Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise

Abstract

We prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Grün, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise.

1 Introduction

In this work, we consider the stochastic thin-film equation

$$\begin{aligned} \mathrm {d}u = - \partial _x \left( M(u) \, \partial _x^3 u\right) \mathrm {d}t + \partial _x \left( \sqrt{M(u)} \circ \mathrm {d}W\right) \quad \text{ in } Q_T, \end{aligned}$$

(1.1)

where $u = u(t,x)$ denotes the height of a thin viscous film depending on the independent variables time $t \in [0,T]$, where $T \in (0,\infty )$ is fixed, and lateral position $x \in \mathbb {T}$, where $\mathbb {T}$ is the one-dimensional torus of length $L {:}{=} \left|\mathbb {T} \right|$, and $Q_T {:}{=} [0,T] \times \mathbb {T}$. Equation (1.1) describes the spreading of viscous thin films driven by capillary forces (acting at the liquid-air interface) and thermal noise and decelerated by friction (in the bulk or at the liquid-solid interface). The function $M : \mathbb {R}\rightarrow [0,\infty )$ is called mobility and the following results apply to the choice $M(r) = \left|r \right|^n$ for $r \in \mathbb {R}$, where $n \in \left[ \frac{8}{3}, 4\right)$. In particular, this covers the physically relevant case of a cubic mobility, that is, $n=3$, modelling no slip at the liquid-solid interface in the underlying stochastic Navier–Stokes equations of which (1.1) is an approximation. The symbol W denotes a Wiener process in the Hilbert space $H^2(\mathbb {T})$.

Since its introduction over 15 years ago, by Davidovitch, Moro, and Stone in [13], and by the fourth author, Mecke, and Rauscher in [26], the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise W in (1.1). Attaining a solution of this problem is the main goal of this work.

We refer to [3, 9, 44] for details on the physical derivation by means of a lubrication approximation and on the relevance of (1.1) in the deterministic case, where $W = 0$ in $[0,T] \times \mathbb {T}$. Stochastic versions of the thin-film equation have been proposed independently in [13] and [26]. The former paper is concerned with the question of how thermal fluctuations enhance the spreading of purely surface-tension driven flow. On the contrary, the paper [26] considers the effect of noise on the stability of liquid films and time-scales of the dewetting process. Therefore, the energy considered in [26] differs from that one of [13] by an additional effective interface potential – giving rise to a so called conjoining-disjoining pressure in the equation. We emphasize that the structure of the noise term in (1.1) is common to [26] and [13]. We further refer to [14] for a more recent derivation of the model including the discussion of detailed-balance conditions.

A first existence result of martingale solutions to stochastic thin-film equations has been obtained in [17] by Fischer and the fourth author of this paper, in the setting of quadratic mobility $M(r) = r^2$, additional conjoining-disjoining pressure, and Itô noise. We also mention the paper [7] by Cornalba who introduced additional nonlocal source terms and in this way obtained results for more general mobilities. In [21], the second and the third author of this paper have studied (1.1) with Stratonovich noise and quadratic mobility $M(r) = r^2$ without conjoining-disjoining pressure. It turns out that non-negative martingale solutions exist that allow for touch down of solutions with complete-wetting boundary conditions. The case of quadratic mobility is special and simpler since in this case the stochastic part in (1.1) becomes linear. This allows to separately treat the deterministic and stochastic parts in (1.1), a fact crucial to the approach in [21], and which fails in the case of non-quadratic mobility.

In this paper, we study the existence of weak (or martingale) solutions to (1.1) in the situation in which the gradient-noise term $\partial _x \left( \sqrt{M(u)} \circ \mathrm {d}W\right)$ is nonlinear in the film height u, in particular covering the situation $M(r) = \left|r \right|^3$. This includes precisely the situation studied in [13] in the complete-wetting regime.

The analysis of the present work is based on a combination of estimates of the surface (excess) energy $\frac{1}{2} \int _{\mathbb {T}} (\partial _x u)^2 \, \mathrm {d}x = \frac{1}{2} \left\Vert \partial _x u \right\Vert _{L^2(\mathbb {T})}^2$ and the (mathematical) entropy $\int _{\mathbb {T}} G_0(u) \, \mathrm {d}x$, where

$$\begin{aligned} G_0(r) = {\left\{ \begin{array}{ll} \frac{r^{2-n}}{(2-n) (1-n)} &{} \text{ for } r > 0, \\ \infty &{} \text{ for } r \leqq 0. \end{array}\right. } \end{aligned}$$

(1.2)

The main difficulty comes from the fact that - in contrast to the case of quadratic mobility and Stratonovich noise - the energy estimate cannot be closed on its own. This is caused by the nonlinear, stochastic conservation law structure of the noise in (1.1). Indeed, this nonlinear structure may lead to the occurrence of shocks and, hence, to the blow up of the energy $\frac{1}{2} \left\Vert \partial _x u \right\Vert _{L^2(\mathbb {T})}^2$. In the light of this, the task becomes to understand if the thin-film operator, that is, the deterministic part in (1.1), has a sufficiently strong regularity-improving effect to compensate the possible energy blow up caused by the stochastic perturbation. Since the thin-film operator degenerates when $u \approx 0$, this requires a control on the smallness of u. Such a control is obtained by the entropy estimate, which explains its importance in the case of non-quadratic mobility. Indeed, in the present work we prove that a blow up of the energy can be ruled out by means of a combination of energy and entropy estimates. Once this importance of the entropy estimate for the construction of weak solutions to (1.1) is understood, the next task is to find approximations to (1.1) which allow for uniform (energy) estimates. In light of the previous discussion, these approximations are chosen in a careful way, compatible with both energy and entropy estimates.

We next give a brief account on the literature for the deterministic thin-film equation: A theory of existence of weak solutions for the deterministic thin-film equation has been developed in [1, 4, 6] and [5, 43, 45] for zero and nonzero contact angles at the intersection of the liquid-gas and liquid-solid interfaces, respectively, while the higher-dimensional version of (1.1) with $W = 0$ in $[0,T] \times \mathbb {T}$ and zero contact angles has been the subject of [11, 32]. For these solutions, a number of quantitative results has been obtained – including optimal estimates on spreading rates of free boundaries, that is, the triple lines separating liquid, gas, and solid, see [2, 18, 30, 34], optimal conditions on the occurrence of waiting time phenomena [12], as well as scaling laws for the size of waiting times [19, 20]. We also refer to [31] for an existence result based on numerical analysis.

A corresponding theory of classical solutions, giving the existence and uniqueness for initial data close to generic solutions or short times, has been developed in [22,23,24,25, 27, 28] for zero contact angles and in [15, 37,38,39,40] for nonzero contact angles in one space dimension, while the higher-dimensional version has been the subject of [29, 36, 47] and [8] for zero and nonzero contact angles, respectively.

The paper is structured as follows: In §2, we introduce the necessary mathematical framework and state our main result on existence of martingale solutions. In §3 we introduce a suitable approximation of (1.1) using a Galerkin scheme, a regularization of the mobility M controlled by a small parameter $\varepsilon$, and a cut-off in $\left\Vert u \right\Vert _{L^\infty (\mathbb {T})}$. The Galerkin scheme only makes use of the energy inequality, which is valid also in the infinite-dimensional setting but ceases to hold as $\varepsilon \searrow 0$. In §4 we then derive an energy-entropy estimate which is uniform in $\varepsilon$ and the cut-off in $\left\Vert u \right\Vert _{L^\infty (\mathbb {T})}$ (the latter is removed at the end of this section). Finally, in §5 the limit $\varepsilon \searrow 0$ is carried out and the existence of martingale solutions to the original problem (1.1) is obtained.

2 Setting and Main Result

2.1 Notation

For a set X and $A \subseteq X$ we write $\mathbb {1}_A : X \rightarrow \{0,1\}$ for the indicator function of A, that is,

$$\begin{aligned} \mathbb {1}_A(x) {:}{=} {\left\{ \begin{array}{ll} 1 &{} \text{ for } x \in A, \\ 0 &{} \text{ for } x \in X \setminus A. \end{array}\right. } \end{aligned}$$

For a measurable set $D\subseteq \mathbb {R}^d$, where $d \in \mathbb {N}$, we write $\left|D \right|$ for its d-dimensional Lebesgue measure. We write $\mathbb {T}{:}{=} \mathbb {R}/ (L \mathbb {Z})$ for the one-dimensional torus of length $L > 0$. For any $T \in [0,\infty )$ we write $Q_T {:}{=} [0,T] \times \mathbb {T}$ for the corresponding parabolic cylinder.

For ${\alpha _1, \alpha _2} \in (0,1)$ and $T > 0$, we introduce the Hölder space $C^{{\alpha _1, \alpha _2}}({Q_T})$ to be the subset of all functions on $Q_T$ which satisfy

$$\begin{aligned}{}[u]_{C^{{\alpha _1, \alpha _2}}({Q_T})}{:}{=}&\sup _{x\in \mathbb {T}} \sup _{{\begin{array}{c} t_1, t_2\in [0,T]\\ t_1 \ne t_2 \end{array}}} \frac{|u(t_1,x)-u(t_2,x)|}{\left|t_1-t_2 \right|^{\alpha _1}} \\&+ \sup _{t\in [0,T]}\sup _{{\begin{array}{c} x_1, x_2 \in \mathbb {T}\\ x_1 \ne x_2 \end{array}}}\frac{\left|u(t,x_1)-u(t,x_2) \right|}{\left|x_1-x_2 \right|^{\alpha _2}}<\infty , \end{aligned}$$

and we set

$$\begin{aligned} \Vert u\Vert _{C^{{\alpha _1, \alpha _2}}({Q_T})} {:}{=} \sup _{(t,x) \in Q_T} |u(t,x)|+[u]_{C^{{\alpha _1, \alpha _2}}({Q_T})}. \end{aligned}$$

For $p \in [1,\infty ]$, a measure space $\left( \Omega ,\mathcal {A},\mu \right)$, and a Banach space $\left( X,\left\Vert \cdot \right\Vert \right)$, we write $L^p\left( \Omega ,\mathcal {A},\mu ;X\right)$ for the X-valued Lebesgue space of $\mu$-measurable functions $\Omega \rightarrow X$ with separable range and finite norm $\left\Vert \cdot \right\Vert _{L^p(\Omega ,\mathcal {A},\mu ;X)}$, where

$$\begin{aligned} \left\Vert v \right\Vert _{L^p(\Omega ,\mathcal {A},\mu ;X)} {:}{=} {\left\{ \begin{array}{ll} \left( \int _\Omega \left|v(y) \right|^p \mathrm {d}\mu (y)\right) ^{\frac{1}{p}} &{} \text{ if } p \in [1,\infty ), \\ {{\mu \text{-ess-sup }}}_{y \in \Omega } \left|v(y) \right| &{} \text{ if } p = \infty , \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} {{\mu \text{-ess-sup }}}_{y \in \Omega } \left|v(y) \right| {:}{=} \inf \left\{ C \in [0,\infty ] : \left\Vert v \right\Vert \leqq C \mu \text{-almost } \text{ everywhere }\right\} \end{aligned}$$

denotes the essential supremum of $\left|v \right|$. For $p = 2$ and a Hilbert space $\left( X,\left( \cdot ,\cdot \right) \right)$, we have $\left\Vert v \right\Vert _{L^2(U,\mathcal {A},\mu ;X)} {:}{=} \sqrt{\left( v,v \right) _{L^2(U,\mathcal {A},\mu ;X)}}$, where the inner product is given by $\left( w_1,w_2 \right) _{L^2(U,\mathcal {A},\mu ;X)} {:}{=} \int _U \left( w_1(y),w_2(y) \right) _X \mathrm {d}\mu (y)$ for $w_1, w_2 \in L^2(U)$. We write $L^p(U,\mathcal {A},\mu ) {:}{=} L^p(U,\mathcal {A},\mu ;\mathbb {R})$ if $X = \mathbb {R}$. If $U \subseteq \mathbb {R}^d$ with $d \in \mathbb {N}$ is Borel measurable, $\mathcal {A}$ is the Borel $\sigma$-algebra $\mathcal {B}(U)$, and $\mu = \lambda _U$ the Lebesgue measure on U, we simply write $L^p(U;X) {:}{=} L^p(U,\mathcal {B}(U),\lambda _U;X)$ and if additionally $X = \mathbb {R}$, we write $L^p(U) {:}{=} L^p(U;\mathbb {R})$. For $v \in L^1(U)$, we write

$$\begin{aligned} \mathcal {A}( v ){:}{=} \frac{1}{\left|U \right|} \int _U v(y) \, \mathrm {d}y \end{aligned}$$

for its average value.

For $k \in \mathbb {N}$, $1 \leqq p \leqq \infty$, and $U \subset \mathbb {R}^d$ with $\partial U \in C^\infty$, we write $W^{k,p}(U;X)$ for the Sobolev space of all $u \in L^p(U;X)$ such that $\partial ^\alpha u \in L^p(U;X)$ for all $\alpha \in \mathbb {N}_0^d$ with $\left|\alpha \right| \leqq k$, where the norm is given by

$$\begin{aligned} \left\Vert u \right\Vert _{W^{k,p}(U;X)} {:}{=} \sum _{\left|\alpha \right| \leqq k} \left\Vert \partial ^\alpha u \right\Vert _{L^p(U;X)}. \end{aligned}$$

For $s \in (0,1)$ and $u : U \rightarrow X$ measurable, we define

$$\begin{aligned}{}[u]_{W^{s,p}(U;X)} {:}{=} \left( \int _U \int _U \frac{\left\Vert u(x) - u(y) \right\Vert ^p}{\left|x-y \right|^{s p}} \, \mathrm {d}x \, \mathrm {d}y\right) ^{\frac{1}{p}}. \end{aligned}$$

For $s \in (0,\infty )$, we define the Sobolev-Slobodeckij space $W^{s,p}(U;X)$ as the space of all $u \in W^{\lfloor s \rfloor ,p}(U;X)$ such thar $[\partial ^\alpha u]_{W^{s-\lfloor s \rfloor ,p}(U;X)} < \infty$ for all $\alpha \in \mathbb {N}_0^d$ with $\left|\alpha \right| = \lfloor s \rfloor$, where the norm is given by

$$\left\Vert u \right\Vert _{W^{s,p}(U;X)} {:}{=} \left\Vert u \right\Vert _{W^{\lfloor s \rfloor ,p}(U;X)} + \sum _{\left|\alpha \right| = \lfloor s \rfloor } [\partial ^\alpha u]_{W^{s-\lfloor s \rfloor ,p}(U;X).}$$

For $k \in \mathbb {N}_0$ we define the periodic Sobolev space $H^k(\mathbb {T})$ as the closure of all smooth $v : \mathbb {T}\rightarrow \mathbb {R}$ such that the norm $\left\Vert v \right\Vert _{H^k(\mathbb {T})}$ is finite, where

$$\left\Vert v \right\Vert _{H^k(\mathbb {T})} {:}{=} \sqrt{\left( v,v \right) _{H^k(\mathbb {T})}}$$

and the inner product is defined as

$$ \left( w_1,w_2 \right) _{H^k(\mathbb {T})} {:}{=} \sum _{j = 0}^k\left( \partial _x^j w_1, \partial _x^j w_2 \right) _{L^2(\mathbb {T})}$$

for all smooth $w_1, w_2 : \mathbb {T}\rightarrow \mathbb {R}$. We define $H^{-k}(\mathbb {T}) {:}{=} \left( H^k(\mathbb {T})\right) '$ as the dual of $H^k(\mathbb {T})$ relative to $L^2(\mathbb {T})$.

For $s \in \mathbb {R}\setminus \mathbb {Z}$ we introduce the fractional Sobolev space $H^s(\mathbb {T})$ as the closure of all smooth $v : \mathbb {T}\rightarrow \mathbb {R}$ such that the norm $\left\Vert v \right\Vert _{H^s(\mathbb {T})}$ is finite, where $\left\Vert v \right\Vert _{H^s(\mathbb {T})} {:}{=} \sqrt{\left( v,v \right) _{H^s(\mathbb {T})}}$ and the inner product is defined as

$$\left( w_1,w_2 \right) _{H^s(\mathbb {T})} {:}{=} \sum _{k \in \mathbb {Z}} \left( 1+\lambda _k^s\right) \hat{w}_1(k) \, \hat{w}_2(k)$$

for all smooth $w_1, w_2 : \mathbb {T}\rightarrow \mathbb {R}$, where

$$\hat{w}_j(k) {:}{=} \frac{1}{\sqrt{L}} \int _\mathbb {T}e_k(x) w_j(x) \, \mathrm {d}x$$

is the discrete Fourier transform with respect to the family $(e_k)_{k \in \mathbb {Z}}$ defined in (2.2a) below and $\lambda _k {\mathop {=}\limits ^{(\hbox {2.2c})}} \frac{4 \pi ^2 k^2}{L^2}$.

For $s \in \mathbb {R}$ we write $H^s_\mathrm {w}(\mathbb {T})$ for the space $H^s(\mathbb {T})$ endowed with the weak topology.

