# Thin Film Equations with Nonlinear Deterministic and Stochastic
Perturbations

Oleksiy Kapustyan<EMAIL_ADDRESS>Department of Mathematics, Taras
Shevchenko National University of Kyiv, Ukraine  Olha Martynyuk
<EMAIL_ADDRESS>Department of Mathematics, Yuriy Fedkovych Chernivtsi
National University, Chernivtsi, Ukraine  Oleksandr Misiats<EMAIL_ADDRESS>Department of Mathematics, Virginia Commonwealth University, Richmond, VA,
23284, USA  Oleksandr Stanzhytskyi<EMAIL_ADDRESS>Department of
Mathematics, Taras Shevchenko National University of Kyiv, Ukraine

###### Abstract

In this paper we consider stochastic thin-film equation with nonlinear drift
terms, colored Gaussian Stratonovych noise, as well as nonlinear colored
Wiener noise. By means of Trotter-Kato-type decomposition into deterministic
and stochastic parts, we couple both of these dynamics via a discrete-in-time
scheme, and establish its convergence to a non-negative weak martingale
solution.

##  1 Introduction.

We consider the nonlinear stochastic thin-film equation with nonlinear drift
coefficients

$du=(-\partial_{x}(u^{2}u_{xxx})+l(u))dt+\partial_{x}(u\circ
dW)+f(u)dW_{1}(t)$ (1.1)

for $(t,x)\in[0,T)\times\mathbb{T}_{L}$, on torus $\mathbb{T}_{L}$, where $T$
and $L$ are positive constants, and $\mathbb{T}_{L}$ denotes the torus on the
interval $[0,L]$ with periodic boundary conditions

$\partial_{x}^{i}u(\cdot,0)=\partial_{x}^{i}u(\cdot,L),\ i=0,1,2,3$

and non-negative initial condition $u(0,x)=u_{0}(x)$. The term
$\partial_{x}(u\circ dW)$ is a stochastic perturbation in Stratonovich form,
and $f(u)dW$ is a Stochastic perturbation of Ito type. Here

$W(t,x):=\sum_{k\in\mathbb{Z}}\lambda_{k}\Psi_{k}(x)\beta^{k}(t),\ \
W_{1}(t,x):=\sum_{k\in\mathbb{Z}}\gamma_{k}\Psi_{k}(x)\beta_{1}^{k}(t),$

where $\\{\Psi_{k}\\}$ is ONB in $H^{2}(\mathbb{T})$, namely

$\Psi_{k}(x):=c_{k}\begin{cases}\cos\left(\frac{2\pi k}{L}x\right),k>0,\
x\in[0,L];\\\ \frac{1}{\sqrt{2}},k=0,\ x\in[0,L];\\\ \sin\left(\frac{2\pi
k}{L}x\right),k<0,\ x\in[0,L],\end{cases}$ (1.2)

with

$c_{k}=\sqrt{\frac{2}{L\left(1+\left(\frac{2\pi
k}{L}\right)^{2}+\left(\frac{2\pi k}{L}\right)^{4}\right)}}.$

In particular, the functions $\\{\Psi_{k}\\}$ are the eigenfunctions of the
Laplace operator with periodic boundary conditions, which satisfy

$\displaystyle\partial_{x}\Psi_{k}=\frac{2\pi k}{L}\Psi_{-k};\
\partial_{x}^{2}\Psi_{k}=-\frac{4\pi^{2}k^{2}}{L^{2}}\Psi_{k};$ (1.3)
$\displaystyle\partial_{x}^{3}\Psi_{k}=-\frac{8\pi^{3}k^{3}}{L^{3}}\Psi_{-k};\
\partial_{x}^{4}\Psi_{k}=\frac{16\pi^{4}k^{4}}{L^{4}}\Psi_{k},\
k\in\mathbb{Z}.$

The processes $\beta^{k}$ and $\beta_{1}^{k}$ are mutually independent
standard real-valued $\mathcal{F}_{t}$-Wiener processes on a complete filtered
probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\},\mathcal{P}),t\in[0,T]$, with a
complete and right-continuous filtration $(\mathcal{F}_{t})_{t\in[0,T]}$. The
coefficients $\lambda_{k}\geq 0$ and $\gamma_{k}$ satisfy the coloring
condition

$\sum_{k\in\mathbb{Z}}(\lambda_{k}^{2}+\gamma_{k}^{2})<\infty.$ (1.4)

Since in this work we will be studying the martingale solutions using the
Skorokhod approach [27] and its generalization [17], without loss of
generality we consider the probability space to be $\Omega=[0,1]$ with
Lebesgue measure. Similarly to [13], may be re-written in Ito’s form

$du=\left(\partial_{x}(-u^{2}u_{xxx})+\frac{1}{2}\sum_{k=1}^{\infty}\lambda_{k}^{2}\Psi_{k}^{2}\partial_{x}(\Psi_{k}u)+l(u)\right)dt+\sum_{k=1}^{\infty}\lambda_{k}(\partial_{x}(\Psi_{k}u))d\beta^{k}\\\
+\sum_{k=1}^{\infty}\gamma_{k}\Psi_{k}f(u)d\beta_{1}^{k}$

Finally, we will assume that the nonlinear drift coefficients are
$l(u)=-|u|^{r-1}u$ for some $r\geq 1$, and $f(u)$ is globally Lipschitz with
$f(0)=0$.

The deterministic equations of type (1.1) arise in modeling the motion of
liquid droplets of thickness $u$, spreading over the solid surface. This model
follows from lubrication theory under the assumption that the dimensions in
the horizontal directions are significantly larger than in the vertical
(normal) one. In this regime the dynamics of the droplet is governed by the
surface tension and limited by viscosity.

In broad terms, the dynamics of the deterministic version of (1.1) is
characterized by the presence of the wetted regions $u>0$. The equation is
parabolic in the interior of these regions, and degenerate on their boundary.
The boundary of the wetted region, in turn, has a finite speed of propagation
[4]. Thus, one may interpret (1.1) as a fourth-order nonlinear free boundary
problem inside a wetted region, which itself evolves in time. Furthermore, the
classical parabolic theory is not applicable to this equation, in particular,
due to the lack of comparison (maximum) principle, which is widely used in the
existence theory of degenerate second order parabolic equations. Bernis and
Friedman [5] used the energy-entropy method to prove the existence of a non-
negative generalized weak solution, which was constructed as a limit of
solutions of a regularized problem. In this work, the authors’ notion of a
weak solution is somewhat “weaker” than usual, since the integral identity in
the definition of the weak solution has to hold not in the entire
$\mathbb{T}$, but only on the subset where $u>0$. In addition, in [5] the
authors analyzed the support of the solution. The existence of more regular
(strong or entropy) solutions was shown in [2], where the authors also studied
the asymptotic in $t$ behavior of the solution. Further properties of
solutions, including convergence to steady states, finite propagation speed,
and waiting time phenomenon, were obtained in [6]. The work [18] studies the
generalized thin film equation with a nonlinear dissipative term $l(u)$, which
models the relation between nonlinear absorption and spatial injection.

In this paper we consider the thin film equation with two different stochastic
perturbations, of Ito and Statonovych type. It is worth mentioning that one
needs to be very careful when dealing with the effect of noise on nonlocal
and/or ill-posed problems. In some cases, e.g. [22], the presence of a even
small stochastic perturbation leads to a finite time blowup while an
unperturbed equation has global solution. In others [26], the effect of the
random perturbation is exactly opposite - it may lead to the existence of a
global solution while the corresponding deterministic equation has a finite
time blowup. The long time behavior of stochastically perturbed evolution
equations is typically described via the existence and properties of invariant
measures, see, e.g. [23], [25], [24], [28], [15], [19], [7].

The stochastic version of thin-film equation, which takes into account the
effect of random forcing when modeling the enhanced spreading of droplets, was
first introduced in [10], with $l=f=0$. In the subsequent work [14] the
authors additionally take the interface potential between fluid and substrate
into account, which prevents the solution $u$ from becoming negative. This
work describes coarsening and de-wetting phenomena. The first rigorous
construction of a non-negative martingale solution of the stochastic thin film
equation with Ito noise and additional interface potential was obtained in
[11]. In [8], the author considered a more general case of the main operator
in the form $-\partial_{x}(u^{n}u_{xxx})$ (referred as more general mobility),
and established the conditions for the existence of a global strong solution
for this problem. In [13] the authors established the existence of a
nonnegative matringale solution for (1.1) by means of Trotter-Kato type
decomposition. The subsequent work [9] establishes the existence and
uniqueness of a weak solution of this problem. To this end, the authors
started with the establishing the existence of a weak solution for the
regularized equation by means of Galiorkin approximations, and proceeded with
passing to the limit in the regularized problem.

The equation (1.1), which is considered in this paper, has significant
differences from the similar models, analyzed by the other authors. In it, we
introduce the nonlinear drift $l(u)$, as well as the nonlinear stochastic Ito
perturbation $f(u)dW_{1}(t)$. Due to the presence of these two terms, the
equation is no longer in divergence form. In other words, if $f\equiv 0$ and
$l\equiv 0$, integrating both sides of (1.1) over $\mathbb{T}$, one gets
$\frac{d}{dt}\int_{\mathbb{T}}udx=0$, implying, in view of nonnegativity of
$u$, the conservation of mass property (almost surely). This property plays a
crucial role in [13]. However, if either $f\neq 0$ or $l\neq 0$, this property
no longer holds. Nevertheless, in this paper we obtain the estimates on the
mass, which, in turn, enable us to obtain the energy estimates similar to
[13]. Our analysis starts with Trotter-Kato decomposition, which separates the
deterministic dynamics from stochastic. The principal difference in our case
is the presence of the nonlinear drift term in the deterministic dynamics,
which prevents us from using the classic results on deterministic thin-film
equations, e.g. [5, 2]. Instead, we obtain the analogs of the main result in
[18] with different boundary conditions. The stochastic dynamics, in contrast
to [13], is nonlinear, which makes its analysis more complicated, especially
in view of the lack of mass preservation. In particular, it requires a
different approach to establish the non-negativity property of the solutions.

The paper is structured as follows: in Section 2 we introduce the notation,
list preliminary results, formulate the main results, and introduce the
decomposition of the dynamics. In Section 3 we describe the deterministic
dynamics in detail. Section 4 is devoted to stochastic dynamics, and finally,
the main result is established in Section 5.

##  2 Preliminaries and main results.

Throughout the paper, we denote

$Q_{T}=(0,T)\times\mathbb{T}_{L}.$

For $u,v\in\mathbb{T}_{L}$, let

$(u,v)_{2}:=\int_{0}^{L}u(x)v(x)dx,\text{ and }\|u\|_{2}:=\sqrt{(u,u)_{2}}.$

Next, for $Q\in\mathbb{R}^{d}$ with $\partial Q\in C^{\infty}$, for
$s\in[0,\infty)$, $p\in[1,\infty)$, let $W^{s,p}(Q)$ be the regular Sobolev
space for $s\in\mathbb{N}$, and Sobolev-Slobodeckij space for non-integer $s$.
For $p=2$ we will denote $W^{s,p}(Q)=H^{s}(Q)=H^{s}$. If $X$ is a Banach
space, the space $C^{k+\alpha}(Q;X)$ is the space of $k$ times differentiable
functions $Q\to X$, whose $k$-th derivatives are Holder-continous with
exponent $\alpha\in(0,1)$ on compact subsets of $Q$. We also denote
$C^{k-}(Q;X)$ to be the space of $k-1$-times differentiable functions, whose
$k-1$-st derivatives are Lipschitz continuous. The space $BC^{0}(Q,X)$ is the
set of bounded continuous functions. The pairings $\langle\cdot,\cdot\rangle$
is the $H^{-1}(\mathbb{T}_{L})-H^{1}(\mathbb{T}_{L})$ pairing in
$L^{2}(\mathbb{T}_{L})$, and $\langle\langle\cdot,\cdot\rangle\rangle$ stands
for the pairing between $L^{2}(\mathbb{T}_{L})$ and $H^{2}(\mathbb{T}_{L})$ in
$H^{1}(\mathbb{T}_{L})$. We denote $H^{1}_{w}(\mathbb{T}_{L})$ to be the space
of $H^{1}(\mathbb{T}_{L})$ functions endowed with the weak topology, induced
by $\|\cdot\|_{1,2}$. The space $\dot{H}^{1}(\mathbb{T}_{L})$ is the Sobolev
space of $H^{1}(\mathbb{T}_{L})$ functions endowed with the norm
$\|\partial_{x}u\|_{L^{2}(\mathbb{T}_{L})}$. In this space the functions which
differ by an additive constant are the same. Denote $L_{2}(U;H)$ to be the
space of Hilbert-Schmidt operators from $U$ to $H$. In this case, for $A\in
L_{2}(U;H)$ we have

$\|A\|^{2}_{L_{2}(U;H)}:=\sum_{n\geq 1}\|Ae_{n}\|^{2}_{H},$

where $\\{e_{n}\\}$ is any orthonormal basis in $U$. The symbol
$\langle\langle\cdot\rangle\rangle_{t}$ denotes the quadratic variation
process. For $u:Q_{T}\to\mathbb{R}$ we denote

$P_{T}:=\\{(t,x)\in\bar{Q}_{T},u(t,x)>0\\}.$

Consider a triple consisting of a filtered probability space
$([0,1],\tilde{\mathcal{F}},\tilde{\mathcal{F}}_{t\in[0,T]},P)$, where
$\tilde{\mathcal{F}}_{t\in[0,T]}$ is a complete right continuous filtration.

######  Definition 2.1.

Let $u_{0}\in H^{1}(\mathbb{T}_{L})$. An $\tilde{\mathcal{F}}_{t}$ adapted
bounded continuous $H^{1}_{w}(\mathbb{T}_{L})$ \- valued process $\tilde{u}$
on $[0,T]$, such that the distributional derivative
$\partial_{x}^{3}\tilde{u}$ is $\tilde{\mathcal{F}}_{t}$ \- adapted with
$\partial_{x}^{3}\tilde{u}\in L^{2}(\\{\tilde{u}>r\\})$ for any $r>0$, and
$\tilde{u}^{2}(\partial_{x}^{3}\tilde{u})$ is in $L^{2}(\\{\tilde{u}>0\\})$
$\mathbb{P}$ \- almost surely, as well as mutually independent standard real-
valued $(\tilde{\mathcal{F}}_{t})$-Wiener processes $\tilde{\beta}^{k}$, is
called a martingale solution of the equation (1.1), if its weak formulation

$\displaystyle(\tilde{u}(t,\cdot),\varphi)_{2}-(u_{0},\varphi)_{2}=\int_{0}^{t}\int_{\\{\tilde{u}(s,\cdot)>0\\}}\tilde{u}^{2}(s,\cdot)(\partial_{x}^{3}\tilde{u}(s,\cdot))\partial_{x}\varphi
dxds$ (2.1)
$\displaystyle-\frac{1}{2}\sum_{k\in\mathbb{Z}}\lambda_{k}^{2}\int_{0}^{t}(\Psi_{k}\partial_{x}(\Psi_{k}\tilde{u}(s,\cdot)),\partial_{x}\varphi)_{2}ds+\int_{0}^{t}(l(\tilde{u}(s,\cdot)),\varphi)_{2}ds$
$\displaystyle-\sum_{k\in\mathbb{Z}}\lambda_{k}\int_{0}^{t}(\Psi_{k}\tilde{u}(s,\cdot),\partial_{x}\varphi)_{2}d\tilde{\beta}^{k}(s)+\sum_{k\in\mathbb{Z}}\lambda_{k}\int_{0}^{t}(\Psi_{k}f(\tilde{u}(s,\cdot)),\varphi)d\tilde{\beta}^{k}(s)$

is satisfied for every $\varphi\in C^{\infty}(\mathbb{T}_{L})$ and $t\in[0,T]$
$\mathbb{P}=\lambda_{[0,1]}$ almost surely.

We have the following result:

######  Theorem 2.2.

(existence of martingale solution) Suppose $u_{0}\in H^{1}(\mathbb{T}_{L})$ is
such that $u_{0}\geq 0$. Then the equation (1.1) with has a martingale
solution $\tilde{u}(t)$ in the sense of Definition 2.1, which is non-negative
a.s. for $t\in[0,T]$, and for any $p\geq 2$ there is $C_{p}>0$ such that

$\mathbb{E}\sup_{t\in[0,T]}\|\tilde{u}(t,\cdot)\|_{H^{1}(\mathbb{T})}^{p}\leq
C_{p}\|u_{0}\|_{H^{1}(\mathbb{T})}^{p}$

for any $p\in[2,\infty)$, where $C<\infty$ is independent on $u_{0}$.

In order to establish this result, we will use the Trotter-Kato type
decomposition of the dynamics into deterministic and stochastic parts. This
method was used by a number of authors, e.g. [13, 21], in particular, to
establish the existence of for SPDEs with local Lipschitz coefficients. To
this end, for fixed $N\geq 1$, we equi-partition the time interval $[0,T]$
into intervals of length $\delta=\frac{T}{N+1}$. Then, for
$t\in[(j-1)\delta,j\delta]$ and for an arbitrary $\varphi\in
C^{\infty}(\mathbb{\mathbb{T}_{L}})$ we define the following dynamics:

Deterministic Dynamics (D) We look for the function $v_{N}$ satisfying

$(v_{N}(t,\cdot),\varphi)_{2}-(v_{N}((j-1)\delta,\cdot),\varphi)_{2}=\int_{(j-1)\delta}^{t}\int_{v_{N}(s,\cdot)>0}v_{N}^{2}(s,\cdot)\partial_{x}^{3}v_{N}^{2}(s,\cdot)(\partial_{x}\varphi)dxds\\\
+\int_{(j-1)\delta}^{t}(l(v_{N}(s,\cdot)),\varphi)_{2}ds.$ (2.2)

Stochastic Dynamics (S) We look for the function $w_{N}$ satisfying

$(w_{N}(t,\cdot),\varphi)_{2}-(w_{N}((j-1)\delta,\cdot),\varphi)_{2}=-\frac{1}{2}\sum_{k\in\mathbb{Z}}\lambda_{k}^{2}\int_{(j-1)\delta}^{t}(\Psi_{k}\partial_{x}(\Psi_{k}w_{N}(s,\cdot)),\partial_{x}\varphi)_{2}ds\\\
-\sum_{k\in\mathbb{Z}}\lambda_{k}\int_{(j-1)\delta}^{t}(\Psi_{k}w_{N}(s,\cdot),\partial_{x}\varphi)_{2}d\beta^{k}(s)+\sum_{k\in\mathbb{Z}}\gamma_{k}\int_{(j-1)\delta}^{t}(\Psi_{k}f(w_{N}(s,\cdot)),\varphi)_{2}d\beta_{1}^{k}(s).$
(2.3)

Deterministic-Stochastic Connection (DS) We set $v_{N}(0):=u_{0}$,
$v_{N}(j\delta,\cdot):=\lim_{t\to j\delta-}w_{N}(t,\cdot)$ and
$w_{N}((j-1)\delta,\cdot):=\lim_{t\to j\delta-}v_{N}(t,\cdot)$ a.s.

