-zero only when $\alpha=\beta$. Write
$a_{ij}^{\alpha}\coloneqq a_{ij}^{\alpha\alpha}$ and $b_{j}^{\alpha}\coloneqq
b_{j}^{\alpha\alpha}$. For fixed $\alpha$, the matrix $a^{\alpha}$ does not
need to be diagonal.

  2. (ii)

The coefficients $a^{\alpha}_{ij}\colon(0,T)\times
D\times\mathbb{R}^{N}\to\mathbb{R}$ and
$b^{\alpha}_{j}\coloneqq(b^{\alpha}_{n,j})_{n\geq 1}\colon(0,T)\times
D\times\mathbb{R}\to\ell^{2}$ are continuous in the last component,
measurable, and uniformly bounded by a constant $\Lambda$.

  3. (iii)

One has $a^{\alpha}_{ij}=a^{\alpha}_{ji}$ and there exists $\lambda>0$ such
that, for all $t\in(0,T)$, $x\in D$, and $y\in\mathbb{R}^{N}$,

$\big{(}a^{\alpha}_{ij}(t,x,y)-\frac{1}{2}b^{\alpha}_{n,i}(t,x,y^{\alpha})b^{\alpha}_{n,j}(t,x,y^{\alpha})\big{)}\xi_{i}^{\alpha}\xi_{j}^{\alpha}\geq\lambda|\xi|^{2},\quad\xi\in\mathbb{R}^{dN}.$

  4. (iv)

The non-linearities $\Phi^{\alpha}_{i}\colon(0,T)\times
D\times\mathbb{R}^{N}\to\mathbb{R}$ and $\phi^{\alpha}\colon(0,T)\times
D\times\mathbb{R}^{N}\to\mathbb{R}$ are locally Lipschitz in the last
component, measurable, and satisfy the following growth condition: there
exists a constant $C$ such that,

(G) $\displaystyle|\Phi(t,x,y)|+|\phi(t,x,y)|$ $\displaystyle\leq
C(1+|y|^{h}),\quad t\in(0,T),x\in D,y\in\mathbb{R}^{N}.$

  5. (v)

The non-linearity $\phi$ is dissipative in the following sense: there exists a
constant $C$ such that one has for all $t\in(0,T)$, $x\in D$,
$y\in\mathbb{R}^{N}$ and $\alpha=1,\dots,N$ that

(51) $\displaystyle\phi^{\alpha}(t,x,y)y^{\alpha}\leq C(|y|^{2}+1).$

  6. (vi)

The stochastic non-linearity $g^{\alpha}\coloneqq(g^{\alpha}_{n})_{n\geq
1}\colon(0,T)\times D\times\mathbb{R}^{N}\to\ell^{2}$ is Lipschitz continuous
in the last component and measurable.

######  Remark 6.2.

Some discussion of the hypotheses is in order.

  1. (i)

When we work with non-linearities of linear growth, see Section 7, then the
diagonal structure assumed in (i) is not needed. This is why (QL) is
formulated for the non-diagonal case.

  2. (ii)

The non-linearities for $a$, $\Phi$ and $\phi$ lead to a coupling between the
components even when working with a diagonal structure.

  3. (iii)

To identify the limit in the stochastic compactness method, it seems crucial
that $b_{j}^{\alpha\beta}$ only depends on the $\beta$th component. In fact,
it is a surprising strength of our approach that this structural assumption is
not needed for $a$.

######  Remark 6.3 (diagonal structure).

Instead of imposing a diagonal structure, we could also strengthen the
coercivity condition as follows: for $q\in[0,2(h-1)+\varepsilon]$ assume that
there exists $\lambda_{q}>0$ such that, for all $t\in(0,T)$, $x\in D$, and
$y\in\mathbb{R}^{N}$,

$|y^{\alpha}|^{q}\Big{(}a^{\alpha}_{ij}(t,x,y)-\frac{1}{2}b^{\alpha}_{n,i}(t,x,y^{\alpha})b^{\alpha}_{n,j}(t,x,y^{\alpha})\Big{)}\xi_{i}^{\alpha}\xi_{j}^{\alpha}\geq\lambda_{q}|y^{\alpha}|^{q}|\xi^{\alpha}|^{2}\quad\text{
for all }\xi\in\mathbb{R}^{dN}.$

For instance, such a condition can be ensure for perturbations of diagonal
systems in which $|a^{\alpha\beta}_{ij}|\leq c(1+|y|^{q})^{-1}$ whenever
$\alpha\neq\beta$, where $c$ is sufficiently small. For clarity of exposition,
we only work with the diagonal case. This case is also the most important one
in reaction–diffusion equations. However, we do not know whether the
structural condition we have made is essential. It is needed at a technical
point in the proof in the identification of the limit in the stochastic
compactness method.

To make precise the homogeneous Dirichlet boundary condition in (QL), we
introduce the following function spaces.

######  Definition 6.4.

Write $\mathrm{H}^{1}_{0}(D)$ for the closure in $\mathrm{H}^{1}(D)$ of
$\mathrm{C}^{\infty}_{0}(D)$, the smooth and compactly supported functions in
$D$. Moreover, for $s\in(0,1)$ put

(52)
$\displaystyle\mathrm{H}^{s}_{0}(D)\coloneqq[\mathrm{L}^{2}(D),\mathrm{H}^{1}_{0}(D)]_{s}\quad\&\quad\mathrm{B}^{s}_{2,p,0}(D)\coloneqq(\mathrm{L}^{2}(D),\mathrm{H}^{1}_{0}(D))_{s,p}.$

######  Remark 6.5.

The spaces $\mathrm{H}^{s}_{0}(D)$ and $\mathrm{B}^{s}_{2,p,0}(D)$ can often
be identified with closed subspaces of the usual Bessel potential and Besov
spaces on $D$.

The main result on the quasi-linear system (QL) reads as follows.

######  Theorem 6.6 (Weak existence).

Let $h>1$. Suppose Assumption 6.1 holds. Then there is $p_{0}>2$, depending
only on $C,\Lambda,\lambda$ from the assumption as well as dimensions, such
that, if $p\in(2,p_{0}]$ and $q\in(ph,\infty)$, then, given
$u_{0}\in\mathrm{L}^{p}_{\mathcal{F}_{0}}(\Omega;\mathrm{B}^{1-\frac{2}{p}}_{2,p,0}(D))\cap\mathrm{L}^{q}_{\mathcal{F}_{0}}(\Omega\times
D)$, there exists a weak solution
$(\widetilde{u},\widetilde{W},\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{\mathbb{P}},(\widetilde{\mathcal{F}}_{t})_{t\geq
0})$ of (QL) in the sense of Definition 6.7. One has the estimates

(53)
$\displaystyle\|\widetilde{u}\|_{\mathrm{L}^{p}(\widetilde{\Omega};\mathrm{C}([0,T];\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D)))}$
$\displaystyle\lesssim_{\theta}1+\|u_{0}\|_{\mathrm{L}^{q}(\Omega;\mathrm{L}^{q}(D))}^{h}+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\frac{2}{p}}_{2,p,0}(D))},$
(54)
$\displaystyle\|\widetilde{u}\|_{\mathrm{L}^{q}(\widetilde{\Omega}\times(0,T)\times
D)}$ $\displaystyle\lesssim 1+\|u_{0}\|_{\mathrm{L}^{q}(\Omega\times D)}^{h}.$
Moreover, for all $\theta\in[0,1/2)$ one has the bound (55)
$\displaystyle\|\widetilde{u}\|_{\mathrm{L}^{p}(\widetilde{\Omega};\mathrm{H}^{\theta,p}(0,T;\mathrm{H}^{1-2\theta}_{0}(D)))}$
$\displaystyle\lesssim_{\theta}1+\|u_{0}\|_{\mathrm{L}^{q}(\Omega;\mathrm{L}^{q}(D))}^{h}+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\frac{2}{p}}_{2,p,0}(D))}.$

Our notion of a weak solution employed in Theorem 6.6 is the following.

######  Definition 6.7.

Call the tuple
$(\widetilde{u},\widetilde{W},\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{\mathbb{P}},(\widetilde{\mathcal{F}}_{t})_{t\geq
0})$ a _weak solution_ of (QL) if $\widetilde{W}=(\widetilde{w}_{n})_{n\geq
1}$ is an $\ell^{2}$-cylindrical Brownian motion on $\widetilde{\Omega}$,
$\widetilde{u}(0)$ has the same distribution as $u_{0}$,
$\widetilde{u}\in\mathrm{L}^{2}(0,T;\mathrm{H}^{1}_{0}(D))\cap\mathrm{C}([0,T];\mathrm{L}^{2}(D))$
almost surely and is $(\widetilde{\mathcal{F}}_{t})$-progressively measurable,
$\Phi^{\alpha}_{i}(\widetilde{u}),\phi^{\alpha}(\widetilde{u})\in\mathrm{L}^{1}(0,T;\mathrm{L}^{1}(D))$,
$g(\widetilde{u})\in\mathrm{L}^{2}(0,T;\mathrm{L}^{2}(D;\ell^{2}))$, and for
all $\xi\in\mathrm{C}^{\infty}_{0}(D)$ and $t\in[0,T]$, one has almost surely

(56)
$\displaystyle\langle\widetilde{u}^{\alpha}(t),\xi\rangle=\langle\widetilde{u}^{\alpha}(0),\xi\rangle$
$\displaystyle+\int_{0}^{t}-\langle
a^{\alpha\beta}_{ij}(\widetilde{u})\partial_{j}\widetilde{u},\partial_{i}\xi\rangle-\langle\Phi^{\alpha}_{i}(\widetilde{u}),\partial_{i}\xi\rangle+\langle\phi^{\alpha}(\widetilde{u}),\xi\rangle\,\mathrm{d}t$
(57) $\displaystyle+\sum_{n\geq 1}\int_{0}^{t}\langle
b^{\alpha\beta}_{n,j}(\widetilde{u})\partial_{j}\widetilde{u},\xi\rangle+\langle
g^{\alpha}_{n}(\widetilde{u}),\xi\rangle\,\mathrm{d}w_{n}.$

The proof of Theorem 6.6 will occupy the rest of this article.

######  Remark 6.8.

In the scalar case $N=1$ with $b=0$ $\phi=0$ a result such as Theorem 6.6 is
proved in [11] in the case of periodic boundary conditions. Moreover, even the
degenerate case and initial data in $L^{1}$ are considered via a different
solution concept called kinetic solutions. In this framework uniqueness is
proved as well. It would be interesting to see if some of these results can be
obtained in the generality of Theorem 6.6.

##  7\. Quasilinear problem with Lipschitz non-linearities

First, we study (QL) under the following set of hypotheses. In Section 8 we
are going to reduce the general case to the current one in virtue of an
approximation procedure.