2.2 Setting

Suppose that we are given a stochastic basis $\left( \Omega ,\mathcal {F},\mathbb {F},\mathbb {P}\right)$, that is, the triple $\left( \Omega ,\mathcal {F},\mathbb {P}\right)$ is a complete probability space and $\mathbb {F}=( \mathcal {F}_t)_{t \in [0,T]}$ is a filtration satisfying the usual conditions. Further suppose that independent real-valued standard $\mathbb {F}$-Wiener processes $\left( \beta _k\right) _{k \in \mathbb {Z}}$ are given. For what follows, we write

$$\begin{aligned} M(r) {:}{=} F_0^2(r), \quad \text{ where } \quad F_0(r) {:}{=} \left|r \right|^{\frac{n}{2}} \quad \text{ for } \quad r \in \mathbb {R}, \end{aligned}$$

(2.1)

and $n \geqq 1$ is a fixed real constant called mobility exponent. Further assume that $\sigma {:}{=} \left( \sigma _k\right) _{k \in \mathbb {N}}$ is an orthogonal family of eigenfunctions for the negative one-dimensional Laplacian $- \Delta = - \partial _x^2$ on $\mathbb {T}$ (that is, periodic boundary conditions are employed). Specifically, we introduce the orthonormal basis $(e_k)_{k=-\infty }^\infty$ of $L^2(\mathbb {T})$ with

$$\begin{aligned} e_k(x) {:}{=} \sqrt{\frac{2}{L}} {\left\{ \begin{array}{ll} \cos \left( \frac{2 \pi k x}{L}\right) &{} \text{ for } k \geqq 1 \text{ and } x \in \mathbb {T}, \\ \frac{1}{\sqrt{2}} &{} \text{ for } k = 0 \text{ and } x \in \mathbb {T}, \\ \sin \left( \frac{2 \pi k x}{L}\right) &{} \text{ for } k \leqq - 1 \text{ and } x \in \mathbb {T}, \end{array}\right. } \end{aligned}$$

(2.2a)

so that, in particular,

$$\begin{aligned} \partial _x e_k = \underbrace{\frac{2 \pi k}{L}}_{= \mathrm {sign}(k) \sqrt{\lambda _k}} e_{-k} \quad \text{ and } \quad - \partial _x^2 e_k = \underbrace{\frac{4 \pi ^2 k^2}{L^2}}_{= \lambda _k} e_k \quad \text{ for } \quad k \in \mathbb {Z}, \end{aligned}$$

(2.2b)

where

$$\begin{aligned} \lambda _k {:}{=} \frac{4 \pi ^2 k^2}{L^2}. \end{aligned}$$

(2.2c)

We then write

$$\begin{aligned} \sigma _k {=}{:} \nu _k e_k \quad \text{ with } \nu _k \in \mathbb {R}\end{aligned}$$

(2.2d)

and assume

$$\begin{aligned} \sum _{k\in \mathbb {Z}} \lambda _k^2 \nu _k^2 < \infty . \end{aligned}$$

(2.2e)

Notice that because of (2.2b), this implies that

$$\begin{aligned} \sum _{k \in \mathbb {Z}} \Vert \sigma _k \Vert _{W^{2,\infty }(\mathbb {T})}^2< \infty . \end{aligned}$$

(2.2f)

Now, we introduce the $H^2(\mathbb {T})$-valued Wiener process

$$\begin{aligned} W(t,x) {:}{=} \sum _{k \in \mathbb {Z}} \sigma _k(x) \beta ^k(t) \quad \text{ for } \quad (t,x) \in [0,T] \times \mathbb {T}. \end{aligned}$$

(2.3)

The stochastic partial differential equation (SPDE) (1.1) thus attains the form

$$\begin{aligned} \mathrm {d}u = \partial _x \left( - F^2_0 (u) \partial ^3_x u\right) \mathrm {d}t + \sum _{k \in \mathbb {Z}} \partial _x \left( \sigma _k F_0 (u) \right) \circ \mathrm {d}\beta ^k \quad \text{ in } [0,T] \times \mathbb {T}. \end{aligned}$$

It is more convenient for the subsequent analysis to rewrite this equation using Itô calculus, leading to a stochastic correction of the drift (in the physics literature sometimes referred to as the spurious drift), that is,

$$\begin{aligned} \mathrm {d}u= & {} \left[ \partial _x \left( - F^2_0 (u) \partial ^3_x u\right) + \frac{1}{2} \sum _{k \in \mathbb {Z}} \partial _x \left( \sigma _k F_0'(u) \partial _x \left( \sigma _k F_0 (u) \right) \right) \right] \mathrm {d}t\nonumber \\&+ \sum _{k \in \mathbb {Z}} \partial _x \left( \sigma _k F_0 (u) \right) \mathrm {d}\beta ^k \end{aligned}$$

(2.4)

in $[0,T] \times \mathbb {T}$.

2.3 Main result and discussion

We have the following notion of weak (or martingale) solutions to (2.4):

Definition 2.1

A weak (or martingale) solution to (2.4) for $\mathcal {F}_0$-measurable initial data $u^{(0)} \in L^2(\Omega ; H^1(\mathbb {T};\mathbb {R}^+_0))$ is a quadruple

$$\begin{aligned} \left\{ ( \tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {F}},\tilde{\mathbb {P}}), \ (\tilde{\beta }_k)_{k \in \mathbb {Z}},\ \tilde{u}^{(0)}, \tilde{u} \right\} \end{aligned}$$

such that $(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {F}},\tilde{\mathbb {P}})$ is a filtered probability space satisfying the usual conditions, $\tilde{u}^{(0)}$ is $\tilde{\mathcal {F}}_0$-measurable and has the same distribution as $u^{(0)}$, $(\tilde{\beta }_k)_{k \in \mathbb {Z}}$ are independent real-valued standard $\tilde{\mathbb {F}}$-Wiener processes, and $\tilde{u}$ is an $\tilde{\mathbb {F}}$-adapted continuous $H^1_\mathrm {w}(\mathbb {T})$-valued process, such that

• (i) $\tilde{\mathbb {E}} \sup _{t \leqq T} \Vert \tilde{u}(t) \Vert ^2_{H^1(\mathbb {T})}< \infty$

• (ii) For almost all $(\tilde{\omega }, t) \in \tilde{\Omega } \times [0,T]$, the weak derivative of third order $\partial ^3_x\tilde{u}$ exists on $\{ \tilde{u}(t) \ne 0\}$ and satisfies $\tilde{\mathbb {E}} \Vert \mathbb {1}_{\{\tilde{u}\ne 0\}} F_0(\tilde{u}) \, \partial _x^3 \tilde{u} \Vert _{L^2(Q_T)}^2< \infty$,

• (iii) For all $\varphi \in C^\infty (\mathbb {T})$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely, we have

  $$\begin{aligned} \left( \tilde{u}(t),\varphi \right) _{L^2(\mathbb {T})}= & {} \left( \tilde{u}^{(0)},\varphi \right) _{L^2(\mathbb {T})}\nonumber \\&+ \int _0^t \int _{\left\{ \tilde{u}(t') > 0\right\} } F_0^2(\tilde{u}(t')) \left( \partial _x^3 \tilde{u}(t')\right) \left( \partial _x \varphi \right) \mathrm {d}x \, \mathrm {d}t' \nonumber \\&- \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t \left( \sigma _k F_0'(\tilde{u}(t')) \partial _x \left( \sigma _k F_0(\tilde{u}(t'))\right) , \partial _x\varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t' \nonumber \\&- \sum _{k \in \mathbb {Z}} \int _0^t \left( \sigma _k F_0(\tilde{u}(t')), \partial _x \varphi \right) _{L^2(\mathbb {T})} \mathrm {d}\beta ^k(t') \end{aligned}$$

  (2.5)

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

The main result of this paper reads as follows:

Theorem 2.2

Let $T \in (0,\infty )$, $n \in \left[ \frac{8}{3},4\right)$, $p> n+2$, $q > 1$ satisfying $q \geqq \max \left\{ \frac{1}{4-n},\frac{n-2}{2n-5}\right\}$. Suppose that

$$\begin{aligned} u^{(0)} \in L^{p}\left( \Omega ,\mathcal {F}_0,\mathbb {P};H^1(\mathbb {T})\right) \end{aligned}$$

such that $u^{(0)} \geqq 0$, $\mathrm {d}\mathbb {P}$-almost surely, $\mathbb {E}\left|{\mathcal {A}(u^{(0)})} \right|^{2pq} < \infty$, and $\mathbb {E}\left\Vert G_0\left( u^{(0)}\right) \right\Vert _{L^1(\mathbb {T})}^{pq} < \infty$. Then (2.4) admits a weak solution

$$\begin{aligned} \left\{ ( \tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {F}},\tilde{\mathbb {P}}), \ (\tilde{\beta }_k)_{k \in \mathbb {N}},\ \tilde{u}^{(0)}, \tilde{u} \right\} \end{aligned}$$

in the sense of Definition 2.1 such that $\tilde{u} \geqq 0$, $\mathrm {d}\tilde{\mathbb {P}} \otimes \mathrm {d}t \otimes \mathrm {d}x$-almost everywhere. This solution satisfies the estimate

$$\begin{aligned}&\tilde{\mathbb {E}} \Big [\sup _{t \in [0,T]} \left\Vert \partial _x \tilde{u}(t) \right\Vert _{L^2(\mathbb {T})}^{p} + \sup _{t \in [0,T]} \left\Vert G_0\left( \tilde{u}(t)\right) \right\Vert _{L^1(\mathbb {T})}^{pq} + \Vert \mathbb {1}_{\{\tilde{u}>0\}}\tilde{u}^{\frac{n}{2}}(\partial _{x}^{3}\tilde{u})\Vert _{L^2(Q_T)}^{p}\nonumber \\&\qquad \qquad + \left\Vert \partial _x^2 \tilde{u} \right\Vert _{L^2(Q_T)}^{2pq}\Big ] \nonumber \\&\quad \leqq C \, \mathbb {E}\left[ 1 + \left|{\mathcal {A}(u^{(0)})} \right|^{2pq} + \left\Vert G_0(u^{(0)}) \right\Vert _{L^1(\mathbb {T})}^{pq} + \left\Vert \partial _x u^{(0)} \right\Vert _{L^2(\mathbb {T})}^{p} \right] , \end{aligned}$$

(2.6)

where $C < \infty$ is a constant depending only on $p,q,\sigma =(\sigma _k)_{k\in \mathbb {Z}}, n,L,$ and T. Moreover,

$$\begin{aligned} \tilde{u}\in L^{p'}(\tilde{\Omega }, \tilde{\mathbb {F}}, \tilde{\mathbb {P}}; {C^{\frac{\gamma }{4},\gamma }}(Q_T)),\quad \text{ for } \text{ all } \quad \gamma \in \left( 0,\tfrac{1}{2}\right) \quad \text{ and } \quad p'\in \left[ 1,\tfrac{2p}{n+2}\right) . \end{aligned}$$

(2.7)

The proof of Theorem  2.2 is given in Section 5.2 below.

Theorem 2.2 is a global existence result for weak solutions to the stochastic thin-film equation (2.4) for a range of mobility exponents, including the cubic one $n = 3$, corresponding to a no-slip condition at the substrate of the underlying stochastic Navier–Stokes equations (see [13] for details on the modelling and a non-rigorous derivation). Therefore, Theorem 2.2 in particular applies to the physically relevant situation considered in [13]. We expect that the limitations $n \geqq \frac{8}{3}$ and $n < 4$ are due to technical reasons and that these restrictions can be potentially removed in future work by making use of so-called $\alpha$-entropies as first introduced in [1]. Similarly, upgrading Theorem 2.2 to cover higher dimensions, as done in [11, 32], would be an interesting direction for future research. Notably, our solutions are non-negative as in [21] but since $\left\Vert G_0\left( \tilde{u}(t)\right) \right\Vert _{L^1(\mathbb {T})}$ is $\mathrm {d}t \otimes \mathrm {d}\tilde{\mathbb {P}}$-almost everywhere finite, by (1.2) it holds $\left|\left\{ \tilde{u}(t) = 0\right\} \right| = 0$ for all $t \in [0,T]$, $\mathrm {d}\tilde{\mathbb {P}}$-almost everywhere. Since the arguments in [21] are purely energetic, the support of the initial data in [21] is not necessarily $\mathbb {T}$ and in general this is not the case for the corresponding solution of the SPDE, either. We expect that it is possible to overcome this constraint also in the situation of this paper by using a renormalization technique, which will be left as an endeavour for future research, too.

3 Galerkin Approximation

In this section, we use the definitions and assumptions of §2.2.

3.1 Setup

We write $V_N= \text {span} \{e_{-N},...,e_N \}$, where the $(e_j)_{j \in \mathbb {Z}}$ are defined as in (2.2a), $N \in \mathbb {N}$, and let $\Pi _N : L^2(\mathbb {T}) \rightarrow V_N$ be the orthogonal projection given by

$$\begin{aligned} \Pi _N v = \sum _{j=-N}^N \left( v, e_j \right) _{L^2(\mathbb {T})} e_j \quad \text{ for } \text{ any } \quad v \in L^2(\mathbb {T}). \end{aligned}$$

(3.1)

It is immediate from (2.2b) that $\partial _x^2 \Pi _N = \Pi _N \partial _x^2$. Furthermore, we obtain for any $v \in L^2(\mathbb {T})$ through integration by parts and with our specific choice of eigenfunctions,

$$\begin{aligned} \Pi _N (\partial _x v)&{\mathop {=}\limits ^{(3.1)}} \sum _{j = -N}^N \left( \partial _x v, e_j \right) _{L^2(\mathbb {T})} e_j = - \sum _{j = -N}^N \left( v,\partial _x e_j \right) _{L^2(\mathbb {T})} e_j \nonumber \\&\! {\mathop {=}\limits ^{(\hbox {2.2b})}} \sum _{j = -N}^N \left( v, e_{-j} \right) _{L^2(\mathbb {T})} \frac{- 2 \pi j}{L} e_j {\mathop {=}\limits ^{(\hbox {2.2b})}} \sum _{j = -N}^N \left( v, e_{-j} \right) _{L^2(\mathbb {T})} \partial _x e_{-j} \nonumber \\&\,\,= \partial _x \left( \Pi _N v\right) . \end{aligned}$$

(3.2)

Let $g : [0,\infty ) \rightarrow [0,1]$ be a smooth function such that $g = 1$ on [0, 1] and $g = 0$ on $[2,\infty )$. Further define $g_R(s) {:}{=} g(s/R)$ for $s \in [0,\infty )$ and set for $\varepsilon \geqq 0$

$$\begin{aligned} F_\varepsilon (r){:}{=} \left( r^2+\varepsilon ^2\right) ^{\frac{n}{4}} \quad \text{ for } r \in \mathbb {R}, \end{aligned}$$

(3.3)

where $n > 0$ is constant. Notably, for $\varepsilon = 0$ the definition (3.3) is consistent with the corresponding expression in (2.1), but we will assume $\varepsilon > 0$ and thus that $F_\varepsilon$ is smooth with $F_\varepsilon (r) \geqq \varepsilon ^{\frac{n}{2}}$ for all $r \in \mathbb {R}$ until §5. We consider the Galerkin scheme, that is, the finite-dimensional stochastic differential equation (SDE)

$$\begin{aligned} \mathrm {d}u_{\varepsilon ,R,N}= & {} \Pi _N \left[ \partial _x\left( -F_\varepsilon ^2(u_{\varepsilon ,R,N}) \partial _x^3 u_{\varepsilon ,R,N}\right) \right] \mathrm {d}t \nonumber \\&+ \frac{1}{2} g_R^2\left( \left\Vert u_{\varepsilon ,R,N} \right\Vert _{L^\infty (\mathbb {T})}\right) \Pi _N \left[ \sum _{k \in \mathbb {Z}} \partial _x \left( \sigma _k F_\varepsilon '(u_{\varepsilon ,R,N}) \partial _x(\sigma _k F_\varepsilon (u_{\varepsilon ,R,N}))\right) \right] \mathrm {d}t \nonumber \\&+ g_R\left( \left\Vert u_{\varepsilon ,R,N} \right\Vert _{L^\infty (\mathbb {T})}\right) \Pi _N \left[ \sum _{k \in \mathbb {Z}} \partial _x\left( \sigma _k F_\varepsilon (u_{\varepsilon ,R,N})\right) \mathrm {d}\beta ^k\right] . \end{aligned}$$

(3.4)

The approximation in (3.4) is three-fold. While applying the projection $\Pi _N$ yields a finite-dimensional SDE, additionally the mobility $F_0^2$ is regularized with $F_\varepsilon ^2$, so that the limiting equation as $N \rightarrow \infty$ is non-degenerate if $\varepsilon > 0$. For technical reasons in what follows, we also cut off the noise with the pre-factor $g_R\left( \left\Vert u_{\varepsilon ,R,N} \right\Vert _{L^\infty (\mathbb {T})}\right)$.

Notice that (3.4) is equivalent to the system on $\mathbb {R}^{2N+1}$

$$\begin{aligned} \mathrm {d}y= [A_1( y)+ A_2(y) ] \, \mathrm {d}t +\sum _{k\in \mathbb {Z}} B^k(y) \, \mathrm {d}\beta ^k(t), \end{aligned}$$

(3.5)

where, with the short-hand notation $v_y(x)= \sum _{j=-N}^N y^j e_j (x)$ for $y \in \mathbb {R}^{2N+1}$,

$$\begin{aligned} A_1^i(y)= & {} \left( F^2_\varepsilon (v_y)\sum _{j=-N}^N y^j \partial ^3_x e_j , \partial _x e_i \right) _{L^2(\mathbb {T})}, \\ A_2^i(y)= & {} -\frac{1}{2} g_R^2\left( \Vert v_y \Vert _{L^\infty (\mathbb {T})}\right) \left( \sum _{k \in \mathbb {Z}} \sigma _k F_\varepsilon '(v_y ) \partial _x(\sigma _k F_\varepsilon (v_y ) , \partial _x e_i \right) _{L^2(\mathbb {T})}, \\ B^k(y)= & {} - \left( g_R\left( \left\Vert v_y \right\Vert _{L^\infty (\mathbb {T})}\right) \sigma _k F_\varepsilon (v_y), \partial _xe_i \right) _{L^2(\mathbb {T})}. \end{aligned}$$

Let us consider on $\mathbb {R}^{2N+1}$ the inner product

$$\begin{aligned} \left( y^1, y^2 \right) _\lambda : = \sum _{j=_N}^N \lambda _j y^j_1 y^j_2, \end{aligned}$$

and denote by $\Vert \cdot \Vert _\lambda$ the corresponding norm. By (2.2b), it is easy to see that for all $y \in \mathbb {R}^{2N+1}$ we have

$$\begin{aligned} \left( A_1(y), y \right) _\lambda =- \Vert F_\varepsilon (v_y) \partial ^3_x v_y \Vert _{L^2(\mathbb {T})} \leqq 0. \end{aligned}$$

In addition, because of the truncation in R and the finite dimensionality, it is easy to see that there exists a constant $C= C(R, N)$ such that for all $y \in \mathbb {R}^{2N+1}$

$$\begin{aligned} \Vert A_2(y) \Vert _\lambda + \sum _{k=1}^ \infty \Vert B^k(y)\Vert _\lambda ^2 \leqq C. \end{aligned}$$

This shows that the system (3.5) is coercive, which combined with the local Lipschitz continuity of the coefficients implies that for any $\mathcal {F}_0$-measurable random variable in $\mathbb {R}^{2N+1}$, there exists a unique solution of (3.5) starting from $y_0$. In particular, (3.4) has a unique solution starting from $u_N^{(0)} {:}{=} \Pi _N u^{(0)}$. Finally, notice that with (3.2) it follows that (3.4) is still in divergence form so that in particular $\mathcal {A}(u_{\varepsilon ,R,N}(t)) = \mathcal {A}(u_{\varepsilon ,R,N}(0))$ for any $t \in [0,T]$.