The dynamics in (D) is deterministic with random initial conditions, while the
dynamics in (S) is purely stochastic. We note that both deterministic and
stochastic dynamics are significantly different from the one considered in
[13] due to the presence of nonlinearities $l$ and $f$ in them respectively.
In particular, $f$ makes the stochastic dynamics nonlinear. In order to
resolve the lack of mass preservation property, our strategy is to show that,
due to the dissipative nature of nonlinearity $l(u)$, the mass in
deterministic dynamics is non-increasing in time, while the stochastic
dynamics preserves the mass on average, i.e.
$\mathbb{E}\int_{\mathbb{T}}w_{N}(x)dx\equiv const.$ We proceed with
establishing the apriori bounds on the solutions of both deterministic and
stochastic problems, which enable us to pass to the limit as $N\to\infty$,
thus establishing that both $v_{N}$ and $w_{N}$ converge to the desired
martingale solution for (1.1).

##  3 Deterministic dynamics.

In this section we consider the deterministic equation

$\begin{cases}u_{t}+\partial_{x}(u^{2}\partial_{x}^{3}u)+|u|^{\lambda-1}u=0\text{
in }(0,T)\times\mathbb{T}_{L}\\\ u(0,x)=u_{0}(x),\end{cases}$ (3.1)

with $\lambda\geq 1$, and nontrivial non-negative $u_{0}\in
H^{1}(\mathbb{T}_{L})$.

######  Definition 3.1.

The function $u\in C(\bar{Q}_{T})\cap L^{\infty}(0,T,H^{1}(\mathbb{T}_{L}))$
is called the weak solution of (3.1) if

  1. 1.

$u\in C^{1,4}(P_{T})$, $u\cdot u_{xxx}\in L^{2}(P_{T})$, where

$P_{T}=\\{(t,x)\in\bar{Q}_{T}:u>0\\};$

  2. 2.

$\int_{Q_{T}}u\cdot\psi_{t}dxdt+\int_{P_{T}}u^{2}u_{xxx}\psi_{x}dxdt-\int_{Q_{T}}|u|^{\lambda-1}u\cdot\psi
dxdt=0$

for all $\psi\in C^{1}(\bar{Q}_{T})$, and $u(t,\cdot)\to u_{0}(\cdot)$ in
$H^{1}(\mathbb{T}_{L})$ as $t\to\infty$.

The equation of type (3.1) was studied in [18] with the nonlinear term in the
form $(|u|^{n}u_{xxx})_{x},\ n\geq 1,$ and with the boundary conditions
$u_{x}|_{x=0,L}=u_{xxx}|_{x=0,L}=0$ on $[0,T]$. However, the main result in
[18] guarantees the existence of the solution for arbitrary $0\leq u_{0}(x)\in
H^{1}(\mathbb{T}_{L})$ only for $n\in(1,2)$. For other values of $n$, the
solution exists only under the initial conditions, which satisfy the entropy
estimates. Therefore, the existence of the solution of (3.1) for any $0\leq
u_{0}(x)\in H^{1}(\mathbb{T}_{L})$ requires extra work. We have the following
result:

######  Theorem 3.2.

The Cauchy problem (3.1) admits a nonnegative solution in $Q_{T}$.
Furthermore,

$\int_{0}^{L}u(t,x)dx\leq\int_{0}^{L}u_{0}(x)dx\text{ on }t\in[0,T].$

###### Proof.

For any $\varepsilon>0$ introduce $f_{\varepsilon}(u)=\frac{u^{6}}{\varepsilon
u^{2}+u^{4}}$. Suppose $u_{0\varepsilon}>0$ in $[0,L]$ and
$u_{0\varepsilon}(x)\in C^{\infty}(\mathbb{T}_{L})$ and $u_{0\varepsilon}\to
u_{0}$ in $H^{1}(\mathbb{T}_{L}),\ \varepsilon\to 0$. Thus, it follows from
the results in [29] that the regularized problem

$\begin{cases}u_{t}+\partial_{x}(f_{\varepsilon}(u)\partial_{x}^{3}u)+|u|^{\lambda-1}u=0\text{
in }(0,T)\times\mathbb{T}_{L}\\\ u(0,x)=u_{0}\varepsilon(x),\end{cases}$ (3.2)

has a unique non-negative smooth (classic) solution $u^{\varepsilon}(t,x)$ in
$Q_{T}$. This fact is proved using compactness arguments and parabolic
Schauder estimates. Multiplying (3.2) by $-u_{xx}^{\varepsilon}$ and
integrating over $(0,L)$ we have

$\frac{1}{2}\frac{d}{dt}\|u_{x}^{2}(t)\|_{2}^{2}+\int_{0}^{L}f_{\varepsilon}(u^{\varepsilon})(u_{xxx}^{\varepsilon})^{2}dx+\lambda\int_{0}^{L}(u^{\varepsilon})^{\lambda-1}(u_{x}^{\varepsilon})^{2}dx=0$
(3.3)

The convergence of $u_{0\varepsilon}$ implies the existence of a constant
$C>0$ such that $\|u_{x}^{\varepsilon}(0)\|\leq C$. Then it follows from
(3.3), Poincare inequality, and the embedding $H^{1}(\mathbb{T}_{L})\in
L^{\infty}(\mathbb{T}_{L})$:

$\|u^{\varepsilon}\|_{L^{\infty}(Q_{T})}\leq C_{1},\ \ C_{1}>0.$ (3.4)

Then (3.4) yields $\forall x_{1}<x_{2}\in[0,L]$

$|u(t,x_{1})-u(t,x_{2})|\leq\int_{x_{1}}^{x_{2}}|u_{x}(t,x)|dx\leq
K|x_{1}-x_{2}|^{\frac{1}{2}}.$ (3.5)

Denote
$h_{\varepsilon}(u^{\varepsilon}):=(f_{\varepsilon}(u^{\varepsilon})u_{xxx}^{\varepsilon})$.
Integrating (3.3) on $(0,T)$ we get

$\frac{1}{2}\|u_{x}^{\varepsilon}(T)\|_{2}^{2}+\int_{Q_{T}}h^{2}_{\varepsilon}(u^{\varepsilon})dxdt+\lambda\int_{Q_{T}}(u^{\varepsilon})^{\lambda-1}(u_{x}^{\varepsilon})^{2}dxdt=\frac{1}{2}\|u_{x}^{\varepsilon}(0)\|_{2}^{2}.$
(3.6)

Since $u^{\varepsilon}$ is bounded in $L^{\infty}(Q_{T})$, so is
$f_{\varepsilon}(u^{\varepsilon})$, thus

$\int_{Q_{T}}h_{\varepsilon}^{2}(u^{\varepsilon})dxdt\leq A\text{ for some
}A>0.$ (3.7)

######  Lemma 3.3.

There exists $M\geq 0$ such that for any $x_{0}\in(0,L)$ and any
$t_{1},t_{2}\in[0,T],\ |t_{1}-t_{2}|\leq 1$

$|u^{\varepsilon}(t_{1},x_{0})-u^{\varepsilon}(t_{2},x_{0})|\leq
M|t_{1}-t_{2}|^{\frac{1}{8}}.$ (3.8)

###### Proof.

In this proof we will follow the main ideas of [5]. We argue by contradiction.
In the identity

$\int_{Q_{T}}u^{\varepsilon}\varphi_{t}dxdt=-\int_{Q_{T}}f_{\varepsilon}(u^{\varepsilon})u^{\varepsilon}_{xxx}\varphi_{x}dxdt+\int_{Q_{T}}l(u^{\varepsilon})\varphi
dxdt.$ (3.9)

we set $\varphi(t,x)=\xi(x)\theta_{\delta}(t)$, where $|\xi(x)|\leq 1$ and

$\xi(x)=\begin{cases}0,\ |x-x_{0}|\geq CM^{2}|t_{2}-t_{1}|^{\frac{1}{4}},\\\
1,\ |x-x_{0}|\leq\frac{1}{2}CM^{2}|t_{2}-t_{1}|^{\frac{1}{4}},\end{cases}$

and the function $|\theta_{\delta}|\leq 1$ is described in [5]. Following [5]
we have

$\int_{Q_{T}}u^{2}\varphi_{t}dxdt\geq C_{3}M^{3}(t_{2}-t_{1})^{\frac{3}{8}},$

while

$\left|\int_{Q_{T}}f_{\varepsilon}(u^{\varepsilon})u^{\varepsilon}_{xxx}\varphi_{x}dxdt+\int_{Q_{T}}l(u^{\varepsilon})\varphi
dxdt\right|\\\
\leq\frac{C_{4}}{M}\left(\int_{Q_{T}}h_{\varepsilon}(u^{\varepsilon})dxdt\right)^{\frac{1}{2}}(t_{2}-t_{1})^{\frac{1}{2}-\frac{1}{8}}+C_{5}\|l\|_{\infty}M(t_{2}-t_{1})^{\frac{1}{2}+\frac{1}{8}}.$

Hence

$M^{3}(t_{2}-t_{1})^{\frac{3}{8}}\leq
C_{6}\left(\frac{1}{M}(t_{2}-t_{1})^{\frac{1}{2}-\frac{1}{8}}+M(t_{2}-t_{1})^{\frac{1}{2}+\frac{1}{8}}\right)\leq
C_{6}\left(\frac{1}{M}+M\right)(t_{2}-t_{1})^{\frac{3}{8}}.$

Thus, any $M\geq 1$ has to satisfy the inequality $M^{2}\leq 2C_{\delta}$,
which is a contradiction. Hence (3.8) follows. ∎

It follows from (3.4), (3.5) and (3.8) that the set $\\{u^{\varepsilon}\\}$ is
relatively compact in $C(\bar{Q}_{T})$, and hence up to a subsequence
$u^{\varepsilon}\to u$ in $C(\overline{Q_{T}})$. Note that all the derivatives
$u_{t}^{\varepsilon},u_{t}^{\varepsilon},u_{x}^{\varepsilon},u_{xx}^{\varepsilon},u_{xxx}^{\varepsilon}$
and $u_{xxxx}^{\varepsilon}$ converge to the corresponding derivatives of $u$
in view of uniform parabolicity of in any compact subset of $P_{T}$. Thus,
passing to the limit in (3.9) as $\varepsilon\to 0$, we deduce that $u$ solves
(3.1), which is also non-negative. Let us show that the mass of the solution
is non-increasing. Integrating the solution of the regularized problem (3.2)
on $[0,L]$, we have

$\frac{d}{dt}\int_{0}^{L}u^{\varepsilon}(t,x)dx+\int_{0}^{L}(u^{\varepsilon}(t,x))^{\lambda}dx=0$

Since $u\geq 0$, we have

$\frac{d}{dt}\int_{0}^{L}u^{\varepsilon}(t,x)dx\leq 0,$

hence, for the regularized problem,

$\int_{0}^{L}u^{\varepsilon}(t,x)dx\leq\int_{0}^{L}u_{0\varepsilon}(x)dx.$
(3.10)

Since $u^{\varepsilon}(t,x)$ converges to $u(t,x)$ uniformly in
$C(\bar{Q}_{T})$, while $u_{0\varepsilon}$ converges to $u_{0}$ in
$H^{1}(\mathbb{T}_{L})$, the non-increasing property of the mass of $u$
follows from (3.10). Hence the proof is complete. ∎

######  Corollary 3.4.

Suppose the conditions of Theorem 3.2 hold. Then for any $p\geq 2$ the
solution $u$ satisfies

$\|\partial_{x}u(t,\cdot)\|_{2}^{p}+2\int_{0}^{t}\|\partial_{x}u(s,\cdot)\|_{2}^{p-2}\int_{u(s,\cdot)>0}u^{2}(s,x)(\partial_{x}^{3}u(s,x))^{2}dxds\leq\|\partial_{x}u_{0}\|_{2}^{p}.$
(3.11)

###### Proof.

Since $u^{\varepsilon}>0$, it follows from (3.2) that

$\|u_{x}^{\varepsilon}(t,\cdot)\|_{2}^{2}+2\int_{Q_{t}}f_{\varepsilon}(u^{\varepsilon})(u^{\varepsilon}_{xxx})^{2}dxds\leq\|u_{x}^{\varepsilon}(0,\cdot)\|_{2}^{2}.$

Thus for any $r>0$ we have

$\|u_{x}^{\varepsilon}(t,\cdot)\|_{2}^{2}+2\int_{0}^{t}\int_{u(s)>r}f_{\varepsilon}(u^{\varepsilon})(u^{\varepsilon}_{xxx})^{2}dxds\leq\|u_{x}^{\varepsilon}(0,\cdot)\|_{2}^{2}.$
(3.12)

Since $u^{\varepsilon}(0)\to u_{0}$ in $H^{1}(\mathbb{T}_{L})$, then
$u_{x}^{\varepsilon}(t)$ is weakly compact in $L^{2}(0,L)$ for any
$t\in[0,T].$ Then for any $t\in[0,T]$ there exists $\varepsilon_{k}(t)\to 0$
as $k\to\infty$ such that for any $\varphi\in C^{\infty}(\mathbb{T}_{L})$ and
some $\xi\in L^{2}(\mathbb{T}_{L})$ we have

$\int_{0}^{L}u_{x}^{\varepsilon_{k}}(t,x)\varphi(x)dx\to\int_{0}^{L}\xi(x)\varphi(x)dx,\
k\to\infty.$

On the other hand, using the uniform convergence $u^{\varepsilon}\to u$ in
$C(Q_{T})$ and Dominated Convergence Theorem, we have

$\int_{0}^{L}u_{x}^{\varepsilon_{k}}(t,x)\varphi(x)dx=-\int_{0}^{L}u^{\varepsilon_{k}}(t,x)\varphi^{{}^{\prime}}(x)dx\to-\int_{0}^{L}u(t,x)\varphi^{{}^{\prime}}(x)dx=\int_{0}^{L}u_{x}(t,x)\varphi(x)dx$

since $u\in H^{1}(\mathbb{T}_{L})$. Thus
$u_{x}^{\varepsilon_{k}}\to^{w}u_{x},\ k\to\infty$ in $L^{2}(\mathbb{T}_{L})$.
By weak lower semi-continuity of the norm,

$\liminf_{k\to\infty}\|u_{x}^{\varepsilon_{k}(t)}(t)\|_{2}^{2}\geq\|u_{x}(t)\|_{2}^{2}.$
(3.13)

Fix $t\in[0,T]$ and denote $\varepsilon_{k}(t)=\varepsilon_{k}$. Due to
uniform convergence of $u$ and all of its derivatives (up to the fourth order)
on any compact subset of $P$ we deduce

$\|u_{x}(t)\|_{2}^{2}+2\int_{0}^{t}\int_{u(s)>r}u^{2}(s)(u_{xxx}(s))^{2}dxds\leq\|u_{x}(0,\cdot)\|_{2}^{2}.$
(3.14)

It follows from (3.14) that for arbitrary $r>0$ (3.11) is satisfied for $p=2$.
The estimate (3.11) follows along the lines of [13], since

$\|\partial_{x}u(t,\cdot)\|_{2}^{p}+\int_{0}^{t}\|\partial_{x}u(s,\cdot)\|_{2}^{p-2}\int_{u(s,\cdot)>0}u^{2}(s,x)(\partial_{x}^{3}u(s,x))^{2}dxds\\\
\leq\sup_{s\in[0,t]}\|\partial_{x}u(s)\|_{2}^{p-2}\left(\|\partial_{x}u(t)\|^{2}_{2}+\int_{0}^{t}\int_{u(s,\cdot)>0}u^{2}(s,x)(\partial_{x}^{3}u(s,x))^{2}dxds\right).$

∎

##  4 Stochastic dynamics.

In this section we consider the stochastic contribution to (1.1) on
$[0,\delta)\times\mathbb{T}_{L}$ for some $0<\delta\leq T$:

$\begin{cases}dw=\frac{1}{2}\sum_{k\in\mathbb{Z}}\lambda_{k}^{2}\partial_{x}(\Psi_{k}\partial_{x}(\Psi_{k}w))dt+\sum_{k\in\mathbb{Z}}\lambda_{k}(\partial_{x}(\Psi_{k}w))d\beta_{k}(t)+\sum_{k\in\mathbb{Z}}\gamma_{k}\Psi_{k}f(w))d\beta_{1}^{k},\\\
w(0)=w_{0}.\end{cases}$ (4.1)

######  Definition 4.1.

Let $w_{0}=w_{0}(x,\omega)\in H^{1}(\mathbb{T}_{L})$ almost surely. A triple
consisting of a filtered probability space
$([0,1],\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_{t})_{t\in[0,\delta)},\mathbf{P})$,
where $(\tilde{\mathcal{F}}_{t})_{t\in[0,\delta)}$ is a complete and right-
continuous filtration, an $(\tilde{\mathcal{F}}_{t})$ \- adapted
$H_{w}^{1}(\mathbb{T}_{L})$\- valued process $\tilde{W}(t)$ and mutually
independent standard real-valued $(\tilde{\mathcal{F}}_{t})$ \- Wiener
processes $\tilde{\beta}^{k}(t)$ and $\tilde{\beta}_{1}^{k}(t)$ is called the
solution of (4.1) if

$\displaystyle(\tilde{w}(t,\cdot),\varphi)_{2}=(w_{0},\varphi)_{2}-\frac{1}{2}\sum_{k\in\mathbb{Z}}\lambda_{k}^{2}\int_{0}^{t}(\Psi_{k}\partial_{x}(\Psi_{k}\tilde{w}(s,\cdot)),\partial_{x}\varphi)_{2}ds$
(4.2)
$\displaystyle-\sum_{k\in\mathbb{Z}}\lambda_{k}\int_{0}^{t}(\Psi_{k}\tilde{w}(s,\cdot)),\partial_{x}\varphi)_{2}d\tilde{\beta}_{k}(s)+\sum_{k\in\mathbb{Z}}\gamma_{k}\int_{0}^{t}(\Psi_{k}f(\tilde{w}(s,\cdot)),\varphi)_{2}d\tilde{\beta}_{1}^{k}(s)$

is satisfied for every $\varphi\in C^{\infty}(\mathbb{T}_{L})$ and
$t\in[0,\delta)$ $\mathbb{P}=\lambda_{[0,1]}$ almost surely.