######  Assumption 7.1.

  1. (i)

The coefficient functions $a^{\alpha\beta}_{ij}\colon(0,T)\times
D\times\mathbb{R}^{N}\to\mathbb{R}$ and
$b^{\alpha\beta}_{j}\coloneqq(b^{\alpha\beta}_{n,j})_{n\geq
1}\colon(0,T)\times D\times\mathbb{R}\to\ell^{2}$ are continuous in the last
component, measurable, and uniformly bounded by a constant $\Lambda$.

  2. (ii)

One has $a^{\alpha\beta}_{ij}=a^{\beta\alpha}_{ji}$ and there exists
$\lambda>0$ such that, for all $t\in(0,T)$, $x\in D$, and
$y\in\mathbb{R}^{N}$,

$\big{(}a^{\alpha\beta}_{ij}(t,x,y)-\frac{1}{2}b^{\gamma\alpha}_{n,i}(t,x,y^{\alpha})b^{\gamma\beta}_{n,j}(t,x,y^{\beta})\big{)}\xi_{i}^{\alpha}\xi_{j}^{\beta}\geq\lambda|\xi|^{2}\quad\text{
for all }\xi\in\mathbb{R}^{dN}.$

  3. (iii)

The non-linearities $\Phi^{\alpha}_{i}\colon(0,T)\times
D\times\mathbb{R}^{N}\to\mathbb{R}$ and $\phi^{\alpha}\colon(0,T)\times
D\times\mathbb{R}^{N}\to\mathbb{R}$, and the stochastic non-linearity
$g^{\alpha}\coloneqq(g^{\alpha}_{n})_{n\geq 1}\colon(0,T)\times
D\times\mathbb{R}^{N}\to\ell^{2}$ are measurable and Lipschitz continuous in
the last component.

Observe that Assumption 7.1 (iii) is the combination of Assumption 6.1 (iv)
with $h=1$ and Assumption 6.1 (vi). Assumption 6.1 (v) becomes obsolete.

We are going to show the following well-posedness result.

######  Theorem 7.2.

Suppose Assumption 7.1 holds. There is $p_{0}>2$, depending only on
$C,\Lambda,\lambda$ from the assumption as well as dimensions, such that if
$p\in(2,p_{0}]$, then, given
$u_{0}\in\mathrm{L}^{p}_{\mathcal{F}_{0}}(\Omega;\mathrm{B}^{1-\frac{2}{p}}_{2,p,0}(D))$,
there exists a weak solution
$(\widetilde{u},\widetilde{W},\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{\mathbb{P}},(\widetilde{\mathcal{F}}_{t})_{t\geq
0})$ of (QL). Moreover, for all $\theta\in[0,1/2)$ one has

(58)
$\displaystyle\|\widetilde{u}\|_{\mathrm{L}^{p}(\widetilde{\Omega};\mathrm{C}([0,T];\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D)))}+\|\widetilde{u}\|_{\mathrm{L}^{p}(\widetilde{\Omega};\mathrm{H}^{\theta,p}(0,T;\mathrm{H}^{1-2\theta}_{0}(D)))}$
$\displaystyle\lesssim_{\theta}1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\frac{2}{p}}_{2,p,0}(D))}.$

###  7.1. Smoothing of the coefficients

Let $\zeta\colon\mathbb{R}\to[0,\infty)$ be a smooth and compactly supported
function with integral equal to $1$ and
$\rho\colon\mathbb{R}^{N}\to[0,\infty)$ the $N$-fold product of $\zeta$. For
$m\geq 1$, write $\zeta_{m}$ and $\rho_{m}$ for the induced mollifier
sequences. Now, define the quasi-linear operators

(59) $\displaystyle{}_{m}A(t)\colon\mathrm{L}^{2}(D)^{N}$
$\displaystyle\to\mathcal{L}(\mathrm{H}^{1}_{0}(D)^{N},\mathrm{H}^{-1}(D)^{N}),$
(60) $\displaystyle{}_{m}B(t)\colon\mathrm{L}^{2}(D)$
$\displaystyle\to\mathcal{L}(\mathrm{H}^{1}_{0}(D)^{N},\mathcal{L}_{2}(\ell^{2},\mathrm{L}^{2}(D)^{N}))$

by

(61) $\displaystyle{}_{m}A(t,v)u$
$\displaystyle=\bigl{(}\partial_{i}({}_{m}a_{ij}^{\alpha\beta}(t,x,R_{m}v)\partial_{j}u^{\beta})\bigr{)}_{\alpha=1}^{N},\qquad(u\in\mathrm{H}^{1}_{0}(D)^{N},v\in\mathrm{L}^{2}(D)^{N}),$
(62) $\displaystyle\langle{}_{m}B(t,w)u,e_{n}\rangle$
$\displaystyle=\bigl{(}{}_{m}b_{n,j}^{\alpha\beta}(t,x,R_{m}w^{\beta})\partial_{j}u^{\beta}\bigr{)}_{\alpha=1}^{N},\qquad(u\in\mathrm{H}^{1}_{0}(D)^{N},w\in\mathrm{L}^{2}(D)),$

where $R_{m}f=\rho_{m}\ast Ef$,
${}_{m}a_{ij}^{\alpha\beta}(t,x,\cdot)=\rho_{m}\ast
a_{ij}^{\alpha\beta}(t,x,\cdot)$ and
${}_{m}b_{n,j}^{\alpha\beta}(t,x,\cdot)=\zeta_{m}\ast
b_{n,j}^{\alpha\beta}(t,x,\cdot)$. Here, $E$ denotes the zero extension of a
given function, and we apply the convolution in the definition of $R_{m}$
component-wise if $f$ is vector-valued.

Consider now the family of equations

(m-QL) $\displaystyle\left\\{\quad\begin{aligned}
\,\mathrm{d}u&=\big{[}{}_{m}A(u)u+\partial_{i}\Phi_{i}(u)+\phi(u)\big{]}\,\mathrm{d}t\\\
&\qquad\qquad+\sum_{n\geq
1}\big{[}\langle{}_{m}B(u)u,e_{n}\rangle+g_{n}(u)\big{]}\,\mathrm{d}w_{n},\ \
&\text{ on }D,\\\ u&=0,\ \ &\text{ on }\partial D,\\\ u(0)&=u_{0},\ \ &\text{
on }D.\end{aligned}\right.$

Following [19, p. 211] we show that the smoothed coefficients satisfy the same
coercivity condition as the original ones.

######  Lemma 7.3.

For fixed $m$, the smoothed coefficients ${}_{m}a_{ij}^{\alpha\beta}$ and
${}_{m}b_{n,j}^{\alpha\beta}$ satisfy Assumptions 7.1 (i) and (ii) with the
same constants.

###### Proof.

Fix $m$. By definition of the convolution, symmetry of
${}_{m}a^{\alpha\beta}_{ij}$ is immediate. The same holds for uniform
boundedness, since the convolution kernel has normalized
$\mathrm{L}^{1}$-norm.

For better readability, suppress dependence on $t$ and $x$ of the
coefficients. Introduce the vector notation
$\vec{b}_{n,j}^{\alpha}(y)\coloneqq\bigl{(}b_{n,j}^{\alpha\beta}(y^{\beta})\bigr{)}_{\beta=1}^{N}$,
likewise for ${}_{m}\vec{b}_{n,j}^{\alpha}(y)$. In this way,
$\sum_{\beta}b^{\alpha\beta}_{n,j}(y^{\beta})\xi_{j}^{\beta}=\vec{b}_{n,j}^{\alpha}(y)\cdot\xi_{j}$
and
$({}_{m}b_{n,j}^{\alpha\beta})_{\beta=1}^{N}=\rho_{m}\ast\vec{b}_{n,j}^{\alpha}$.

Observe first that for $y\in\mathbb{R}^{N}$,

(63)
$\displaystyle\sum_{n}\sum_{i,j}\sum_{\alpha,\beta,\gamma}{}_{m}b^{\gamma\alpha}_{n,i}(y^{\alpha}){}_{m}b^{\gamma\beta}_{n,j}(y^{\beta})\xi_{i}^{\alpha}\xi_{j}^{\beta}=\sum_{n}\sum_{\gamma}\Bigl{|}\sum_{j}{}_{m}\vec{b}^{\gamma}_{j}(y)\cdot\xi_{j}\Bigr{|}_{2}^{2},$

where $|\cdot|_{2}$ is the Euclidean norm on $\mathbb{R}^{N}$. Write
$\bm{x}\colon\mathbb{R}^{N}\ni x\mapsto x$ for the identity map. Using that
the convolution kernel is positive with normalized integral, Jensen’s
inequality gives

(64)
$\displaystyle\sum_{n}\sum_{\gamma}\Bigl{|}\sum_{j}{}_{m}\vec{b}^{\gamma}_{j}(y)\cdot\xi_{j}\Bigr{|}_{2}^{2}$
$\displaystyle\leq\sum_{n}\sum_{\gamma}\Bigl{|}\sum_{j}\vec{b}^{\gamma}_{j}(\bm{x})\cdot\xi_{j}\Bigr{|}_{2}^{2}\ast\rho_{m}(y)$
(65)
$\displaystyle=\sum_{n}\sum_{i,j}\sum_{\alpha,\beta,\gamma}\Bigl{(}b^{\gamma\alpha}_{n,i}(\bm{x}^{\alpha})b^{\gamma\beta}_{n,j}(\bm{x}^{\beta})\xi_{i}^{\alpha}\xi_{j}^{\beta}\Bigr{)}\ast\rho_{m}(y).$

Hence, it follows from Assumptions 7.1 (ii) and positivity of the convolution
kernel that

(66)
$\displaystyle\quad\Big{(}{}_{m}a^{\alpha\beta}_{ij}(y)-\frac{1}{2}{}_{m}b^{\gamma\alpha}_{n,i}(y^{\alpha}){}_{m}b^{\gamma\beta}_{n,j}(y^{\beta})\Big{)}\xi_{i}^{\alpha}\xi_{j}^{\beta}$
(67)
$\displaystyle\geq\int_{\mathbb{R}^{N}}\Big{(}a^{\alpha\beta}_{ij}(z)-\frac{1}{2}b^{\gamma\alpha}_{n,i}(z^{\alpha})b^{\gamma\beta}_{n,j}(z^{\beta})\Big{)}\xi_{i}^{\alpha}\xi_{j}^{\beta}\rho_{m}(y-z)\,\mathrm{d}z\geq\lambda|\xi|^{2}.\qed$

Using [1] we show that for each $m$, problem (m-QL) admits a unique solution.