3.2 Energy estimate for the Galerkin scheme

Lemma 3.1

Suppose $p \in [2,\infty )$, $u^{(0)} \in L^p\left( \Omega ,\mathcal {F}_0,\mathbb {P};H^1(\mathbb {T})\right)$, and $n > 0$. Let $u_{\varepsilon ,R,N}$ be the unique solution to (3.4) with initial data $u_N^{(0)}$. Then $u_{\varepsilon ,R,N}$ satisfies

$$\begin{aligned}&\mathbb {E}\sup _{t \leqq T } \left\Vert \partial _x u_{\varepsilon ,R,N}(t) \right\Vert _{L^2(\mathbb {T})}^p + \mathbb {E}\left\Vert F_\varepsilon (u_{\varepsilon ,R,N}) \, \partial _x^3 u_{\varepsilon ,R,N} \right\Vert _{L^2(Q_T)}^p \nonumber \\&\quad \leqq C \left( 1 + \mathbb {E}\left\Vert \partial _x u^{(0)} \right\Vert ^p_{L^2(\mathbb {T})}\right) , \end{aligned}$$

(3.6)

where $C < \infty$ is a constant depending only on $\varepsilon$, R, p, $\sigma = \left( \sigma _k\right) _{k \in \mathbb {Z}}$, n, and T (but not on N).

Proof

For convenience, we drop the dependence on $\varepsilon$, R, and N in the notation and simply write u and $\gamma _u(t) {:}{=} g_R\left( \left\Vert u(t) \right\Vert _{L^\infty (\mathbb {T})}\right)$. Applying Itô’s formula to (3.4), we have, $\mathrm {d}\mathbb {P}$-almost surely,

$$\begin{aligned}&\frac{1}{2} \left\Vert \partial _x u(t) \right\Vert _{L^2(\mathbb {T})}^2 - \frac{1}{2} \left\Vert \partial _x u(0) \right\Vert _{L^2(\mathbb {T})}^2 \\&\quad = - \int _0^t \left( \partial _x u(t'), \partial _x \left( \Pi _N \partial _x \left( F_\varepsilon ^2(u(t')) \partial _x^3 u(t')\right) \right) \right) _{L^2(\mathbb {T})} \mathrm {d}t' \\&\qquad + \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t {\gamma _u^2}(t') \left( \partial _x u(t'), \partial _x \left( \Pi _N \partial _x \left( \sigma _k F_\varepsilon '(u(t')) \partial _x \left( \sigma _k F_\varepsilon (u(t'))\right) \right) \right) \right) _{L^2(\mathbb {T})} \mathrm {d}t' \\&\qquad + \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t {\gamma _u^2}(t') \left\Vert \partial _x \left( \Pi _N \partial _x \left( \sigma _k F_\varepsilon (u(t'))\right) \right) \right\Vert _{L^2(\mathbb {T})}^2 \mathrm {d}t' \\&\qquad + \sum _{k \in \mathbb {Z}} \int _0^t \gamma _u(t') \left( \partial _x u(t'), \partial _x \left( \Pi _N \partial _x \left( \sigma _k F_\varepsilon (u(t'))\right) \right) \right) _{L^2(\mathbb {T})} \mathrm {d}\beta ^k(t') \end{aligned}$$

for all $t \in [0,T]$. Since $\Pi _N$ is an orthogonal projection, it furthermore holds $\left( v, \Pi _N w \right) _{L^2(\mathbb {T})} = \left( \Pi _N v, w \right) _{L^2(\mathbb {T})}$ for any $v, w \in L^2(\mathbb {T})$. Since $\Pi _N u = u$ and $\left\Vert \Pi _N v \right\Vert _{L^2(\mathbb {T})} \leqq \left\Vert v \right\Vert _{L^2(\mathbb {T})}$ for any $v \in L^2(\mathbb {T})$, we obtain with the help of (3.2) the simplification

$$\begin{aligned}&\frac{1}{2} \left\Vert \partial _x u(t) \right\Vert _{L^2(\mathbb {T})}^2 - \frac{1}{2} \left\Vert \partial _x u^{(0)} \right\Vert _{L^2(\mathbb {T})}^2 \\&\quad \leqq - \int _0^t \left( \partial _x u(t'), \partial _x^2 \left( F_\varepsilon ^2(u(t')) \, \partial _x^3 u(t')\right) \right) _{L^2(\mathbb {T})} \mathrm {d}t' \\&\qquad + \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t {\gamma _u^2}(t') \left( \partial _x u(t'), \partial _x^2 \left( \sigma _k F_\varepsilon '(u(t')) \, \partial _x \left( \sigma _k F_\varepsilon (u(t'))\right) \right) \right) _{L^2(\mathbb {T})} \mathrm {d}t' \\&\qquad + \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t {\gamma _u^2}(t') \left\Vert \partial _x^2 \left( \sigma _k F_\varepsilon (u(t'))\right) \right\Vert _{L^2(\mathbb {T})}^2 \mathrm {d}t' \\&\qquad + \sum _{k \in \mathbb {Z}} \int _0^t \gamma _u(t') \left( \partial _x u(t'), \partial _x^2 \left( \sigma _k F_\varepsilon (u(t'))\right) \right) _{L^2(\mathbb {T})} \mathrm {d}\beta ^k(t'). \end{aligned}$$

Integration by parts gives for the terms to the right of the inequality

$$\begin{aligned}&\left( \partial _x u, \partial _x^2 \left( F_\varepsilon ^2(u) \, \partial _x^3 u\right) \right) _{L^2(\mathbb {T})} = \left\Vert F_\varepsilon (u) \, \partial _x^3 u \right\Vert _{L^2(\mathbb {T})}^2,\\&\frac{1}{2} \int _{\mathbb {T}} (\partial _x u) \partial _x^2 \left( \sigma _k F_\varepsilon '(u) \partial _x (\sigma _k F_\varepsilon (u))\right) \mathrm {d}x \\&\quad = \int _{\mathbb {T}} \sigma _k^2 \left( - \tfrac{1}{2} (F_\varepsilon ')^2(u) \, (\partial _x^2 u)^2 + \tfrac{1}{6} \left( (F_\varepsilon ')^2\right) ''(u) \, (\partial _x u)^4\right) \mathrm {d}x \\&\qquad + \int _{\mathbb {T}} \left( \partial _x(\sigma _k^2)\right) \left( \tfrac{1}{16} (F_\varepsilon ^2)'''(u) + \tfrac{5}{12} \left( (F_\varepsilon ')^2\right) '(u)\right) (\partial _x u)^3 \, \mathrm {d}x \\&\qquad + \int _{\mathbb {T}} \left( \partial _x^2 (\sigma _k^2)\right) \left( \tfrac{1}{4} (F_\varepsilon ')^2(u) + \tfrac{3}{16} (F_\varepsilon ^2)''(u)\right) (\partial _x u)^2 \, \mathrm {d}x \\&\qquad - \frac{1}{8} \int _{\mathbb {T}} \left( \partial _x^4 (\sigma _k^2)\right) F_\varepsilon ^2(u) \, \mathrm {d}x, \\&\frac{1}{2} \int _{\mathbb {T}} \left( \partial _x^2(\sigma _k F_\varepsilon (u))\right) ^2 \mathrm {d}x \\&\quad = \int _{\mathbb {T}} \sigma _k^2 \left( \tfrac{1}{2} \, (F_\varepsilon ')^2(u) \, (\partial _x^2 u)^2 + \left( \tfrac{1}{2} (F_\varepsilon '')^2(u) - \tfrac{1}{6} \left( (F_\varepsilon ')^2\right) ''(u)\right) (\partial _x u)^4\right) \mathrm {d}x \\&\qquad - \frac{1}{6} \int _{\mathbb {T}} \left( \partial _x (\sigma _k^2)\right) \left( (F_\varepsilon ')^2\right) '(u) \, (\partial _x u)^3 \, \mathrm {d}x \\&\qquad + \int _{\mathbb {T}} \left( (\partial _x \sigma _k)^2 - 2 \sigma _k \, (\partial _x^2 \sigma _k)\right) (F_\varepsilon ')^2(u) \, (\partial _x u)^2 \, \mathrm {d}x \\&\qquad + \frac{1}{2} \int _{\mathbb {T}} \sigma _k \, (\partial _x^4 \sigma _k) \, F_\varepsilon ^2(u) \, \mathrm {d}x, \\&\int _{\mathbb {T}} (\partial _x u) \partial _x^2\left( \sigma _k F_\varepsilon (u)\right) \mathrm {d}x = \int _{\mathbb {T}} \sigma _k \, F_\varepsilon (u) \, \partial _x^3 u \, \mathrm {d}x, \end{aligned}$$

so that we can infer that

$$\begin{aligned}&\frac{1}{2} \left\Vert \partial _x u(t) \right\Vert _{L^2(\mathbb {T})}^2 - \frac{1}{2} \left\Vert \partial _x u(0) \right\Vert _{L^2(\mathbb {T})}^2 \\&\quad \leqq - \left\Vert F_\varepsilon (u) \, \partial _x^3 u \right\Vert _{L^2(Q_t)}^2 \\&\qquad +\frac{1}{6} \sum _{k \in \mathbb {Z}} \int _0^t {\gamma _u^2} \int _{\mathbb {T}} \sigma _k^2 \, (F_\varepsilon '')^2(u) \, (\partial _x u)^4 \, \mathrm {d}x \, \mathrm {d}t' \\&\qquad + \frac{1}{16} \sum _{k \in \mathbb {Z}} \int _0^t {\gamma _u^2} \int _{\mathbb {T}} \left( \partial _x (\sigma _k^2)\right) \left( (F_\varepsilon ^2)'''(u) + 4 \left( (F_\varepsilon ')^2\right) '(u)\right) (\partial _x u)^3 \, \mathrm {d}x \, \mathrm {d}t' \\&\qquad + \frac{3}{16} \sum _{k \in \mathbb {Z}} \int _0^t {\gamma _u^2} \int _{\mathbb {T}} \left( 8 \left( (\partial _x \sigma _k)^2 - \sigma _k (\partial _x^2 \sigma _k)\right) (F_\varepsilon ')^2(u) \right. \\&\qquad \left. + \left( \partial _x^2 (\sigma ^2)\right) (F_\varepsilon ^2)''(u)\right) (\partial _x u)^2 \, \mathrm {d}x \, \mathrm {d}t' \\&\qquad + \frac{1}{8} \sum _{k \in \mathbb {Z}} \int _0^t {\gamma _u^2} \int _{\mathbb {T}} \left( 4 \sigma _k \, \partial _x^4 \sigma _k - \partial _x^4 (\sigma _k^2)\right) F_\varepsilon ^2(u) \, \mathrm {d}x \, \mathrm {d}t' \\&\qquad + \sum _{k \in \mathbb {Z}} \int _0^t \gamma _u \int _{\mathbb {T}} \sigma _k \, F_\varepsilon (u) \, \partial _x^3 u \, \mathrm {d}x \, \mathrm {d}\beta ^k(t'). \end{aligned}$$

For $j, \ell \in \mathbb {N}_0$ with $j + \ell \leqq 4$ we have

$$\begin{aligned} \sum _{k \in \mathbb {Z}} \left\Vert (\partial _x^j \sigma _k) (\partial _x^\ell \sigma _k) \right\Vert _{L^\infty (\mathbb {T})} {\mathop {\leqq }\limits ^{(\hbox {2.2a}), (\hbox {2.2b}), (\hbox {2.2d})}}&\frac{2}{L} \sum _{k \in \mathbb {Z}} \lambda _k^{\frac{j+\ell }{2}} \nu _k^2 \\&\leqq \frac{2}{L} \sum _{k \in \mathbb {Z}} \left( 1+\lambda _k^2\right) \nu _k^2 {\mathop {<}\limits ^{(\hbox {2.2c}), (\hbox {2.2e})}} \infty . \end{aligned}$$

This and our control of $\left\Vert u \right\Vert _{L^\infty (\mathbb {T})}$ via the cut-off function $\gamma _u$ imply, together with (3.3), that

$$\begin{aligned}&\frac{1}{2} \left\Vert \partial _x u(t) \right\Vert _{L^2(\mathbb {T})}^2 - \frac{1}{2} \left\Vert \partial _x u^{(0)} \right\Vert _{L^2(\mathbb {T})}^2 \\&\quad \leqq - \left\Vert F_\varepsilon (u) \, \partial _x^3 u \right\Vert _{L^2(Q_t)}^2 \\&\qquad + C_{\varepsilon ,R,\sigma ,n} \left( 1 + \int _0^t {\gamma _u^2} \int _{\mathbb {T}} \left( (\partial _x u)^4 + \left|\partial _x u \right|^3 + (\partial _x u)^2\right) \mathrm {d}x \, \mathrm {d}t'\right) \\&\qquad + \sum _{k \in \mathbb {Z}} \int _0^t \gamma _u \int _{\mathbb {T}} \sigma _k \, F_\varepsilon (u) \, \partial _x^3 u \, \mathrm {d}x \, \mathrm {d}\beta ^k(t'). \end{aligned}$$

Now, note that $\left|\partial _x u \right|^3 \leqq \frac{1}{2} (\partial _x u)^4 + \frac{1}{2} (\partial _x u)^2$. Furthermore, if $\gamma _u>0$, then we have, through integration by parts, that

$$\begin{aligned} \int _{\mathbb {T}} (\partial _xu )^4 \, \mathrm {d}x= & {} - 3 \int _{\mathbb {T}} u ( \partial _x u)^2 \partial ^2_x u \, \mathrm {d}x = - \frac{3}{2} \int _{\mathbb {T}} (\partial _x u^2) \, (\partial _x u) \, \partial _x^2 u \, \mathrm {d}x \\= & {} \frac{3}{2} \int _{\mathbb {T}} u^2 \, ( \partial _x^2 u)^2 \, \mathrm {d}x + \frac{3}{2} \int _{\mathbb {T}} u^2 \, (\partial _x u) \, \partial _x^3 u \, \mathrm {d}x \\\leqq & {} C_R \left( \int _{\mathbb {T}} (\partial _x^2 u)^2 \, \mathrm {d}x + \int _{\mathbb {T}} \left|\partial _x u \right| \left|\partial _x^3 u \right| \mathrm {d}x\right) \leqq C_R \int _{\mathbb {T}} \left|\partial _x u \right| \left|\partial _x^3 u \right| \mathrm {d}x. \end{aligned}$$

Consequently, by Young’s inequality, we have

$$\begin{aligned} C_{\varepsilon ,R,\sigma ,n} \, \gamma _u \int _{\mathbb {T}} (\partial _xu )^4 \, \mathrm {d}x \leqq \frac{\varepsilon ^n}{2} \, \left\Vert \partial ^3_x u \right\Vert _{L^2(\mathbb {T})}^2 + C_{\varepsilon ,R,\sigma ,n} \, \gamma _u \left\Vert \partial _x u \right\Vert _{L^2(\mathbb {T})}^2, \end{aligned}$$

so that

$$\begin{aligned}&\frac{1}{2} \left\Vert \partial _x u(t) \right\Vert _{L^2(\mathbb {T})}^2 + \frac{1}{4} \int _0^t \int _{\mathbb {T}} F_\varepsilon ^2(u) \, (\partial _x^3 u)^2 \, \mathrm {d}x \, \mathrm {d}t' \\&\quad \leqq \frac{1}{2} \left\Vert \partial _x u^{(0)} \right\Vert _{L^2(\mathbb {T})}^2 + C_{\varepsilon ,R,\sigma ,n} + \sum _{k \in \mathbb {Z}} \int _0^t \gamma _u \int _{\mathbb {T}} \sigma _k \, F_\varepsilon (u) \, \partial _x^3 u \, \mathrm {d}x \, \mathrm {d}\beta ^k(t') \\&\qquad + C_{\varepsilon ,R,\sigma ,n} \int _0^t \left\Vert \partial _x u(t') \right\Vert ^2 \mathrm {d}t'. \end{aligned}$$

Let us set

$$\begin{aligned} \tau _m = \inf \left\{ t \geqq 0 : \Vert \partial _x u(t) \Vert ^2_{L^2(\mathbb {T})} + \int _0^t \int _{\mathbb {T}} F_\varepsilon ^2(u) \, (\partial _x^3 u)^2 \, \mathrm {d}x \, \mathrm {d}t' >m \right\} \wedge T . \end{aligned}$$

By replacing t with $t \wedge \tau _m$ in the above inequality, raising to the power $\frac{p}{2}$, taking expectations, and using Grönwall’s lemma, we conclude that

$$\begin{aligned}&\mathbb {E}\sup _{t \in [0,\tau _m]} \left\Vert \partial _x u(t) \right\Vert _{L^2(\mathbb {T})}^p + \mathbb {E}\left\Vert \mathbb {1}_{[0, \tau _m]} F_\varepsilon (u) \, \partial _x^3 u \right\Vert _{L^2(Q_T)}^p\nonumber \\&\quad \leqq C_{\varepsilon ,R,p,\sigma ,n,T} \left( 1 + \mathbb {E}\left\Vert \partial _x u^{(0)} \right\Vert _{L^2(\mathbb {T})}^p + \mathbb {E}\sup _{t \in [0,T]} \left|\sum _{k \in \mathbb {Z}} \int _0^{t\wedge \tau _m} \gamma _u \int _{\mathbb {T}} \sigma _k \, F_\varepsilon (u) \, \partial _x^3 u \, \mathrm {d}x \, \mathrm {d}\beta ^k(t') \right|^{\frac{p}{2}}\right) ,\nonumber \\ \end{aligned}$$

(3.7)