######  Theorem 4.2.

For any $p\in[2,\infty)$ and for any $w_{0}\in
L^{p}([0,1],\mathcal{F}_{0},\mathbb{P},H^{1}(\mathbb{T}_{L}))$, $w_{0}\geq 0$
a.s., there exists a non-negative solution $w(t,x,\omega)$ of (4.2) with the
initial condition $w_{0}$, and satisfies the a-priori estimates

$\mathbb{E}\sup_{t\in[0,\delta)}\|w(t,\cdot)\|_{1,2}^{p}\leq
C_{1}\mathbb{E}\|w_{0}\|_{1,2}^{p};$ (4.3)
$\lim_{t\to\delta}\sup\mathbb{E}\|\partial_{x}w(t,\cdot)\|_{2}^{p}\leq
e^{C_{2}\delta}\left(\mathbb{E}\|\partial_{x}w_{0}\|_{2}^{p}+C_{3}\delta\mathbb{E}\left(\int_{0}^{L}w_{0}dx\right)^{p}\right),$
(4.4)

where $C_{1},C_{2},C_{3}<\infty$ are independent of $\delta,w$ and $w_{0}$.

###### Proof.

Step 1: Regularization of the problem. For any $\varepsilon\in(0,1]$ we
consider the viscous regularization of the problem (4.1):

$dw^{\varepsilon}=\left(\frac{1}{2}\sum_{k\in\mathbb{Z}}\lambda_{k}^{2}\partial_{x}(\Psi_{k}\partial_{x}(\Psi_{k}w^{\varepsilon}))+\varepsilon\partial_{x}^{2}w^{\varepsilon}\right)dt+\sum_{k\in\mathbb{Z}}\lambda_{k}(\partial_{x}(\Psi_{k}w^{\varepsilon}))d\beta^{k}+\sum_{k\in\mathbb{Z}}\gamma_{k}\Psi_{k}f(w^{\varepsilon}))d\beta_{1}^{k},$
(4.5)

with the initial condition $w^{\varepsilon}(0)=w_{0}$. Denote
$A^{\varepsilon}:H^{2}(\mathbb{T}_{L})\to L^{2}(\mathbb{T}_{L})$ to be

$A^{\varepsilon}w:=\frac{1}{2}\sum_{k\in\mathbb{Z}}\lambda_{k}^{2}\partial_{x}(\Psi_{k}\partial_{x}(\Psi_{k}w^{\varepsilon}))+\varepsilon\partial_{x}^{2}w^{\varepsilon},$
(4.6)

and the diagonal Hilbert-Schmidt-Valued operator

$B:H^{2}(\mathbb{T}_{L})\to L^{2}(H^{2}(\mathbb{T}_{L})\times
H^{2}(\mathbb{T}_{L});H^{1}(\mathbb{T}_{L})):=L^{2}$

is given by

$B(w)\begin{pmatrix}u\\\ v\end{pmatrix}=(B_{1}(w),B_{2}(w))\begin{pmatrix}u\\\
v\end{pmatrix}:=B_{1}(w)u+B_{2}(w)v,$

where

$B_{1}(w)u=\sum_{k\in\mathbb{Z}}\lambda_{k}(u,\Psi_{k})_{2,2}(\partial_{x}(\Psi_{k}w))$

and

$B_{2}(w)v=\sum_{k\in\mathbb{Z}}\gamma_{k}(v,\Psi_{k})_{2,2}\Psi_{k}f(w)$

Since $\begin{pmatrix}\Psi_{k}\\\ 0\end{pmatrix}$ and $\begin{pmatrix}0\\\
\Psi_{k}\end{pmatrix}$ form the basis in $H^{2}(\mathbb{T}_{L})\times
H^{2}(\mathbb{T}_{L})$, then

$\|B(w)\|^{2}_{L^{2}}:=\|B_{1}(w)\Psi_{k}\|_{H^{1}(\mathbb{T}_{L})}^{2}+\|B_{2}(w)\Psi_{k}\|_{H^{1}(\mathbb{T}_{L})}^{2}\\\
=\sum_{k\in
Z}\lambda_{k}^{2}\int_{0}^{L}\left[(\partial_{x}(\Psi_{k}w))^{2}+(\partial_{x}^{2}(\Psi_{k}w))^{2}\right]\,dx\\\
+\sum_{k\in
Z}\gamma_{k}^{2}\int_{0}^{L}\left[(\Psi_{k}f(w))^{2}+(\partial_{x}(\Psi_{k}f(w)))^{2}\right]\,dx.$

This way the equation (4.1) can be re-written in the abstract form

$dw^{\varepsilon}=A^{\varepsilon}w^{\varepsilon}dt+(Bw^{\varepsilon})d\overline{W}_{H^{2}(\mathbb{T}_{L})},$
(4.7)

where

$\overline{W}_{H^{2}(\mathbb{T}_{L})}:=(W,W_{1}):=\left(\sum_{k\in\mathbb{Z}}\beta^{k}\Psi_{k},\sum_{k\in\mathbb{Z}}\beta_{1}^{k}\Psi_{k}\right)$

is a cylindrical $\mathcal{F}_{t}$-Wiener process in
$H^{2}(\mathbb{T}_{L})\times H^{2}(\mathbb{T}_{L})$.

######  Proposition 4.3.

For any $p\in[2,\infty)$ and for any initial condition $w_{0}\in
L^{p}([0,1],\mathcal{F}_{0},\mathbb{P},H^{1}(\mathbb{T}_{L}))$, the equation
(4.5) has a unique variational solution $w^{\varepsilon}$ satisfying

$\mathbb{E}\left(\sup_{t\in[0,\delta)}\|w^{\varepsilon}(t,\cdot)\|_{1,2}^{p}+\int_{0}^{\delta}\|w^{\varepsilon}(t,\cdot)\|_{2,2}^{2}dt\right)<\infty.$
(4.8)

Furthermore,

  1. 1.

$\mathbb{E}\sup_{t\in[0,\delta)}\|w^{\varepsilon}(t,\cdot)\|_{1,2}^{2}\leq
C_{1}\mathbb{E}\|w_{0}\|_{1,2}^{p},$ (4.9)

  2. 2.

If $w_{0}\geq 0$ a.s., then $w^{\varepsilon}(t,x)\geq 0$ a.s.

  3. 3.

If $w^{\varepsilon}(t,x)\geq 0$ a.s., then

$\lim_{t\to\delta}\mathbb{E}\|\partial_{x}w^{\varepsilon}\|_{2}^{p}\leq
e^{C_{2}\delta}\left(\mathbb{E}\|\partial_{x}w_{0}\|_{2}^{p}+C_{3}\delta\mathbb{E}\left|\int_{0}^{L}w_{0}dx\right|^{p}\right)$
(4.10)

where $C_{1},C_{2},C_{3}$ are independent of $\varepsilon,\delta$ and $w_{0}$.

###### Proof.

The conclusion of proposition 4.3 follows from Theorem 5.1.3 in [20] under the
following conditions for any $u,v$ and $w$ in $H^{2}(\mathbb{T}_{L})$:

(H1) (Hemicontinuity) The map $\lambda\to\langle\langle
A^{\varepsilon}(u+\lambda v),w\rangle\rangle$ is continuous on $\mathbb{R}$

(H2’) (Local monotonicity)

$2\langle\langle
A^{\varepsilon}(u-v),u-v\rangle\rangle+\|B(u)-B(v)\|^{2}_{\mathcal{L}_{2}(H^{2}(\mathbb{T}_{L})\times
H^{2}(\mathbb{T}_{L}),H^{1}(\mathbb{T}_{L}))}\leq\rho(v)\|u-v\|_{H^{1}(\mathbb{T}_{L})}^{2}$

for some non-negative, mearurable, hemicontinuous and locally bounded function
on $V$.

(H3) (Coercivity) There are constants $C_{0}\in\mathbb{R}$, $\alpha>1$ and
$\theta>0$ such that

$2\langle\langle
A^{\varepsilon}(v),v\rangle\rangle+\|B(v)\|^{2}_{\mathcal{L}_{2}(H^{2}(\mathbb{T}_{L})\times
H^{2}(\mathbb{T}_{L}),H^{1}(\mathbb{T}_{L}))}\leq
C_{0}\|v\|_{H^{1}(\mathbb{T}_{L})}^{2}-\theta\|v\|_{H^{2}(\mathbb{T}_{L})}^{\alpha}$

(H4’) (Growth)

$\|A(v)\|_{L^{2}(\mathbb{T}_{L})}^{\frac{\alpha}{\alpha-1}}\leq
C_{0}\|v\|^{\alpha}_{H^{2}(\mathbb{T}_{L})}(1+\|v\|_{H^{1}(\mathbb{T}_{L})}^{\beta})$

Note that the above conditions make use of the Gelfand triple $V\subset
H\subset V^{*}$, where $V^{*}=L^{2}(\mathbb{T}_{L}),\ H=H^{1}(\mathbb{T}_{L})$
and $V=H^{2}(\mathbb{T}_{L})$.

The condition (H1) (hemicontinuity) is straightforward.

Next, we check the condition (H2’). We have

$\displaystyle 2$ $\displaystyle\langle\langle
A^{\varepsilon}(u-v),u-v\rangle\rangle+\|B(u)-B(v)\|^{2}_{L^{2}(H^{2}(\mathbb{T}_{L}),H^{1}(\mathbb{T}_{L}))}=2\langle
A^{\varepsilon}(u-v),u-v\rangle$ $\displaystyle+$ $\displaystyle
2\langle\partial_{x}A^{\varepsilon}(u-v),\partial_{x}(u-v)\rangle+\|B(u)-B(v)\|^{2}_{\mathcal{L}_{2}(H^{2}(\mathbb{T}_{L})\times
H^{2}(\mathbb{T}_{L});L^{2}(\mathbb{T}_{L}))}$ $\displaystyle+$
$\displaystyle\|B(u)-B(v)\|^{2}_{\mathcal{L}_{2}(H^{2}(\mathbb{T}_{L})\times
H^{2}(\mathbb{T}_{L});\dot{H}^{1}(\mathbb{T}_{L}))}$ $\displaystyle=$
$\displaystyle 2\langle
A^{\varepsilon}(u-v),u-v\rangle+2\langle\partial_{x}A^{\varepsilon}(u-v),\partial_{x}(u-v)\rangle$
$\displaystyle+$
$\displaystyle\sum_{k}\lambda_{k}^{2}\int_{0}^{L}(\partial_{x}(\Psi_{k}(u-v)))^{2}\,dx+\sum_{k}\lambda_{k}^{2}\int_{0}^{L}(\partial^{2}_{x}(\Psi_{k}(u-v)))^{2}\,dx$
$\displaystyle+$
$\displaystyle\sum_{k}\gamma_{k}^{2}\int_{0}^{L}(\Psi_{k}(f(u)-f(v)))^{2}\,dx+\sum_{k}\gamma_{k}^{2}\int_{0}^{L}(\partial_{x}(\Psi_{k}(f(u)-f(v))))^{2}\,dx$
$\displaystyle:=$ $\displaystyle J_{1}+J_{2}+J_{3}+J_{4}+J_{5}+J_{6}.$

Let us analyze every term in (4) separately. Integrating by parts, we have

$\displaystyle
J_{1}:=-\sum_{k}\lambda_{k}^{2}\int_{0}^{L}\Psi_{k}^{2}(\partial_{x}(u-v)^{2})dx-\frac{1}{4}\sum_{k}\lambda_{k}^{2}\int_{0}^{L}(\partial_{x}\Psi_{k}^{2})(\partial_{x}(u-v))^{2}dx$
$\displaystyle-$ $\displaystyle
2\varepsilon\int_{0}^{L}(\partial_{x}(u-v))^{2}dx=-\sum_{k}\lambda_{k}^{2}\int_{0}^{L}\Psi_{k}^{2}(\partial_{x}(u-v))^{2}\,dx+\frac{1}{4}\sum_{k}\int_{0}^{L}(\partial_{x}^{2}\Psi_{k}^{2})(u-v)^{2}\,dx$
$\displaystyle-$ $\displaystyle
2\varepsilon\int_{0}^{L}(\partial_{x}(u-v))^{2}dx\leq-\sum_{k}\lambda_{k}^{2}\int_{0}^{L}\Psi_{k}^{2}(\partial_{x}(u-v)^{2})dx+C\|u-v\|_{2}^{2},$

for some $C>0$, independent of $\varepsilon$. Here we used (1.4) and (1.3).
Next, following [13], we have

$\displaystyle
J_{2}:=-\sum_{k}\lambda_{k}^{2}\int_{0}^{L}\Psi_{k}^{2}(\partial_{x}^{2}(u-v)^{2})dx+\frac{5}{4}\sum_{k}\lambda_{k}^{2}\int_{0}^{L}(\partial_{x}^{2}\Psi_{k})^{2}(\partial_{x}(u-v))^{2}dx$
$\displaystyle-$
$\displaystyle\frac{1}{4}\sum_{k}\lambda_{k}^{2}\int_{0}^{L}(\partial_{x}^{4}\Psi_{k}^{2})(\partial_{x}(u-v))^{2}dx-2\varepsilon\int_{0}^{L}(\partial_{x}^{2}(u-v))^{2}\,dx$
$\displaystyle\leq$
$\displaystyle-\sum_{k}\lambda_{k}^{2}\int_{0}^{L}\Psi_{k}^{2}(\partial_{x}^{2}(u-v)^{2})dx+C\|u-v\|_{1,2}^{2}-2\varepsilon\int_{0}^{L}(\partial_{x}^{2}(u-v))^{2}\,dx,$

where, once again, $C>0$ is independent of $\varepsilon$.

$\displaystyle
J_{3}=\sum_{k}\lambda_{k}^{2}\int_{0}^{L}[(\partial_{x}\Psi_{k})(u-v)\Psi_{k}\partial_{x}(u-v)]^{2}\,dx$
$\displaystyle\leq
2\left(\sum_{k}\lambda_{k}^{2}\int_{0}^{L}\Psi^{2}_{k}(\partial_{x}(u-v))^{2}\,dx-\sum_{k}\lambda_{k}^{2}\int_{0}^{L}\Psi_{k}(\partial_{x}^{2}\Psi_{k})(u-v)^{2}\,dx\right).$
(4.14)

We proceed with the estimate for $J_{4}$:

$\displaystyle
J_{4}=\sum_{k}\lambda_{k}^{2}\int_{0}^{L}\Psi_{k}^{2}(\partial_{x}^{2}(u-v))^{2}\,dx+\sum_{k}\lambda_{k}^{2}\int_{0}^{L}(2(\partial_{x}\Psi_{k})^{2}-4\Psi_{k}(\partial_{k}^{2}\Psi_{K}))(\partial_{x}(u-v))^{2}\,dx$
(4.15) $\displaystyle+$
$\displaystyle\sum_{k}\lambda_{k}^{2}\int_{0}^{L}\Psi_{k}(\partial_{x}^{4}\Psi_{k})(u-v)^{2}\,dx\leq\sum_{k}\lambda_{k}^{2}\int_{0}^{L}\Psi_{k}^{2}(\partial_{x}^{2}(u-v))^{2}\,dx+C\|u-v\|_{1,2}^{2},$

where $C>0$ is independent of $\varepsilon$.

The estimate of the fifth term uses the Lipschitz condition:

$J_{5}\leq\sum_{k}\gamma_{k}^{2}\int_{0}^{L}(|\Psi_{k}|K|u-v|)^{2}\,dx\leq
C\|u-v\|_{2}^{2},$ (4.16)

where, once again, $C$ is independent of $\varepsilon$. In order to estimate
$J_{6}$, consider

$\int_{0}^{L}\left(\partial_{x}(\Psi_{k}(f(u)-f(v)))\right)^{2}\,dx=\int_{0}^{L}\left[\partial_{x}\Psi_{k}\,(f(u)-f(v))+\Psi_{k}(f^{\prime}(u)u_{x}-f^{\prime}(v)v_{x})\right]^{2}\,dx\\\
\leq
C\|u-v\|_{2}^{2}+C\int_{0}^{L}|f^{\prime}(u)u_{x}-f^{\prime}(v)v_{x}|^{2}\,dx.$
(4.17)

Since $f^{\prime}(u)$ is both Lipschitz and bounded, we have

$|f^{\prime}(u)u_{x}-f^{\prime}(v)v_{x}|\leq K|u_{x}-v_{x}|+K(u-v)|v_{x}|.$

Taking into account that $v_{x}$ is absolutely continuous due to the embedding
$H^{1}\subset C^{0,1/2}$, we have $\sup_{x\in\mathbb{T}_{L}}|v_{x}|\leq
C_{1}\|v\|_{2,2}$. Therefore,

$\int_{0}^{L}|f^{\prime}(u)u_{x}-f^{\prime}(v)v_{x}|\,dx\leq
C\|u-v\|_{1,2}^{2}+C\|u-v\|_{2}^{2}\|v\|_{2,2}^{2}.$ (4.18)

In view of (4.17) and (4.18), we deduce

$J_{6}\leq C\|u-v\|_{1,2}^{2}+C\|u-v\|_{2}^{2}\|v\|_{2,2}^{2}.$ (4.19)

Combining the estimates for $J_{1}-J_{6}$, we get

$2\langle\langle
A^{\varepsilon}(u-v),u-v\rangle\rangle+\|B(u)-B(v)\|^{2}_{L^{2}}\leq
C(1+\|v\|_{2,2}^{2})\|u-v\|_{1,2}^{2}-2\varepsilon\|u-v\|_{2,2}^{2}$ (4.20)

which implies that the condition (H2’) (local monotonicity) holds. Note that
the constant $C>0$ in (4.20) is independent of $u,v$ and $\varepsilon$.

Let us now check the coercivity condition (H3) from Theorem 5.1.3 [20]. In a
similar way as before, we get

$2\langle\langle
A^{\varepsilon}u,u\rangle\rangle+\|B_{1}(u)\|^{2}_{L^{2}(H^{2}(\mathbb{T}_{L}),H^{1}(\mathbb{T}_{L}))}\leq
C\|u\|^{2}_{1,2}-2\varepsilon\|u\|^{2}_{2,2}$

for some constant $C>0$ independent of $u\in H^{2}(\mathbb{T}_{L})$ and
$\varepsilon$. Let us proceed with the estimate for $B_{2}$:

$\|B_{2}(u)\|^{2}_{L^{2}(H^{2}(\mathbb{T}_{L}),H^{1}(\mathbb{T}_{L}))}=\sum_{k}\gamma_{k}^{2}\int_{0}^{L}(\Psi_{k}f(u))^{2}\,dx+\sum_{k}\gamma_{k}^{2}\int_{0}^{L}(\partial_{x}(\Psi_{k}f(u)))^{2}\,dx:=I_{1}+I_{2}.$

We have

$I_{1}\leq C\|u\|_{2}^{2}$ (4.21)

which follows from the linear growth condition on $f(u)$. As for $I_{2}$,

$I_{2}=\sum_{k}\gamma_{k}^{2}\int_{0}^{L}[\partial_{x}(\Psi_{k})f(u)+\Psi_{k}f^{\prime}(u)u_{x}]^{2}\,dx\leq
C\|u\|_{2}^{2}+C\|u_{x}\|_{2}^{2}=C\|u\|_{1,2}^{2},$ (4.22)

where, once again, we use the fact that $f^{\prime}(u)$ is bounded. Thus,
combining the estimates (4.21)-(4.22), we conclude

$2\langle\langle
A^{\varepsilon}u,u\rangle\rangle+\|B(u)\|^{2}_{\mathcal{L}_{2}(H^{2}(\mathbb{T}_{L})\times
H^{2}(\mathbb{T}_{L}),H^{1}(\mathbb{T}_{L}))}\leq
C\|u\|^{2}_{1,2}-2\varepsilon\|u\|^{2}_{2,2},$ (4.23)

hence the condition (H3) holds.