######  Lemma 7.4.

Suppose Assumption 7.1 holds. Then, for each $m$, there is a unique strong
solution
$u_{m}\in\mathrm{L}^{2}(\Omega\times(0,T);\mathrm{H}^{1}_{0}(D))\cap\mathrm{L}^{2}(\Omega;\mathrm{C}([0,T];\mathrm{L}^{2}(D)))$
to (m-QL).

###### Proof.

Fix $m$. Write
$\|\cdot\|_{\mathcal{L}_{2}}=\|\cdot\|_{\mathcal{L}_{2}(\ell^{2},\mathrm{L}^{2}(D))}$.
The assertion follows from [1, Thm. 3.5] applied with $A_{0}(v)u\coloneqq
A_{0}(t,v)u\coloneqq{}_{m}A(t,v)u$ and $B_{0}(v)u\coloneqq
B_{0}(t,v)u\coloneqq{}_{m}B(t,v)u$, the non-linearities $F(v)\coloneqq
F(t,v)\coloneqq\partial_{i}\Phi_{i}(t,v)+\phi(t,v)$ and $G(v)e_{n}\coloneqq
G(t,v)e_{n}\coloneqq g_{n}(t,v)$, and the cylindrical Brownian motion
$We_{n}\coloneqq w_{n}$.

Indeed, note first that, in the light of Lemma 7.3, all assertions from
Assumption 7.1 remain valid for the problem (m-QL). Besides boundedness of
$A_{0}$ and $B_{0}$, and linear growth of $F$ and $G$ in the respective norms,
which readily follow from uniform boundedness of the coefficients in the first
two cases, and linear growth of the non-linearities in the last two cases, we
have to check the regularity conditions

(68)
$\displaystyle\|A_{0}(u)w-A_{0}(v)w\|_{\mathrm{H}^{-1}(D)}+\|B_{0}(u)w-B_{0}(v)w\|_{\mathcal{L}_{2}}\lesssim\|u-v\|_{\mathrm{L}^{2}(D)}\|w\|_{\mathrm{H}^{1}_{0}(D)},$

for all $u,v\in\mathrm{L}^{2}(D)$ and $w\in\mathrm{H}^{1}_{0}(D)$, and

(69)
$\displaystyle\|F(u)-F(v)\|_{\mathrm{H}^{-1}(D)}+\|G(u)-G(v)\|_{\mathcal{L}_{2}}\lesssim\|u-v\|_{\mathrm{L}^{2}(D)},$

for all $u,v\in\mathrm{L}^{2}(D)$, as well as the following coercivity
condition: there exists $\theta,\eta>0$ and $M\geq 0$ such that a.s.

(70) $\displaystyle\langle
A_{0}(v)u,u\rangle-(\nicefrac{{1}}{{2}}+\eta)\|B_{0}(v)u\|_{\mathcal{L}_{2}}^{2}\geq\theta\|u\|_{\mathrm{H}^{1}_{0}(D)}^{2}-M\|u\|_{\mathrm{L}^{2}(D)}^{2},$

for all $u\in\mathrm{H}^{1}_{0}(D)$ and $v\in\mathrm{L}^{2}(D)$.

The second regularity condition, (69), follows immediately from Lischitz
continuity of $\Phi$, $\phi$, and $g$. For brevity, we present the regularity
condition (68) only for $A_{0}$. The calculation for $B_{0}$ is similar. Fix
an integer $k>\nicefrac{{d}}{{2}}$. Using the Sobolev embedding and Young’s
convolution inequality, calculate

(71) $\displaystyle\|A_{0}(u)w-A_{0}(v)w\|_{\mathrm{H}^{-1}(D)}$
$\displaystyle\leq\|{}_{m}a^{\alpha\beta}_{ij}(R_{m}u)-{}_{m}a^{\alpha\beta}_{ij}(R_{m}v)\|_{\infty}\|\partial_{j}w^{\beta}\|_{\mathrm{L}^{2}(D)}$
(72)
$\displaystyle\lesssim\|{}_{m}a^{\alpha\beta}_{ij}\|_{\mathrm{Lip}}\|R_{m}u-R_{m}v\|_{\infty}\|\partial_{j}w^{\beta}\|_{\mathrm{L}^{2}(D)}$
(73)
$\displaystyle\lesssim_{m}\|a^{\alpha\beta}_{ij}\|_{\infty}\|R_{m}u-R_{m}v\|_{\mathrm{W}^{k,2}(D)}\|\partial_{j}w^{\beta}\|_{\mathrm{L}^{2}(D)}$
(74)
$\displaystyle\lesssim\|\rho_{m}\|_{\mathrm{W}^{k,1}(D)}\|u-v\|_{\mathrm{L}^{2}(D)}\|w\|_{\mathrm{H}^{1}_{0}(D)}$
(75)
$\displaystyle\lesssim_{m}\|u-v\|_{\mathrm{L}^{2}(D)}\|w\|_{\mathrm{H}^{1}_{0}(D)}.$

To complete the proof, note first that Assumption 7.1 (ii) implies (70) with
$\eta=0$ and $\theta\coloneqq\lambda\eqqcolon M$. Therefore, since
$B_{0}(v)\colon\mathrm{H}^{1}_{0}(D)^{N}\to\mathrm{L}^{2}(D)^{N}$ is bounded
(uniformly in $v$), we get (70) with $\eta>0$ if we replace $\theta$ by
$\nicefrac{{\theta}}{{2}}$. ∎

###  7.2. Higher integrability of semilinear equations

It will turn out useful in the sequel to consider a semi-linear version of
(QL). More precisely, consider

(SL) $\displaystyle\left\\{\quad\begin{aligned}
\,\mathrm{d}u^{\alpha}&=\big{[}\partial_{i}(\bm{a}^{\alpha\beta}_{ij}\partial_{j}u^{\beta})+\partial_{i}\Phi^{\alpha}_{i}(u)+\phi^{\alpha}(u)\big{]}\,\mathrm{d}t\\\
&\qquad\qquad+\sum_{n\geq
1}\big{[}\bm{b}^{\alpha\beta}_{n,j}\partial_{j}u^{\beta}+g^{\alpha}_{n}(u)\big{]}\,\mathrm{d}w_{n},\
\ &\text{ on }D,\\\ u^{\alpha}&=0,\ \ &\text{ on }\partial D,\\\
u^{\alpha}(0)&=u^{\alpha}_{0},\ \ &\text{ on }D.\end{aligned}\right.$

We impose the following assumption, which is similar to Assumption 7.1.

######  Assumption 7.5.

  1. (i)

The coefficients $\bm{a}^{\alpha\beta}_{ij}\colon\Omega\times(0,T)\times
D\to\mathbb{R}$ as well as
$\bm{b}^{\alpha\beta}_{j}\coloneqq(\bm{b}^{\alpha\beta}_{n,j})_{n\geq
1}\colon\Omega\times(0,T)\times D\to\ell^{2}$ are progressively measurable and
uniformly bounded by a constant $\Lambda$.

  2. (ii)

One has $\bm{a}^{\alpha\beta}_{ij}=\bm{a}^{\beta\alpha}_{ji}$ and there exists
$\lambda>0$ such that, for all $t\in(0,T),x\in D$, one has almost surely

$\Big{(}\bm{a}^{\alpha\beta}_{ij}(t,x)-\frac{1}{2}\bm{b}^{\gamma\alpha}_{n,i}(t,x)\bm{b}^{\gamma\beta}_{n,j}(t,x)\Big{)}\xi_{i}^{\alpha}\xi_{j}^{\beta}\geq\lambda|\xi|^{2}\quad\text{
for all }\xi\in\mathbb{R}^{dN}.$

  3. (iii)

The deterministic non-linearities
$\Phi^{\alpha}_{i}\colon\mathbb{R}^{N}\to\mathbb{R}$ and
$\phi^{\alpha}\colon\mathbb{R}^{N}\to\mathbb{R}$ and the stochastic non-
linearity $g^{\alpha}\coloneqq(g^{\alpha}_{n})_{n\geq
1}\colon\mathbb{R}^{N}\to\ell^{2}$ are Lipschitz continuous.

A prototypical example for Assumption 7.5 is
$\bm{a}^{\alpha\beta}_{ij}(t,x)\coloneqq a^{\alpha\beta}_{ij}(t,x,u(t,x))$ and
$\bm{b}^{\alpha\beta}_{n,j}(t,x)\coloneqq
b^{\alpha\beta}_{n,j}(t,x,u^{\beta}(t,x))$, where $a^{\alpha\beta}_{ij}$ and
$b^{\alpha\beta}_{n,j}$ are coefficients for (QL) subject to Assumption 7.1
and the progressively measurable function $u\colon\Omega\times(0,T)\times
D\to\mathbb{R}^{N}$ is frozen.

It is well-known that (SL) admits a unique variational solution in
$\mathrm{L}^{2}(\Omega\times(0,T);\mathrm{H}^{1}_{0}(D))\cap\mathrm{L}^{2}(\Omega;\mathrm{C}([0,T];\mathrm{L}^{2}(D)))$.
As a consequence of Theorem 2.2, the main result of the first part of this
article, we can improve integrability in time of this variational solution to
$p>2$.

######  Proposition 7.6 (Higher integrability for (SL)).

Suppose Assumption 7.5 holds. Then there is some $p_{0}>2$ such that for any
$p\in(2,p_{0}]$ and any
$u_{0}\in\mathrm{L}^{p}_{\mathcal{F}_{0}}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))$
there is a unique solution
$u\in\mathrm{L}^{p}(\Omega\times(0,T);\mathrm{H}^{1}_{0}(D))\cap\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D)))$
to (SL) satisfying the estimate

(76)
$\displaystyle\|u\|_{\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D)))}\lesssim
1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))},$

and for all $\theta\in[0,\nicefrac{{1}}{{2}})$ the estimate

(77)
$\displaystyle\|u\|_{\mathrm{L}^{p}(\Omega;\mathrm{H}^{\theta,p}(0,T;\mathrm{H}^{1-2\theta}_{0}(D)))}\lesssim_{\theta}1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))}.$

If $\theta\in(\nicefrac{{1}}{{p}},\nicefrac{{1}}{{2}})$, then one has moreover
the estimate

(78)
$\displaystyle\|u\|_{\mathrm{L}^{p}(\Omega;\mathrm{C}^{\theta-\nicefrac{{1}}{{p}}}([0,T];\mathrm{H}^{1-2\theta}_{0}(D)))}\lesssim_{\theta}1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))}.$

###### Proof.

Step 1: Existence of variational solution and bootstrapping integrability.