The Burkholder-Davis-Gundy inequality and the Cauchy-Schwarz inequality imply

$$\begin{aligned}&C_{\varepsilon ,R,p,\sigma ,n,T} \, \mathbb {E}\sup _{t \in [0,T]} \left|\sum _{k \in \mathbb {Z}} \int _0^{t\wedge \tau _m} \gamma _u \int _{\mathbb {T}} \sigma _k \, F_\varepsilon (u) \, \partial _x^3 u \, \mathrm {d}x \, \mathrm {d}\beta ^k \right|^{\frac{p}{2}} \\&\,\,\,\leqq C_{\varepsilon ,R,p,\sigma ,n,T} \, \mathbb {E}\left( \sum _{k \in \mathbb {Z}} \int _0^{\tau _m} \gamma _u^2 \left( \int _{\mathbb {T}} \sigma _k \, F_\varepsilon (u) \, \partial _x^3 u \, \mathrm {d}x\right) ^2 \mathrm {d}t'\right) ^{\frac{p}{4}} \\&{\mathop {\leqq }\limits ^{(\hbox {2.2f})}} C_{\varepsilon ,R,p,\sigma ,n,T} \, \mathbb {E}\left( \int _0^{\tau _m} \gamma _u^2 \int _{\mathbb {T}} F_\varepsilon ^2(u) \, (\partial _x^3 u)^2 \, \mathrm {d}x \, \mathrm {d}t'\right) ^{\frac{p}{4}} \\&\,\,\,\leqq C_{\varepsilon ,R,p,\sigma ,n,T} + \frac{1}{2} \, \mathbb {E}\left\Vert \mathbb {1}_{[0, \tau _m]} F_\varepsilon (u) \, \partial _x^3 u \right\Vert _{L^2(Q_T)}^p, \end{aligned}$$

which shows that the last term at the right hand side of (3.7) can be dropped. The claim then follows by letting $m \rightarrow \infty$ and using Fatou’s lemma. $\square$

3.3 Passage to the limit in the Galerkin scheme

Let us consider the equation

$$\begin{aligned} \mathrm {d}u_{\varepsilon ,R}= & {} \partial _x\left( -F_\varepsilon ^2(u_{\varepsilon ,R}) \partial _x^3u_{\varepsilon ,R}\right) \mathrm {d}t \nonumber \\&+ \frac{1}{2} \sum _{k \in \mathbb {Z}} {g_R^2}\left( \left\Vert u_{\varepsilon ,R} \right\Vert _{L^\infty (\mathbb {T})}\right) \partial _x \left( \sigma _k F_\varepsilon '(u_{\varepsilon ,R}) \partial _x(\sigma _k F_\varepsilon (u_{\varepsilon ,R}))\right) \mathrm {d}t \nonumber \\&+ \sum _{k \in \mathbb {Z}} g_R\left( \left\Vert u_{\varepsilon ,R} \right\Vert _{L^\infty (\mathbb {T})}\right) \left( \partial _x\left( \sigma _k F_\varepsilon (u_{\varepsilon ,R})\right) \right) \mathrm {d}\beta ^k, \end{aligned}$$

(3.8a)

$$\begin{aligned} u_{\varepsilon ,R}(0,\cdot )= & {} u^{(0)}. \end{aligned}$$

(3.8b)

Definition 3.2

Let $R\in (0,\infty ]$. A weak (or martingale) solution to (3.8) is a quadruple

$$\begin{aligned} \left\{ ( \hat{\Omega }, \hat{\mathcal {F}},\hat{\mathbb {F}},\hat{\mathbb {P}}), (\hat{\beta }_k)_{k \in \mathbb {Z}}, \hat{u}^{(0)}, \hat{u}_{\varepsilon ,R} \right\} \end{aligned}$$

such that $( \hat{\Omega }, \hat{\mathcal {F}},\hat{\mathbb {F}},\hat{\mathbb {P}})$ is a filtered probability space satisfying the usual conditions, $\hat{u}^{(0)}$ is $\hat{\mathcal {F}}_0$-measurable and has the same distribution as $u^{(0)}$, $(\hat{\beta }_k)_{k \in \mathbb {Z}}$ are independent real-valued standard $\hat{\mathbb {F}}$-Wiener processes, and $\hat{u}_{\varepsilon ,R}$ is an $\hat{\mathbb {F}}$-adapted continuous $H^1(\mathbb {T})$-valued process, such that

• (i) $\hat{\mathbb {E}} \Vert \hat{u}_{\varepsilon ,R}\Vert ^2_{L^\infty (0,T; H^1(\mathbb {T}))}< \infty$ and for almost all $(\hat{\omega }, t) \in \hat{\Omega } \times [0,T]$, the weak derivative of third order $\partial ^3_x\hat{u}_{\varepsilon ,R}$ exists and satisfies

  $$ \hat{\mathbb {E}} \Vert F_\varepsilon (\hat{u}_{\varepsilon ,R}) \, \partial _x^3 \hat{u}_{\varepsilon ,R} \Vert ^2_{L^2(Q_T)}< \infty ,$$

• (ii) For all $\varphi \in C^\infty (\mathbb {T})$, $\mathrm {d}\hat{\mathbb {P}}$-almost surely, we have

  $$\begin{aligned}&\left( \hat{u}_{\varepsilon ,R}(t),\varphi \right) _{L^2(\mathbb {T})} \\&\quad = \left( \hat{u}^{(0)},\varphi \right) _{L^2(\mathbb {T})} + \int _0^t \int _{\left\{ \hat{u}_{\varepsilon ,R}(t') > 0\right\} } F_\varepsilon ^2(\hat{u}_{\varepsilon ,R}(t')) \left( \partial _x^3 \hat{u}_{\varepsilon ,R}(t')\right) \left( \partial _x \varphi \right) \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\qquad - \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t g_R^2\left( \left\Vert \hat{u}_{\varepsilon ,R}(t') \right\Vert _{{{L^\infty (\mathbb {T})}}}\right) \left( \sigma _k F_\varepsilon '(\hat{u}_{\varepsilon ,R}(t')) \partial _x \left( \sigma _k F_\varepsilon (\hat{u}_{\varepsilon ,R}(t'))\right) , \partial _x\varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t' \nonumber \\&\qquad - \sum _{k \in \mathbb {Z}} \int _0^t g_R\left( \left\Vert \hat{u}_{\varepsilon ,R}(t') \right\Vert _{{L^\infty (\mathbb {T})}}\right) \left( \sigma _k F_\varepsilon (\tilde{u}_{\varepsilon ,R}(t')), \partial _x \varphi \right) _{L^2(\mathbb {T})} \mathrm {d}\beta ^k(t') \end{aligned}$$

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

Remark 3.3

• 1. Note that Definition 3.2 covers also the case that the cutoff by $g_R$ is not active – just by formally setting $R=\infty .$

• 2. (Mass conservation) In the situation of Definition 3.2 by setting $\varphi = 1$ in (ii) it follows that

  $$\begin{aligned} \int _{\mathbb {T}} \hat{u}_{\varepsilon ,R}(t,x) \, \mathrm {d}x = \int _{\mathbb {T}} \hat{u}^{(0)}(x) \, \mathrm {d}x {=}{:} L \mathcal {A}({\hat{u}}^{(0)}) \quad \text{ for } \quad t \in [0,T], \quad \mathrm {d}\hat{\mathbb {P}}\text{-almost } \text{ surely. } \end{aligned}$$

  Hence, by Poincaré’s inequality there exists a constant $C_L < \infty$, only depending on L, such that we have

  $$\begin{aligned} \left\Vert \hat{u}_{\varepsilon ,R}(t) \right\Vert _{L^2(\mathbb {T})} \leqq C_L \left( \left\Vert \partial _x \hat{u}_{\varepsilon ,R}(t) \right\Vert _{L^2(\mathbb {T})} +| \mathcal {A}(\hat{u}^{(0)})| \right) \, \text{ for } \, t \in [0,T], \end{aligned}$$

  (3.9)

  $\mathrm {d}\hat{\mathbb {P}}\text{-almost } \text{ surely. }$

Proposition 3.4

For $n \in (0,4]$, $p \geqq n+2$, and $u^{(0)} \in L^p\left( \Omega ;\mathcal {F}_0,\mathbb {P};H^1(\mathbb {T})\right)$, problem (3.8) admits a weak solution in the sense of Definition 3.2.

Proof

Let $(\Omega , \mathcal {F}, \mathbb {F}, \mathbb {P})$ be a filtered probability space carrying a sequence $\left( \beta ^k\right) _{k=1}^\infty$ of independent $\mathbb {F}$-Wiener processes and on this probability space let $u_{\varepsilon , R, N}$ be the unique –probabilistically– strong solution of (3.4). From now on, since $\varepsilon$ and R are fixed, we drop them and we write $u_N$ instead of $u_{\varepsilon , R, N}$ in order to simplify the notation. By Lemma 3.1 we have that $u_N$ satisfies the bound

$$\begin{aligned} \mathbb {E}\sup _{t \in [0,T]} \left\Vert \partial _x u_N(t) \right\Vert _{L^2(\mathbb {T})}^p + \mathbb {E}\left\Vert F_\varepsilon (u_N) \partial _x^3 u_N \right\Vert _{L^2(Q_T)}^p \leqq C, \end{aligned}$$

(3.10)

where $C < \infty$ is independent of N. Let us introduce the notation $\gamma _w(t) {:}{=} g_R\left( \left\Vert w(t) \right\Vert _{L^\infty (\mathbb {T})}\right)$, and let us decompose $u_N$ as $u_N= u^{(1)}_N+u^{(2)}_N$, where

$$\begin{aligned} u^{(1)}_N(t) {:}{=}&u_N^{(0)}+ \int _0^t \partial _x \left( \Pi _N \left( -F_\varepsilon ^2(u_N(t')) \partial _x^3 u_N(t')\right) \right) \mathrm {d}t' \\&+ \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t {\gamma _{\hat{u}_N}^2(t')} \, \partial _x \left( \Pi _N \left( \sigma _k F_\varepsilon '(u_N(t')) \partial _x(\sigma _k F_\varepsilon (u_N(t')))\right) \right) \mathrm {d}t' \end{aligned}$$

and

$$\begin{aligned} u^{(2)}_N(t') {:}{=} \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}_N}(t') \, \partial _x \left( \Pi _N \left( \sigma _k F_\varepsilon (u_N(t'))\right) \right) \mathrm {d}\beta ^k(t'), \end{aligned}$$

(recall that we can interchange the projection operator and the derivative by virtue of (3.2)). Let $\alpha \in \left( 0, \frac{1}{2}\right)$ such that $\alpha > \frac{1}{p}$. By Sobolev’s embedding and Young’s inequality, we have

$$\begin{aligned}&\sup _{N \in \mathbb {N}} \mathbb {E}\left\Vert u^{(1)}_N \right\Vert ^2 _{W^{\alpha ,p}\left( 0,T;H^{-1}(\mathbb {T})\right) } \\&\quad \leqq C \sup _{N \in \mathbb {N}} \mathbb {E}\left\Vert u^{(1)}_N \right\Vert ^2 _{W^{1,2}\left( 0,T;H^{-1}(\mathbb {T})\right) } \\&\quad \leqq C \sup _{N \in \mathbb {N}} \left( \mathbb {E}\left\Vert u_N^{(0)} \right\Vert _{L^2(\mathbb {T})}^2 + \mathbb {E}\left\Vert F^2_\varepsilon (u_N) \partial ^3_x u_N \right\Vert _{L^2(Q_T)}^2\right) \\&\qquad + C \sup _{N \in \mathbb {N}} \sum _{k \in \mathbb {Z}} \mathbb {E}\left\Vert \sigma _k F_\varepsilon '(u_N) \partial _x(\sigma _k F_\varepsilon (u_N)) \right\Vert ^2_{L^2(Q_T)} \\&\quad \leqq C \left( \left\Vert u^{(0)} \right\Vert _{L^2(\mathbb {T})}^2 + \sup _{N \in \mathbb {N}} \mathbb {E}\left( \Vert F_\varepsilon (u_N)\Vert _{L^\infty (\mathbb {T})}^2 \left\Vert F_\varepsilon (u_N) \partial ^3_x u_N \right\Vert _{L^2(Q_T)}^2\right) \right) \\&\qquad + C \sup _{N \in \mathbb {N}} \sum _{k \in \mathbb {Z}} \mathbb {E}\left\Vert \sigma _k F_\varepsilon '(u_N) \partial _x(\sigma _k F_\varepsilon (u_N)) \right\Vert ^2_{L^2(Q_T)} \\&\quad {\mathop {\leqq }\limits ^{(3.3), (\hbox {2.2f})}} C \left( 1+ \sup _{N \in \mathbb {N}} \mathbb {E}\sup _{t \in [0,T]} \left\Vert u_N \right\Vert _{H^1(\mathbb {T})}^{n+2}\right) \\&\qquad + C \sup _{N \in \mathbb {N}} \mathbb {E}\left\Vert F_\varepsilon (u_N) \partial ^3_x u_N \right\Vert _{L^2(Q_T)}^{n+2} {\mathop {<}\limits ^{(3.10)}} \infty , \end{aligned}$$

where we have used $2n-2 \leqq n+2$. By [16, Lemma 2.1] we get

$$\begin{aligned} \sup _{N \in \mathbb {N}} \mathbb {E}\left\Vert u^{(2)}_N \right\Vert ^p_{W^{\alpha , p}\left( 0,T;H^{-1}(\mathbb {T})\right) }\leqq & {} C \sup _{N \in \mathbb {N}} \int _0^T \mathbb {E}\left\Vert u_N(t) \right\Vert _{H^1(\mathbb {T})}^p \, \mathrm {d}t \\\leqq & {} C \sup _{N \in \mathbb {N}} \sup _{t \in [0,T]} \mathbb {E}\left\Vert u_N(t) \right\Vert _{H^1(\mathbb {T})}^p {\mathop {<}\limits ^{(3.10)}} \infty . \end{aligned}$$

From these two estimates we have that

$$\begin{aligned} \sup _{N \in \mathbb {N}} \mathbb {E}\left\Vert u_N \right\Vert _{W^{\alpha , p}\left( 0,T;H^{-1}(\mathbb {T})\right) \cap L^\infty \left( 0,T;H^1(\mathbb {T})\right) }< \infty . \end{aligned}$$

(3.11)

Let us set

$$\begin{aligned} \beta (t) {:}{=} \sum _{k\in \mathbb {Z}} 2^{-\left|k \right|} \beta ^k(t) \, \mathfrak {e}_k, \end{aligned}$$

where $(\mathfrak {e}_k)_{k \in \mathbb {Z}}$ is the standard orthonormal basis of $\ell ^2(\mathbb {Z})$. We now fix ${s} \in \left( \frac{1}{2}, 1\right)$. By [48, §8, Corollary 5] we have that the embedding

$$\begin{aligned} W^{\alpha , p}\left( 0,T;H^{-1}(\mathbb {T})\right) \cap L^\infty \left( 0,T;H^1(\mathbb {T})\right) \hookrightarrow C\left( [0,T];H^{{s}}(\mathbb {T})\right) \end{aligned}$$

is compact. Combining this with (3.10) and (3.11), it follows that for each $\delta >0$ a compact set $K_\delta \subset \mathcal {Z}{:}{=} C\left( [0,T];H^{{s}}(\mathbb {T})\right) \times \mathcal {Y}\times \ell ^2(\mathbb {Z})$ exists, where $\mathcal {Y}$ denotes the linear space $L^2\left( 0,T;H^3(\mathbb {T})\right)$ endowed with the weak topology, such that

$$\begin{aligned} \sup _{N \in \mathbb {N}} \mathbb {P}\left\{ \left( u_N, u_N , \beta \right) \in K_\delta \right\} \geqq 1-\delta . \end{aligned}$$

By  [35, Theorem 2]  (Prokhorov’s theorem for non-metric spaces), there  exist$\mathcal {Z}$-valued random variables $(\hat{u}_N, \hat{\theta }_N, \hat{\beta }_N )$, $(\hat{u},\hat{\theta }, \hat{\beta })$, for $N \in \mathbb {N}$, on a probability space $(\hat{\Omega }, \hat{\mathcal {F}}, \hat{\mathbb {P}})$ such that in $\mathcal {Z}$,

$$\begin{aligned} (\hat{u}_N, \hat{\theta }_N, \hat{\beta }_N ) \rightarrow (\hat{u}, \hat{\theta }, \hat{\beta } ) \quad \text{ as } \quad N \rightarrow \infty , \quad \mathrm {d}\hat{\mathbb {P}}\text{-almost } \text{ surely }, \end{aligned}$$

(3.12)

and for each $N \in \mathbb {N}$, as random variables in $\mathcal {Z}$

$$\begin{aligned} ( \hat{u}_N, \hat{\theta }_N, \hat{\beta }_N) \sim (u_N, u_N, \beta ). \end{aligned}$$

(3.13)

It follows that

$$\begin{aligned} \hat{\theta }_N = \hat{u}_N \quad \text{ and } \quad \hat{\theta } = \hat{u}. \end{aligned}$$

(3.14)

We set $\hat{u}^{(0)} {:}{=} \hat{u}(0,\cdot )$. Let $\hat{\mathbb {F}} = (\hat{\mathcal {F}}_t)_{t \in [0,T]}$ be the augmented filtration of

$$\begin{aligned} \mathcal {G}_t{:}{=} \sigma \left( \hat{u}(t'), \hat{\beta }(t'); t' \leqq t\right) \end{aligned}$$

and let

$$\begin{aligned} \hat{\beta }^k(t){:}{=} 2^{\left|k \right|} \big (\hat{\beta }(t),\mathfrak {e}_k\big )_{\ell ^2(\mathbb {Z})}. \end{aligned}$$

It follows that $\hat{\beta }^k$, $k \in \mathbb {Z}$, are mutually independent, standard, real-valued$\hat{\mathcal {F}}_t$-Wiener processes (see, for example, [21, Proposition 5.3] or [10, Proof of Proposition 5.4] or [17, Lemma 5.7]). We claim that the probability space $(\hat{\Omega }, \hat{\mathcal {F}}, \hat{\mathbb {F}}, \mathbb {P})$ with $\hat{\mathcal {F}} {:}{=} \hat{\mathcal {F}}_T$, together with the Wiener processes $(\hat{\beta }_k)_{k\in \mathbb {Z}}$ and the process $\hat{u}$ set up a weak solution of (3.8a3.8b).