Finally, let us check the condition (H4’). For any $u\in
H^{2}(\mathbb{T}_{L})$ and $\varphi\in C^{\infty}(\mathbb{T}_{L})$,
integrating by parts we have

$|\langle
A^{\varepsilon}u,\varphi\rangle|=\frac{1}{2}\sum_{k}\lambda_{k}^{2}\left|\int_{0}^{L}\Psi_{k}(\partial_{x}(\Psi_{k}u))(\partial_{x}\varphi)dx\right|+\varepsilon\left|\int_{0}^{L}(\partial_{x}u)(\partial_{x}\varphi)\,dx\right|\\\
+\frac{1}{2}\sum_{k}\lambda_{k}^{2}\left|\int_{0}^{L}\left(\partial_{x}(\Psi_{k}(\partial_{x}(\Psi_{k}u)))\right)(\partial_{x}^{2}\varphi)\,dx\right|+\varepsilon\left|\int_{0}^{L}(\partial_{x}^{2}u)(\partial_{x}^{2}\varphi)\,dx\right|\leq
C\|u\|_{2,2}\|\varphi\|_{2,2},$

for some $C>0$ and $\varepsilon<1$. We are now in position to apply Banach-
Steinhaus theorem to obtain

$\|A^{\varepsilon}u\|_{L^{2}(\mathbb{T}_{L})}\leq C\|u\|_{2,2}.$

Therefore, the conditions of Theorem 5.1.3 [20] hold, and the regularized
problem (4.5) admits the unique solution for all $\varepsilon\in(0,1]$. We now
proceed with the estimate (4.9):

$\|w^{\varepsilon}(t)\|_{1,2}^{2}-\|w_{0}\|_{1,2}^{2}=2\sum_{k}\lambda_{k}\int_{0}^{t}(\partial_{x}(\Psi_{k}w^{\varepsilon}(s)),w^{\varepsilon}(s))_{1,2}d\beta^{k}(s)\\\
+2\sum_{k}\gamma_{k}\int_{0}^{t}(\Psi_{k}f(w^{\varepsilon}(s)),w^{\varepsilon}(s))_{1,2}d\beta_{1}^{k}(s)+\int_{0}^{t}(2\langle\langle
A^{\varepsilon}w^{\varepsilon}(s),w^{\varepsilon}(s)\rangle\rangle+\|Bw^{\varepsilon}(s)\|_{L^{2}}^{2})\,ds$
(4.24)

for all $t\in[0,\delta)$ almost surely. Furthermore, for $p\geq 4$, using
Ito’s formula for $\|u\|_{1,2}^{p}$, we have

$\displaystyle\|w^{\varepsilon}(t)\|_{1,2}^{p}-\|w_{0}\|_{1,2}^{p}=p\sum_{k}\lambda_{k}\int_{0}^{t}\|w^{\varepsilon}(s)\|_{1,2}^{p-2}(\partial_{x}(\Psi_{k}w^{\varepsilon}(s)),w^{\varepsilon}(s))_{1,2}\,d\beta^{k}(s)$
(4.25) $\displaystyle+$ $\displaystyle
p\sum_{k}\gamma_{k}\int_{0}^{t}\|w^{\varepsilon}(s)\|_{1,2}^{p-2}(\Psi_{k}f(w^{\varepsilon}(s)),w^{\varepsilon}(s))_{1,2}d\beta_{1}^{k}(s)$
$\displaystyle+$
$\displaystyle\frac{p}{2}\int_{0}^{t}\|w^{\varepsilon}(s)\|_{1,2}^{p-2}(2\langle\langle
A^{\varepsilon}w^{\varepsilon}(s),w^{\varepsilon}(s)\rangle\rangle+\|B(w^{\varepsilon}(s))\|_{L_{2}}^{2})\,ds$
$\displaystyle+$
$\displaystyle\frac{p(p-2)}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\|w^{\varepsilon}(s)\|_{1,2}^{p-4}(\partial_{x}(\Psi_{k}w^{\varepsilon}(s)),w^{\varepsilon}(s))^{2}_{1,2}\,ds$
$\displaystyle+$
$\displaystyle\frac{p(p-2)}{2}\sum_{k}\gamma_{k}^{2}\int_{0}^{t}\|w^{\varepsilon}(s)\|_{1,2}^{p-4}(\Psi_{k}f(w^{\varepsilon}(s)),w^{\varepsilon}(s))^{2}_{1,2}\,ds$

where we used the definition of the norm $\|B(u)\|_{L_{2}}$. Next, using
Burkholder-Davis-Gundy inequality:

$\displaystyle\mathbb{E}\sup_{t^{\prime}\in[0,t)}\left|\int_{0}^{t^{\prime}}\|w^{\varepsilon}(s)\|_{1,2}^{p-2}(\partial_{x}(\Psi_{k}w^{\varepsilon}(s)),w^{\varepsilon})_{1,2}\,d\beta^{k}(s)\right|$
(4.26) $\displaystyle\leq$ $\displaystyle
3\mathbb{E}\left(\int_{0}^{t}\|w^{\varepsilon}(s)\|_{1,2}^{2p-4}\partial_{x}(\Psi_{k}w^{\varepsilon}(s),w^{\varepsilon}(s))_{1,2}^{2}\,ds\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C\mathbb{E}\left(\int_{0}^{t}\|w^{\varepsilon}(s)\|_{1,2}^{2p}\right)^{\frac{1}{2}}\leq
C\mathbb{E}\sqrt{\sup_{s\in[0,t]}\|w^{\varepsilon}(s)\|_{1,2}^{p}\,\int_{0}^{t}\|w^{\varepsilon}(s)\|_{1,2}^{p}\,ds}$
$\displaystyle\leq$
$\displaystyle\nu\mathbb{E}\sup_{s\in[0,t]}\|w^{\varepsilon}(s)\|_{1,2}^{p}+\frac{C}{\nu}\mathbb{E}\int_{0}^{t}\|w^{\varepsilon}(s)\|_{1,2}^{p}\,ds,$

where $\nu>0$ can be chosen arbitrarily small, and $C$ is independent of
$\nu$. Note that the expected values in (4.26) are finite due to (4.8).

In a similar way, we get

$\mathbb{E}\sup_{\tau\in[0,t]}\left|\int_{0}^{\tau}\|w^{\varepsilon}(s)\|_{1,2}^{p-2}(\Psi_{k}f(w^{\varepsilon}(s)),w^{\varepsilon}(s))_{1,2}d\beta_{1}^{k}(s)\right|\\\
\leq
3\mathbb{E}\left(\int_{0}^{t}\|w^{\varepsilon}(s)\|_{1,2}^{2p-4}(\Psi_{k}f(w^{\varepsilon}(s)),w^{\varepsilon}(s))_{1,2}\,ds\right)^{\frac{1}{2}}.$
(4.27)

But

$(\Psi_{k}f(w^{\varepsilon}(s)),w^{\varepsilon}(s))_{1,2}=\int_{0}^{L}\Psi_{k}f(w^{\varepsilon}(s))w^{\varepsilon}(s)\,ds\\\
+\int_{0}^{L}\partial_{x}(\Psi_{k}f(w^{\varepsilon}(s)))\partial_{x}w^{\varepsilon}(s)\,ds:=I_{1}+I_{2}.$
(4.28)

The estimate for $I_{1}$ follows from the conditions imposed on $f$, namely

$|I_{1}|\leq C\|w^{\varepsilon}(s)\|_{2}^{2}.$

Similarly,

$|I_{2}|\leq\left|\int_{0}^{L}[\partial_{x}(\Psi_{k})f(w^{\varepsilon}(s))+\Psi_{k}f^{\prime}(w^{\varepsilon}(s))\partial_{x}(w^{\varepsilon}(s))]\partial_{x}w^{\varepsilon}(s)\,ds\right|\leq
C\|w^{\varepsilon}\|_{1,2}^{2}.$

Thus, the expression in (4.27) is estimated with

$\mathbb{E}\sup_{\tau\in[0,t]}\left|\int_{0}^{\tau}\|w^{\varepsilon}(s)\|_{1,2}^{p-2}(\Psi_{k}f(w^{\varepsilon}(s)),w^{\varepsilon}(s))_{1,2}d\beta_{1}^{k}(s)\mathbb{E}\|w^{\varepsilon}(s)\|\right|\leq
C\int_{0}^{t}\mathbb{E}\|w^{\varepsilon}(s)\|_{1,2}^{2}\,ds$ (4.29)

The third term in (4.25), using the same approach as we used in verification
of coercivity (H3), can be bound as follows:

$\frac{p}{2}\int_{0}^{t}\|w^{\varepsilon}(s)\|_{1,2}^{p-2}(2\langle\langle
A^{\varepsilon}w^{\varepsilon}(s),w^{\varepsilon}(s)\rangle\rangle+\|B(w^{\varepsilon}(s))\|_{L_{2}}^{2})\,ds\leq\frac{pc}{2}\int_{0}^{t}\mathbb{E}\|w_{0}^{\varepsilon}(s)\|_{1,2}^{p}\,ds.$
(4.30)

Furthermore, using the same estimates, the last two terms in (4.25) can be
estimated the same way as in (4.30). Hence, combining (4.25)- (4.27), (4.29)
and (4.30), for sufficiently small $\nu$ we have

$\mathbb{E}\sup_{s\in[0,t]}\|w^{\varepsilon}(s)\|_{1,2}^{p}\leq
C\left(\mathbb{E}\|w_{0}\|_{1,2}^{p}+\int_{0}^{t}\mathbb{E}\sup_{\tau\in[0,s]}\|w^{\varepsilon}(\tau)\|_{1,2}^{p}\right),$
(4.31)

where $C>0$ only depends on $L$. The bound (4.9) now follows from Gronwall’s
inequality for $p=2$ and for $p\geq 4$. If $p\in(2,4)$, the bound follows from
Young inequality and the interpolation inequality

$\|u\|_{\tilde{p}}^{\tilde{p}}\leq\|u\|_{p_{0}}^{p_{0}(1-\theta)}\|u\|_{p_{1}}^{p_{1}\theta},\
\text{ where }\tilde{p}=(1-\theta)p_{0}+\theta p_{1},\ \theta\in(0,1),$

taking $p_{0}=2$ and $p_{1}=4$.
Let us proceed with the proof of the second statement of Proposition 4.3. In
order to simplify the notation, in this part of the proof we are going to omit
the subscript $\varepsilon$. Introduce

$u^{+}=\begin{cases}u,\ u\geq 0;\\\ 0,\ u<0,\end{cases}\ \text{ and }\
u^{-}=\begin{cases}u,\ u<0;\\\ 0,\ u\geq 0\end{cases}$

This way $u=u^{+}+u^{-}$. The function $z=-u$ satisfies

$dz=\left(\frac{1}{2}\sum_{k}\lambda_{k}^{2}\partial_{x}(\Psi_{k}\partial_{x}(\Psi_{k}z))+\varepsilon\partial_{x}^{2}z\right)\,dt+\sum_{k}\lambda_{k}\partial_{x}(\Psi_{k}z)\,d\beta^{k}(t)\\\
-\sum_{k\in\mathbb{Z}}\gamma_{k}\Psi_{k}f(-z)d\beta_{1}^{k}(t),$ (4.32)

with $z(0)=-u_{0}:=z_{0}.$ Let $\varphi(y)=(y^{+})^{2}$ for any
$y\in\mathbb{R}$. Now, let $\psi(y)$ be a $C^{\infty}$ function, such that

$\psi(y)=\begin{cases}0,\ y\in(-\infty,1];\\\ 1,\ y\in[2,\infty).\end{cases}$

Set $\varphi_{n}(y)=y^{2}\psi(ny)$. Then

$\lim_{n\to\infty}\varphi_{n}(y)=\varphi(y)$ (4.33)

uniformly in $y\in\mathbb{R}$. Furthermore, for any $y\in\mathbb{R}$,

$\varphi^{\prime}_{n}(y)\to 2y^{+}\text{ and
}\varphi_{n}^{{}^{\prime\prime}}(y)\to 2\chi_{y>0},\ n\to\infty.$ (4.34)

Furthermore,

$0\leq\varphi_{n}(y)\leq\varphi(y),\ 0\leq\varphi_{n}^{{}^{\prime}}(y)\leq
Cy,\ \|\varphi_{n}^{{}^{\prime\prime}}(y)\|\leq C,$ (4.35)

for all $y\geq 0$ and $n\geq 1$. Next, we apply Ito’s formula to
$\int_{0}^{L}\varphi_{n}(z(t))dx$:

$\displaystyle\int_{0}^{L}\varphi_{n}(z(t))dx=\int_{0}^{L}\varphi_{n}(z_{0}(x))dx+\sum_{k}\lambda_{k}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime}}_{n}(z(s))\partial_{x}(\Psi_{k}z(s))dxd\beta^{k}(s)$
$\displaystyle-\sum_{k}\gamma_{k}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime}}_{n}(z(s))\Psi_{k}f(-z)dxd\beta_{1}^{k}(s)$
$\displaystyle+\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime}}_{n}(z(s))\left(\frac{1}{2}\sum_{k}\lambda_{k}^{2}\partial_{x}(\Psi_{k}\partial_{x}(\Psi_{k}z(s)))+\varepsilon\partial_{x}^{2}z\right)\,dxds$
$\displaystyle+\frac{1}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z(s))(\partial_{x}(\Psi_{k}z(s)))^{2}\,dxds$
$\displaystyle+\frac{1}{2}\sum_{k}\gamma_{k}^{2}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z(s))(\Psi_{k}f(-z(s)))^{2}\,dxds.$
(4.36)

Integrating by parts, we have

$\displaystyle\int_{0}^{L}\varphi^{{}^{\prime}}_{n}(z(s))\partial_{x}(\Psi_{k}\partial_{x}(\Psi_{k}z(s)))\,dx=-\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z(s))z_{x}\Psi_{k}\partial_{x}(\Psi_{k}z(s))\,dx$
(4.37)
$\displaystyle=-\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z(s))z_{x}\Psi_{k}(\partial_{x}(\Psi_{k})z+\Psi_{k}z_{x})\,dx=-\int_{0}^{L}\varphi^{{}^{\prime\prime}}(z(s))z_{x}\Psi_{k}\partial_{x}(\Psi_{k})z\,dx$
(4.38)
$\displaystyle-\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z(s))z_{x}^{2}\Psi_{k}^{2}\,dx.$
(4.39)

Since

$\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z(s))\left(\partial_{x}(\Psi_{k}z)\right)^{2}\,dx=\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z(s))\left(\partial_{x}(\Psi_{k})\right)^{2}z^{2}\,dx+\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z(s))\Psi_{k}^{2}z_{x}^{2}\,dx\\\
+2\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z(s))\,\partial_{x}(\Psi_{k})\Psi_{k}zz_{x}\,dx,$

the expression (4.35) has the form

$\displaystyle\int_{0}^{L}\varphi_{n}(z(t))\,dx=\int_{0}^{L}\varphi_{n}(z_{0})\,dx+\sum_{k}\lambda_{k}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime}}_{n}(z(s))\partial_{x}(\Psi_{k}z(s))\,dx\,d\beta^{k}(s)$
$\displaystyle-\sum_{k}\gamma_{k}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime}}_{n}(z(s))\Psi_{k}f(-z)\,dxd\beta_{1}^{k}(s)-\frac{1}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z)\Psi_{k}(\partial_{x}\Psi_{k})zz_{x}\,dxds$
$\displaystyle-\frac{1}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z)\Psi_{k}^{2}z_{x}^{2}\,dxds-\varepsilon\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z)z_{x}^{2}\,dxds$
$\displaystyle+\frac{1}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z)\left[(\partial_{x}\Psi_{k})^{2}z^{2}+2(\partial_{x}\Psi_{k})\Psi_{k}zz_{x}+\Psi_{k}^{2}z_{k}^{2}\right]\,dxds$
$\displaystyle+\frac{1}{2}\sum_{k}\gamma_{k}^{2}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z)(\Psi_{k}f(-z(s)))^{2}\,dxds$
$\displaystyle=\int_{0}^{L}\varphi_{n}(z_{0})\,dx+\sum_{k}\lambda_{k}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime}}_{n}(z(s))\partial_{x}(\Psi_{k}z(s))\,dx\,d\beta^{k}(s)$
$\displaystyle-\sum_{k}\gamma_{k}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime}}_{n}(z(s))\Psi_{k}f(-z)\,dxd\beta_{1}^{k}(s)-\frac{1}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z)\Psi_{k}(\partial_{x}\Psi_{k})zz_{x}\,dxds$
$\displaystyle-\varepsilon\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z)z_{x}^{2}\,dxds+\frac{1}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z)\left[(\partial_{x}\Psi_{k})^{2}z^{2}+2(\partial_{x}\Psi_{k})\Psi_{k}zz_{x}\right]\,dxds$
$\displaystyle+\frac{1}{2}\sum_{k}\gamma_{k}^{2}\int_{0}^{t}\int_{0}^{L}\varphi^{{}^{\prime\prime}}_{n}(z)(\Psi_{k}f(-z(s)))^{2}\,dxds.$
(4.40)

Taking the expected value in (4), we may use (4.33), (4.35), the dominated
convergence theorem, and (4.9), to pass to the limit as $n\to\infty$:

$\displaystyle\mathbb{E}\int_{0}^{L}(z^{+}(t,x))^{2}\,dx=\mathbb{E}\int_{0}^{L}(z^{+}_{0}(x))^{2}\,dx-2\varepsilon\int_{0}^{t}\mathbb{E}\left(\int_{0}^{L}\chi_{\\{z(s)>0\\}}z_{x}^{2}dx\right)ds$
$\displaystyle+\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\mathbb{E}\left(\int_{0}^{L}\chi_{\\{z(s)>0\\}}(\partial_{x}\Psi_{k})^{2}z^{2}(s)dx\right)\,ds$
$\displaystyle+\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\mathbb{E}\left(\int_{0}^{L}\chi_{\\{z(s)>0\\}}(\partial_{x}\Psi_{k})\Psi_{k}zz_{x}dx\right)\,ds$
$\displaystyle+\sum_{k}\gamma_{k}^{2}\int_{0}^{t}\mathbb{E}\left(\int_{0}^{L}\chi_{\\{z(s)>0\\}}(\Psi_{k}f(-z))^{2}dx\right)\,ds$
$\displaystyle=\mathbb{E}\int_{0}^{L}(z^{+}_{0}(x))^{2}\,dx-2\varepsilon\int_{0}^{t}\mathbb{E}\left(\int_{0}^{L}(z^{+}_{x})^{2}dx\right)ds$
$\displaystyle+\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\mathbb{E}\left(\int_{0}^{L}(\partial_{x}\Psi_{k})^{2}(z^{+}(s))^{2}dx\right)\,ds$
$\displaystyle+\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\mathbb{E}\left(\int_{0}^{L}\Psi_{k}(\partial_{x}\Psi_{k})z^{+}z^{+}_{x}dx\right)\,ds$
$\displaystyle+\sum_{k}\gamma_{k}^{2}\int_{0}^{t}\mathbb{E}\left(\int_{0}^{L}(\Psi_{k}f(-z^{+}))^{2}dx\right)\,ds$
$\displaystyle:=\mathbb{E}\int_{0}^{L}(z^{+}_{0}(x))^{2}\,dx-2\varepsilon\int_{0}^{t}\mathbb{E}\left(\int_{0}^{L}(z^{+}_{x})^{2}dx\right)ds+J_{1}+J_{2}+J_{3}.$

Using (1.4) and (1.3), we have

$J_{1}\leq C\int_{0}^{t}\mathbb{E}\int_{0}^{L}(z^{+}(s))^{2}dxds.$ (4.41)

In order to estimate $J_{2}$, integrating by parts, we have

$\int_{0}^{L}\Psi_{k}(\partial_{x}\Psi_{k})z^{+}z^{+}_{x}dx=\frac{1}{4}\int_{0}^{L}\partial_{x}(\Psi_{k}^{2})\,\partial_{x}(z^{+}(s))^{2}dx=-\frac{1}{4}\int_{0}^{L}\partial_{x}^{2}(\Psi_{k}^{2})(z^{+}(s))^{2}dx.$

This way

$J_{2}\leq C\int_{0}^{t}\mathbb{E}\int_{0}^{L}(z^{+}(s))^{2}dxds.$ (4.42)

In order to estimate $J_{3}$, we make use of the conditions on $f(u)$. We have

$J_{3}\leq\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\mathbb{E}\left(\int_{0}^{L}K^{2}\Psi_{k}^{2}(z^{+}(s))^{2}dx\right)\,ds\leq
C\int_{0}^{t}\mathbb{E}\int_{0}^{L}(z^{+}(s))^{2}dx\,ds.$ (4.43)

It follows from (4)-(4.43) we obtain

$\mathbb{E}\int_{0}^{L}(z^{+}(t,x))^{2}\,dx\leq\mathbb{E}\int_{0}^{L}(z_{0}^{+}(x))^{2}\,dx-2\varepsilon\int_{0}^{t}\mathbb{E}\int_{0}^{L}(z^{+}(s))^{2}dxds\\\
+C_{1}\int_{0}^{t}\mathbb{E}\left(\int_{0}^{L}(z^{+}(s))^{2}\,dx\right)ds.$
(4.44)

Hence, by Gronwall’s inequality, we have

$\int_{0}^{L}(z^{+}(t,x))^{2}\,dx\leq
e^{C_{1}t}\mathbb{E}\int_{0}^{L}(z_{0}^{+}(x))^{2}\,dx=0,$

since $z_{0}^{+}=0$ a.s. Therefore, $z(t,x)\leq 0$ a.s., and hence $u(t,x)\geq
0$ a.s. Thus the second statement of Proposition 4.3 follows. In order to
prove the third statement, we need the following Lemma:

######  Lemma 4.4.

Let $w^{\varepsilon}(t,x)$ be a non-negative solution of (4.5). Then

$\mathbb{E}\left(\int_{0}^{L}w^{\varepsilon}(t,x)dx\right)^{p}\leq
e^{Ct}\mathbb{E}\left(\int_{0}^{L}w_{0}(x)dx\right)^{p}$ (4.45)

for some $C>0$ and $p\geq 2$.

###### Proof.

Consider $F:L^{2}(\mathbb{T}_{L})\to\mathbb{R}$ given by

$F(w):=\left(\int_{0}^{L}w(x)dx\right)^{p}.$

By Ito’s formula,

$\displaystyle\left(\int_{0}^{L}w^{\varepsilon}(t)\,dx\right)^{p}-\left(\int_{0}^{L}w_{0}\,dx\right)^{p}=\frac{1}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}p\left(\int_{0}^{L}w^{\varepsilon}(s)\,dx\right)^{p-1}$
$\displaystyle\times\int_{0}^{L}\partial_{x}(\Psi_{k}\partial_{x}(\Psi_{k}w^{\varepsilon}(s)))\,dx\,ds+\int_{0}^{t}p\left(\int_{0}^{L}w^{\varepsilon}(s)\,dx\right)^{p-1}\int_{0}^{L}\varepsilon(\partial_{x}^{2}w^{\varepsilon}(s))\,dx\,ds$
$\displaystyle+\sum_{k}\lambda_{k}\int_{0}^{t}\left(p\left(\int_{0}^{L}w^{\varepsilon}(s)\,dx\right)^{p-1}\int_{0}^{L}\partial_{x}(\Psi_{k}w^{\varepsilon}(s))dx\right)\,d\beta^{k}(s)$
$\displaystyle+\sum_{k}\gamma_{k}\int_{0}^{t}\left(p\left(\int_{0}^{L}w^{\varepsilon}(s)\,dx\right)^{p-1}\int_{0}^{L}\Psi_{k}f(w^{\varepsilon}(s))\,dx\right)d\beta_{1}^{k}(s)$
$\displaystyle+\frac{1}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\left(p(p-1)\left(\int_{0}^{L}w^{\varepsilon}(s)\,dx\right)^{p-2}\left(\int_{0}^{L}\partial_{x}(\Psi_{k}w^{\varepsilon}(s))dx\right)^{2}\right)\,ds$
$\displaystyle+\frac{1}{2}\sum_{k}\gamma_{k}^{2}\int_{0}^{t}\left(p(p-1)\left(\int_{0}^{L}w^{\varepsilon}(s)\,dx\right)^{p-2}\left(\int_{0}^{L}\Psi_{k}f(w^{\varepsilon}(s))dx\right)^{2}\right)\,ds$

The first, the second, the third and the fifth terms in this expansion are
zero due to the periodic boundary conditions. Thus, taking the expected value,
we have

$\mathbb{E}\left(\int_{0}^{L}w^{\varepsilon}(t)\,dx\right)^{p}=\mathbb{E}\left(\int_{0}^{L}w_{0}\,dx\right)^{p}\\\
+\frac{1}{2}\sum_{k}\gamma_{k}^{2}\int_{0}^{t}\left(p(p-1)\mathbb{E}\left(\int_{0}^{L}w^{\varepsilon}(s)\,dx\right)^{p-2}\left(\int_{0}^{L}\Psi_{k}f(w^{\varepsilon}(s))dx\right)^{2}\right)\,ds$
(4.46)

Since

$\left|\int_{0}^{L}\Psi_{k}f(w^{\varepsilon}(s))\,dx\right|\leq
C\int_{0}^{L}|w^{\varepsilon}(s))|\,dx=C\int_{0}^{L}w^{\varepsilon}(s))\,dx$

due to the condition (1.3), the linear growth condition on $f(u)$, and non-
negativity of $w^{\varepsilon}(s)$, we have

$\mathbb{E}\left(\int_{0}^{L}w^{\varepsilon}(t)\,dx\right)^{p}\leq\mathbb{E}\left(\int_{0}^{L}w_{0}\,dx\right)^{p}+C\int_{0}^{t}\mathbb{E}\left(\int_{0}^{L}w^{\varepsilon}(t)\,dx\right)^{p}\,ds,$

and thus (4.45) follows from Gronwall Lemma. This completes the proof of the
Lemma. ∎

We now obtain the inequality (4.10). Using Ito’s formula we get for
$t\in[0,\delta)$

$\displaystyle\|\partial_{x}w^{\varepsilon}(t)\|_{2}^{2}-\|\partial_{x}w_{0}\|_{2}^{2}=2\sum_{k\in\mathbb{Z}}\lambda_{k}\int_{0}^{t}\left(\partial_{x}^{2}(\Psi_{k}w^{\varepsilon}(s)),\partial_{x}w^{\varepsilon}(s)\right)_{2}\,d\beta^{k}(s)$
$\displaystyle+2\sum_{k}\gamma_{k}\int_{0}^{t}\left(\partial_{x}(\Psi_{k}f(w^{\varepsilon}(s))),\partial_{x}w^{\varepsilon}(s)\right)d\beta_{1}^{k}(s)$
$\displaystyle+\int_{0}^{t}\left(2\langle\partial_{x}A^{\varepsilon}w^{\varepsilon}(s),\partial_{x}w^{\varepsilon}(s)\rangle+\left\|Bw^{\varepsilon}(s)\right\|^{2}_{L_{2}(H^{2}(\mathbb{T}_{L}),\dot{H}^{1}(\mathbb{T}_{L})}\right)\,ds$

For $p\geq 4$, using Ito’s formula one again, we have

$\displaystyle\|\partial_{x}w^{\varepsilon}(t)\|_{2}^{p}-\|\partial_{x}w_{0}\|_{2}^{p}=p\sum_{k\in\mathbb{Z}}\lambda_{k}\int_{0}^{t}\|\partial_{x}w^{\varepsilon}\|_{2}^{p-2}\left(\partial_{x}^{2}(\Psi_{k}w^{\varepsilon}(s)),\partial_{x}w^{\varepsilon}(s)\right)_{2}\,d\beta^{k}(s)$
$\displaystyle+p\sum_{k}\gamma_{k}\int_{0}^{t}\|\partial_{x}w^{\varepsilon}\|_{2}^{p-2}\left(\partial_{x}(\Psi_{k}f(w^{\varepsilon}(s))),\partial_{x}w^{\varepsilon}(s)\right)d\beta_{1}^{k}(s)$
$\displaystyle+\frac{p}{2}\int_{0}^{t}\|\partial_{x}w^{\varepsilon}\|_{2}^{p-2}\left(2\langle\partial_{x}A^{\varepsilon}w^{\varepsilon}(s),\partial_{x}w^{\varepsilon}(s)\rangle+\left\|Bw^{\varepsilon}(s)\right\|^{2}_{L_{2}(H^{2}(\mathbb{T}_{L}),\dot{H}^{1}(\mathbb{T}_{L})}\right)\,ds$
$\displaystyle+\frac{p(p-2)}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\|\partial_{x}w^{\varepsilon}\|_{2}^{p-4}\left(\partial_{x}^{2}(\Psi_{k}w^{\varepsilon}(s)),\partial_{x}w^{\varepsilon}(s)\right)_{2}^{2}\,ds$
$\displaystyle+\frac{p(p-2)}{2}\sum_{k}\gamma_{k}^{2}\int_{0}^{t}\|\partial_{x}w^{\varepsilon}\|_{2}^{p-4}\left(\partial_{x}(\Psi_{k}f(w^{\varepsilon}(s))),\partial_{x}w^{\varepsilon}(s)\right)_{2}^{2}\,ds$

Taking the expected value yields

$\displaystyle\mathbb{E}\|\partial_{x}w^{\varepsilon}(t)\|_{2}^{p}-\mathbb{E}\|\partial_{x}w_{0}\|_{2}^{p}$
(4.47)
$\displaystyle=\frac{p}{2}\mathbb{E}\int_{0}^{t}\|\partial_{x}w^{\varepsilon}\|_{2}^{p-2}\left(2\langle\partial_{x}A^{\varepsilon}w^{\varepsilon}(s),\partial_{x}w^{\varepsilon}(s)\rangle+\left\|Bw^{\varepsilon}(s)\right\|^{2}_{L_{2}(H^{2}(\mathbb{T}_{L}),\dot{H}^{1}(\mathbb{T}_{L})}\right)\,ds$
$\displaystyle+\frac{p(p-2)}{2}\sum_{k}\lambda_{k}^{2}\mathbb{E}\int_{0}^{t}\|\partial_{x}w^{\varepsilon}\|_{2}^{p-4}\left(\partial_{x}^{2}(\Psi_{k}w^{\varepsilon}(s)),\partial_{x}w^{\varepsilon}(s)\right)_{2}^{2}\,ds$
$\displaystyle+\frac{p(p-2)}{2}\sum_{k}\gamma_{k}^{2}\mathbb{E}\int_{0}^{t}\|\partial_{x}w^{\varepsilon}\|_{2}^{p-4}\left(\partial_{x}(\Psi_{k}f(w^{\varepsilon}(s))),\partial_{x}w^{\varepsilon}(s)\right)_{2}^{2}\,ds$

Integrating by parts,

$\left|\left(\partial_{x}^{2}(\Psi_{k}w^{\varepsilon}(s)),\partial_{x}w^{\varepsilon}(s)\right)_{2}\right|=\left|\int_{0}^{L}(\partial_{x}^{2}\Psi_{k})w^{\varepsilon}\partial_{x}w^{\varepsilon}\,dx+\frac{3}{2}\int_{0}^{L}(\partial_{x}\Psi_{k})(\partial_{x}w^{\varepsilon}(s))^{2}\,dx\right|\\\
\leq
C(\|w^{\varepsilon}(s)\|_{2}\|w_{x}^{\varepsilon}(s)\|_{2}+\|w_{x}^{\varepsilon}\|_{2}^{2})=C\|w^{\varepsilon}(s)\|_{1,2}\|w_{x}^{2}(s)\|_{2}$
(4.48)

Similarly,

$\displaystyle(\partial_{x}(\Psi_{k}f(w^{\varepsilon}(s))),w^{\varepsilon}_{x}(s))_{2}=\int_{0}^{L}[\partial_{x}(\Psi_{k})f(w^{\varepsilon}(s))w_{x}^{\varepsilon}(s)+\Psi_{k}f^{\prime}(w^{\varepsilon}(s))w_{x}^{\varepsilon}(s)]\,dx$
$\displaystyle\leq
C\int_{0}^{L}w^{\varepsilon}(s)w^{\varepsilon}_{x}(s)\,dx+C\int_{0}^{L}(w_{x}^{\varepsilon}(s))^{2}\,dx\leq
C\|w^{\varepsilon}(s)\|_{1,2}\|w_{x}^{2}(s)\|_{2},$ (4.49)

where we used (1.3), nonnegativity of $w^{\varepsilon}$, and the linear growth
of $f(u)$. Using the inequality

$2\langle\partial_{x}A^{\varepsilon}u,\partial_{x}u\rangle+\|B(u)\|^{2}_{\mathcal{L}_{2}(H^{2}(\mathbb{T}_{L})\times
H^{2}(\mathbb{T}_{L}),\dot{H}_{1}(\mathbb{T}_{L}))}\leq
C\|u\|^{2}_{1,2}-2\varepsilon\|\partial_{x}u\|_{1,2}^{2},$

which we obtained in (4.23), as well as (4.48) and (4), we get

$\mathbb{E}\|\partial_{x}w^{\varepsilon}(t)\|_{2}^{p}-\mathbb{E}\|\partial_{x}w_{0}\|_{2}^{p}\leq
C\frac{p}{2}\mathbb{E}\int_{0}^{t}\|\partial_{x}w^{\varepsilon}\|_{2}^{p-2}\|w^{\varepsilon}(s)\|_{1,2}^{2}\,ds\\\
+C\frac{p(p-2)}{2}\sum_{k}(\lambda_{k}^{2}+\gamma_{k}^{2})\mathbb{E}\int_{0}^{t}\|\partial_{x}w^{\varepsilon}\|_{2}^{p-2}\|w^{\varepsilon}(s)\|_{1,2}^{2}\,ds.$
(4.50)

By Poincare inequality

$\|w^{\varepsilon}(s)\|_{1,2}\leq
C\left(\|\partial_{x}w^{\varepsilon}(s)\|_{2}+\left|\int_{0}^{L}w^{\varepsilon}(s)\,dx\right|\right)$

we have

$\mathbb{E}\|\partial_{x}w^{\varepsilon}(t)\|_{2}^{p}-\mathbb{E}\|\partial_{x}w_{0}\|_{2}^{p}\leq
C_{1}\int_{0}^{t}\mathbb{E}\|\partial_{x}w^{\varepsilon}(s)\|_{2}^{p}\,ds+C_{2}\int_{0}^{t}\mathbb{E}\|w_{x}^{\varepsilon}\|_{2}^{p-2}\left(\int_{0}^{L}w^{\varepsilon}(s)\,dx\right)^{2}\,ds$
(4.51)

Using Young’s inequality,

$\mathbb{E}\|\partial_{x}w^{\varepsilon}(t)\|_{2}^{p}\leq\mathbb{E}\|\partial_{x}w^{\varepsilon}(0)\|_{2}^{p}+C_{3}\int_{0}^{t}\mathbb{E}\|\partial_{x}w^{\varepsilon}(s)\|_{2}^{p}\,ds+C_{4}\int_{0}^{t}\mathbb{E}\left(\int_{0}^{L}w^{\varepsilon}(s)\,dx\right)^{p}\,ds$

Hence using the estimate (4.45), we have

$\mathbb{E}\|\partial_{x}w^{\varepsilon}(t)\|_{2}^{p}\leq\mathbb{E}\|\partial_{x}w^{\varepsilon}(0)\|_{2}^{p}+C_{3}\int_{0}^{t}\mathbb{E}\|\partial_{x}w^{\varepsilon}(s)\|_{2}^{p}\,ds+C_{4}e^{\delta}\mathbb{E}\left(\int_{0}^{L}w^{\varepsilon}_{0}\,dx\right)^{p}\,\delta$

The conclusion of Proposition 4.3 now follows from Gronwall’s inequality.

∎

Step 2. We now return to the proof of Theorem 4.2. Proposition 4.3 implies
that the initial value problem (4.5) has a unique solution
$w^{\varepsilon}(t)$ for any $\varepsilon>0$. Thus, for the sake of proving
Theorem 4.2, one needs to pass to the limit in (4.5) as $\varepsilon\to 0$ and
to show that $w^{\varepsilon}(t)$ converges to $w(t)$, which solves (4.1). We
shall need the analog of Lemma 4.4 [13]:

######  Lemma 4.5.

For any $p\in[2,\infty),\nu>0$ and $q\in[p,\infty)$ there exists a constant
$C>0$ such that for all $\gamma\in(0,1)$ we have

$w^{\varepsilon}\in
L^{p}(\Omega,[0,1],\mathcal{F},\lambda_{[0,1]};B_{q}^{\frac{\gamma}{2}-\nu,q}([0,\delta),B_{q}^{\frac{1}{2}-2\gamma,q}(\mathbb{T}_{L})))$

and

$\mathbb{E}\|w^{\varepsilon}\|^{p}_{B_{q}^{\frac{\gamma}{2}-\nu,q}([0,\delta),B_{q}^{\frac{1}{2}-2\gamma,q}(\mathbb{T}_{L}))}\leq
C\mathbb{E}\|w_{0}\|_{1,2}^{p}$ (4.52)

with $C$ independent of $\varepsilon$.