Consider the progressively measurable maps

(79) $\displaystyle A_{0}(t,u)$
$\displaystyle\coloneqq(\partial_{i}(\bm{a}^{\alpha\beta}_{ij}(t,x)\partial_{j}u^{\beta}))_{\alpha=1}^{N},$
(80) $\displaystyle B_{0}(t,u)e_{n}$
$\displaystyle\coloneqq(\bm{b}^{\alpha\beta}_{n,j}(t,x)\partial_{j}u^{\beta})_{\alpha=1}^{N},$
(81) $\displaystyle F(u)$
$\displaystyle\coloneqq(\partial_{i}\Phi^{\alpha}_{i}(u)+\phi^{\alpha}(u))_{\alpha=1}^{N},$
(82) $\displaystyle G(u)e_{n}$
$\displaystyle\coloneqq(g^{\alpha}_{n}(u))_{\alpha=1}^{N}.$

The well-posedness result [15, Thm. 2.4] applied with $A(t,v)\coloneqq
A_{0}(t,v)+F(v)$ and $B(t,v)e_{n}\coloneqq B_{0}(t,v)e_{n}+G(u)e_{n}$ yields a
unique strong solution

$u\in\mathrm{L}^{p}(\Omega;\mathrm{L}^{2}(0,T;\mathrm{H}^{1}_{0}(D)))\cap\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];\mathrm{L}^{2}(D)))$

together with the estimate

(83)
$\displaystyle\|u\|_{\mathrm{L}^{p}(\Omega;\mathrm{L}^{2}(0,T;\mathrm{H}^{1}_{0}(D)))}+\|u\|_{\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];\mathrm{L}^{2}(D)))}\lesssim
1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{L}^{2}(D))}$

for some $p>2$ depending only on the quantities from Assumption 7.5, provided
we can verify the hypotheses listed in [15, Ass. 2.1]. Using boundedness of
the coefficients and linear growth of the non-linearities, (H4) and (H5) with
$f\equiv 1$ follow readily. As in the proof of Lemma 7.4 one deduces (H3) with
$p>2$ sufficiently close to $2$ from Assumption 7.5 (ii) and boundedness of
$B_{0}$. Condition (H2) is void for $A_{0}(t,u)$ and $B_{0}(t,u)$ in virtue of
(H3) and linearity, and is fulfilled for $F(u)$ and $G(u)$ by Lipschitz
continuity. Likewise, (H1) is immediate from linearity and Lipschitz
continuity.

Step 2: bootstrapping time-regularity.

A particular consequence of (83) is
$F(u)\in\mathrm{L}^{p}(\Omega\times(0,T);\mathrm{L}^{2}(D))$ and
$G(u)\in\mathrm{L}^{p}(\Omega\times(0,T);\mathcal{L}_{2}(\ell^{2},\mathrm{L}^{2}(D)))$,
with the estimate

(84)
$\displaystyle\|F(u)\|_{\mathrm{L}^{p}(\Omega\times(0,T);\mathrm{L}^{2}(D))}+\|G(u)\|_{\mathrm{L}^{p}(\Omega\times(0,T);\mathcal{L}_{2}(\ell^{2},\mathrm{L}^{2}(D)))}\lesssim
1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{L}^{2}(D))}.$

Consequently, if we freeze $u$ in the semi-linearities of (SL), that is to
say, if we consider (SL) as a linear problem with right-hand sides $f=F(u)$
and $g=G(u)$, then Theorem 2.2 becomes applicable (of course we can restrict
$p$ from Step 1 further so that the smallness condition in the theorem is
verified) and yields a unique solution
$v\in\mathrm{L}^{p}(\Omega\times(0,T);\mathrm{H}^{1}_{0}(D))\cap\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D)))$.
For the first estimate in the theorem calculate

(85)
$\displaystyle\|v\|_{\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D)))}$
(86) $\displaystyle\lesssim{}$
$\displaystyle\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))}+\|F(u)\|_{\mathrm{L}^{p}(\Omega\times(0,T),\mathrm{L}^{2}(D))}+\|G(u)\|_{\mathrm{L}^{p}(\Omega\times(0,T),\mathcal{L}_{2}(\ell^{2},\mathrm{L}^{2}(D)))}$
(87) $\displaystyle\lesssim{}$ $\displaystyle
1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))},$

where we used (84) and the embedding
$\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D)\subseteq\mathrm{L}^{2}(D)$ in
the last step. The other estimates in the theorem follow also from Theorem 2.2
using the same argument. By uniqueness with $p=2$, $u=v$, which completes the
proof. ∎

###  7.3. Uniform bounds with $p>2$

Recall the family of solutions

$(u_{m})_{m}\subseteq\mathrm{L}^{2}(\Omega;\mathrm{C}([0,T];\mathrm{L}^{2}(D)))\cap\mathrm{L}^{2}(\Omega\times(0,T);\mathrm{H}^{1}_{0}(D))$

to the approximate problems (m-QL) from Lemma 7.4. Using the result from
Section 7.2, we derive uniform $\mathrm{L}^{p}$-bounds for some $p>2$.

######  Proposition 7.7.

Suppose Assumption 7.1 holds. Then there is some $p_{0}>2$ such that for all
$m$, for all $p\in(2,p_{0}]$, and any
$u_{0}\in\mathrm{L}^{p}_{\mathcal{F}_{0}}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))$
there is a unique solution

$u_{m}\in\mathrm{L}^{p}(\Omega\times(0,T);\mathrm{H}^{1}_{0}(D))\cap\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D)))$

to (m-QL) satisfying the estimate

(88)
$\displaystyle\|u_{m}\|_{\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D)))}\lesssim
1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))},$

and for all $\theta\in[0,\nicefrac{{1}}{{2}})$ the estimate

(89)
$\displaystyle\|u_{m}\|_{\mathrm{L}^{p}(\Omega;\mathrm{H}^{\theta,p}(0,T;\mathrm{H}^{1-2\theta}_{0}(D)))}\lesssim_{\theta}1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))}.$

If $\theta\in(\nicefrac{{1}}{{p}},\nicefrac{{1}}{{2}})$, then one has moreover
the estimate

(90)
$\displaystyle\|u_{m}\|_{\mathrm{L}^{p}(\Omega;\mathrm{C}^{\theta-\nicefrac{{1}}{{p}}}([0,T];\mathrm{H}^{1-2\theta}_{0}(D)))}\lesssim_{\theta}1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))}.$

###### Proof.

Fix some $m$. As already mentioned in Section 7.2, we can consider the
coefficients
$\bm{a}^{\alpha\beta}_{ij}(t,x)\coloneqq{}_{m}a^{\alpha\beta}_{ij}(t,x,u_{m}(t,x))$
and
$\bm{b}^{\alpha\beta}_{n,j}(t,x)\coloneqq{}_{m}b^{\alpha\beta}_{n,j}(t,x,u_{m}^{\beta}(t,x))$,
which then satisfy Assumption 7.5. Now by Proposition 7.6, (SL) with these
coefficients admits a unique solution $v$ that satisfies the bounds claimed in
the proposition. But since

(91) $\displaystyle v\in{}$
$\displaystyle\mathrm{L}^{p}(\Omega\times(0,T);\mathrm{H}^{1}_{0}(D))\cap\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D)))$
(92) $\displaystyle\subseteq{}$
$\displaystyle\mathrm{L}^{2}(\Omega\times(0,T);\mathrm{H}^{1}_{0}(D))\cap\mathrm{L}^{2}(\Omega;\mathrm{C}([0,T];\mathrm{L}^{2}(D))),$

it follows $v=u_{m}$ by uniqueness for $p=2$ stated in Lemma 7.4, which
completes the proof. ∎

###  7.4. Stochastic compactness argument

In Theorem 1.3 we have already seen that a general tightness result can be
deduced from a priori estimates such as the ones of Proposition 7.7. In the
application to the quasi-linear system we need a slight variation of the
tightness result in which a certain weak compactness is also taken into
account.

In this section we suppose Assumption 7.1 holds, and we fix $p_{0}$,
$p\in(2,p_{0}]$, and $(u_{m})_{m\geq 1}$ as in Proposition 7.6. Put

(93)
$\displaystyle\mathcal{X}_{u}\coloneqq\mathrm{C}([0,T];\mathrm{L}^{2}(D))\cap\mathrm{L}^{p}_{\mathrm{w}}(0,T;\mathrm{H}^{1}_{0}(D)),\quad\mathcal{X}_{W}\coloneqq\mathrm{C}([0,T];\mathcal{U}_{0}),\quad\mathcal{X}\coloneqq\mathcal{X}_{u}\times\mathcal{X}_{W}.$

With the subscript “$\mathrm{w}$” we indicate that the space
$\mathrm{L}^{p}(0,T;\mathrm{H}^{1}_{0}(D))$ is equipped with the weak
topology. As a consequence of separability, this has no consequence for
questions of measurability. The space $\mathcal{U}_{0}\supseteq\mathcal{U}$ is
chosen in such a way that the cylindrical Brownian motion converges almost
surely. The random vectors $(u_{m},W)$ take values in $\mathcal{X}$. Write
$\mathcal{L}_{m}$ for their (joint) laws.

We claim that the family $(\mathcal{L}_{m})_{m}$ is tight. It is sufficient to
show tightness of the laws $\mathcal{L}(u_{m})$ on $\mathcal{X}_{u}$. Let us
emphasise that no “diagonal structure” of the coefficients is needed.

######  Lemma 7.8 (Tightness).

The family of laws $(\mathcal{L}(u_{m}))_{m}$ on $\mathcal{X}_{u}$ is tight.

###### Proof.

Let $\theta\in(\nicefrac{{1}}{{p}},\nicefrac{{1}}{{2}})$. For brevity, put

$\mathcal{X}_{u}^{\theta,p}\coloneqq\mathrm{C}^{\theta-\nicefrac{{1}}{{p}}}([0,T];\mathrm{H}^{1-2\theta}(D))\cap\mathrm{L}^{p}(0,T;\mathrm{H}^{1}_{0}(D))$

and write $\mathrm{B}_{R}$ for the open ball of radius $R$ in
$\mathcal{X}_{u}^{\theta,p}$. First, by Chebyshev’s inequality and Proposition
7.7,

(94)
$\displaystyle\begin{split}\mathbb{P}(\|u_{m}\|_{\mathcal{X}_{u}^{\theta,p}}\geq
R)&\leq R^{-p}\mathbb{E}\|u_{m}\|_{\mathcal{X}_{u}^{\theta,p}}^{p}\\\
&\lesssim
R^{-p}\Bigl{(}1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))}^{p}\Bigr{)}\to
0\quad\text{ as }R\to\infty.\end{split}$

Second, we claim that the closure of $\mathrm{B}_{R}$ is compact in
$\mathcal{X}_{u}$. Indeed, pre-compactness in
$\mathrm{C}([0,T];\mathrm{L}^{2}(D))$ follows from the vector-valued Arzéla-
Ascoli theorem [21, Theorem III.3.1], taking into account the compact
embedding $\mathrm{H}^{1}_{0}(D)\subseteq\mathrm{L}^{2}(D)$, and compactness
in the space $\mathrm{L}^{p}_{\mathrm{w}}(0,T;\mathrm{H}^{1}_{0}(D))$ is
clear, for $\mathrm{B}_{R}$ is bounded in the norm topology. ∎

Therefore, Prokhorov’s theorem allows to pass to a weakly convergent
subsequence (for convenience, we use the same symbol for this subsequence).
Second, the Jakubowski–Skorohod theorem [8, Thm. 2.7.1] gives the following
almost sure convergence.