Notice that Definition 3.2 (i) is satisfied because of (3.10), (3.12), (3.13), and Fatou’s lemma. Hence, we only have to prove Definition 3.2 (ii) and the continuity of $\hat{u}$ as a process with values in in $H^1(\mathbb {T})$. Let us set

$$\begin{aligned} M(\hat{u}, t) {:}{=}&\hat{u}(t)- \hat{u}(0,\cdot )- \int _0^t \left( \partial _x\left( -F_\varepsilon ^2(\hat{u}(t')) \partial _x^3 \hat{u}(t') \right) \right) \mathrm {d}t'\\&+ \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t {\gamma _{\hat{u}}^2(t')} \left( \partial _x \left( \sigma _k F_\varepsilon '(\hat{u}(t')) \partial _x(\sigma _k F_\varepsilon (\hat{u}(t')))\right) \right) \mathrm {d}t' \end{aligned}$$

and for $v \in \{ \hat{u}_N, u_N\}$

$$\begin{aligned} M_N(v, t)&{:}{=} v(t)- v(0)- \int _0^t \Pi _N \left( \partial _x\left( - F_\varepsilon ^2 (v(t')) \partial _x^3 v(t') \right) \right) \mathrm {d}t' \\&\quad + \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t {\gamma _{v}^2(t')} \Pi _N \left( \partial _x \left( \sigma _k F_\varepsilon '(v(t')) \partial _x(\sigma _k F_\varepsilon (v(t')))\right) \right) \mathrm {d}t'. \end{aligned}$$

Fix an arbitrary $l \in \mathbb {Z}$. We will show that for any $\varphi \in H^{-1}(\mathbb {T})$, the processes

$$\begin{aligned} M^1(\hat{u}, t)&{:}{=} (M(\hat{u}, t), \varphi )_{H^{-1}(\mathbb {T})}, \\ M^2( \hat{u} , t)&{:}{=} ( M( \hat{u} , t), \varphi )^2_{H^{-1}(\mathbb {T})} - \sum _{k\in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2(t') \left( \partial _x\left( \sigma _k F_\varepsilon (\hat{u}(t'))\right) , \varphi \right) _{H^{-1}(\mathbb {T})}^2 \, \mathrm {d}t', \\ M^{3}( \hat{u} , t)&{:}{=} \hat{\beta }^l(t)( M( \hat{u} , t), \varphi )_{H^{-1}(\mathbb {T})} - \int _0^t \gamma _{\hat{u}}(t') \left( \partial _x\left( \sigma _l F_\varepsilon (\hat{u}(t'))\right) , \varphi \right) _{H^{-1}(\mathbb {T})} \mathrm {d}t' \end{aligned}$$

are continuous $\hat{\mathcal {F}}_t$-martingales. We first show that they are continuous $\mathcal {G}_t$-martingales. Let us further assume for now that $\varphi \in \bigcup _{N \in \mathbb {N}} V_N$, and for $i=1,2,3$ and $v \in \{ u_N , \hat{u}_N\}$, let us also define the processes $M^i_N(v, t)$ as $M^i( \hat{u}, t)$, but with $\hat{u}$, $M( \hat{u}, t)$, $\partial _x( \sigma _k F_\varepsilon (\hat{u}))$ replaced by v, $M_N(v,t)$, $\Pi _N \partial _x( \sigma _k F_\varepsilon (v))$, respectively. Let us fix $t' < t$ and let $\Phi$ be a bounded continuous function on

$$C\left( [0,t'];H^{-1}(\mathbb {T})\right) \times C\left( [0, t'] ; \ell ^2(\mathbb {Z})\right) .$$

We have that

$$\begin{aligned} \left( M_N(u_N, t), \varphi \right) _{H^{-1}(\mathbb {T})} {\mathop {=}\limits ^{(3.4)}} \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{u_N}(t'') \left( \Pi _N \partial _x \left( \sigma _k F_\varepsilon (u_N(t'')) \right) , \varphi \right) _{H^{-1}(\mathbb {T})} \, \mathrm {d}\beta ^k(t''). \end{aligned}$$

It follows that the $M^i_N(u_N, t)$ are continuous $\mathcal {F}_t$-martingales. Hence,

$$\begin{aligned} \mathbb {E}\left[ \Phi (u_N|_{[0,t']}, \beta |_{[0,t']} )\big (M_N^i(u_N , t)-M_N^i(u_N, t')\big )\right] = 0, \end{aligned}$$

which, combined with (3.13), gives that

$$\begin{aligned} \hat{\mathbb {E}} \big [\Phi (\hat{u}_N|_{[0,t']},\hat{\beta }_N |_{[0,t']} ) \big (M_N^i(\hat{u}_N, t)- M_N^i(\hat{u}_N, t')\big )\big ] =0. \end{aligned}$$

(3.15)

Next, notice that since $\varphi \in V_M$ for some M, we have, for all $N >M$,

$$\begin{aligned}&\int _0^t \left( \Pi _N \partial _x\left( -F_\varepsilon ^2(\hat{u}_N(t')) \partial _x^3 \hat{u}_N(t')\right) , \varphi \right) _{H^{-1}(\mathbb {T})} \mathrm {d}t' \\&\quad = \int _0^T\int _{\mathbb {T}} \mathbb {1}_{[0,t]}(t') \, F_\varepsilon ^2 (\hat{u}_N(t')) (\partial _x^3 \hat{u}_N(t')) \, \partial _x (I-\Delta )^{-1} \varphi \, \mathrm {d}x \, \mathrm {d}t'. \end{aligned}$$

By (3.12) we have that $\mathrm {d}\hat{\mathbb {P}}$-almost surely $\left\Vert \hat{u}_N - \hat{u} \right\Vert _{C(Q_T)} \rightarrow 0$ as $N \rightarrow \infty$, which, in particular, implies that, $\mathrm {d}\hat{\mathbb {P}}$-almost surely in $L^2(Q_T)$,

$$\begin{aligned} \mathbb {1}_{[0,t]} F_\varepsilon ^2(\hat{u}_N)\rightarrow \mathbb {1}_{[0,t]} F^2_\varepsilon (\hat{u}) \quad \text{ as } \quad N \rightarrow \infty . \end{aligned}$$

Since in addition from (3.12) and (3.14) we have that, $\mathrm {d}\hat{\mathbb {P}}$-almost surely in $L^2(Q_T)$,

$$\begin{aligned} \partial ^3_x \hat{u}_N \rightharpoonup \partial ^3_x \hat{u} \quad \text{ as } \quad N \rightarrow \infty , \end{aligned}$$

one easily deduces that for each $t \in [0,T]$, $\mathrm {d}\hat{\mathbb {P}}$-almost surely,

$$\begin{aligned} \left( M_N(\hat{u}_N ,t), \varphi \right) _{H^{-1}(\mathbb {T})} \rightarrow \left( M(\hat{u} , t), \varphi \right) _{H^{-1}(\mathbb {T})} \quad \text{ as } \quad N \rightarrow \infty . \end{aligned}$$

(3.16)

In addition, we have

$$\begin{aligned} \gamma _{\hat{u}_N}^2 \left( \Pi _N \partial _x \left( \sigma _k F_\varepsilon \left( \hat{u}_N \right) \right) , \varphi \right) _{H^{-1}(\mathbb {T})}^2 = \gamma _{\hat{u}_N}^2 \left( \sigma _k F_\varepsilon \left( \hat{u}_N \right) , \partial _x (I-\Delta )^{-1}\varphi \right) _{L^2(\mathbb {T})}^2, \end{aligned}$$

which combined with (3.12) (uniform convergence in (t, x)) implies that, $\mathrm {d}\hat{\mathbb {P}}$-almost surely as $N \rightarrow \infty$,

$$\begin{aligned}&\int _0^t \gamma _{\hat{u}_N}^2(t') \left( \Pi _N \partial _x \left( \sigma _k F_\varepsilon \left( \hat{u}_N(t') \right) \right) , \varphi \right) _{H^{-1}(\mathbb {T})}^2 \, \mathrm {d}t'\\&\quad \rightarrow \int _0^t \gamma _{\hat{u}_N}^2(t') \left( \sigma _k F_\varepsilon \left( \hat{u}(t') \right) , \partial _x (I-\Delta )^{-1}\varphi \right) _{L^2(\mathbb {T})}^2 \mathrm {d}t'\\&\quad = \int _0^t \gamma _{\hat{u}_N}^2(t') \left( \partial _x \left( \sigma _k F_\varepsilon \left( \hat{u}(t') \right) \right) , \varphi \right) _{H^{-1}(\mathbb {T})}^2 \, \mathrm {d}t'. \end{aligned}$$

Hence, we have in particular that $M^2_N(\hat{u}_N, t) \rightarrow M^2(\hat{u}, t)$ as $N \rightarrow \infty$ in probability. Similarly one shows that $M^3_N(\hat{u}_N, t) \rightarrow M^3(\hat{u}, t)$. Therefore, for each $t \in [0,T]$ we have that $M^i_N(\hat{u}_N, t) \rightarrow M^i(\hat{u}, t)$ in probability for $i\in \{1,2,3\}$. Moreover, for $q {:}{=} \frac{2 p}{n} > 2$, we have that

$$\begin{aligned}&\sup _{N\in \mathbb {N}} \hat{\mathbb {E}} \left|\left( M_N(\hat{u}_N ,t), \varphi \right) _{H^{-1}(\mathbb {T})} \right|^q\\&\quad = \sup _{N\in \mathbb {Z}} \mathbb {E}\left|\sum _{k \in \mathbb {Z}} \int _0^t \gamma _{u_N}^2(t') \left( \sigma _k F_\varepsilon (u_N(t')) , \partial _x (I-\Delta )^{-1}\varphi \right) _{L^2(\mathbb {T})} \mathrm {d}\beta ^k(t') \right|^q \\&\quad \leqq C \sup _{N\in \mathbb {N}} \mathbb {E}\left( \int _0^t \sum _{k=1}^\infty \gamma _{u_N}^4(t') \left( \sigma _k F_\varepsilon (u_N(t')) , \partial _x (I-\Delta )^{-1}\varphi ) \right) _{L^2(\mathbb {T})}^2 \mathrm {d}t' \right) ^{\frac{q}{2}} \\&\quad \leqq C \left( \sum _{k \in \mathbb {Z}} \left\Vert \sigma _k \right\Vert _{L^2(\mathbb {T})}^2\right) ^{\frac{q}{2}} \left\Vert \partial _x (I-\Delta )^{-1}\varphi \right\Vert _{L^\infty (\mathbb {T})}^q \left( 1+ \sup _{N\in \mathbb {N}} \mathbb {E}\left\Vert u_N \right\Vert _{L^\infty (Q_T)}^{\frac{q n}{2}} \right) \\&\,\, {\mathop {\leqq }\limits ^{(\hbox {2.2e})}} C \left( 1+ \sup _{N\in \mathbb {N}} \mathbb {E}\sup _{t' \in [0,T]} \left\Vert u_N(t') \right\Vert _{H^1(\mathbb {T})}^p\right) {\mathop {<}\limits ^{(3.10)}} \infty , \end{aligned}$$

where in the last step we have used $\frac{q n}{2} = p$, Sobolev’s inequality, and (3.8a3.8b) combined with conservation of mass. Similarly, for $q {:}{=} \frac{2 p}{n} > 2$,

$$\begin{aligned}&\sup _{N\in \mathbb {N}} \hat{\mathbb {E}} \left( \sum _{k\in \mathbb {Z}} \int _0^t \gamma _{\hat{u}_N}^2(t') \left( \partial _x\left( \sigma _k F_\varepsilon (\hat{u}_N(t'))\right) , \varphi \right) _{H^{-1}(\mathbb {T})}^2 \mathrm {d}t'\right) ^{\frac{q}{2}} \\&\quad \leqq C \sup _{N\in \mathbb {N}} \mathbb {E}\left( \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}_N}^2(t') \left( \sigma _k F_\varepsilon (u_N(t')) , \partial _x (I-\Delta )^{-1}\varphi \right) _{L^2(\mathbb {T})}^2 \mathrm {d}t'\right) ^{\frac{q}{2}} \\&\,\, {\mathop {\leqq }\limits ^{(\hbox {2.2e})}} C \left( 1+ \sup _{N\in \mathbb {N}} \mathbb {E}\sup _{t' \in [0,T]} \left\Vert u_N(t') \right\Vert _{H^1(\mathbb {T})}^p\right) {\mathop {<}\limits ^{(3.10)}} \infty , \end{aligned}$$

from which one deduces that for each $i =1,2,3$ and $t \in [0,T]$, the $M^i_N(\hat{u}_N, t)$ are uniformly integrable in $\hat{\omega } \in \hat{\Omega }$. Hence, we can pass to the limit in (3.15) to obtain

$$\begin{aligned} \hat{\mathbb {E}}\left[ \Phi \big (\hat{u}|_{[0,t']}, \hat{\beta }|_{[0,t']}\big ) \big (M^i(\hat{u}, t)- M^i(\hat{u}, t')\big )\right] = 0. \end{aligned}$$

(3.17)

In addition, using the continuity of $M^i(\hat{u}, t)$ in $\varphi$, the uniform integrability in $\hat{\Omega }$, and the fact that $\bigcup _N V_N$ is dense in $H^{-1}(\mathbb {T})$, it follows that (3.17) holds also for all $\varphi \in H^{-1}(\mathbb {T})$. Hence, for all $\varphi \in H^{-1}(\mathbb {T})$, $i = 1,2,3$, one can see that the $\hat{M}^i(\hat{u} , t)$ are continuous $\mathcal {G}_t$-martingales having finite $\frac{q}{2}$-moments, where $q {:}{=} \frac{2 p}{n}$. In particular, by Doob’s maximal inequality, they are uniformly integrable (in $t \in [0,T]$), which combined with continuity (in $t \in [0,T]$) implies that they are also $\hat{\mathcal {F}}_t$-martingales. By [33, Proposition A.1] we obtain that $\mathrm {d}\hat{\mathbb {P}}$-almost surely, for all $\varphi \in H^{-1}(\mathbb {T})$, $t \in [0,T]$,

$$\begin{aligned}&\left( \hat{u}(t), \varphi \right) _{H^{-1}(\mathbb {T})} \nonumber \\&\quad = \left( \hat{u}(0,\cdot ), \varphi \right) _{H^{-1}(\mathbb {T})}+\int _0^t \left( \partial _x \left( -F_\varepsilon ^2(\hat{u}(t')) \partial _x^3 \hat{u}(t') \right) , \varphi \right) _{H^{-1}(\mathbb {T})} \mathrm {d}t' \nonumber \\&\qquad + \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t {\gamma _{\hat{u}}^2(t')} \left( \partial _x \left( \sigma _k F_\varepsilon '(\hat{u}(t')) \partial _x(\sigma _k F_\varepsilon (\hat{u}(t')))\right) , \varphi \right) _{H^{-1}(\mathbb {T})} \mathrm {d}t' \nonumber \\&\qquad +\sum _{k\in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}(t') \left( \partial _x\left( \sigma _k F_\varepsilon (\hat{u}(t'))\right) , \varphi \right) _{H^{-1}(\mathbb {T})} \mathrm {d}\hat{\beta }^k(t') \end{aligned}$$

(3.18)

Choosing $\varphi {:}{=} (I-\Delta ) \psi$ in (3.18) for $\psi \in C^\infty (\mathbb {T})$, we obtain that, for $\mathrm {d}\hat{\mathbb {P}} \otimes \mathrm {d}t$-almost all $(\hat{\omega }, t) \in \hat{\Omega } \times [0,T]$,

$$\begin{aligned} \left( \hat{u}(t), \psi \right) _{L^2(\mathbb {T})}= & {} (u^{(0)}, \psi )_{H^{-1}(\mathbb {T})} + \int _0^t \left( F_\varepsilon ^2 (\hat{u}(t')) \partial _x^3 \hat{u}(t') ,\partial _x \psi \right) _{L^2(\mathbb {T})} \mathrm {d}t' \\&- \frac{1}{2} \int _0^t {\gamma _{\hat{u}}^2(t')} \left( \sigma _k F_\varepsilon '(\hat{u}(t')) \partial _x(\sigma _k F_\varepsilon (\hat{u}(t'))), \partial _x \psi \right) _{L^2(\mathbb {T})} \mathrm {d}t' \\&+ \sum _{k\in \mathbb {N}} \int _0^t \gamma _{\hat{u}}(t') \left( \sigma _k F_\varepsilon (\hat{u}(t')), \partial _x \psi \right) _{L^2(\mathbb {T})} \mathrm {d}\hat{\beta }^k(t'). \end{aligned}$$

By [41, Theorem 3.2] we have that $\hat{u}$ is an $\hat{\mathbb {F}}$-adapted continuous $L^2(\mathbb {T})$-valued process and therefore the above equality is satisfied $\mathrm {d}\hat{\mathbb {P}}$-almost surely, for all $t\! \in \! [0,T]$. Moreover, from the above and the fact that $\hat{u}$ satisfies Definition  3.2 (i), it follows that for all $\psi \in C^\infty (\mathbb {T})$, for almost all $(\hat{\omega }, t ) \in \hat{\Omega } \times (0,T)$, we have

$$\begin{aligned} \left( \partial _x \hat{u}(t), \psi \right) _{L^2(\mathbb {T})}= & {} \left( \partial _x u^{(0)}, \psi \right) _{L^2(\mathbb {T})} + \int _0^t {}_{H^{-{2}}(\mathbb {T})}\left\langle v^*(t'), \psi \right\rangle _{H^2(\mathbb {T})} \mathrm {d}t' \\&+ \left( M( \hat{u}, t), \psi \right) _{L^2(\mathbb {T})}, \end{aligned}$$

where

$$\begin{aligned} v^*{:}{=} \Delta \left( -F_\varepsilon ^2 (\hat{u}) \partial _x^3 \hat{u} \right) \end{aligned}$$

is a predictable $H^{-2}(\mathbb {T})$-valued process such that $v^* \in L^2((0, T) ; H^{-2}(\mathbb {T}))$ with probability one, $M(\hat{u}, \cdot )$ is an $L^2(\mathbb {T})$-valued martingale, and the duality between $H^2(\mathbb {T})$ and $H^{-2}(\mathbb {T})$ is given by means of the inner product in $L^2(\mathbb {T})$. Hence, $\partial _x \hat{u}$ also satisfies the conditions of [41, Theorem 3.2] with the choices $V = H^2(\mathbb {T})$ and $H=L^2(\mathbb {T})$. Consequently, $\partial _x \hat{u}$ is also continuous $L^2(\mathbb {T})$-valued. This finishes the proof. $\square$

4 A-Priori Estimates

In this section, we use the definitions and assumptions of §2.2.