The definition of Besov space $B_{q}^{s,p}(\Omega,X)$, with
$\Omega\in\mathbb{R}^{d}$ and Banach space $X$ can be found in [3, 30].

###### Proof.

For any $\varepsilon>0$ and $\varphi\in C^{\infty}(\mathbb{T}_{L})$ we have

$\displaystyle(w^{\varepsilon}(t),\varphi)_{2}=(w_{0},\varphi)-\frac{1}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}(\Psi_{k}\partial_{x}(\Psi_{k}w^{\varepsilon}(s)),\partial_{x}\varphi)_{2},\,ds-\varepsilon\int_{0}^{t}(\partial_{x}w^{\varepsilon}(s),\partial_{x}\varphi)_{2}\,ds$
(4.53)
$\displaystyle-\sum_{k}\lambda_{k}\int_{0}^{t}(\Psi_{k}w^{\varepsilon}(s),\partial_{x}\varphi)_{2}d\beta^{k}(s)+\sum_{k}\gamma_{k}\int_{0}^{t}(\Psi_{k}f(w^{\varepsilon}(s)),\varphi)_{2}\,d\beta_{1}^{k}(s)$

Introduce the following notation:

$w_{1}^{\varepsilon}=\frac{1}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}\partial_{x}(\Psi_{k}\partial_{x}(\Psi_{k}w^{\varepsilon}(s)))\,ds-\varepsilon\int_{0}^{t}\partial_{x}^{2}w^{\varepsilon}(s)\,ds$
$w_{2}^{\varepsilon}=\sum_{k}\lambda_{k}\int_{0}^{t}\partial_{x}(\Psi_{k}w^{\varepsilon}(s))\,d\beta^{k}(s)$
$w_{3}^{\varepsilon}=\sum_{k}\gamma_{k}\int_{0}^{t}\Psi_{k}f(w^{\varepsilon}(s))\,d\beta_{1}^{k}(s)$

Using (4.53), for $0\leq t_{1}\leq t_{2}<\delta$

$\|w_{1}^{\varepsilon}(t_{2})-w_{1}^{\varepsilon}(t_{1})\|^{p}_{H^{-1}(\mathbb{T}_{L})}\leq
C\left(\sum_{k}\lambda_{k}^{2}\int_{t_{1}}^{t_{2}}(\|w^{\varepsilon}(s)\|_{1,2}+\varepsilon\|w^{\varepsilon}(s)\|_{1,2})\right)^{p}\\\
\leq C|t_{2}-t_{1}|^{p}\sup_{t\in[0,\delta)}\|w^{\varepsilon}(s)\|_{1,2}^{p}$

almost surely. Then from (4.8) we obtain

$\|w_{1}^{\varepsilon}\|_{L^{p}([0,1],\mathcal{F},\lambda_{[0,1]};C^{1-}([0,\delta];H^{-1}(\mathbb{T}_{L}))}\leq
C\mathbb{E}\|w_{0}\|_{1,2}^{p}$ (4.54)

We next estimate Sobolev-Slobodeckij norm [12] (Lemma 2.1) with
$\alpha=\frac{1}{2}-\nu<\frac{1}{2}$:

$\left(\mathbb{E}\|w_{2}^{\varepsilon}\|^{p}_{W^{\frac{1}{2}-\nu,q}([0,\delta);L^{2}(\mathbb{T}_{L}))}\right)^{q}\leq\mathbb{E}\|w_{2}^{\varepsilon}\|^{q}_{W^{\frac{1}{2}-\nu,q}([0,\delta);L^{2}(\mathbb{T}_{L}))}\\\
\leq
C\mathbb{E}\int_{0}^{\delta}\left(\sum_{k\in\mathbb{Z}}\lambda_{k}^{2}\|\partial_{x}(\Psi_{k}w^{\varepsilon}(s))\|_{2}^{2}\right)^{\frac{q}{2}}\,ds\leq
C\delta\mathbb{E}\sup_{t\in[0,\delta)}\|w^{\varepsilon}(t)\|_{1,2}^{q}\leq
C\delta\mathbb{E}\|w_{0}\|_{1,2}^{q}$ (4.55)

Here we used (4.9). An analogous estimate, based on the linear growth of
$f(u)$, will hold for $w_{3}^{\varepsilon}$. The rest of the proof of the
Lemma follows the lines of Lemma 4.4 [13]. ∎

We now proceed with the proof of Theorem 4.2. Let us choose the following
parameters in (4.52): $\kappa\in[0,\frac{1}{2})$,
$\nu\in\left(0,\frac{\kappa}{2}\right)$, $p=2$, $q\in[2,\infty)$. Using the
inequality (4.52), and Markov’s inequality, for $R>0$ we have

$\lambda_{[0,1]}\\{\|w^{\varepsilon}\|_{B_{q}^{\frac{\kappa}{2}-\nu,q}([0,\delta);B^{\frac{1}{2}-2\kappa,q}(\mathbb{T}_{L}))}>R\\}\leq\frac{C^{2}}{R^{2}}\mathbb{E}\|w_{0}\|_{1,2}^{p}\to
0\text{ as }R\to\infty.$

Thus the set

$\\{w:\|w^{\varepsilon}\|_{B_{q}^{\frac{\kappa}{2}-\nu,q}([0,\delta);B^{\frac{1}{2}-2\kappa,q}(\mathbb{T}_{L}))}\leq
R\\}$

is compact in $BC^{0}([0,\delta)\times\mathbb{T}_{L})$ when
$\kappa<\frac{1}{4}$ and
$q>\max\\{\frac{2}{\kappa-2\nu},\frac{2}{1-4\kappa}\\}$, [1], Th.4.4. The set
of measures

$\mu_{\varepsilon}(A)=\lambda_{[0,1]}\\{\omega:w^{\varepsilon}(t,x,\omega)\in
A\\},\ A\subset BC^{0}([0,\delta)\times\mathbb{T}_{L})$

is tight in $BC^{0}([0,\delta)\times\mathbb{T}_{L})$. Thus, by Skorokhod’s
Theorem [27], there exists a random variable $\tilde{w}^{\varepsilon}$, and
Wiener process $\tilde{\overline{W}}^{\varepsilon}(t)$ on
$([0,1],B([0,1],\lambda_{[0,1]})$ such that
$(\tilde{w}^{\varepsilon},\tilde{\overline{W}}^{\varepsilon})\sim(\omega^{\varepsilon},\bar{W})$
and $\tilde{w}^{\varepsilon}\to\tilde{w}$ a.s. in
$BC^{0}([0,\delta)\times\mathbb{T}_{L})$,
$\tilde{\overline{W}}^{\varepsilon}\to\bar{W}$ a.s. in
$BC^{0}([0,\delta);H^{2}(\mathbb{T}_{L}))$ as $\varepsilon\to 0$. Let us show
that $\tilde{w}$ is the solution of the limiting equation (4.1) with Wiener
process $\tilde{W}$. Introduce the following real-valued processes:

$\displaystyle\beta^{k}(t)=\lambda_{k}^{-1}(W(t),\Psi_{k})_{2,2},$
$\displaystyle\beta^{k}_{1}(t)=\gamma_{k}^{-1}(W_{1}(t),\Psi_{k})_{2,2},$
$\displaystyle\tilde{\beta}^{k}(t)=\lambda_{k}^{-1}(\tilde{W}(t),\Psi_{k})_{2,2},$
$\displaystyle\tilde{\beta}^{k}_{\varepsilon}(t)=\lambda_{k}^{-1}(\tilde{W}^{\varepsilon}(t),\Psi_{k})_{2,2},$
$\displaystyle\beta^{k}_{1,\varepsilon}(t)=\gamma_{k}^{-1}(\tilde{W}_{1}^{\varepsilon}(t),\Psi_{k})_{2,2},$
$\displaystyle\beta^{k}(t)=\gamma_{k}^{-1}(\tilde{W}_{1}(t),\Psi_{k})_{2,2}.$

Additionally, we denote

$\bar{W}^{\varepsilon}=(\tilde{W}^{\varepsilon},\tilde{W}_{1}^{\varepsilon}),\
\bar{W}=(\tilde{W},\tilde{W}_{1})$

Similarly to Proposition 5.3 [13] we can show that that the processes
$\tilde{\beta}^{k}$ and $\tilde{\beta}^{k}_{1}$ are mutually independent
standard real-valued $\tilde{F}_{t}$-Wiener processes. Here

$\tilde{F}_{t}=\sigma((\tilde{w}(s),\tilde{\overline{W}}(s)),s\in[0,t])$

Since
$\mathcal{L}(w^{\varepsilon},\bar{W})=\mathcal{L}(\tilde{w}^{\varepsilon},\tilde{\overline{W}}^{\varepsilon})$,
using (4.53) we have

$\displaystyle(\tilde{w}^{\varepsilon}(t),\varphi)-(w_{0},\varphi)=-\frac{1}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}(\Psi_{k}\partial_{x}(\Psi_{k}\tilde{w}^{\varepsilon}(s)),\partial_{x}\varphi)_{2}\,ds-\varepsilon\int_{0}^{t}(\partial_{x}\tilde{w}^{\varepsilon}(s),\partial_{x}\varphi)_{2}\,ds$
$\displaystyle-\sum_{k}\lambda_{k}\int_{0}^{t}(\Psi_{k}w^{\varepsilon}(s),\partial_{x}\varphi)_{2}d\tilde{\beta}_{\varepsilon}^{k}(s)+\sum_{k}\gamma_{k}\int_{0}^{t}(\Psi_{k}f(\tilde{w}^{\varepsilon}(s)),\varphi)_{2}d\tilde{\beta}_{1,\varepsilon}^{k}(s).$
(4.56)

Passing to the limit as $\varepsilon\to 0$, we obtain the desired result. In
view of the convergence
$(\tilde{w}^{\varepsilon}(t),\varphi)\to(\tilde{w},\varphi)_{2},\varepsilon\to
0$, we have

$\displaystyle\sum_{k}\lambda_{k}^{2}\int_{0}^{t}(\Psi_{k}\partial_{x}(\Psi_{k}\tilde{w}^{\varepsilon}(s)),\partial_{x}\varphi)_{2}\,ds=-\sum_{k}\lambda_{k}^{2}\int_{0}^{t}(\tilde{w}^{\varepsilon}(s),\Psi_{k}\partial_{x}(\Psi_{k}\partial_{x}\varphi)_{2})_{2}\,ds$
$\displaystyle\to-\sum_{k}\lambda_{k}^{2}\int_{0}^{t}(\tilde{w}(s),\Psi_{k}\partial_{x}(\Psi_{k}\partial_{x}\varphi)_{2})_{2}\,ds=\sum_{k}\lambda_{k}^{2}\int_{0}^{t}(\Psi_{k}\partial_{x}(\Psi_{k}\tilde{w}(s)),\partial_{x}\varphi)_{2}\,ds\text{
as }\varepsilon\to 0.$

Integrating by parts, we have

$\varepsilon\int_{0}^{t}\partial_{x}\tilde{w}^{\varepsilon}(s),\partial_{x}\varphi)\,ds=-\varepsilon\int_{0}^{t}(\tilde{w}^{\varepsilon}(s),\partial_{x}^{2}\varphi)_{2}\to
0,\ \varepsilon\to 0,$

since there exists a random constant $C(\omega)>0$ such that

$\sup_{t\in[0,\delta),\
x\in\mathbb{T}_{L}}|\tilde{w}^{\varepsilon}(t,x,\omega)|\leq C(\omega)\text{
a.s. }$

Let us show the convergence of the two last terms in (4). To this end, we
rewrite the in the form

$\sum_{k}\left[\int_{0}^{t}\gamma_{k}(\Psi_{k}f(\tilde{w}^{\varepsilon}(s)),\varphi)_{2}d\tilde{\beta}_{1,\varepsilon}^{k}-\lambda_{k}\int_{0}^{t}(\Psi_{k}w^{\varepsilon}(s),\partial_{x}\varphi)_{2}d\tilde{\beta}_{\varepsilon}^{k}(s)\right]=(\tilde{w}^{\varepsilon}(t),\varphi)_{2}-(w_{0},\varphi)_{2}\\\
+\varepsilon\int_{0}^{t}(\partial_{x}(\tilde{w}^{\varepsilon}(s)),\partial_{x}\varphi)_{2}\,ds+\frac{1}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}(\Psi_{k}\partial_{x}(\Psi_{k}\tilde{w}^{\varepsilon}(s)),\partial_{x}\varphi)_{2}\,ds=\tilde{M}_{\varepsilon}(t)$
(4.57)

Thus $\tilde{M}_{\varepsilon}(t)$ is a martingale with respect to sigma-
algebra
$\tilde{F}_{t,\varepsilon}=\sigma((\tilde{w}^{\varepsilon}(s),\tilde{W}(s)),s\in[0,t])$.
Similarly to Lemma 5.7 [13] one can show that the quadratic variation of the
process satisfies the following bound:

$\langle\langle\tilde{M}\rangle\rangle_{t}=\sum_{k}\gamma_{k}^{2}\int_{0}^{t}(\Psi_{k}f(\tilde{w}^{\varepsilon}(s)),\varphi)_{2}^{2}\,ds+\sum_{k}\lambda_{k}^{2}\int_{0}^{t}(\Psi_{k}\tilde{w}^{\varepsilon}(s),\partial_{x}\varphi)_{2}^{2}\,ds\\\
\leq
C\|\varphi\|_{1,2}^{2}\int_{0}^{t}\|\tilde{w}^{\varepsilon}(s)\|_{2}^{2}\,ds.$
(4.58)

This way

$\mathbb{E}\left(\langle\langle\tilde{M}\rangle\rangle_{t}\right)^{p}\leq
Ct^{p}\|\varphi\|_{1,2}^{2p}\mathbb{E}\sup_{s\in[0,\delta)}(\|\tilde{w}^{\varepsilon}(s)\|_{1,2}^{2p})\leq
C\delta^{p}\|\varphi\|_{1,2}^{2p}\mathbb{E}\|w_{0}\|_{1,2}^{2p}\text{ for
}p\geq 1,$

where we used (4.9) and the fact that the distributions for
$\tilde{w}^{\varepsilon}$ and $w^{\varepsilon}$ coincide. Thus
$\tilde{M}_{\varepsilon}$ is a square integrable martingale. Thus, it follows
from (4.57) that

$\tilde{M}_{\varepsilon}(t)\to\tilde{M}(t)=(\tilde{w}(t),\varphi)_{2}-(w_{0},\varphi)+\frac{1}{2}\sum_{k}\lambda_{k}^{2}\int_{0}^{t}(\Psi_{k}\partial_{x}(\Psi_{k}\tilde{w}(s)),\partial_{x}\varphi)_{2}\,ds$

almost surely as $\varepsilon\to 0.$ It remains to show that

$\tilde{M}(t)=-\sum_{k\in\mathbb{Z}}\lambda_{k}\int_{0}^{t}(\Psi_{k}\tilde{w}(s),\partial\varphi)_{2}d\tilde{\beta}^{k}(s)+\sum_{k}\gamma_{k}\int_{0}^{t}(\Psi_{k}f(\tilde{w}(s)),\varphi)_{2}d\tilde{\beta}_{1}^{k}(s).$
(4.59)

To this end, we will use [16], Proposition A1. $\tilde{M}_{\varepsilon}$ is a
square integrable martingale, which is expressed via the stochastic integral
(4.57). For $0\leq t_{1}\leq t<\delta$ and any bounded continuous
$\tilde{F}_{\varepsilon,t}$-measurable function

$\Phi:BC^{0}([0,t_{1}]\times\mathbb{T}_{L})\times
BC^{0}([0,t_{1}];H^{2}(\mathbb{T}_{L}))\times[0,1]\to\mathbb{R}$

we have

$\mathbb{E}[(\tilde{M}_{\varepsilon}(t)-\tilde{M}_{\varepsilon}(t_{1}))\tilde{\Phi}_{\varepsilon}]=0$
(4.60)
$\mathbb{E}[((\tilde{M}_{\varepsilon}(t))^{2}-(\tilde{M}_{\varepsilon}(t_{1}))^{2}-\sum_{k}\lambda_{k}^{2}\int_{t_{1}}^{t}(\Psi_{k}\tilde{w}^{\varepsilon}(s),\partial_{x}\varphi)_{2}^{2}\,ds\\\
-\sum_{k}\gamma_{k}^{2}\int_{t_{1}}^{t}(\Psi_{k}f(\tilde{w}^{\varepsilon}(s)),\varphi)_{2}^{2}\,ds)\tilde{\Phi}_{\varepsilon}]=0$
(4.61)
$\mathbb{E}\left[\left(\tilde{\beta}_{\varepsilon}^{k}(t)\tilde{M}_{\varepsilon}(t)-\tilde{\beta}_{\varepsilon}^{k}(t_{1})\tilde{M}_{\varepsilon}(t_{1})+\lambda_{k}\int_{t_{1}}^{t}(\Psi_{k}\tilde{w}^{\varepsilon}(s),\partial_{x}\varphi)_{2}\,ds\right)\tilde{\Phi}_{\varepsilon}\right]=0$
$\mathbb{E}\left[\left(\tilde{\beta}_{1,\varepsilon}^{k}(t)\tilde{M}_{\varepsilon}(t)-\tilde{\beta}_{1,\varepsilon}^{k}(t_{1})\tilde{M}_{\varepsilon}(t_{1})-\gamma_{k}\int_{t_{1}}^{t}(\Psi_{k}w^{\varepsilon}(s),\partial_{x}\varphi)_{2}\,ds\right)\tilde{\Phi}_{\varepsilon}\right]=0$

where

$\tilde{\Phi}_{\varepsilon}=\Phi\left(\tilde{w}^{\varepsilon}|_{[0,t_{1}]},\tilde{W}^{\varepsilon}|_{[0,t_{1}]}\right).$

We may now pass to the limit as $\varepsilon\to 0$ in (4.61) the same way as
in Lemma 5.7 [13]. Similar identities hold for the limiting processes
$\tilde{w}(t)$, $\tilde{M}(t)$, $\tilde{\beta}^{k}(t)$ and
$\tilde{\beta}^{k}_{1}$. Thus (4.61) follows from [16], Proposition A1. Thus
we have the existence of the solution of the initial value problem (4.1). Non-
negativity of this solution follows from non-negativity
$\tilde{w}^{\varepsilon}(t)$. The bounds (4.3) follow from (4.9) and (4.10),
which are uniform in $\varepsilon$. Similarly we have the estimate for its
mass from (4.45). This completes the proof of Theorem 4.2. ∎

##  5 Proof of Theorem 2.2.

###  5.1 Estimates for the approximate solutions.

Let $\\{j\delta,j=1,...,N+1\\}$ with $\delta=\frac{T}{N+1}$ be a partition of
$[0,T]$. According to Theorem 3.2, on every interval $[(j-1)\delta,j\delta)$
there exists a non-negative solution $v_{N}$ of the equation (2.2) (in the
sense of Definition 3.1) for any nonnegative $H^{1}(\mathbb{T}_{L})$ initial
condition. Similarly, it follows from Theorem 4.2 that on every interval
$[(j-1)\delta,j\delta]$ there is a nonnegative martingale solution
$(\tilde{w}_{j},\tilde{W}_{j})$ of (2.3) for any nonnegative initial process
from $L^{p}([0,1],\mathcal{F}_{0},\mathcal{P},H^{1}(\mathbb{T}_{L}))$, in the
sense of Definition 4.1. Here $\mathcal{P}=\lambda_{[0,1]}$,

$\tilde{W}_{j}=\left(\sum_{k\in\mathbb{Z}}\tilde{\beta}_{j}^{k}\Psi_{k},\sum_{k\in\mathbb{Z}}\tilde{\beta}_{1,j}^{k}\Psi_{k}\right)$

and the processes $\tilde{\beta}_{j}^{k}$ and $\tilde{\beta}_{1,j}^{k}$ are
defined analogously to $\tilde{\beta}^{k}$ and $\tilde{\beta}_{1}^{k}$ via
(4.54) on the interval $[(j-1)\delta,j\delta]$.