######  Lemma 7.9 (Skorohod).

There exists a probability space
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$, a sequence of
$\mathcal{X}$-valued random variables $(\widetilde{u}_{m},\widetilde{W}_{m})$,
and a limiting $\mathcal{X}$-valued random variable
$(\widetilde{u},\widetilde{W})$, such that

  1. (i)

for each $m$, the law of $(\widetilde{u}_{m},\widetilde{W}_{m})$ under
$\tilde{\mathbb{P}}$ coincides with the law $\mathcal{L}_{m}$, and

  2. (ii)

one has $\tilde{\mathbb{P}}$-almost surely that
$\widetilde{u}_{m}\to\widetilde{u}$ in $\mathrm{C}([0,T];\mathrm{L}^{2}(D))$,
$\nabla\widetilde{u}_{m}\to\nabla\widetilde{u}$ weakly in
$\mathrm{L}^{p}(0,T;\mathrm{L}^{2}(D))$, and
$\widetilde{W}_{m}\to\widetilde{W}$ in $\mathrm{C}([0,T];\mathcal{U}_{0})$ as
$m\to\infty$.

######  Corollary 7.10.

The inclusion

$(\widetilde{u}_{m})_{m}\subseteq\mathrm{L}^{p}(\widetilde{\Omega};\mathrm{C}([0,T];\mathrm{L}^{2}(D)))\cap\mathrm{L}^{p}(\widetilde{\Omega}\times(0,T);\mathrm{H}^{1}_{0}(D))$

holds with the uniform bound

(95)
$\displaystyle\sup_{m}\left(\|\widetilde{u}_{m}\|_{\mathrm{L}^{p}(\widetilde{\Omega};\mathrm{C}([0,T];\mathrm{L}^{2}(D)))}+\|\widetilde{u}_{m}\|_{\mathrm{L}^{p}(\widetilde{\Omega}\times(0,T);\mathrm{H}^{1}_{0}(D))}\right)\lesssim
1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))}.$

According to [11, Lem. 4.8], $\widetilde{W}$ is a cylindrical Brownian motion.
Fix $\alpha=1,\dots,N$ and $\xi\in\mathrm{C}^{\infty}_{c}(D)$. Define

(96) $\displaystyle\widetilde{M}^{\alpha}(t)$
$\displaystyle=\langle\widetilde{u}^{\alpha}(t),\xi\rangle-\langle\widetilde{u}^{\alpha}(0),\xi\rangle+\int_{0}^{t}\langle
a^{\alpha\beta}_{ij}(\widetilde{u})\partial_{j}\widetilde{u}^{\beta},\partial_{i}\xi\rangle\,\mathrm{d}r$
(97)
$\displaystyle\qquad+\int_{0}^{t}\langle\Phi^{\alpha}_{i}(\widetilde{u}),\partial_{i}\xi\rangle\,\mathrm{d}r-\int_{0}^{t}\langle\phi^{\alpha}(\widetilde{u}),\xi\rangle\,\mathrm{d}r.$

As presented in the proof of [11, Prop. 4.7], well-posedness of (QL) under
Assumption 7.1, that is to say, validity of Theorem 7.2, follows directly from
the following lemma.

######  Lemma 7.11.

The processes

(98) $\displaystyle\begin{aligned} \widetilde{M},&\
\widetilde{M}^{2}-\sum_{n\geq 1}\int_{0}^{\cdot}\big{[}\langle
b^{\alpha\beta}_{n,j}(\widetilde{u}^{\beta})\partial_{j}\widetilde{u}^{\beta},\xi\rangle+\langle
g^{\alpha}_{n}(\widetilde{u}),\xi\rangle\big{]}^{2}\,\mathrm{d}r,\\\ &\
\text{and}\ \widetilde{M}w_{n}-\int_{0}^{\cdot}\langle
b^{\alpha\beta}_{n,j}(\widetilde{u}^{\beta})\partial_{j}\widetilde{u}^{\beta},\xi\rangle+\langle
g^{\alpha}_{n}(\widetilde{u}),\xi\rangle\,\mathrm{d}r\end{aligned}$

are $(\widetilde{\mathcal{F}}_{t})$-martingales, where
$\widetilde{\mathcal{F}}_{t}\coloneqq\sigma(\\{\widetilde{u}(s),\widetilde{w}_{n}(s)\colon
s\leq t,n\geq 1\\})$.

###### Proof.

In a first step, we showcase the general strategy for the process
$\widetilde{M}$. Afterwards, we explain the necessary changes for the two
remaining processes.

Step 1: $\widetilde{M}$ is an $(\widetilde{\mathcal{F}}_{t})$-martingale.

Introduce the processes

(99) $\displaystyle M^{\alpha}_{m}(t)$ $\displaystyle=\langle
u^{\alpha}_{m}(t),\xi\rangle-\langle
u^{\alpha}_{0},\xi\rangle+\int_{0}^{t}\langle{}_{m}a^{\alpha\beta}_{ij}(R_{m}u_{m})\partial_{j}u^{\beta}_{m},\partial_{i}\xi\rangle\,\mathrm{d}r$
(100)
$\displaystyle\qquad+\int_{0}^{t}\langle\Phi^{\alpha}_{i}(u_{m}),\partial_{i}\xi\rangle\,\mathrm{d}r-\int_{0}^{t}\langle\phi^{\alpha}(u_{m}),\xi\rangle\,\mathrm{d}r,$
(101) $\displaystyle\widetilde{M}^{\alpha}_{m}(t)$
$\displaystyle=\langle\widetilde{u}^{\alpha}_{m}(t),\xi\rangle-\langle\widetilde{u}^{\alpha}_{m}(0),\xi\rangle+\int_{0}^{t}\langle{}_{m}a^{\alpha\beta}_{ij}(R_{m}\widetilde{u}_{m})\partial_{j}\widetilde{u}^{\beta}_{m},\partial_{i}\xi\rangle\,\mathrm{d}r$
(102)
$\displaystyle\qquad+\int_{0}^{t}\langle\Phi^{\alpha}_{i}(\widetilde{u}_{m}),\partial_{i}\xi\rangle\,\mathrm{d}r-\int_{0}^{t}\langle\phi^{\alpha}(\widetilde{u}_{m}),\xi\rangle\,\mathrm{d}r.$

For each $m$, $M^{\alpha}_{m}$ is an $(\mathcal{F}_{t})$-martingale because
$u_{m}$ is a solution to (m-QL).

Fix $\alpha$ and $s\leq t$ throughout the proof. Let $\rho_{s}$ denote the
canonical restriction
$\mathrm{C}([0,T];\mathcal{X})\to\mathrm{C}([0,s];\mathcal{X})$ and fix any
continuous function $\gamma\colon\mathrm{C}([0,s];\mathcal{X})\to[0,1]$. One
has that

$\gamma_{m}\coloneqq\gamma(\rho_{s}(u_{m},W))$

is $\mathcal{F}_{s}$-measurable. Therefore, conditioning and the martingale
property yield

(103)
$\displaystyle\mathbb{E}\Bigl{(}\gamma_{m}[M^{\alpha}_{m}(t)-M^{\alpha}_{m}(s)]\Bigr{)}=0.$

We claim that $\gamma_{m}$, $M^{\alpha}_{m}(t)$, and $M^{\alpha}_{m}(s)$
depend on $u$ in a measurable way. Indeed, these quantities even depend
continuously on $u$ if we equip $\mathrm{L}^{p}(0,T;\mathrm{H}^{1}_{0}(D))$
with the strong topology in the definition of $\mathcal{X}$. As already
mentioned, passing to the weak topology preserves measurability, which gives
the claim. Therefore, with Lemma 7.9 (i) deduce from (103) that

(104)
$\displaystyle\widetilde{\mathbb{E}}\Bigl{(}\widetilde{\gamma}_{m}[\widetilde{M}^{\alpha}_{m}(t)-\widetilde{M}^{\alpha}_{m}(s)]\Bigr{)}=0,$

where $\widetilde{\gamma}_{m}$ and $\widetilde{M}^{\alpha}_{m}$ are defined by
the same expressions as $\gamma_{m}$ and $M^{\alpha}_{m}$, but with $u_{m}$
replaced by $\widetilde{u}_{m}$ and $W$ replaced by $\widetilde{W}$.

Next, we take the limit $m\to\infty$ in (104). To do so, we appeal to Vitali’s
convergence theorem. Owing to Corollary 7.10, we have for instance

(105)
$\displaystyle\widetilde{\mathbb{E}}\left|\int_{0}^{t}\langle{}_{m}a^{\alpha\beta}_{ij}(R_{m}\widetilde{u}_{m})\partial_{j}\widetilde{u}^{\beta}_{m},\partial_{i}\xi\rangle\,\mathrm{d}r\right|^{p}$
$\displaystyle\lesssim
T^{p-1}\widetilde{\mathbb{E}}\int_{0}^{T}\|\nabla\widetilde{u}_{m}\|_{\mathrm{L}^{2}(D)}^{p}$
(106) $\displaystyle\lesssim
1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))}^{p},$

which justifies the application of Vitali’s convergence theorem. It remains to
determine the almost-sure limit of
$\widetilde{\gamma}_{m}[\widetilde{M}^{\alpha}_{m}(t)-\widetilde{M}^{\alpha}_{m}(t)]$.
On the one hand, by continuity of $\gamma\circ\rho_{s}$ on $\mathcal{X}$,
deduce $\widetilde{\gamma}_{m}\to\widetilde{\gamma}$ almost surely from Lemma
7.9 (ii). On the other hand, we claim that
$\widetilde{M}^{\alpha}_{m}(t)\to\widetilde{M}^{\alpha}(t)$ almost surely.
Convergence of the first two terms of $\widetilde{M}^{\alpha}_{m}(t)$ is
immediate. For the fourth and fifth term, use that $\Phi$ and $\phi$ are
Lipschitz continuous along with almost sure convergence in
$\mathrm{C}([0,T];\mathrm{L}^{2}(D))\subseteq\mathrm{L}^{1}(0,T;\mathrm{L}^{2}(D))$.
The most challenging term is the third. Rewrite