4.1 Entropy estimate

For $r \in \mathbb {R}$, let us set

$$\begin{aligned} G_\varepsilon (r)= \int _r^\infty \int _{r'}^\infty \frac{1}{F_\varepsilon ^2 (r'')} \, \mathrm {d}r'' \mathrm {d}r' \quad \text{ and } \quad H_\varepsilon (r)= \int _r^\infty \frac{1}{F_\varepsilon (r')}\, \mathrm {d}r', \end{aligned}$$

(4.1)

where $F_\varepsilon (r)$ was introduced in (3.3). We collect some properties of $F_\varepsilon$, $G_\varepsilon$, and $H_\varepsilon$ that we will need later on.

Lemma 4.1

Let $n > 2$. Then there exists a constant $C_n < \infty$, only depending on n, such that for all $r \in \mathbb {R}$ and all $\varepsilon \in (0,1)$ we have

$$\begin{aligned} \left|\ln F_\varepsilon (r) \right| \leqq C_n \left( G_\varepsilon (r) + \left|r \right| + 1\right) . \end{aligned}$$

Proof

Suppose first that $r \geqq 0$. We have

$$\begin{aligned} \left|\ln F_\varepsilon (r) \right|&{\mathop {\leqq }\limits ^{(3.3)}} \frac{n}{4} \left|\ln \left( r^2+\varepsilon ^2\right) \right| \leqq \frac{n}{4} \left( \ln 2+ 2 \left|\ln (r+\varepsilon ) \right| \right) \\&\,\,\leqq C_n \left( r+\varepsilon + (r+\varepsilon )^{2-n} + 1\right) . \end{aligned}$$

Then, notice that

$$\begin{aligned} (r+\varepsilon )^{2-n}= (n-1)(n-2)\int _r^\infty \int _{r'}^\infty \frac{1}{(r''+\varepsilon )^n} \, \mathrm {d}r'' \, \mathrm {d}r' {\mathop {\leqq }\limits ^{(4.1)}} C_n G_\varepsilon (r), \end{aligned}$$

(4.2)

since $(r''+\varepsilon )^{-n} \leqq \left( (r'')^2+\varepsilon ^2\right) ^{-\frac{n}{2}} = F_\varepsilon ^{-2}(r'')$. This proves the inequality when $r \geqq 0$. If $r <0$, let us first consider the case $r^2+\varepsilon ^2 \geqq 1$. In this case, we have

$$\begin{aligned} \left|\ln F_\varepsilon (r) \right| {\mathop {\leqq }\limits ^{(3.3)}} \frac{n}{4} \left|\ln \left( r^2+\varepsilon ^2\right) \right| \leqq \frac{n}{4} \ln \left( \left|r \right|+\varepsilon \right) ^2 \leqq \frac{n}{2} \left( \left|r \right|+\varepsilon \right) , \end{aligned}$$

due to $\left( \left|r \right|+\varepsilon \right) ^2 \geqq r^2+\varepsilon ^2 \geqq 1$. This again shows the desired inequality. If $0 \leqq r^2+\varepsilon ^2 \leqq 1$, then

$$\begin{aligned} \left|\ln F_\varepsilon (r) \right| {\mathop {\leqq }\limits ^{(3.3)}}&\frac{n}{4} \left|\ln \varepsilon ^2 \right| \leqq C_n \varepsilon ^{2-n} \leqq C_n \int _0^\infty \int _0^\infty \frac{1}{(r''+\varepsilon )^n} \, \mathrm {d}r'' \, \mathrm {d}r'\\ {\mathop {\leqq }\limits ^{(4.1)}}&C_n G_\varepsilon (0) \leqq C_n G_\varepsilon (r), \end{aligned}$$

since $G_\varepsilon$ is decreasing. This finishes the proof. $\square$

Lemma 4.2

Let $n> 2$. Then there exists a constant $C_n < \infty$, only depending on n, such that for all $r \in \mathbb {R}$ and all $\varepsilon \in (0,1)$ we have

$$\begin{aligned} H_\varepsilon ^2(r) \leqq C_n G_\varepsilon (r). \end{aligned}$$

Proof

Let us first look at the case $r\geqq 0$. We have

$$\begin{aligned} H^2_\varepsilon (r)&{\mathop {\leqq }\limits ^{(3.3)}} 2^{\frac{n}{2}} \left( \int _r^\infty \frac{1}{(r'+\varepsilon )^{\frac{n}{2}}} \, \mathrm {d}r'\right) ^2 = \frac{2^{\frac{n}{2} + 2}}{(n-2)^2} (r+\varepsilon )^{2-n} \nonumber \\&\,\,\leqq 2^{\frac{n}{2} + 2} \frac{n-1}{n-2} \int _r^\infty \int _{r'}^\infty \frac{1}{(r''+\varepsilon )^n} \, \mathrm {d}r'' \, \mathrm {d}r' \nonumber \\&{\mathop {\leqq }\limits ^{(3.3)}} 2^{\frac{n}{2} + 2} \frac{n-1}{n-2} G_\varepsilon (r), \end{aligned}$$

(4.3)

where for the last inequality we have used (4.2). Hence, we only have to check the case $r<0$. In this case we have

$$\begin{aligned} H_\varepsilon (r) = \int _r^0 \frac{1}{F_\varepsilon (r')} \, \mathrm {d}r'+ H_\varepsilon (0) {\mathop {\leqq }\limits ^{(3.3)}} 2 H_\varepsilon (0) \end{aligned}$$

since $F_\varepsilon$ is even. Therefore,

$$\begin{aligned} H_\varepsilon ^2(r) \leqq 4 H_\varepsilon ^2(0) \leqq 2^{\frac{n}{2} + 4} \frac{n-1}{n-2} G_\varepsilon (r), \end{aligned}$$

where we have used (4.3) and the fact that $G_\varepsilon$ is decreasing. This finishes the proof. $\square$

Lemma 4.3

(Entropy Estimate) Suppose that $n \in (2,4]$, $T \in (0,\infty )$, $p \geqq 1$, and $u^{(0)} \in L^p\left( \Omega ;\mathcal {F}_0,\mathbb {P};H^1(\mathbb {T})\right)$. For a weak solution of problem (3.8) in the sense of Definition 3.2 it holds that

$$\begin{aligned}&\hat{\mathbb {E}} \sup _{t \in [0,T]} \left\Vert G_\varepsilon \left( \hat{u}_{\varepsilon ,R}(t)\right) \right\Vert _{L^1(\mathbb {T})}^p + \hat{\mathbb {E}} \left\Vert \partial ^2_x \hat{u}_{\varepsilon ,R} \right\Vert _{L^2(Q_T)}^{2p} \nonumber \\&\quad \leqq C \, \hat{\mathbb {E}} \left( \left\Vert G_\varepsilon \left( \hat{u}^{(0)}\right) \right\Vert _{L^1(\mathbb {T})}^p + \left| \mathcal {A}\left( {\hat{u}}^{(0)}\right) \right|^{2p}+1\right) , \end{aligned}$$

(4.4)

where $C< \infty$ is a constant depending only on p, $\sigma = \left( \sigma _k\right) _{k \in \mathbb {Z}}$, and T.

Proof

For the convenience of the reader, we simply write $\hat{u}$ instead of $\hat{u}_{\varepsilon ,R}$. By Itô’s formula [42] we have

$$\begin{aligned} \int _{\mathbb {T}} G_\varepsilon \left( \hat{u}(t)\right) \, \mathrm {d}x {\mathop {=}\limits ^{(\hbox {3.8a})}}&\int _{\mathbb {T}} G_\varepsilon \left( \hat{u}^{(0)}\right) \mathrm {d}x + \int _0^t \int _{\mathbb {T}} G_\varepsilon ''( \hat{u}) F_\varepsilon ^2(\hat{u}) (\partial ^3_x \hat{u}) \, \partial _x \hat{u} \, \mathrm {d}x \, \mathrm {d}t' \\&+ \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} G_\varepsilon '\left( \hat{u}\right) \partial _x \left( \sigma _k F_\varepsilon ' (\hat{u}) \, \partial _x (\sigma _k F_\varepsilon (\hat{u}))\right) \mathrm {d}x \, \mathrm {d}t' \\&+ \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} G_\varepsilon ''(\hat{u}) \left( \partial _x (\sigma _k F_\varepsilon (\hat{u}))\right) ^2 \mathrm {d}x \, \mathrm {d}t' \\&+ \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}} \int _{\mathbb {T}} G_\varepsilon '(\hat{u}) \left( \partial _x (\sigma _k F_\varepsilon (\hat{u}))\right) \mathrm {d}x \, \mathrm {d}\hat{\beta }^k(t'), \end{aligned}$$

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, where $\gamma _{\hat{u}}(t') {:}{=} g_R\left( \left\Vert \hat{u}(t') \right\Vert _{L^\infty (\mathbb {T})}\right)$, so that after integration by parts we get

$$\begin{aligned} \int _{\mathbb {T}} G_\varepsilon \left( \hat{u}(t)\right) \, \mathrm {d}x {\mathop {=}\limits ^{(4.1)}}&\int _{\mathbb {T}} G_\varepsilon \left( \hat{u}^{(0)}\right) \mathrm {d}x - \int _0^t \int _{\mathbb {T}} (\partial ^2_x \hat{u})^2 \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&+ \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} (\partial _x \sigma _k) \, F_\varepsilon ^{-1}(\hat{u}) \left( \partial _x (\sigma _k F_\varepsilon (\hat{u}))\right) \mathrm {d}x \, \mathrm {d}t' \nonumber \\&- \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}}\sigma _k F_\varepsilon ^{-1}(\hat{u}) \, (\partial _x \hat{u}) \, \mathrm {d}x \, \mathrm {d}\hat{\beta }^k(t'), \end{aligned}$$

(4.5)

$\mathrm {d}\hat{\mathbb {P}}$-almost surely. Then we have for all $\delta >0$

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _{\mathbb {T}} (\partial _x \sigma _k) \, F_\varepsilon ^{-1}(\hat{u}) \left( \partial _x (\sigma _k F_\varepsilon (\hat{u}))\right) \mathrm {d}x \\&\quad = \sum _{k \in \mathbb {Z}} \int _{\mathbb {T}} (\partial _x \sigma _k)^2 \, \mathrm {d}x - \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _{\mathbb {T}} (\partial ^2_x\sigma _k^2) \, \ln F_\varepsilon (\hat{u}) \, \mathrm {d}x \\&\quad {\mathop {\leqq }\limits ^{(2.2)}} C_{\sigma ,\delta } \left( 1 + \left\Vert G_\varepsilon (\hat{u}) \right\Vert _{L^1(\mathbb {T})}\right) + \delta \left\Vert \hat{u} \right\Vert _{L^2(\mathbb {T})}^2, \end{aligned}$$

where for the last inequality we have used Lemma 4.1. Moreover, since $\partial _x \hat{u}$ has zero average, we get from Poincaré’s inequality using conservation of mass (cf. Remark 3.3),

$$\begin{aligned} \left\Vert \hat{u}(t) \right\Vert _{L^2(\mathbb {T})}&\leqq C_L \left( \left\Vert \partial _x \hat{u}(t) \right\Vert _{L^2(\mathbb {T})}+ \left|\int _{\mathbb {T}} \hat{u}(t) \, \mathrm {d}x \right|\right) \\&\leqq C_L \left( \left\Vert \partial _x^2 \hat{u}(t) \right\Vert _{L^2(\mathbb {T})}+ \left| \mathcal {A}({\hat{u}}^{(0)}) \right|\right) . \end{aligned}$$

Consequently, for any $\delta > 0$ we have

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _{\mathbb {T}} (\partial _x \sigma _k) \, F_\varepsilon ^{-1}(\hat{u}) \left( \partial _x (\sigma _k F_\varepsilon (\hat{u}))\right) \mathrm {d}x\\&\quad \leqq C_{L,\sigma ,\delta } \left( 1 + \left| \mathcal {A}({\hat{u}}^{(0)}) \right|^2 + \left\Vert G_\varepsilon (\hat{u}(t)) \right\Vert _{L^1(\mathbb {T})}\right) + \delta \left\Vert \partial _x^2 u \right\Vert _{L^2(\mathbb {T})}^2 , \end{aligned}$$

Using this in (4.5), choosing $\delta > 0$ small, rearranging, and taking the p-th power gives

$$\begin{aligned} \left\Vert G_\varepsilon (\hat{u}(t)) \right\Vert _{L^1(\mathbb {T})}^p + \left\Vert \partial ^2_x \hat{u} \right\Vert _{L^2(Q_t)}^{2p}\leqq & {} C_{L,T,\sigma ,p} \left( \left\Vert G_\varepsilon \left( \hat{u}^{(0)}\right) \right\Vert _{L^1(\mathbb {T})}^p+ \left|\mathcal {A}\left( \hat{u}^{(0)}\right) \right|^{2p}+1\right) \\&+ C_p \int _0^t \left\Vert G_\varepsilon (\hat{u}(t')) \right\Vert _{L^1(\mathbb {T})}^p \, \mathrm {d}t' + C_p \left|M(t) \right|^p, \end{aligned}$$

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, where

$$\begin{aligned} M(t) {:}{=} - \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}}\sigma _k F_\varepsilon ^{-1}(\hat{u}) \, (\partial _x \hat{u}) \, \mathrm {d}x \, \mathrm {d}\hat{\beta }^k(t') \end{aligned}$$

is the martingale from (4.5). Notice that $G_\varepsilon ( \hat{u} (t))$ is a continuous $L^1(\mathbb {T})$-valued process and let us set

$$\begin{aligned} \tau _m= \inf \{ t>0 : \Vert G_\varepsilon ( \hat{u}(t)) \Vert _{L^1(\mathbb {T})} + \int _0^t \Vert \partial ^2_x \hat{u} \Vert ^2_{L^2(\mathbb {T})} \, \mathrm {d}t' > m \}\wedge T . \end{aligned}$$

Taking suprema up to $\tau _m \wedge t'$, for $t' \in [0,T]$, in the above inequality and expectation, we obtain by virtue of the Burkholder-Davis-Gundy inequality

$$\begin{aligned}&\hat{\mathbb {E}} \sup _{t \in [0, t']} \left\Vert G_\varepsilon (\hat{u}(t\wedge \tau _m)) \right\Vert _{L^1(\mathbb {T})}^p + \hat{\mathbb {E}} \left\Vert \partial ^2_x \hat{u} \right\Vert _{L^2(Q_{t' \wedge \tau _m})}^{2p}\nonumber \\&\quad \leqq C_{L,\sigma ,p} \, \hat{\mathbb {E}} \left( \left\Vert G_\varepsilon \left( \hat{u}^{(0)}\right) \right\Vert _{L^1(\mathbb {T})}^p+ \left| \mathcal {A}({\hat{u}}^{(0)}) \right|^{2p}+1\right) \nonumber \\&\qquad + C_p \, \hat{\mathbb {E}} \int _0^{t'} \left\Vert G_\varepsilon (\hat{u}(t'' \wedge \tau _m)) \right\Vert _{L^1(\mathbb {T})}^p \, \mathrm {d}t'' + C_p \, \hat{\mathbb {E}} \langle M \rangle _{t' \wedge \tau _m} ^{\frac{p}{2}}. \end{aligned}$$

(4.6)

Next, by integration by parts, we have

$$\begin{aligned} \hat{\mathbb {E}} \langle M \rangle _{t' \wedge \tau _m}^{\frac{p}{2}}&= \hat{\mathbb {E}} \left( \sum _{k \in \mathbb {Z}} \int _0^{t' \wedge \tau _m} \gamma _{\hat{u}}^2 \left( \int _{\mathbb {T}} (\partial _x\sigma _k) \, H_\varepsilon (\hat{u}) \, \mathrm {d}x \right) ^2 \, \mathrm {d}t'' \right) ^{\frac{p}{2}} \\&{\mathop {\leqq }\limits ^{(2.2)}} C_\sigma \, \hat{\mathbb {E}} \left( \sum _{k \in \mathbb {Z}} \int _0^{t' \wedge \tau _m} \int _{\mathbb {T}} \left( H_\varepsilon (\hat{u})\right) ^2 \mathrm {d}x \, \mathrm {d}t'' \right) ^{\frac{p}{2}}. \end{aligned}$$

By Lemma 4.2 we see that

$$\begin{aligned} \hat{\mathbb {E}} \langle M \rangle _{t' \wedge \tau _m}^{\frac{p}{2}}&\leqq C_{\sigma ,n} \, \hat{\mathbb {E}} \left( \int _0^{t' \wedge \tau _m} \left\Vert G_\varepsilon (\hat{u}(t'')) \right\Vert _{L^1(\mathbb {T})} \, \mathrm {d}t''\right) ^{\frac{p}{2}} \\&\leqq C_{\sigma ,n,p,T} \left( 1 + \hat{\mathbb {E}} \int _0^{t'} \left\Vert G_\varepsilon (\hat{u}(t'' \wedge \tau _m)) \right\Vert _{L^1(\mathbb {T})}^p \, \mathrm {d}t''\right) . \end{aligned}$$

Using this and rearranging in (4.6), we have the desired inequality by virtue of Grönwall’s inequality and Fatou’s lemma. $\square$

4.2 Uniform energy estimate

The following auxiliary result is convenient for deriving an energy estimate.

Lemma 4.4

For $\varepsilon \in (0,1)$, $n \in (2,\infty )$, and $r \in \mathbb {R}$ we have

$$\begin{aligned} \left|\int _1^r (F_\varepsilon '')^2(r') \, \mathrm {d}r' \right| \leqq C_n {\left\{ \begin{array}{ll} 1 + \left|r \right|^{n-3} &{} \text { if } n>3, \\ C_{\vartheta }\left( 1 + \left|r \right|^{\vartheta } + G_\varepsilon ^{{\vartheta }}(r)\right) &{} \text { if } n=3 \\ 1 + G_\varepsilon ^{\frac{3-n}{n-2}}(r) &{} \text{ if } n \in \left( 2, 3\right) . \end{array}\right. } \end{aligned}$$

(4.7)

for any ${\vartheta } > 0$, where $C_n < \infty$ only depends on n and $C_{\vartheta }$ only depends on ${\vartheta }$.