Let us now describe the construction of the solution $w_{N}$ on $[0,T]$. In
view of Theorem 4.2 there is a martingale solution $w_{N}^{1}$ on
$[0,\delta)$, with the initial condition $w_{N}^{0}(0):=v_{N}(\delta-0)$,
defined on
$(\Omega^{1}=[0,1],\mathcal{F}^{1},\mathcal{F}^{1}_{t},\lambda^{1}_{[0,1]})$
with the Wiener process $\tilde{\overline{W}}_{1}$ and filtration
$\tilde{F}_{t}^{1}=\sigma(\tilde{\overline{W}}_{1}(s),s\leq t)$ for
$t\in[0,\delta)$. Introduce the new probability space $\Omega^{2}:=[0,1]\times
0,1]=\\{(\omega_{1},\omega_{2}),\omega_{1}\in[0,1],\omega_{2}\in[0,1]\\},$
with the measure
$\lambda^{2}=\lambda_{[0,1]}(\omega_{1})\lambda_{[0,1]}(\omega_{2})$,
$\mathcal{F}^{2}=\mathcal{F}^{1}(\omega_{1})\times\mathcal{F}^{2}(\omega_{2})$.
We define the new Wiener process $\tilde{\overline{W}}^{\prime}(t,\omega_{2})$
and then construct the Wiener process
$\tilde{\overline{W}}(t,\omega_{1},\omega_{2})$ on $[0,2\delta)$ as follows:

$\tilde{\overline{W}}(t,\omega_{1},\omega_{2})=\begin{cases}\tilde{\overline{W}}_{1}(t,\omega_{1}),\
t\in[0,\delta),\\\
\tilde{\overline{W}}^{\prime}(t-\delta,\omega_{2})+\tilde{\overline{W}}_{1}(\delta,\omega_{1}),\
t\in[\delta,2\delta).\end{cases}$

For any fixed $\omega_{1}$, the equation (2.3) has a martingale solution
$\eta(t)$ on $[\delta,2\delta)$ with the initial condition
$\eta(\delta,\omega_{1},\omega_{2})=v_{N}(2\delta-0,\omega_{1})$. This way the
equation (2.3) has the following non-negative martingale solution on
$[0,2\delta)$:

$W_{N}^{2}(t,\omega_{1},\omega_{2})=\begin{cases}W_{N}^{1}(t,\omega_{1}),\
t\in[0,\delta),\\\ \eta(t,\omega_{1},\omega_{2}),\
t\in[\delta,2\delta).\end{cases}$

with filtration
$\mathcal{F}_{t}^{2}=\sigma((\tilde{\overline{W}}(s),W_{N}^{2}(s)),s\leq t)$
with $t\in[0,2\delta)$. In other words, we have the martingale solution of
(2.3) on $[0,2\delta)$ with some Wiener process
$\tilde{\overline{W}}_{2}(t,\omega_{1},\omega_{2})$, defined on $\Omega^{2}$
for $t\in[0,2\delta)$. Continuing this procedure, we have the existence of a
martingale solution $w_{N}(t)$ on $[0,T)$, defined on $\Omega^{N}$ with some
Wiener process

$\overline{W}_{N}(t)=\left(\sum_{k\in\mathbb{Z}}\beta_{N}^{k}\Psi_{k},\sum_{k\in\mathbb{Z}}\beta_{1,N}^{k}\Psi_{k}\right)$
(5.1)

Then the equation (2.3) can be rewritten in the form

$(w_{N}(t,\cdot),\varphi)_{2}-(w_{N}((j-1)\delta,\varphi)_{2}=-\frac{1}{2}\sum_{k\in\mathbb{Z}}\lambda_{k}^{2}\int_{(j-1)\delta}^{t}(\Psi_{k}\partial_{x}(\Psi_{k}w_{N}(s,\cdot)),\partial_{x}\varphi)_{2}\,ds\\\
-\sum_{k\in\mathbb{Z}}\lambda_{k}\int_{(j-1)\delta}^{t}(\Psi_{k}w_{N}(s,\cdot),\partial_{x}\varphi)_{2}\,d\beta_{N}^{k}(s)+\sum_{k\in\mathbb{Z}}\gamma_{k}\int_{(j-1)\delta}^{t}(\Psi_{k}f(w_{N}(s,\cdot)),\varphi)_{2}\,d\beta_{1,N}^{k}(s)$
(5.2)

Using Theorems (3.2) and (4.2) the limits $w_{N}((j-1)\delta,\cdot)=\lim_{t\to
j\delta}v_{N}(t,\cdot)$ and $v_{N}(j\delta,\cdot)=\lim_{t\to
j\delta}w_{N}(t,\cdot)$ exist almost surely. Furthermore,

$\mathbb{E}\|\partial_{x}w_{N}((j-1)\delta,\cdot)\|_{2}^{p}<\infty$

and

$\|\partial_{x}v_{N}(j\delta,\cdot)\|_{2}^{p}<\infty$

for $j\in\\{1,...,N+1\\}$. In a similar way to [13], we define the
concatenated approximate solution
$u_{N}:[0,T)\times\mathbb{T}_{L}\times[0,1]\to[0,\infty)$ by

$u_{N}(t,\cdot):=\begin{cases}v_{N}(2t-(\gamma-1)\delta,\cdot),\
t\in[(j-1)\delta,(j-\frac{1}{2})\delta),\\\ w_{N}(2t-j\delta,\cdot),\
t\in[(j-\frac{1}{2})\delta,j\delta),\ \ j=1,...,N+1.\end{cases}$ (5.3)

It follows from Theorem 3.2 that the mass of $v_{N}$ is non-increasing, that
is

$\int_{0}^{L}v_{N}(t,x)\,dx\leq\int_{0}^{L}v_{N}((j-1)\delta,x)\,dx,\text{ for
}t\in[(j-1)\delta,j\delta)$ (5.4)

Similarly, by Theorem 4.2 we get

$\mathbb{E}\left(\int_{0}^{L}w_{N}(t,x)\,dx\right)^{p}\leq
e^{Ct}\mathbb{E}\left(\int_{0}^{L}w_{N}((j-1)\delta,x)\,dx\right)^{p},\
t\in[(j-1)\delta,j\delta).$ (5.5)

It follows from (2.2), (2.3), (5.4) and (5.5), for $t\in[(j-1)\delta,j\delta)$

$\mathbb{E}\left|\int_{0}^{L}w_{N}(t,x)\,dx\right|^{p}\leq e^{C\delta
j}\left|\int_{0}^{L}u_{0}(x)\,dx\right|^{p},$

and

$\mathbb{E}\left|\int_{0}^{L}v_{N}(t,x)\,dx\right|^{p}\leq e^{C\delta
j}\left|\int_{0}^{L}u_{0}(x)\,dx\right|^{p}.$

So for all $t\in[0,T)$ we obtain

$\mathbb{E}\left|w_{N}(t,x)\,dx\right|^{p}+\mathbb{E}\left|v_{N}(t,x)\,dx\right|^{p}+\mathbb{E}\left|u_{N}(t,x)\,dx\right|^{p}\leq
A\left|u_{0}(x)\,dx\right|^{p}$ (5.6)

for some positive constant $A$.

######  Proposition 5.1.

For any $p\in[2,\infty)$, there exists a constant $C>0$ such that for all
$N\in\mathbb{N}$ we have

$v_{N},w_{N}\text{ and }u_{N}\in
L^{p}([0,1],\mathcal{F},\lambda_{[0,1]},L^{\infty}([0,T];H^{1}(\mathbb{T}_{L})))$

with

${\rm esssup}_{t\in[0,T)}\|u_{N}(t)\|_{1,2}^{p}+{\rm
esssup}_{t\in[0,T)}\|u_{N}(t)\|_{1,2}^{p}+{\rm
esssup}_{t\in[0,T)}\|u_{N}(t)\|_{1,2}^{p}\\\
+\mathbb{E}\int_{0}^{T}\|v_{N}(t)\|_{1,2}^{p-2}\int_{v_{N}(t)>0}(v_{N}\partial_{x}^{3}v_{N})^{2}\,dx\,dt\leq
C\|u_{0}\|_{1,2}^{p}.$ (5.7)

###### Proof.

Similarly to [13], using (3.11) and (4.3) we get

$\mathbb{E}\|\partial_{x}v_{N}(j\delta,\cdot)\|_{2}^{p}\leq
e^{C_{2}j\delta}\left(\|\partial_{x}u_{0}\|_{2}^{p}+C_{3}j\delta\left(\int_{0}^{L}u_{0}(x)\,dx\right)^{p}\right)$
(5.8) $\mathbb{E}\|\partial_{x}w_{N}(j\delta,\cdot)\|_{2}^{p}\leq
e^{C_{2}j\delta}\left(\|\partial_{x}u_{0}\|_{2}^{p}+C_{3}j\delta\left(\int_{0}^{L}u_{0}(x)\,dx\right)^{p}\right)$
(5.9)

for $j=0,...,N.$ Using (5.6), (5.8), (5.9) and Poincare inequality, we have

$\mathbb{E}\|v_{N}(j\delta,\cdot)\|_{1,2}^{p}+\mathbb{E}\|w_{N}(j\delta,\cdot)\|_{1,2}^{p}\leq
C\|u_{0}\|_{1,2}^{p},\ j=0,...,N.$ (5.10)

for some $C>0$. Denote

$Y(t):=\|v_{N}(t)\|_{1,2}^{p-2}\int_{v_{N}(t)>0}(v_{N}\partial_{x}^{3}v_{N})^{2}\,dx.$

It follows from (3.11)

$\int_{0}^{\delta}Y(t)\,dt\leq\|\partial_{x}u_{0}\|_{2}^{p}\|\partial_{x}v_{N}(\delta-0)\|_{2}^{p}$
$\displaystyle\mathbb{E}\int_{\delta}^{2\delta}Y(t)\,dt\leq\mathbb{E}\|\partial_{x}v_{N}(\delta+0)\|_{2}^{p}-\mathbb{E}\|\partial_{x}v_{N}(2\delta-0)\|_{2}^{p}$
$\displaystyle=\mathbb{E}\|\partial_{x}w_{N}(\delta-0)\|_{2}^{p}-\mathbb{E}\|\partial_{x}v_{N}(2\delta-0)\|_{2}^{p}$
$\displaystyle\leq
e^{C_{2}\delta}\mathbb{E}\|\partial_{x}v_{N}(\delta-0)\|_{2}^{p}+e^{C_{2}\delta}C_{3}\delta\mathbb{E}\left|\int_{0}^{L}v_{N}(\delta-0)\,dx\right|^{p}-\mathbb{E}\|\partial_{x}v_{N}(2\delta-0)\|_{2}^{p}$

This way

$\mathbb{E}\int_{0}^{2\delta}Y(t)\,dt\leq\|\partial_{x}u_{0}\|_{2}^{p}+(e^{C_{2}\delta}-1)\mathbb{E}\|\partial_{x}v_{N}(\delta-0)\|_{1,2}^{p}+C_{4}\delta\|v_{N}(\delta-0)\|_{1,2}^{p}\\\
-\mathbb{E}\|\partial_{x}v_{N}(2\delta-0)\|_{2}^{p}$ (5.11)

for all $j=0,...,N.$ It follows from (5.11) and (5.10) we have

$\mathbb{E}\int_{0}^{2\delta}Y(t)\,dt\leq\|\partial_{x}u_{0}\|_{2}^{p}+C(e^{C_{2}\delta}-1)\|u_{0}\|_{1,2}^{p}+C_{5}\delta\|u_{0}\|_{1,2}^{p}-\mathbb{E}\|\partial_{x}v_{N}(2\delta-0)\|_{2}^{p}.$
(5.12)

We continue in a similar way:

$\mathbb{E}\int_{2\delta}^{3\delta}Y(t)\,dt\leq\mathbb{E}\|\partial_{x}v_{N}(2\delta+0)\|_{2}^{p}-\mathbb{E}\|\partial_{x}v_{N}(3\delta-0)\|_{2}^{p}=\mathbb{E}\|\partial_{x}w_{N}(2\delta-0)\|_{2}^{p}\\\
-\mathbb{E}\|\partial_{x}v_{N}(3\delta-0)\|_{2}^{p}\leq
e^{C_{2}\delta}\mathbb{E}\|\partial_{x}v_{N}(2\delta-0)\|_{2}^{p}-\mathbb{E}\|\partial_{x}v_{N}(3\delta-0)\|_{2}^{p}.$

Thus

$\displaystyle\mathbb{E}\int_{0}^{3\delta}Y(t)\,dt\leq\|\partial_{x}u_{0}\|_{2}^{p}+C(e^{C_{2}\delta}-1)\|u_{0}\|_{1,2}^{p}+C_{5}\delta\|u_{0}\|_{1,2}^{p}$
$\displaystyle+(e^{C_{2}\delta}-1)\mathbb{E}\|\partial_{x}v_{N}(2\delta-0)\|_{2}^{p}+C_{5}\delta\|u_{0}\|_{1,2}^{p}-\mathbb{E}\|\partial_{x}v_{N}(3\delta-0)\|_{2}^{p}$
$\displaystyle\leq\|\partial_{x}u_{0}\|_{2}^{p}+C(e^{C_{2}\delta}-1)\|u_{0}\|_{1,2}^{p}+C_{5}\delta\|u_{0}\|_{1,2}^{p}+C(e^{C_{2}\delta}-1)\|u_{0}\|_{1,2}^{p}$

Continuing the process on the intervals $[(j-1)\delta,j\delta]$, and using the
inequality (4.3), Theorem 4.2 and Poincare inequality, we obtain (5.7). ∎

######  Proposition 5.2.

For any $p\in[2,\infty),\varepsilon>0$,
$\kappa\in(2\varepsilon,2/p)\cap(2\varepsilon,1/2]$ and
$q\in(2/(\kappa-2\varepsilon),\infty)$, there exists $C>0$ such that for all
$N\in\mathbb{N}$ we have

$u_{N}\in
L^{p}\left([0,1],\mathcal{F},\lambda_{[0,1]},B^{\frac{\kappa}{2}-\varepsilon,q}\left([0,T],B_{q}^{\frac{1}{2}-2\kappa,q}(\mathbb{T}_{L})\right)\right)$

and

$\mathbb{E}\|u_{N}\|^{p}_{B^{\frac{\kappa}{2}-\varepsilon,q}\left([0,T],B_{q}^{\frac{1}{2}-2\kappa,q}(\mathbb{T}_{L})\right)}\leq
C\|u_{0}\|^{p}_{1,2}(1+\|u_{0}\|^{kp}_{1,2})$ (5.13)

###### Proof.

In order to proceed with the proof of this proposition, we start with the
corresponding results for $v_{N}$ and $w_{N}$.

######  Lemma 5.3.

For any $p\in[2,\infty),\varepsilon>0$ and $q\in[p,\infty)$, there exists a
constant $C>0$ such that for all $N\in\mathbb{N}$, $j\in\\{1,...,N+1\\}$ and
$\kappa\in(0,\frac{2}{p})$ we have

$v_{N}\in
L^{p}\left([0,1],\mathcal{F},\lambda_{[0,1]},B_{q}^{\frac{\kappa}{2}-\varepsilon,q}\left([(j-1)\delta,j\delta),B_{q}^{\frac{1}{2}-2\kappa,q}(\mathbb{T}_{L})\right)\right)$

and

$\mathbb{E}\left(\sum_{j=1}^{N+1}\|v_{N}\|^{p}_{B_{q}^{\frac{\kappa}{2}-\varepsilon,q}\left([(j-1)\delta,j\delta),B_{q}^{\frac{1}{2}-2\kappa,q}(\mathbb{T}_{L})\right)}\right)\leq\|u_{0}\|^{(\kappa+1)p}_{L^{2}}(1+\|u_{0}\|^{\lambda-2}_{L^{2}})^{\kappa
p}$ (5.14)

###### Proof.

Using Definition 3.1, for $(j-1)\delta\leq t_{1}\leq t_{2}<j\delta$ we have

$(v_{N}(t_{2},\cdot)-v_{N}(t_{1},\cdot),\varphi)_{2}=\int_{t_{1}}^{t_{2}}\int_{\\{v_{N}(t,\cdot)>0\\}}v_{N}^{2}(\partial_{x}^{3}v_{N})(\partial_{x}\varphi)dxdt+\int_{t_{1}}^{t_{2}}\int_{0}^{L}l(v_{N})\varphi\,dxdt,$

where $\varphi\in C^{\infty}(\mathbb{T}_{L})$. Then
$v_{N}(t_{2},\cdot)-v_{N}(t_{1},\cdot)$ generates a linear continuous
functional on $H^{1}(\mathbb{T}_{L})$. Let us proceed with the estimate of its
norm:

$\|v_{N}(t_{2},\cdot)-v_{N}(t_{1},\cdot)\|_{H^{-1}(\mathbb{T}_{L})}\leq\int_{t_{1}}^{t_{2}}\left(\int_{\\{v_{N}(t,\cdot)>0\\}}v_{N}^{4}(\partial_{x}^{3}v_{N})^{2}dx+\int_{0}^{L}l^{2}(v_{N})\,dx\right)^{\frac{1}{2}}\,dt$

We proceed with the estimate of the last term:

$\int_{t_{1}}^{t_{2}}\left(\int_{0}^{L}l^{2}(v_{N})\,dx\right)^{\frac{1}{2}}=\int_{t_{1}}^{t_{2}}\left(\int_{0}^{L}v_{N}^{2\lambda}\,dx\right)^{\frac{1}{2}}\\\
\leq CL^{\frac{1}{2}}{\rm
esssup}_{t\in[(j-1)\delta,j\delta)}\sup_{x\in\mathbb{T}_{L}}|v_{N}(t,x)|^{\lambda}(t_{2}-t_{1})=C_{1}{\rm
esssup}_{t\in[0,T)}\|v_{N}\|_{1,2}^{\lambda}(t_{2}-t_{1})$ (5.15)

where we used the Sobolev embedding. Then, using (5.7), we have

$\|v_{N}\|_{L^{2}([0,1],\mathcal{F},\lambda_{[0,1]};C^{\frac{1}{2}}([(j-1)\delta,j\delta];H^{-1}(\mathbb{T}_{L})))}\leq
C(\|u_{0}\|_{1,2}^{2}+\|u_{0}\|_{1,2}^{\lambda}).$

The rest of the proof follows the lines of Lemma 4.3 from [13] ∎

We now proceed with the corresponding result for $w_{N}$.