(107)
$\displaystyle\int_{0}^{t}\langle{}_{m}a^{\alpha\beta}_{ij}(R_{m}\widetilde{u}_{m})\partial_{j}\widetilde{u}^{\beta}_{m},\partial_{i}\xi\rangle\,\mathrm{d}r-\int_{0}^{t}\langle
a^{\alpha\beta}_{ij}(\widetilde{u})\partial_{j}\widetilde{u}^{\beta},\partial_{i}\xi\rangle\,\mathrm{d}r$
(108) $\displaystyle={}$
$\displaystyle\int_{0}^{t}\langle\partial_{j}\widetilde{u}^{\beta}_{m},({}_{m}a^{\alpha\beta}_{ij}(R_{m}\widetilde{u}_{m})-a^{\alpha\beta}_{ij}(\widetilde{u}))\partial_{i}\xi\rangle\,\mathrm{d}r+\int_{0}^{t}\langle\partial_{j}(\widetilde{u}^{\beta}_{m}-\widetilde{u}^{\beta}),a^{\alpha\beta}_{ij}(\widetilde{u})\partial_{i}\xi\rangle\,\mathrm{d}r$
(109) $\displaystyle={}$ $\displaystyle\textrm{I}+\textrm{II}.$

We use the convergence properties from Lemma 7.9 (ii) as follows to conclude:
using the weak convergence of $\nabla\widetilde{u}_{m}$ we see that
$\textrm{II}\to 0$ as $m\to\infty$ almost surely. Moreover,
$\nabla\widetilde{u}_{m}$ is uniformly bounded in
$\mathrm{L}^{p}(0,T;\mathrm{L}^{2}(D))\subseteq\mathrm{L}^{2}(0,t;\mathrm{L}^{2}(D))$
almost surely. Consequently, I goes to zero if
$({}_{m}a^{\alpha\beta}_{ij}(R_{m}\widetilde{u}_{m})-a^{\alpha\beta}_{ij}(\widetilde{u}))\partial_{i}\xi\to
0$ in $\mathrm{L}^{2}(0,t;\mathrm{L}^{2}(D))$. Split further

(110)
$\displaystyle{}_{m}a^{\alpha\beta}_{ij}(R_{m}\widetilde{u}_{m})-a^{\alpha\beta}_{ij}(\widetilde{u})={}({}_{m}a^{\alpha\beta}_{ij}(R_{m}\widetilde{u}_{m})-a^{\alpha\beta}_{ij}(R_{m}\widetilde{u}_{m}))+(a^{\alpha\beta}_{ij}(R_{m}\widetilde{u}_{m})-a^{\alpha\beta}_{ij}(\widetilde{u})).$

Since the coefficients are uniformly bounded and $\xi$ is smooth and compactly
supported, we have Vitali’s convergence theorem in $(r,x)$ at our disposal. Up
to passing to another subsequence, we can suppose
$R_{m}\widetilde{u}_{m}\to\widetilde{u}$ pointwise almost everywhere in
$(r,x)$. Then the claim follows by continuity of $a^{\alpha\beta}_{ij}$ and
${}_{m}a^{\alpha\beta}_{ij}$ in conjunction with the pointwise convergence
${}_{m}a^{\alpha\beta}_{ij}\to a^{\alpha\beta}_{ij}$.

In conclusion, we deduce

(111)
$\displaystyle\widetilde{\mathbb{E}}\Bigl{(}\widetilde{\gamma}[\widetilde{M}^{\alpha}(t)-\widetilde{M}^{\alpha}(s)]\Bigr{)}=0.$

But (111) remains valid if we replace $\widetilde{\gamma}$ by _any_ bounded
and $\widetilde{\mathcal{F}}_{s}$-measurable function in virtue of the
monotone class theorem, since $\widetilde{\mathcal{F}}_{s}$ is by definition
the $\sigma$-field induced by $\widetilde{u}$ and $\widetilde{W}$. It follows
that $\widetilde{M}^{\alpha}$ is an $(\widetilde{\mathcal{F}}_{t})$-martingale
as claimed.

Step 2: the remaining two processes.

The general strategy is as in Step 1 and we only discuss the necessary
modifications. The martingale property for the processes related to $u_{m}$ is
again clear. We follow the proof of Step 1 until we reach the analogues of
(104). The first step in the identification of the limit, the moment condition
(105), can be reused with the following observations: first of all, since the
moment condition for one term holds with $p>2$, any product appearing in
$(\widetilde{M}_{m}^{\alpha})^{2}$ satisfies a moment condition with
$\nicefrac{{p}}{{2}}>1$ in virtue of Hölder’s inequality. The same is true for
$\widetilde{M}_{m}w_{k}$, of course. Likewise, one has

(112)
$\displaystyle\widetilde{\mathbb{E}}\left|\int_{0}^{t}\langle{}_{m}b^{\alpha\beta}_{n,j}(R_{m}\widetilde{u}_{m}^{\beta})\partial_{j}\widetilde{u}_{m}^{\beta},\xi\rangle^{2}\,\mathrm{d}r\right|^{\nicefrac{{p}}{{2}}}$
$\displaystyle\lesssim
T^{\nicefrac{{p}}{{2}}-1}\widetilde{\mathbb{E}}\int_{0}^{T}\|\nabla\widetilde{u}_{m}\|_{\mathrm{L}^{2}(D)}^{p}$
(113) $\displaystyle\lesssim
1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))}^{p}.$

It remains to verify almost sure convergence, for instance

(114)
$\displaystyle\int_{0}^{t}\langle{}_{m}b^{\alpha\beta}_{n,j}(R_{m}\widetilde{u}_{m}^{\beta})\partial_{j}\widetilde{u}_{m}^{\beta},\xi\rangle^{2}\,\mathrm{d}r\to\int_{0}^{t}\langle
b^{\alpha\beta}_{n,j}(\widetilde{u}^{\beta})\partial_{j}\widetilde{u}^{\beta},\xi\rangle^{2}\,\mathrm{d}r,$

for fixed $n$. First, we claim that we can replace
${}_{m}b^{\alpha\beta}_{n,j}(R_{m}\widetilde{u}_{m}^{\beta})$ by
$b^{\alpha\beta}_{n,j}(\widetilde{u}^{\beta})$ on the left-hand side of (114).
Indeed, the calculation is similar to the treatment of term I in Step 1, but
one has to use uniform boundedness of $\nabla\widetilde{u}_{m}$ in
$\mathrm{L}^{p}(0,T;\mathrm{L}^{2}(D))$ with $p>2$. Next, define the auxiliary
function

(115)
$\displaystyle\bar{b}_{n,j}^{\alpha\beta}(t,x,y)\coloneqq\int_{0}^{y}b_{n,j}^{\alpha\beta}(t,x,z)\,\mathrm{d}z,\qquad(y\in(0,T)).$

Let us stress that this argument heavily relies on the structural assumption
on $b$, and is the reason why we cannot allow full non-linear dependence on
all components of $u$ as is the case for $a$.

By the chain rule,
$\partial_{j}\bar{b}_{n,j}^{\alpha\beta}(\widetilde{u}_{m})=b_{n,j}^{\alpha\beta}(\widetilde{u}_{m})\partial_{j}\widetilde{u}_{m}$.
Therefore, (114) reduces to

(116)
$\displaystyle\int_{0}^{t}\langle\bar{b}^{\alpha\beta}_{n,j}(\widetilde{u}_{m}^{\beta}),\partial_{j}\xi\rangle^{2}\,\mathrm{d}r\to\int_{0}^{t}\langle\bar{b}^{\alpha\beta}_{n,j}(\widetilde{u}^{\beta}),\partial_{j}\xi\rangle^{2}\,\mathrm{d}r.$

Using Vitali’s convergence theorem once more, we only need to show convergence
for a fixed time $r\in(0,t)$. Now since
$\widetilde{u}_{m}^{\beta}(r)\to\widetilde{u}^{\beta}$ in $\mathrm{L}^{2}(D)$,
we can conclude the proof by passing to another subsequence and using
continuity of $\bar{b}^{\alpha\beta}_{n,j}$. ∎

######  Remark 7.12.

We stress that (105) critically uses $p$-integrability of
$\nabla\widetilde{u}_{m}$ in $\widetilde{\Omega}\times(0,T)$. That such a
bound is available for $p>2$ is a novelty of our approach.

##  8\. Quasilinear problem with growing non-linearities

We reduce the general case with growing non-linearities to the Lipschitz case
from the last section. This happens using a truncation argument. We use the
following classical result from convex analysis (see [9, Proposition 5.3]) to
include systems in a neat way.

######  Lemma 8.1 (Phelps).

Let $C$ be a non-empty, closed, convex subset of a Hilbert space $K$. Then the
projection onto $C$ is Lipschitz continuous with Lipschitz constant $1$.

With the preceding lemma, we can consider truncated non-linearities $\Phi\circ
P$ and $\phi\circ P$, where $P$ is a projection to a bounded and convex set.
As in Section 7.2, we would like to consider $f=\Phi_{i}(P(u))+\phi(P(u))$ as
a fixed right-hand side. However, that $f$ is an admissible right-hand side is
more difficult with growing non-linearities. We will take care of this in
Proposition 8.4. The next two lemmas are a preparation for the last-mentioned
proposition.

######  Lemma 8.2 ([12, Lem. 8]).

For $q>2$ and $m\geq 1$, the function $\psi_{m}\colon\mathbb{R}\to\mathbb{R}$
given by

(117) $\displaystyle\psi_{m}(\xi)=\begin{cases}|\xi|^{q},&|\xi|\leq m,\\\
m^{q-2}\Bigl{[}\frac{q(q-1)}{2}\xi^{2}-q(q-2)m|\xi|+\frac{(q-1)(q-2)}{2}m^{2},&|\xi|>m\end{cases}$

is twice continuously differentiable, has bounded second derivative, and
satisfies the following properties:

(118) $\displaystyle|\psi_{m}^{\prime}(\xi)|$
$\displaystyle\leq|\xi|\psi_{m}^{\prime\prime}(\xi),$ (119)
$\displaystyle\frac{\psi_{m}^{\prime}(\xi)}{\xi}$ $\displaystyle\geq
0,\mathrlap{\quad(\xi\neq 0),}$ (120)
$\displaystyle\psi_{m}^{\prime\prime}(\xi)$ $\displaystyle\leq
q(q-1)(1+\psi_{m}(\xi)),$ (121)
$\displaystyle\xi^{2}\psi_{m}^{\prime\prime}(\xi)$ $\displaystyle\leq
q(q-1)\psi_{m}(\xi),$ (122) $\displaystyle\psi_{m}^{\prime\prime}(\xi_{1})$
$\displaystyle\leq\psi_{m}^{\prime\prime}(\xi_{2}),\mathrlap{\quad(|\xi_{1}|\leq|\xi_{2}|).}$

Moreover, $\psi_{m}\to|\cdot|^{q}$ and
$\psi_{m}^{\prime\prime}\to|\cdot|^{q-2}$ pointwise.