Proof

First note that

$$\begin{aligned}&F_\varepsilon '(r){\mathop {=}\limits ^{(3.3)}} \tfrac{n}{2} r (r^2+\varepsilon ^2)^{\frac{n}{4} -1},\\&F_\varepsilon ''(r) \,= \tfrac{n}{2} (r^2+\varepsilon ^2)^{\frac{n}{4} -1} +n( \tfrac{n}{4} -1) r^2 (r^2+\varepsilon ^2)^{\frac{n}{4} -2}, \end{aligned}$$

so that because of $\frac{r^2}{r^2+\varepsilon ^2} \leqq 1$ we have

$$\begin{aligned} (F_\varepsilon '')^2(r) \leqq C_n (r^2+\varepsilon ^2)^{\frac{n}{2} -2}. \end{aligned}$$

This implies

$$\begin{aligned} \left|\int _1^r (F_\varepsilon '')^2(r') \, \mathrm {d}r' \right|&\leqq C_n \int _1^r (r')^{n-4} \, \mathrm {d}r' \leqq C_n {\left\{ \begin{array}{ll} 1+ r^{n-3} &{} \text { if } n \ne 3,\\ \ln r &{} \text { if } n=3, \end{array}\right. } \quad \text{ for } r \geqq 1, \\ \left|\int _1^r (F_\varepsilon '')^2(r') \, \mathrm {d}r' \right|&\leqq C_n \int _{r }^1 (r')^{n-4} \, \mathrm {d}r' \leqq C_n {\left\{ \begin{array}{ll} 1 + r^{n-3} &{} \text { if } n\ne 3,\\ - \ln r &{} \text { if } n=3, \end{array}\right. } \quad \text{ for } \varepsilon \leqq r< 1, \\ \left|\int _1^r (F_\varepsilon '')^2(r') \, \mathrm {d}r' \right|&\leqq C_n \left( \int _\varepsilon ^1 (r')^{n-4} \, \mathrm {d}r' + \int _{-\varepsilon }^\varepsilon \varepsilon ^{n-4} \, \mathrm {d}r' \right) \\&\leqq C_n {\left\{ \begin{array}{ll} 1+\varepsilon ^{n-3} &{} \text { if } n\ne 3,\\ -\ln \varepsilon + 2 &{} \text { if } n=3,\end{array}\right. } \quad \text{ for } -\varepsilon \leqq r< \varepsilon , \\ \left|\int _1^r (F_\varepsilon '')^2(r') \, \mathrm {d}r' \right|&\leqq C_n \left( \int _\varepsilon ^1 (r')^{n-4} \, \mathrm {d}r' + \int _{-\varepsilon }^\varepsilon \varepsilon ^{n-4} \, \mathrm {d}r' + \int _{\varepsilon }^{-r} (r')^{n-4} \, \mathrm {d}r'\right) \\&\leqq C_n {\left\{ \begin{array}{ll} 1 + \varepsilon ^{n-3} + \left|r \right|^{n-3} &{} \text { if } n \ne 3,\\ -2\ln \varepsilon +2+\ln (-r) &{} \text { if } n=3, \end{array}\right. } \quad \text{ for } r < -\varepsilon , \end{aligned}$$

so that, because of $\varepsilon \in (0,1)$,

$$\begin{aligned} \left|\int _1^r \left( F_\varepsilon ''(r')\right) ^2 \mathrm {d}r' \right| \leqq C_n {\left\{ \begin{array}{ll} \left|r \right|^{n-3} \mathbb {1}_{\left\{ \left|r \right| \geqq \varepsilon \right\} } + 1 &{} \text { if } n > 3, \\ \left|\ln \left|r \right| \right| \mathbb {1}_{\left\{ \left|r \right| \geqq \varepsilon \right\} } + (1-\ln \varepsilon ) \mathbb {1}_{\left\{ r< \varepsilon \right\} } &{} \text { if } n=3,\\ \left( 1+\left|r \right|^{n-3}\right) \mathbb {1}_{\left\{ r \geqq \varepsilon \right\} } + \left( 1+\varepsilon ^{n-3}\right) \mathbb {1}_{\left\{ r< \varepsilon \right\} } &{} \text { if } n< 3. \end{array}\right. } \end{aligned}$$

Furthermore, notice that

$$\begin{aligned} G_\varepsilon (r) {\mathop {=}\limits ^{(4.1)}} \int _r^\infty \int _{r'}^\infty \frac{\mathrm {d}r' \, \mathrm {d}r''}{\left( (r'')^2 + \varepsilon ^2\right) ^{\frac{n}{2}}} \geqq 2^{- \frac{n}{2}} \int _r^\infty \int _{r'}^\infty \frac{\mathrm {d}r'' \, \mathrm {d}r'}{(r'')^n} \geqq C_n r^{2-n} \quad \end{aligned}$$

$\text{ for } r \geqq \varepsilon$ and

$$\begin{aligned} G_\varepsilon (r)= & {} \varepsilon ^{2-n} \int _{\frac{r}{\varepsilon }}^\infty \int _{r'}^\infty \frac{\mathrm {d}r' \, \mathrm {d}r''}{\left( (r'')^2 + 1\right) ^{\frac{n}{2}}} > \varepsilon ^{2-n} \int _1^\infty \int _{r'}^\infty \frac{\mathrm {d}r' \, \mathrm {d}r''}{\left( (r'')^2 + 1\right) ^{\frac{n}{2}}} \\\geqq & {} C_n \varepsilon ^{2-n} \quad \text{ for } r < \varepsilon , \end{aligned}$$

so that we may infer that (4.7) holds true. $\square$

Lemma 4.5

For $\varepsilon \in (0,1)$, $n \in (2,\infty )$, and $r \in \mathbb {R}$ we have

$$\begin{aligned} \left|(F_\varepsilon ^2)'''(r) + 4 \left( (F_\varepsilon ')^2\right) '(r) \right| \leqq C_n {\left\{ \begin{array}{ll} 1 + \left|r \right|^{n-3} &{} \text{ if } n \in [3,\infty ), \\ G_\varepsilon ^{\frac{3-n}{n-2}}(r) &{} \text{ if } n \in \left[ \frac{5}{2},3\right) , \end{array}\right. } \end{aligned}$$

(4.8)

where $C_n < \infty$ only depends on n.

Proof

We compute

$$\begin{aligned} (F_\varepsilon ')^2(r) {\mathop {=}\limits ^{(3.3)}} \tfrac{n^2}{4} \, r^2 \, (r^2+\varepsilon ^2)^{\frac{n}{2} -2} \end{aligned}$$

which implies that

$$\begin{aligned} \left( (F_\varepsilon ')^2\right) '(r) = \tfrac{n^2}{2} \, r \, (r^2+\varepsilon ^2)^{\tfrac{n}{2} -2} + (\tfrac{n^3}{4}-n^2) \, r^3 \, (r^2 + \varepsilon ^2)^{\frac{n}{2} -3}, \end{aligned}$$

so that

$$\begin{aligned} \left| \left( (F_\varepsilon ')^2\right) '(r) \right| \leqq C_n r(r^2+\varepsilon ^2)^{\frac{n}{ 2}-2} \leqq C_n (r^2+\varepsilon ^2)^{\frac{n-3}{2}}. \end{aligned}$$

Furthermore,

$$\begin{aligned} (F_\varepsilon ^2)'(r) = n r \, (r^2+\varepsilon ^2)^{\frac{n}{2} -1} \end{aligned}$$

which gives

$$\begin{aligned} (F_\varepsilon ^2)''(r) = n \, (r^2+\varepsilon ^2)^{\frac{n}{2} -1} + n(n-2) \, r^2 \, (r^2+\varepsilon ^2)^{\frac{n}{2} -2} \end{aligned}$$

and

$$\begin{aligned} (F_\varepsilon ^2)'''(r) = 3n(n-2) \, r \, (r^2+\varepsilon ^2)^{\frac{n}{2} -2} +n(n-2)(n-4) \, r^3 \, (r^2+\varepsilon ^2)^{\frac{n}{2} -3}, \end{aligned}$$

whence

$$\begin{aligned} \left|(F_\varepsilon ^2)'''(r) \right| \leqq C_n (r^2+\varepsilon ^2)^{\frac{n-3}{2}}. \end{aligned}$$

Because of $(r^2+\varepsilon ^2)^{\frac{n-3}{2}} \leqq C_n \left( 1 + \left|r \right|^{n-3}\right)$, the first part of (4.8) is immediate. For $n \in \left( \frac{5}{2},3\right)$ we use

$$\begin{aligned} (r^2+\varepsilon ^2)^{\frac{2-n}{2}}&= \left( n-2\right) \int _r^\infty r' \left( (r')^2+\varepsilon ^2\right) ^{-\frac{n}{2}} \mathrm {d}r' \leqq \left( n-2\right) \int _r^\infty \left( (r')^2+\varepsilon ^2\right) ^{\frac{1-n}{2}} \mathrm {d}r' \\&= (n-2) \, (n-1) \int _r^\infty \int _{r'}^\infty r'' \left( (r'')^2+\varepsilon ^2\right) ^{-\frac{n+1}{2}} \mathrm {d}r'' \, \mathrm {d}r' \\&{\mathop {\leqq }\limits ^{(3.3)}} (n-2) \, (n-1) \int _r^\infty \int _{r'}^\infty \frac{1}{F_\varepsilon ^2(r'')} \, \mathrm {d}r'' \, \mathrm {d}r' {\mathop {=}\limits ^{(4.1)}} (n-2) \, (n-1) \, G_\varepsilon (r), \end{aligned}$$

so that (4.8) also holds true in this parameter range, too. $\square$

Lemma 4.6

Suppose $n \in \left[ \frac{8}{3}, 4\right)$, $T \in (0,\infty )$, $p \geqq 1$,

$$u^{(0)} \in L^p\left( \Omega ;\mathcal {F}_0,\mathbb {P};H^1(\mathbb {T})\right) ,$$

$\varepsilon \in (0,1]$, and let $q>1$ such that $q \geqq \max \left\{ \frac{1}{4-n},\frac{n-2}{2n-5}\right\}$. Then, for any weak solution of problem (3.8) in the sense of Definition 3.2 it holds

$$\begin{aligned}&\hat{\mathbb {E}} \left[ \sup _{t \in [0,T]} \left\Vert \partial _x \hat{u}_{\varepsilon ,R}(t) \right\Vert _{L^2(\mathbb {T})}^p + \left\Vert F_\varepsilon (\hat{u}_{\varepsilon ,R}) \, \partial _x^3 \hat{u}_{\varepsilon ,R} \right\Vert _{L^2(Q_T)}^p\right] \nonumber \\&\quad \leqq C \ \hat{\mathbb {E}} \left( 1 +| \mathcal {A}(\hat{u}_{\varepsilon , R}^{(0)})|^{\frac{np}{2}}+ \left\Vert \partial _x \hat{u}^{(0)} \right\Vert _{L^2(\mathbb {T})}^p\right) \nonumber \\&\qquad + C \left( \hat{\mathbb {E}} \sup _{t \in [0,T]} \left\Vert G_\varepsilon (\hat{u}_{\varepsilon ,R}(t)) \right\Vert _{L^1(\mathbb {T})}^{p} \mathbb {1}_{\left[ \frac{8}{3},3\right] }(n) + \left\Vert \partial _x^2 \hat{u}_{\varepsilon ,R} \right\Vert _{L^2(Q_T)}^{2pq}\right) , \end{aligned}$$

(4.9)

where $C< \infty$ is a constant depending only on p, q, $\sigma = (\sigma _k)_{k \in \mathbb {Z}}$, n, L, and T.

Proof

For convenience of the reader, we write $\hat{u}$ instead of $\hat{u}_{\varepsilon ,R}$. By Itô’s formula (see, for example, [42]) we have

$$\begin{aligned} \frac{1}{2} \left\Vert \partial _x \hat{u}(t) \right\Vert _{L^2(\mathbb {T})}^2 {\mathop {=}\limits ^{(\hbox {3.8a})}}&\frac{1}{2} \left\Vert \partial _x \hat{u}^{(0)} \right\Vert _{L^2(\mathbb {T})}^2 - \int _0^t \int _{\mathbb {T}} F_\varepsilon ^2(\hat{u}) (\partial _x^3 \hat{u})^2 \,\mathrm {d}x \, \mathrm {d}t' \\&+ \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \frac{1}{2}(\partial _x \hat{u}) \, \partial _x^2 (\sigma _k F_\varepsilon '(\hat{u}) \partial _x (\sigma _k F_\varepsilon (\hat{u}))) \, \mathrm {d}x \, \mathrm {d}t' \\&+ \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t\gamma _{\hat{u}}^2 \int _{\mathbb {T}} \left( \partial _x^2 (\sigma _k F_\varepsilon (\hat{u}))\right) ^2 \mathrm {d}x \, \mathrm {d}t' \\&+ \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}} \int _{\mathbb {T}} (\partial _x \hat{u}) \, \partial _x^2(\sigma _k F_\varepsilon (\hat{u})) \, \mathrm {d}x \, \mathrm {d}\hat{\beta }^k(t'), \end{aligned}$$

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, where we write $\gamma _{\hat{u}}(t) {:}{=} g_R\left( \left\Vert \hat{u}(t) \right\Vert _{L^\infty (\mathbb {T})}\right)$. The same reasoning as in the proof of Lemma 3.1 leads to

$$\begin{aligned}&\frac{1}{2} \left\Vert \partial _x \hat{u}(t) \right\Vert _{L^2(\mathbb {T})}^2\nonumber \\&\quad = \frac{1}{2} \left\Vert \partial _x \hat{u}^{(0)} \right\Vert _{L^2(\mathbb {T})}^2 - \int _0^t \int _{\mathbb {T}} F_\varepsilon ^2(\hat{u}) \, (\partial _x^3 \hat{u})^2 \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\qquad +\frac{1}{6} \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \sigma _k^2 \, (F_\varepsilon '')^2(\hat{u}) \, (\partial _x \hat{u})^4 \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\qquad + \frac{1}{16} \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \left( \partial _x (\sigma _k^2)\right) \left( (F_\varepsilon ^2)'''(\hat{u}) + 4 \left( (F_\varepsilon ')^2\right) '(\hat{u})\right) (\partial _x \hat{u})^3 \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\qquad + \frac{3}{16} \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \left( 8 \left( (\partial _x \sigma _k)^2 - \sigma _k (\partial _x^2 \sigma _k)\right) (F_\varepsilon ')^2(\hat{u}) \right. \nonumber \\&\qquad \qquad \left. + \left( \partial _x^2 (\sigma _k^2)\right) (F_\varepsilon ^2)''(\hat{u})\right) (\partial _x \hat{u})^2 \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\qquad + \frac{1}{8} \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \left( 4 \sigma _k \, \partial _x^4 \sigma _k - \partial _x^4 (\sigma _k^2)\right) F_\varepsilon ^2(\hat{u}) \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\qquad + \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}} \int _{\mathbb {T}} \sigma _k \, F_\varepsilon (\hat{u}) \, \partial _x^3 \hat{u} \, \mathrm {d}x \, \mathrm {d}\hat{\beta }^k(t'), \end{aligned}$$

(4.10)

$\mathrm {d}\hat{\mathbb {P}}$-almost surely.

$(\partial _x \hat{u})^4$-term We first focus on estimating the term

$$\sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \sigma _k^2 \, (F_\varepsilon '')^2(\hat{u}) \, (\partial _x \hat{u})^4 \mathrm {d}x \, \mathrm {d}t'$$

and note that through integration by parts we have

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \sigma _k^2 \, (F_\varepsilon '')^2(\hat{u}) \, (\partial _x \hat{u})^4 \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\quad {\mathop {\leqq }\limits ^{(2.2)}} C_\sigma \int _0^t \int _{\mathbb {T}} (F_\varepsilon '')^2(\hat{u}) \, (\partial _x \hat{u})^4 \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\quad = - 3 C_\sigma \int _0^t \int _{\mathbb {T}} \left( \int _1^{\hat{u}} (F_\varepsilon '')^2(r) \, \mathrm {d}r\right) (\partial _x \hat{u})^2 \, (\partial _x^2 \hat{u}) \, \mathrm {d}x \, \mathrm {d}t'. \end{aligned}$$

(4.11)

With help of the Cauchy-Schwarz and Hölder’s inequality, we have

$$\begin{aligned}&- 3 \int _{\mathbb {T}} \left( \int _1^{\hat{u}} \left( F_\varepsilon ''(r)\right) ^2 \mathrm {d}r\right) (\partial _x \hat{u})^2 \, (\partial _x^2 \hat{u}) \, \mathrm {d}x \\&\quad \leqq 3 \left( \int _{\mathbb {T}} \left( \int _1^{\hat{u}} \left( F_\varepsilon ''(r)\right) ^2 \mathrm {d}r\right) ^2 \mathrm {d}x\right) ^{\frac{1}{2}} \left\Vert \partial _x \hat{u} \right\Vert _{L^\infty (\mathbb {T})}^2 \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(\mathbb {T})}. \end{aligned}$$

Next, since $\partial _x \hat{u}$ has zero average, we use that

$$\begin{aligned} \left\Vert \partial _x \hat{u} \right\Vert _{L^\infty (\mathbb {T})} \leqq C \left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})}^{\frac{1}{2}} \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(\mathbb {T})}^{\frac{1}{2}}. \end{aligned}$$

(4.12)

In the case $n \in \left[ \frac{8}{3},3\right]$, we deduce, with help of Lemma 4.4, that