######  Lemma 5.4.

For any $p\in[2,\infty),\varepsilon>0$ and $q\in[p,\infty)$, there exists a
constant $C>0$ such that for all $N\in\mathbb{N}$, $j\in\\{1,...,N+1\\}$ and
$\gamma\in(0,1)$ we have

$w_{N}\in
L^{p}\left([0,1],\mathcal{F},\lambda_{[0,1]},B_{q}^{\frac{\gamma}{2}-\varepsilon,q}\left([(j-1)\delta,j\delta),B_{q}^{\frac{1}{2}-2\gamma,q}(\mathbb{T}_{L})\right)\right)$

and

$\mathbb{E}\left(\sum_{j=1}^{N+1}\|w_{N}\|^{p}_{B_{q}^{\frac{\gamma}{2}-\varepsilon,q}\left([(j-1)\delta,j\delta),B_{q}^{\frac{1}{2}-2\gamma,q}(\mathbb{T}_{L})\right)}\right)^{\frac{p}{q}}\leq\|u_{0}\|_{1,2}^{p}$
(5.16)

The proof of Lemma 5.4 is analogous to the proof of Lemma 4.2 [13], and hence
the proof of Proposition 5.2 follows the lines of Proposition 4.2 [13].

###  5.2 Proof of the main result: Theorem 2.2

This subsection is devoted to the proof of Theorem 2.2. While the proof is
conceptually close to the proof of Theorem 1.2 [13], we are going to highlight
the differences, induced by the presence of nonlinear terms $l(u)$ and $f(u)$.

######  Proposition 5.5.

Denote

$\mathcal{X}_{u}:=BC^{0}([0,T]\times\mathbb{T}_{L})$
$\mathcal{X}_{J}:=L^{2}([0,T]\times\mathbb{T}_{L})\text{ (with weak topology)
}$ $\mathcal{X}_{W}:=BC^{0}([0,T];H^{2}(\mathbb{T}_{L})\times
H^{2}(\mathbb{T}_{L}))$

Then there exist random variables $\tilde{u}$,
$\tilde{u}_{N}:[0,1]\to\mathcal{X}_{u}$, $J_{N},J:[0,1]\to\mathcal{X}_{J}$ and
$\tilde{\overline{W}}^{\prime}_{N},\tilde{\overline{W}}:[0,1]\to\mathcal{X}_{W}$
with
$(\tilde{u}_{N},\tilde{J}_{N},\tilde{\overline{W}}^{\prime}_{N})\sim(u_{N},J_{N},\overline{W}_{N})$,
where $J_{N}:=\chi_{v_{N}>0}v_{N}^{2}(\partial_{x}^{3}v_{N})$. Furthermore,
there are subsequences (still indexed with $N$), such that
$\tilde{u}_{N}(\omega)\to\tilde{u}(\omega)$ in $\mathcal{X}_{u}$,
$\tilde{J}_{N}(\omega)\rightharpoonup\tilde{J}(\omega)$ in $\mathcal{X}_{J}$
and
$\tilde{\overline{W}}^{\prime}_{N}(\omega)\rightharpoonup\tilde{\overline{W}}(\omega)$
in $\mathcal{X}_{W}$ for every $\omega\in[0,1]$ as $N\to\infty$.

###### Proof.

Once we introduce

$\overline{W}_{N}^{0}(t,\cdot):=\begin{cases}\overline{W}_{N}((j-1)\delta,\cdot)\
\text{ for }t\in[(j-1)\delta,(j-\frac{1}{2})\delta)\\\
\overline{W}_{N}(2t-j\delta,\cdot)\ \text{ for
}t\in[(j-\frac{1}{2})\delta,j\delta)\end{cases}$

and

$\tilde{\overline{W}}_{N}^{0}(t,\cdot):=\begin{cases}\tilde{\overline{W}}^{\prime}_{N}((j-1)\delta,\cdot)\
\text{ for }t\in[(j-1)\delta,(j-\frac{1}{2})\delta)\\\
\tilde{\overline{W}}^{\prime}_{N}(2t-j\delta,\cdot)\ \text{ for
}t\in[(j-\frac{1}{2})\delta,j\delta)\end{cases}$

the rest of the proof becomes analogous to the proof of Proposition 5.2 [13].
∎

In a similar way we get the analog of Proposition 5.3 [13]. In particular, the
analogs of (5.4a)-(5.4c) in [13] in our case are

$\displaystyle\beta_{N}^{k}(t)=\lambda_{k}^{-1}(W_{N}^{0}(t),\Psi_{k})_{2,2}$
$\displaystyle\beta_{1,N}^{k}(t)=\gamma_{k}^{-1}(W_{1,N}^{0}(t),\Psi_{k})_{2,2}$
$\displaystyle\tilde{\beta}_{N}^{k}(t)=\lambda_{k}^{-1}(\tilde{W}_{N}(t),\Psi_{k})_{2,2}$
$\displaystyle\tilde{\beta}_{1,N}^{k}(t)=\gamma_{k}^{-1}(\tilde{W}_{1,N}(t),\Psi_{k})_{2,2}$
$\displaystyle\tilde{\beta}^{k}(t)=\lambda_{k}^{-1}(\tilde{W}(t),\Psi_{k})_{2,2}$
$\displaystyle\tilde{\beta_{1}}(t)=\gamma_{k}^{-1}(\tilde{W_{1}}(t),\Psi_{k})_{2,2}$

######  Corollary 5.6.

For $\tilde{u}_{N}$, $\tilde{v}_{N}$, $\tilde{w}_{N}$ and $\tilde{u}$ in
Proposition 5.5 we have

$\displaystyle\|\tilde{u}_{N}-u\|_{BC^{0}([0,T]\times\mathbb{T}_{L})}\to 0;$
$\displaystyle\|\tilde{v}_{N}-u\|_{L^{\infty}([0,T]\times\mathbb{T}_{L})}\to
0;$
$\displaystyle\|\tilde{w}_{N}-\tilde{u}\|_{L^{\infty}([0,T]\times\mathbb{T}_{L})}\to
0$

as $N\to\infty$ $\lambda_{[0,1]}$ almost surely.

The proof of this corollary is identical to Corrollary 5.4 [13]. Furthermore,
we have the analog of Proposition 5.5 [13]:

######  Proposition 5.7.

Let $\tilde{u}_{N}$ and $\tilde{u}$ be given in Proposition 5.5. Then there
are subsequences of $\tilde{u}_{N}$, $\tilde{v}_{N}$ and $\tilde{w}_{N}$,
still indexed with $N$, such that for any $p\in[2,\infty)$ we have
$\tilde{u}_{N}\rightharpoonup^{*}\tilde{u}$,
$\tilde{v}_{N}\rightharpoonup^{*}\tilde{u}$ and
$\tilde{w}_{N}\rightharpoonup^{*}\tilde{u}$ in
$L^{p}([0,1],L^{\infty}([0,T],H^{1}(\mathbb{T}_{L}))$ as $N\to\infty$.
Furthermore,

$\mathbb{E}{\rm esssup}_{t\in[0,T)}\|\tilde{u}(t)\|_{1,2}^{p}\leq
C\|u_{0}\|_{1,2}^{p}$ (5.17)

for some positive constant $C$ independent of $\tilde{u}$ and $u_{0}$.

Hence $\tilde{u}$ is a bounded continuous $H_{w}^{1}(\mathbb{T}_{L})$ \-
valued process.

######  Proposition 5.8.

Let $\tilde{u}_{N},\tilde{u},\tilde{J}_{N}$ and $\tilde{J}$ be as in
Proposition 5.5. Then the distributional derivative
$\partial_{x}^{3}\tilde{u}$ satisfies $\partial_{x}^{3}\tilde{u}\in
L^{2}(\\{\tilde{u}>r\\})$ for any $r>0$. Furthermore,
$\tilde{J}_{N}=\chi_{\tilde{v}_{N}>0}\tilde{v}_{N}^{2}(\partial_{x}^{3}\tilde{v}_{N})$
and $\tilde{J}=\chi_{\tilde{u}>0}\tilde{u}_{N}^{2}(\partial_{x}^{3}\tilde{})$
$\lambda_{[0,1]}$ almost surely.

We may proceed with the scheme (2.2) - (2.3) to conclude that for $t\in[0,T)$
and $\delta=\frac{T}{N+1}$

$\displaystyle(v_{N}(t),\varphi)_{2}-(u_{0},\varphi)_{2}=(v_{N}(t),\varphi)_{2}+\sum_{j=1}^{\left[\frac{t}{\delta}\right]}\left(-(v_{N}(j\delta,\cdot),\varphi)_{2}+\lim_{s\to
j\delta}(w_{N}(s,\cdot),\varphi)_{2}\right)$
$\displaystyle+\sum_{j=1}^{\left[\frac{t}{\delta}\right]}(\lim_{s\to
j\delta}(v_{N}(s,\cdot),\varphi)_{2}-(w_{N}((j-1)\delta,\cdot),\varphi)_{2}-(v_{N}(0,\cdot),\varphi)_{2}$
$\displaystyle=(v_{N}(t),\varphi)_{2}-\left(v_{N}\left(\left[\frac{t}{\delta}\right]\delta,\cdot\right),\varphi\right)_{2}+\sum_{j=1}^{\left[\frac{t}{\delta}\right]}(\lim_{s\to
j\delta}(v_{N}(s,\cdot),\varphi)_{2}-(v_{N}(j-1)\delta,\cdot),\varphi)_{2}$
$\displaystyle+\sum_{j=1}^{\left[\frac{t}{\delta}\right]}(\lim_{s\to
j\delta}(w_{N}(s,\cdot),\varphi)_{2}-(w_{N}(j-1)\delta,\cdot),\varphi)_{2}=\int_{0}^{t}\int_{\\{v_{N}(s,\cdot)>0\\}}v_{N}^{2}(\partial_{x}^{3}v_{N})(\partial_{x}\varphi)\,dxds$
$\displaystyle-\frac{1}{2}\sum_{k\in\mathbb{Z}}\lambda_{k}^{2}\int_{0}^{\left[\frac{t}{\delta}\right]\delta}(\Psi_{k}\partial_{x}(\Psi_{k}w_{N}(s,\cdot)),\partial_{x}\varphi)_{2}\,ds+\int_{0}^{t}(l(v_{N}),\varphi)_{2}\,ds$
$\displaystyle-\sum_{k\in\mathbb{Z}}\lambda_{k}\int_{0}^{\left[\frac{t}{\delta}\right]\delta}(\Psi_{k}w_{N}(s,\cdot),\partial_{x}\varphi)_{2}\,d\beta_{N}^{k}(s)+\sum_{k\in\mathbb{Z}}\gamma_{k}\int_{0}^{\left[\frac{t}{\delta}\right]\delta}(\Psi_{k}f(w_{N}(s,\cdot)),\varphi)_{2}\,d\beta_{1,N}^{k}(s)$

for all $\varphi\in C^{\infty}(\mathbb{T}_{L})$. Changing the stochastic basis
to

$([0,1],\mathcal{F},(\mathcal{F})_{t\in[0,T]},\lambda_{[0,1]})$

for $\tilde{u}_{N},\tilde{v}_{N}$ and $\tilde{w}_{N}$ we obtain

$\displaystyle(\tilde{v}_{N}(t,\cdot),\varphi)_{2}-(u_{0},\varphi)_{2}=\int_{0}^{t}\int_{\\{\tilde{v}_{N}(s,\cdot)>0\\}}\tilde{v}_{N}^{2}(\partial_{x}^{3}\tilde{v}_{N})(\partial_{x}\varphi)\,dxds$
(5.18)
$\displaystyle-\frac{1}{2}\sum_{k\in\mathbb{Z}}\lambda_{k}^{2}\int_{0}^{\left[\frac{t}{\delta}\right]\delta}(\Psi_{k}\partial_{x}(\Psi_{k}\tilde{w}_{N}(s,\cdot)),\partial_{x}\varphi)_{2}\,ds+\int_{0}^{t}(l(\tilde{v}_{N}),\varphi)_{2}\,ds$
$\displaystyle-\sum_{k\in\mathbb{Z}}\lambda_{k}\int_{0}^{\left[\frac{t}{\delta}\right]\delta}(\Psi_{k}\tilde{w}_{N}(s,\cdot),\partial_{x}\varphi)_{2}\,d\tilde{\beta}_{N}^{k}(s)+\sum_{k\in\mathbb{Z}}\gamma_{k}\int_{0}^{\left[\frac{t}{\delta}\right]\delta}(\Psi_{k}f(\tilde{u}_{N}(s,\cdot)),\varphi)_{2}\,d\tilde{\beta}_{1,N}^{k}(s)$

######  Lemma 5.9.

Suppose
$\tilde{u}_{N},\tilde{v}_{N},\tilde{w}_{N},\tilde{u},\tilde{v},\tilde{w}$ are
given in Propositions 5.5 and 5.7. Then for any $\varphi\in
C^{\infty}(\mathbb{T}_{L})$ and $t\in[0,T)$, there are subsequences, still
denoted with $N$, such that

$(\tilde{v}_{N}(t,\cdot),\varphi)_{2}\to(\tilde{u}(t,\cdot),\varphi)_{2},$
(5.19)
$\int_{0}^{t}\int_{\\{\tilde{v}_{N}(s,\cdot)>0\\}}\tilde{v}_{N}^{2}(\partial_{x}^{3}\tilde{v}_{N})(\partial_{x}\varphi)\,dxds\to\int_{0}^{t}\int_{\\{\tilde{u}(s,\cdot)>0\\}}\tilde{u}^{2}(\partial_{x}^{3}\tilde{u})(\partial_{x}\varphi)\,dxds$
(5.20)
$\sum_{k\in\mathbb{Z}}\lambda_{k}^{2}\int_{0}^{\left[\frac{t}{\delta}\right]\delta}(\Psi_{k}\partial_{x}(\Psi_{k}\tilde{w}_{N}(s,\cdot)),\partial_{x}\varphi)_{2}\,ds\to\sum_{k\in\mathbb{Z}}\lambda_{k}^{2}\int_{0}^{\left[\frac{t}{\delta}\right]\delta}(\Psi_{k}\partial_{x}(\Psi_{k}\tilde{u}(s,\cdot)),\partial_{x}\varphi)_{2}\,ds$
(5.21)
$\int_{0}^{t}(l(\tilde{v}_{N}),\varphi)\,ds\to\int_{0}^{t}(l(\tilde{u}),\varphi)\,ds$
(5.22)
$-\sum_{k\in\mathbb{Z}}\lambda_{k}\int_{0}^{\left[\frac{t}{\delta}\right]\delta}(\Psi_{k}\tilde{u}_{N}(s,\cdot),\partial_{x}\varphi)_{2}\,d\tilde{\beta}_{N}^{k}(s)+\sum_{k\in\mathbb{Z}}\gamma_{k}\int_{0}^{\left[\frac{t}{\delta}\right]\delta}(\Psi_{k}f(\tilde{u}_{N}(s,\cdot)),\varphi)_{2}\,d\tilde{\beta}_{1,N}^{k}(s)\to\\\
-\sum_{k\in\mathbb{Z}}\lambda_{k}\int_{0}^{t}(\Psi_{k}\tilde{u}(s,\cdot),\partial_{x}\varphi)_{2}\,d\tilde{\beta}^{k}(s)+\sum_{k\in\mathbb{Z}}\gamma_{k}\int_{0}^{t}(\Psi_{k}f(\tilde{u}(s,\cdot)),\varphi)_{2}\,d\tilde{\beta}_{1}^{k}(s)$
(5.23)

###### Proof.

The convergences (5.19) - (5.21) can be established the same way as in Lemma
5.7 [13]. Furthermore, the continuity of $l(u)$ implies
$(l(\tilde{v}_{N}),\varphi)_{2}\to(l(\tilde{u}),\varphi)_{2}$ as $N\to\infty$
almost surely. Thus

$\mathbb{E}\int_{0}^{t}\int_{0}^{L}((l(\tilde{v}_{N})-l(\tilde{u})))^{2}\,dxds\leq
2\mathbb{E}\int_{0}^{t}\int_{0}^{L}l^{2}(\tilde{v}_{N})\varphi^{2}\,dxds+2\mathbb{E}\int_{0}^{t}\int_{0}^{L}l^{2}(\tilde{u})\varphi^{2}\,dxds:=J_{1}+J_{2}.$

Using (5.7) and Sobolev embedding, we have

$J_{1}\leq C_{1}\mathbb{E}{\rm
esssup}\sup_{x\in\mathbb{T}_{L}}|l^{2}(\tilde{v}_{N}(t,x))|\leq
C_{2}\mathbb{E}{\rm esssup}\|v_{N}\|_{1,2}^{2\lambda}\leq
C_{3}\|u_{0}\|_{1,2}^{2\lambda}.$

In view of Proposition 5.7, the same estimate holds for $J_{2}$ as well. Thus,
by Vitali’s convergences theorem, we have (5.22). Finally, the convergence
(5.23) can be shown the same way as the convergence of the last two terms in
(4) in the proof of Theorem 4.2. Hence the proof of Lemma 5.9 follows. ∎

We may now use Lemma 5.9 to pass to the limit as $N\to\infty$ in (5.18), which
concludes the proof of Theorem 2.2. ∎

## Acknowledgments

The research of Oleksandr Misiats was supported by Simons Collaboration Grant
for Mathematicians No. 854856. The work of Oleksandr Stanzhytskyi and Oleksiy
Kapustyan was partially supported by the National Research Foundation of
Ukraine No. F81/41743 and Ukrainian Government Scientific Research Grant No.
210BF38-01.

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