The following technical lemma relates the dissipativity of $\phi$ stated in
Assumption 6.1 (v) with the auxiliary function $\psi_{m}$.

######  Lemma 8.3.

Let $v\in\mathbb{R}^{N}$. For all $q\in(2,\infty)$ and $m\geq 1$ one has for
$\psi_{m}$ defined as in Lemma 8.2 the estimate

(123)
$\displaystyle\sum_{\alpha}\psi_{m}^{\prime}(v^{\alpha})\phi^{\alpha}(v)\lesssim_{q}1+\sum_{\beta}\psi_{m}(v^{\beta}).$

###### Proof.

Let $v\in\mathbb{R}^{N}$ and fix $\alpha$ for the moment. We can assume
$v^{\alpha}\neq 0$. Calculate using (119), Assumption 6.1 (v), and (118) that

(124)
$\displaystyle\psi_{m}^{\prime}(v^{\alpha})\phi^{\alpha}(v)=\frac{\psi_{m}^{\prime}(v^{\alpha})}{v^{\alpha}}v^{\alpha}\phi^{\alpha}(v)\lesssim\frac{\psi_{m}^{\prime}(v^{\alpha})}{v^{\alpha}}(1+|v|^{2})\lesssim\psi_{m}^{\prime\prime}(v^{\alpha})(1+|v|^{2}).$

Using (120), the term $\psi_{m}^{\prime\prime}(v^{\alpha})$ can be controlled
by $1+\psi_{m}(v^{\alpha})$. Likewise,
$\psi_{m}^{\prime\prime}(v^{\alpha})|v^{\alpha}|^{2}$ is controlled by
$\psi_{m}(v^{\alpha})$ owing to (121). It remains to consider
$\psi_{m}^{\prime\prime}(v^{\alpha})|v^{\beta}|^{2}$. We distinguish the cases
$|v^{\alpha}|\leq|v^{\beta}|$ and $|v^{\alpha}|\geq|v^{\beta}|$. In the first
case, we use that $\psi_{m}^{\prime\prime}$ is increasing, (122), to reduce to
the known case $\psi_{m}^{\prime\prime}(v^{\beta})|v^{\beta}|^{2}$. Likewise,
we reduce in the second case using that $|\cdot|^{2}$ is increasing. Finally,
summing in $\alpha$ gives the claim. ∎

######  Proposition 8.4 (bootstrapping integrability).

Suppose Assumption 6.1 holds. Let
$u_{0}\in\mathrm{L}^{q}_{\mathcal{F}_{0}}(\Omega\times D)$ with
$q\in[2,\infty)$ fixed and let
$v\in\mathrm{L}^{2}(\Omega\times(0,T);\mathrm{H}^{1}_{0}(D))\cap\mathrm{L}^{2}(\Omega;\mathrm{C}([0,T];\mathrm{L}^{2}(D)))$
be a solution to (QL) with initial datum $u_{0}$. Then one has

(125) $\displaystyle\|v\|_{\mathrm{L}^{q}(\Omega\times(0,T)\times
D)}^{q}+\mathbb{E}\int_{0}^{T}\int_{D}|v|^{q-2}|\nabla
v|^{2}\,\mathrm{d}x\,\mathrm{d}s\lesssim
1+\|u_{0}\|_{\mathrm{L}^{q}(\Omega;\mathrm{L}^{q}(D))}^{q}.$

###### Proof.

Fix $\alpha$ and let $m\geq 1$, $t\in[0,T]$. Define the linear functional
$v\mapsto\int_{D}\psi_{m}(v)\,\mathrm{d}x$ on $\mathrm{L}^{2}(D)$. Since
$\psi_{m}^{\prime\prime}$ is bounded and, taking (118) into account,
$\psi_{m}^{\prime}$ is of linear growth, the Itô formula from [27, Thm. 4.2]
is applicable and yields

(126) $\displaystyle\int_{D}\psi_{m}(v^{\alpha}(t))\,\mathrm{d}x$
$\displaystyle=\int_{D}\psi_{m}(v^{\alpha}(0))\,\mathrm{d}x$ (127)
$\displaystyle\quad-\int_{0}^{t}\int_{D}\psi_{m}^{\prime\prime}(v^{\alpha})a_{ij}^{\alpha}(v)\partial_{i}v^{\alpha}\partial_{j}v^{\alpha}\,\mathrm{d}x\,\mathrm{d}s$
(128)
$\displaystyle\quad-\int_{0}^{t}\int_{D}\psi_{m}^{\prime\prime}(v^{\alpha})\Phi_{i}^{\alpha}(v)\partial_{i}v^{\alpha}\,\mathrm{d}x\,\mathrm{d}s$
(129)
$\displaystyle\quad+\int_{0}^{t}\int_{D}\psi_{m}^{\prime}(v^{\alpha})\phi^{\alpha}(v)\,\mathrm{d}x\,\mathrm{d}s$
(130)
$\displaystyle\quad+\sum_{n}\int_{0}^{t}\int_{D}\psi_{m}(v^{\alpha})\bigl{[}b_{n,j}^{\alpha}(v^{\alpha})\partial_{j}v^{\alpha}+g_{n}^{\alpha}(v)\bigr{]}\,\mathrm{d}x\,\mathrm{d}w_{n}$
(131)
$\displaystyle\quad+\sum_{n}\frac{1}{2}\int_{0}^{t}\int_{D}\psi_{m}^{\prime\prime}(v^{\alpha})\Bigl{|}\sum_{j}b_{n,j}^{\alpha}(v^{\alpha})\partial_{j}v^{\alpha}+g_{n}^{\alpha}(v)\Bigr{|}^{2}\,\mathrm{d}x\,\mathrm{d}s$
(132)
$\displaystyle\quad=\textrm{I}-\textrm{II}-\textrm{III}+\textrm{IV}+\textrm{V}+\textrm{VI}.$

If we interpret $\Phi^{\alpha}_{i}(v)$ as a mapping
$u\mapsto\Phi^{\alpha}_{i}(t,x,v^{1},\dots,v^{\alpha-1},u,v^{\alpha+1},\dots,v^{N})$,
then $\textrm{III}=0$ follows from the divergence theorem, since $v^{\alpha}$
vanishes on the boundary of $D$.

Use the inequality
$\nicefrac{{1}}{{2}}|b+g|^{2}\leq(\nicefrac{{1}}{{2}}+\varepsilon)b^{2}+C_{\varepsilon}g^{2}$
for all $\varepsilon>0$ in VI to get

(133) VI
$\displaystyle\leq(\nicefrac{{1}}{{2}}+\varepsilon)\sum_{n}\int_{0}^{t}\int_{D}\psi_{m}^{\prime\prime}(v^{\alpha})\Bigl{|}\sum_{j}b_{n,j}^{\alpha}(v^{\alpha})\partial_{j}v^{\alpha}\Bigr{|}^{2}\,\mathrm{d}x\,\mathrm{d}s$
(134)
$\displaystyle\qquad+C_{\varepsilon}\sum_{n}\int_{0}^{t}\int_{D}\psi_{m}^{\prime\prime}(v^{\alpha})|g_{n}^{\alpha}(v)|^{2}\,\mathrm{d}x\,\mathrm{d}s$
(135) $\displaystyle=\textrm{VI}(b)+\textrm{VI}(g).$

By the “diagonal structure” of $a$ and $b$ stated in Assumption 6.1 (i), the
coercivity condition in Assumption 6.1 (iii) can be applied componentwise.
Therefore, by choosing $\varepsilon$ small enough (similar to the proof of
Lemma 7.4), we get the lower bound

(136)
$\displaystyle\textrm{II}-\textrm{VI}(b)\geq\frac{\lambda}{2}\int_{0}^{t}\int_{D}\psi_{m}^{\prime\prime}(v^{\alpha})|\nabla
v^{\alpha}|^{2}\,\mathrm{d}x\,\mathrm{d}s.$

This is the only argument that uses Assumption 6.1 (i), see Remark 6.3 for
further discussion. Hence, we can absorb $-\textrm{II}+\textrm{VI}(b)$ into
the left-hand side, to give

(137)
$\displaystyle\begin{split}&\int_{D}\psi_{m}(v^{\alpha})\,\mathrm{d}x+\frac{\lambda}{2}\int_{0}^{t}\int_{D}\psi_{m}^{\prime\prime}(v^{\alpha})|\nabla
v^{\alpha}|^{2}\,\mathrm{d}x\,\mathrm{d}s\\\
\leq{}&\int_{D}\psi_{m}(v^{\alpha}(0))\,\mathrm{d}x\\\
&+\int_{0}^{t}\int_{D}\psi_{m}^{\prime}(v^{\alpha})\phi^{\alpha}(v)\,\mathrm{d}x\,\mathrm{d}s\\\
&+\sum_{n}\int_{0}^{t}\int_{D}\psi_{m}(v^{\alpha})\bigl{[}b_{n,j}^{\alpha}(v^{\alpha})\partial_{j}v^{\alpha}+g_{n}^{\alpha}(v)\bigr{]}\,\mathrm{d}x\,\mathrm{d}w_{n}\\\
&+C_{\varepsilon}\sum_{n}\int_{0}^{t}\int_{D}\psi_{m}^{\prime\prime}(v^{\alpha})|g_{n}^{\alpha}(v)|^{2}\,\mathrm{d}x\,\mathrm{d}s.\end{split}$

The first term on the right-hand side of (137) is controlled by
$\|u_{0}\|_{\mathrm{L}^{q}(D)}^{q}$ by definition of $\psi_{m}$. We come to
the second term. Applying Lemma 8.3 with $v=v(s,x)$, estimate

(138)
$\displaystyle\int_{0}^{t}\int_{D}\psi_{m}^{\prime}(v^{\alpha})\phi^{\alpha}(v)\,\mathrm{d}x\,\mathrm{d}s\lesssim
1+\int_{0}^{t}\int_{D}\sum_{\beta}\psi_{m}(v^{\beta})\,\mathrm{d}x\,\mathrm{d}s.$

Since $g$ is Lipschitz and hence of linear growth, we get
$\psi_{m}^{\prime\prime}(v^{\alpha})|g_{n}^{\alpha}(v)|^{2}\lesssim\psi_{m}^{\prime\prime}(v^{\alpha})(1+|v|^{2})$,
so the proof of Lemma 8.3 reveals also

(139)
$\displaystyle\int_{0}^{t}\int_{D}\psi_{m}^{\prime\prime}(v^{\alpha})|g_{n}^{\alpha}(v)|^{2}\,\mathrm{d}x\,\mathrm{d}s\lesssim
1+\int_{0}^{t}\int_{D}\sum_{\beta}\psi_{m}(v^{\beta})\,\mathrm{d}x\,\mathrm{d}s.$

So far, we have in summary

(140)
$\displaystyle\begin{split}&\int_{D}\psi_{m}(v^{\alpha})\,\mathrm{d}x+\frac{\lambda}{2}\int_{0}^{t}\int_{D}\psi_{m}^{\prime\prime}(v^{\alpha})|\nabla
v^{\alpha}|^{2}\,\mathrm{d}x\,\mathrm{d}s\\\
\lesssim{}&1+\|u_{0}\|_{\mathrm{L}^{q}(D)}^{q}+\int_{0}^{t}\int_{D}\sum_{\beta}\psi_{m}(v^{\beta})\,\mathrm{d}x\,\mathrm{d}s\\\
&\quad+\sum_{n}\int_{0}^{t}\int_{D}\psi_{m}(v^{\alpha})\bigl{[}b_{n,j}^{\alpha}(v^{\alpha})\partial_{j}v^{\alpha}+g_{n}^{\alpha}(v)\bigr{]}\,\mathrm{d}w_{n}\,\mathrm{d}s.\end{split}$

We take the expectation, so that the stochastic integral vanishes, and sum in
$\alpha$ afterwards, to find

(141)
$\displaystyle\begin{split}&\mathbb{E}\int_{D}\sum_{\alpha}\psi_{m}(v^{\alpha})\,\mathrm{d}x+\frac{\lambda}{2}\mathbb{E}\int_{0}^{t}\int_{D}\sum_{\alpha}\psi_{m}^{\prime\prime}(v^{\alpha})|\nabla
v^{\alpha}|^{2}\,\mathrm{d}x\,\mathrm{d}s\\\
\lesssim{}&1+\mathbb{E}\|u_{0}\|_{\mathrm{L}^{q}(D)}^{q}+\mathbb{E}\int_{0}^{t}\int_{D}\sum_{\alpha}\psi_{m}(v^{\alpha})\,\mathrm{d}x\,\mathrm{d}s.\end{split}$

Note that for all $t\in[0,T]$ one has
$\mathbb{E}\int_{0}^{t}\int_{D}\sum_{\alpha}\psi_{m}(v^{\alpha}(s))\,\mathrm{d}x\,\mathrm{d}s$
is finite, for $\psi_{m}(v(s))\leq|v(s)|^{2}+C$. Therefore, Gronwall’s
inequality is applicable with
$t\mapsto\mathbb{E}\int_{D}\sum_{\alpha}\psi_{m}(v^{\alpha}(t))\,\mathrm{d}x$
and yields

(142)
$\displaystyle\mathbb{E}\int_{D}\sum_{\alpha}\psi_{m}(v^{\alpha})\,\mathrm{d}x\lesssim
1+\mathbb{E}\|u_{0}\|_{\mathrm{L}^{q}(D)}^{q}$

for all $t\in(0,T)$. Integrate this bound in $t$, use Fubini’s theorem, and
take (in virtue of the pointwise convergence of $\psi_{m}$ from Lemma 8.2) the
limit $m\to\infty$ to conclude the first claim of the proposition. Finally,
plug (142) back into (141) to deduce

(143)
$\displaystyle\frac{\lambda}{2}\mathbb{E}\int_{0}^{t}\int_{D}\sum_{\alpha}\psi_{m}^{\prime\prime}(v^{\alpha})|\nabla
v|^{2}\,\mathrm{d}x\,\mathrm{d}s\lesssim{}$ $\displaystyle
1+\mathbb{E}\|u_{0}\|_{\mathrm{L}^{q}(D)}^{q}.$

This time using the pointwise convergence of $\psi_{m}^{\prime\prime}$ stated
in Lemma 8.2, take once more the limit $m\to\infty$. Afterwards, take the
limit $t\to T$ to conclude the second claim in the proposition. ∎

######  Remark 8.5.

Even more is true in the last proposition: if we take first the supremum over
$t$ in (140) and then expectations, a calculation based on the
Burkholder–Davis–Gundy inequality even yields the estimate

(144)
$\displaystyle\|v\|_{\mathrm{L}^{q}(\Omega;\mathrm{C}([0,T];\mathrm{L}^{q}(D)))}\lesssim
1+\|u_{0}\|_{\mathrm{L}^{q}(\Omega;\mathrm{L}^{q}(D))}.$

We will not need this extra information and decided therefore to stick to the
shorter proof that leads to Proposition 8.4.

###### Proof of Theorem 6.6.

Write $P_{R}$ for the projection onto the closed ball of radius $R$ around $0$
in $\mathbb{R}^{N}$. Then, for $R>0$, define the non-linearities
${}_{R}\phi^{\alpha}(t,x,y)\coloneqq\phi^{\alpha}(t,x,P_{R}(y))$ and
${}_{R}\Phi^{\alpha}_{i}(t,x,y)\coloneqq\Phi^{\alpha}_{i}(t,x,P_{R}(y))$. They
are Lipschitz continuous and satisfy Assumption 6.1 (iv) uniformly in $R$ in
virtue of Lemma 8.1. Consider the equation

(R-QL) $\displaystyle\left\\{\quad\begin{aligned}
\,\mathrm{d}u^{\alpha}&=\big{[}\partial_{i}(a^{\alpha\beta}_{ij}(u)\partial_{j}u^{\beta})+\partial_{i}({}_{R}\Phi^{\alpha}_{i}(u))+{}_{R}\phi^{\alpha}(u)\big{]}\,\mathrm{d}t\\\
&\qquad\qquad+\sum_{n\geq
1}\big{[}b^{\alpha\beta}_{n,j}(u^{\beta})\partial_{j}u^{\beta}+g^{\alpha}_{n}(u)\big{]}\,\mathrm{d}w_{n},\
\ &\text{ on }D,\\\ u^{\alpha}&=0,\ \ &\text{ on }\partial D,\\\
u^{\alpha}(0)&=u^{\alpha}_{0},\ \ &\text{ on }D.\end{aligned}\right.$

Problem (R-QL) fulfills Assumption 7.1. Therefore, by the well-posedness
result of Section 7, there are $p_{0}>2$ and solutions

$u_{R}\in\mathrm{L}^{p}(\Omega\times(0,T);\mathrm{H}^{1}_{0}(D))\cap\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D)))$

to (R-QL).

Apply Proposition 8.4 to $u_{R}$ to find
$\|u_{R}\|_{\mathrm{L}^{q}(\Omega\times(0,T)\times D)}\lesssim
1+\|u_{0}\|_{\mathrm{L}^{q}(\Omega\times D)}$. With the growth condition for
the non-linearities we find, keeping $ph\leq q$ in mind,

(145)
$\displaystyle\|{}_{R}\phi^{\alpha}(u_{R})\|_{\mathrm{L}^{p}(\Omega\times(0,T);L^{2}(D))}+\|{}_{R}\Phi^{\alpha}_{i}(u_{R})\|_{\mathrm{L}^{p}(\Omega\times(0,T);L^{2}(D))}$
(146) $\displaystyle\lesssim{}$ $\displaystyle
1+\|u_{R}\|^{h}_{L^{q}(\Omega\times(0,T)\times D)}$ (147)
$\displaystyle\lesssim{}$ $\displaystyle 1+\|u_{0}\|^{h}_{L^{q}(\Omega\times
D)}.$

Therefore, we can freeze $u_{R}$, ${}_{R}\Phi$ and ${}_{R}\phi$ in (R-QL) to
obtain, as in Section 7.3, a linear problem to which Theorem 2.2 applies. This
results in the uniform bounds

(148)
$\displaystyle\|u_{R}\|_{\mathrm{L}^{p}(\Omega;\mathrm{C}([0,T];\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D)))}\lesssim
1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))}+\|u_{0}\|^{h}_{L^{q}(\Omega\times
D)},$

and for all $\theta\in[0,\nicefrac{{1}}{{2}})$ the estimate

(149)
$\displaystyle\|u_{R}\|_{\mathrm{L}^{p}(\Omega;\mathrm{H}^{\theta,p}(0,T;\mathrm{H}^{1-2\theta}_{0}(D)))}\lesssim_{\theta}1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))}+\|u_{0}\|^{h}_{L^{q}(\Omega\times
D)}.$

If $\theta\in(\nicefrac{{1}}{{p}},\nicefrac{{1}}{{2}})$, then one has moreover
the estimate

(150)
$\displaystyle\|u_{R}\|_{\mathrm{L}^{p}(\Omega;\mathrm{C}^{\theta-\nicefrac{{1}}{{p}}}([0,T];\mathrm{H}^{1-2\theta}_{0}(D)))}\lesssim_{\theta}1+\|u_{0}\|_{\mathrm{L}^{p}(\Omega;\mathrm{B}^{1-\nicefrac{{2}}{{p}}}_{2,p,0}(D))}+\|u_{0}\|^{h}_{L^{q}(\Omega\times
D)}.$

Then, the conclusion using the stochastic compactness argument works
analogous, with one additional argument: first, by (145), the terms
corresponding to $\Phi$ and $\phi$ in (99) satisfy the same moment condition
as before. Second, for the almost sure convergence, decompose

(151)
$\displaystyle\phi(\widetilde{u})-{}_{R}\phi(\widetilde{u}_{R})=(\phi(\widetilde{u})-\phi(\widetilde{u}_{R}))+(\phi(\widetilde{u}_{R})-{}_{R}\phi(\widetilde{u}_{R})),$

likewise for ${}_{R}\Phi_{i}$. As seen before, by Vitali’s convergence
theorem, it suffices to show $(r,x)$ almost everywhere convergence of
$\langle{}_{R}\phi(\widetilde{u}_{R}),\xi\rangle$ and
$\langle{}_{R}\Phi_{i}(\widetilde{u}_{R}),\partial_{i}\xi\rangle$, and upon
passing to a subsequence, we can assume that $\widetilde{u}_{R}$ converges to
$\widetilde{u}$ for almost every $(s,x)$. Therefore, the second term on the
right-hand side of (151) goes to zero since $\widetilde{u}_{R}$ is a bounded
sequence for fixed $(s,x)$, so that eventually $\phi$ and ${}_{R}\phi$
coincide. For the first term on the right-hand side of (151), we upgrade
almost everywhere convergence of the subsequence to
$\mathrm{L}^{ph}((0,T)\times D)$ convergence using Vitali’s theorem (recall
that $q>ph)$. Then the claim follows by continuity of the Nemytski operator
[14, Thm. 3.2.24], that is to say, continuity of the operator
$v\mapsto\phi(t,x,v)$ from $\mathrm{L}^{ph}((0,T)\times
D)\to\mathrm{L}^{p}((0,T)\times D)$. Here, we use that $\phi$ is a
Carathéodory function satisfying the uniform growth condition stated in
Assumption 6.1 (iv). ∎

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