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _{\mathbb {T}} \sigma _k^2 (F_\varepsilon '')^2(\hat{u}) \, (\partial _x \hat{u})^4 \, \mathrm {d}x \\&\!\!\! {\mathop {\leqq }\limits ^{(4.11), (4.12)}} C_\sigma \left( \int _{\mathbb {T}} \left( \int _1^{\hat{u}} (F_\varepsilon '')^2(r) \, \mathrm {d}r\right) ^2 \mathrm {d}x\right) ^{\frac{1}{2}} \left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})} \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(\mathbb {T})}^2 \\&\quad {\mathop {\leqq }\limits ^{(4.7)}} C_{n,{\vartheta }} \left( 1 + \left\Vert \hat{u} \right\Vert _{L^1(\mathbb {T})} + \left\Vert G_\varepsilon (\hat{u}) \right\Vert _{L^1(\mathbb {T})}\right) ^{\max \left\{ \frac{3-n}{n-2}, {\vartheta }\right\} } \left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})} \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(\mathbb {T})}^2 \\&\quad \quad \leqq \! C_{n,{\vartheta }, L} \left( 1 \!+\! \left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})}\!+\! | \mathcal {A}(\hat{u}^{(0)})| \!+\! \left\Vert G_\varepsilon (\hat{u}) \right\Vert _{L^1(\mathbb {T})}\right) ^{\max \left\{ \frac{3-n}{n-2}, {\vartheta }\right\} } \left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})} \\&\quad \qquad \times \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(\mathbb {T})}^2, \end{aligned}$$

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, where ${\vartheta } > 0$ and we have applied conservation of mass (cf. Remark 3.3) and the second Poincaré inequality. Integration in time yields for any ${\vartheta } > 0$,

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \sigma _k^2 (F_\varepsilon '')^2(\hat{u}) \, (\partial _x \hat{u})^4 \, \mathrm {d}x \, \mathrm {d}t' \\&\quad \leqq C_{\sigma ,n,{\vartheta }, L } \left( 1 + \sup _{t' \in [0,t]} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})} + | \mathcal {A}(\hat{u}^{(0)})| + \sup _{t' \in [0,t]} \left\Vert G_\varepsilon (\hat{u}(t')) \right\Vert _{L^1(\mathbb {T})}\right) ^{\max \left\{ \frac{3-n}{n-2}, {\vartheta }\right\} } \\&\qquad \times \sup _{t' \in [0,t]} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})} \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^2, \end{aligned}$$

$\mathrm {d}\hat{\mathbb {P}}$-almost surely. Applying Young’s inequality and confining ${\vartheta }$ to the interval (0, 1) yields for any $\delta > 0$

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \sigma _k^2 (F_\varepsilon '')^2(\hat{u}) \, (\partial _x \hat{u})^4 \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\quad \leqq \delta \sup _{t' \in [0,t]} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^2 \nonumber \\&\qquad + C_{\sigma ,n,{\vartheta },\delta } \left( 1 + | \mathcal {A}(\hat{u}^{(0)})| ^2 + \sup _{t' \in [0,t]} \left\Vert G_\varepsilon (\hat{u}(t')) \right\Vert _{L^1(\mathbb {T})}^2 + \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^{\max \left\{ \frac{4(n-2)}{2n-5}, \frac{4}{1-{\vartheta }}\right\} }\right) ,\nonumber \\ \end{aligned}$$

(4.13)

$\mathrm {d}\hat{\mathbb {P}}$-almost surely.

In the case $n\in (3,4)$ we get with Lemma 4.4 and Hölder’s inequality

$$\begin{aligned}&\left|\int _{\mathbb {T}}\left( \int _1^{\hat{u}} (F_\varepsilon ''(r))^2 \mathrm {d}r\right) (\partial _x \hat{u})^2 \, (\partial _x^2 \hat{u}) \, \mathrm {d}x \right| \\&\quad \,\,{\mathop {\leqq }\limits ^{(4.7)}} C_n \left( {\int _{\mathbb {T}}}\left( 1+\left|\hat{u} \right|^{2n-6}\right) \mathrm {d}x\right) ^{\frac{1}{2}} \left\Vert \partial _x \hat{u} \right\Vert _{L^{\infty }(\mathbb {T})}^2 \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(\mathbb {T})}\\&\quad \, {\mathop {\leqq }\limits ^{(4.12)}} C_{n,L} \left( 1 + \left\Vert \hat{u} \right\Vert _{L^\infty (\mathbb {T})}^{n-3}\right) \left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})}\left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(\mathbb {T})}^2 \\&\quad \quad \leqq C_{n,L} \left( 1+\left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})}^{n-3}+ | \mathcal {A}(\hat{u}^{(0)})| ^{n-3}\right) \left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})}\left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(\mathbb {T})}^2,\qquad \end{aligned}$$

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, where we have employed mass conservation (Remark 3.3) and the Sobolev embedding in the last line. Hence,

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \sigma _k^2 (F_\varepsilon '')^2(\hat{u}) \, (\partial _x \hat{u})^4 \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\quad \,\, {\mathop {\leqq }\limits ^{(4.11)}} C_{n,L} \left( 1+ \sup _{t' \in [0,t]} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^{n-2} +|\mathcal {A}(\hat{u}^{(0)})| ^{n-2}\right) \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^2 \nonumber \\&\quad \quad \,\leqq \delta \sup _{t' \in [0,t]} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^2 + C_{n,L,\delta } \left( 1+ |\mathcal {A}(\hat{u}^{(0)})| ^2+ \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^{\frac{4}{4-n}}\right) , \nonumber \\ \end{aligned}$$

(4.14)

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, where we have used Young’s inequality in the last estimate. Combining (4.13) and (4.14), we end up with the bound

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \sigma _k^2 (F_\varepsilon '')^2(\hat{u}) \, (\partial _x \hat{u})^4 \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\quad \leqq \delta \sup _{t' \in [0,T]} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^2 \nonumber \\&\qquad + C_{\sigma ,n,\delta , {\vartheta } } \left( 1 +|\mathcal {A}(\hat{u}^{(0)})| ^2+ \sup _{t' \in [0,t]} \left\Vert G_\varepsilon (\hat{u}(t')) \right\Vert _{L^1(\mathbb {T})}^2 \mathbb {1}_{\left[ \frac{8}{3},3\right] }(n)\right) \nonumber \\&\qquad + C_{\sigma ,n,\delta , {\vartheta } } \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^{\max \left\{ \frac{4}{4-n},\frac{4(n-2)}{2n-5},\frac{4}{1-{\vartheta }}\right\} }, \end{aligned}$$

(4.15)

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, where $\delta > 0$ and ${\vartheta } \in (0,1)$ are free parameters.

$(\partial _x \hat{u})^3$-term We have

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \left( \partial _x (\sigma _k^2)\right) \left( (F_\varepsilon ^2)'''(\hat{u}) + 4 \left( (F_\varepsilon ')^2\right) '(\hat{u})\right) (\partial _x \hat{u})^3 \, \mathrm {d}x \, \mathrm {d}t' \\&\quad {\mathop {\leqq }\limits ^{(2.2)}} C_\sigma \int _0^t \int _{\mathbb {T}} \left|(F_\varepsilon ^2)'''(\hat{u}) + 4 \left( (F_\varepsilon ')^2\right) '(\hat{u}) \right| \left|\partial _x \hat{u} \right|^3 \mathrm {d}x \, \mathrm {d}t'. \end{aligned}$$

This implies

$$\begin{aligned}&\int _{\mathbb {T}} \left|(F_\varepsilon ^2)'''(\hat{u}) + 4 \left( (F_\varepsilon ')^2\right) '(\hat{u}) \right| \left|\partial _x \hat{u} \right|^3 \, \mathrm {d}x \\&\quad \leqq \int _{\mathbb {T}} \left|(F_\varepsilon ^2)'''(\hat{u}) + 4 \left( (F_\varepsilon ')^2\right) '(\hat{u}) \right| \mathrm {d}x \left\Vert \partial _x \hat{u} \right\Vert _{L^\infty (\mathbb {T})}^3 \\&\, {\mathop {\leqq }\limits ^{(4.12)}} C \int _{\mathbb {T}} \left|(F_\varepsilon ^2)'''(\hat{u}) + 4 \left( (F_\varepsilon ')^2\right) '(\hat{u}) \right| \mathrm {d}x \left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})}^{\frac{3}{2}} \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(\mathbb {T})}^{\frac{3}{2}}, \end{aligned}$$

and Lemma 4.5 yields

$$\begin{aligned} \int _{\mathbb {T}} \left|(F_\varepsilon ^2)'''(\hat{u}) + 4 \left( (F_\varepsilon ')^2\right) '(\hat{u}) \right| \mathrm {d}x \leqq C_{n,L} {\left\{ \begin{array}{ll} \left( 1 + \int _{\mathbb {T}} \left|\hat{u} \right|^{n-3} \mathrm {d}x\right) &{} \text{ if } n \in [3,4), \\ \int _{\mathbb {T}} G_\varepsilon ^{\frac{3-n}{n-2}}(\hat{u}) \, \mathrm {d}x &{} \text{ if } n \in \left[ \tfrac{8}{3}, 3\right) , \end{array}\right. } \end{aligned}$$

that is,

$$\begin{aligned} \int _{\mathbb {T}} \left|(F_\varepsilon ^2)'''(\hat{u}) + 4 \left( (F_\varepsilon ')^2\right) '(\hat{u}) \right| \mathrm {d}x \leqq C_{n,L} {\left\{ \begin{array}{ll} \left( 1 + \left\Vert \hat{u} \right\Vert _{L^1(\mathbb {T})}^{n-3}\right) &{} \text{ if } n \in [3,4), \\ \left\Vert G_\varepsilon (\hat{u}) \right\Vert _{L^1(\mathbb {T})}^{\frac{3-n}{n-2}} &{} \text{ if } n \in \left[ \tfrac{8}{3}, 3\right) . \end{array}\right. } \end{aligned}$$

We further use that

$$\begin{aligned} \left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})}^2 = \int _{\mathbb {T}} (\partial _x \hat{u})^2 \, \mathrm {d}x = - \int _{\mathbb {T}} \hat{u} \, \partial _x^2 \hat{u} \, \mathrm {d}x \leqq \left\Vert \hat{u} \right\Vert _{L^2(\mathbb {T})} \, \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(\mathbb {T})}. \end{aligned}$$

(4.16)

For $n \in [3,4)$, by employing mass conservation (Remark 3.3) and the Sobolev embedding, we get, for any $\delta > 0$, that

$$\begin{aligned}&\int _0^t \int _{\mathbb {T}} \left|(F_\varepsilon ^2)'''(\hat{u}) + 4 \left( (F_\varepsilon ')^2\right) '(\hat{u}) \right| \left|\partial _x \hat{u} \right|^3 \mathrm {d}x \, \mathrm {d}t' \\&\quad \,\, \leqq C_{n,L} \int _0^t \left( 1 + \left\Vert \hat{u}(t') \right\Vert _{L^1(\mathbb {T})}^{n-3}\right) \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^{\frac{3}{2}} \left\Vert \partial _x^2 \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^{\frac{3}{2}} \mathrm {d}t' \\&\!\!\quad {\mathop {\leqq }\limits ^{(4.16)}} C_{n,L} \int _0^t \left( 1 + \left\Vert \hat{u}(t') \right\Vert _{L^1(\mathbb {T})}^{n-3}\right) \left\Vert u(t') \right\Vert _{L^2(\mathbb {T})}^{\frac{1}{2}} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^{\frac{1}{2}} \left\Vert \partial _x^2 \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^2 \mathrm {d}t' \\&\quad \,\, \leqq C_{n,L} \left( 1 + \sup _{t' \in [0,t]} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^{n-2}+ |\mathcal {A}(\hat{u}^{(0)})| ^{n-2} \right) \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^2 \\&\,\,\quad \leqq \delta \sup _{t' \in [0,t]} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^2 + C_{n,L,\delta }\left( 1+|\mathcal {A}(\hat{u}^{(0)})| ^2+ \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)} ^{\frac{4}{4-n}} \right) , \end{aligned}$$

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, where we have applied Young’s inequality in the last step.

For $n \in \left[ \frac{8}{3}, 3\right)$, with an analogous reasoning we obtain, for any $\delta > 0$,

$$\begin{aligned}&\int _0^t \int _{\mathbb {T}} \left|(F_\varepsilon ^2)'''(\hat{u}) + 4 \left( (F_\varepsilon ')^2\right) '(\hat{u}) \right| \left|\partial _x \hat{u} \right|^3 \mathrm {d}x \, \mathrm {d}t' \\&\qquad \leqq C_{n,L} \int _0^t \left\Vert G_\varepsilon (\hat{u}(t')) \right\Vert _{L^1(\mathbb {T})}^{\frac{3-n}{n-2}} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^{\frac{3}{2}} \left\Vert \partial _x^2 \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^{\frac{3}{2}} \mathrm {d}t' \\&\,\quad {\mathop {\leqq }\limits ^{(4.16)}} C_{n,L} \int _0^t \left\Vert G_\varepsilon (\hat{u}(t')) \right\Vert _{L^1(\mathbb {T})}^{\frac{3-n}{n-2}} \left\Vert u(t') \right\Vert _{L^2(\mathbb {T})}^{\frac{1}{2}} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^{\frac{1}{2}} \left\Vert \partial _x^2 \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^2 \mathrm {d}t' \\&\qquad \leqq C_{n,L} \, \sup _{t' \in [0,t]} \left\Vert G_\varepsilon (\hat{u}(t')) \right\Vert _{L^1(\mathbb {T})}^{\frac{3-n}{n-2}} \left( 1 + \sup _{t' \in [0,t]} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}+ | \mathcal {A}(\hat{u} ^{(0)})|\right) \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^2 \\&\qquad \leqq \delta \sup _{t' \in [0,t]} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^2 \\&\qquad \quad + C_{n,L,\delta } \left( 1+\sup _{t' \in [0,t]} \left\Vert G_\varepsilon (\hat{u}(t')) \right\Vert _{L^1(\mathbb {T})}^2 +| \mathcal {A}(\hat{u} ^{(0)})|^2+ \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^{\frac{4(n-2)}{2n-5}}\right) , \end{aligned}$$

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, where we have applied Young’s inequality again.

Altogether, for $n\in \left[ \frac{8}{3},4\right)$, we obtain

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \left( \partial _x (\sigma _k^2)\right) \left( (F_\varepsilon ^2)'''(\hat{u}) + 4 \left( (F_\varepsilon ')^2\right) '(\hat{u})\right) (\partial _x \hat{u})^3 \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\quad \leqq \delta \sup _{t' \in [0,t]} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^2 \nonumber \\&\qquad + C_{n,L,T,\delta } \left( 1 +|\mathcal {A}(\hat{u}^{(0)})| ^2+ \sup _{t' \in [0,t]} \left\Vert G_\varepsilon (\hat{u}(t')) \right\Vert _{L^1(\mathbb {T})}^2 \mathbb {1}_{\left[ \frac{8}{3},3\right] }(n)\right) \nonumber \\&\qquad + C_{n,L,T,\delta } \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^{\max \left\{ \frac{4}{4-n},\frac{4(n-2)}{2n-5}\right\} }, \end{aligned}$$

(4.17)

$\mathrm {d}\hat{\mathbb {P}}$-almost surely.

$(\partial _x \hat{u})^2$-term We start out with

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \left( 8 \left( (\partial _x \sigma _k)^2 - \sigma _k (\partial _x^2 \sigma _k)\right) (F_\varepsilon ')^2(\hat{u}) + \left( \partial _x^2 (\sigma _k^2)\right) (F_\varepsilon ^2)''(\hat{u})\right) (\partial _x \hat{u})^2 \, \mathrm {d}x \, \mathrm {d}t' \\&\quad {\mathop {\leqq }\limits ^{(2.2)}} C_\sigma \int _0^t \int _{\mathbb {T}} \left( (F_\varepsilon ')^2(\hat{u}) + \left|(F_\varepsilon ^2)''(\hat{u}) \right|\right) (\partial _x \hat{u})^2 \, \mathrm {d}x \, \mathrm {d}t'. \end{aligned}$$

Next, we compute

$$\begin{aligned} (F_\varepsilon ')^2(r) {\mathop {=}\limits ^{(3.3)}} \tfrac{ n^2}{4} \, r^2 \, (r^2+\varepsilon ^2)^{\frac{n}{2} -2} \lesssim |r|^{n-2} \end{aligned}$$

and

$$\begin{aligned} \left|(F_\varepsilon ^2)''(r) \right| {\mathop {=}\limits ^{(3.3)}} \left|n (r^2+\varepsilon ^2)^{\frac{n}{2} -1} + n(n-2) \, r^2 \, (r^2+\varepsilon ^2)^{\frac{n}{2} -2} \right| \leqq C_n(\left|r \right|^{n-2} + 1), \end{aligned}$$

so that

$$\begin{aligned} \int _{\mathbb {T}} \left( (F_\varepsilon ')^2(\hat{u}) + \left|(F_\varepsilon ^2)''(\hat{u}) \right|\right) (\partial _x \hat{u})^2 \, \mathrm {d}x \leqq C_n \int _{\mathbb {T}} \left( 1+\left|\hat{u} \right|^{n-2}\right) (\partial _x \hat{u})^2 \, \mathrm {d}x. \end{aligned}$$

Similarly as before, we estimate using conservation of mass (Remark 3.3) and that $\partial _x\hat{u}$ has zero average,

$$\begin{aligned}&\int _{\mathbb {T}} \left( 1+\left|\hat{u} \right|^{n-2}\right) (\partial _x \hat{u})^2 \, \mathrm {d}x \\&\quad \leqq \left( 1 + \left\Vert \hat{u} \right\Vert _{L^\infty (\mathbb {T})}^{n-2}\right) \left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})}^2 \\&\quad {\leqq } C_{L} \left( 1 + \left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})}^{n-2}+|\mathcal {A}(\hat{u} ^{(0)} ) |^{n-2}\right) \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(\mathbb {T})}^2, \end{aligned}$$

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, so that

$$\begin{aligned}&\int _0^t \int _{\mathbb {T}} \left( 1+\left|\hat{u} \right|^{n-2}\right) (\partial _x \hat{u})^2 \, \mathrm {d}x \, \mathrm {d}t' \\&\quad \leqq C_{L,T} \left( 1 + \sup _{t' \in [0,t]} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^{n-2}+|\mathcal {A}(\hat{u} ^{(0)} ) |^{n-2}\right) \left( 1 + \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^2\right) , \end{aligned}$$

$\mathrm {d}\hat{\mathbb {P}}$-almost surely. Hence, by Young’s inequality we arrive at

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \left( 8 \left( (\partial _x \sigma _k)^2 - \sigma _k (\partial _x^2 \sigma _k)\right) (F_\varepsilon ')^2(\hat{u}) + \left( \partial _x^2 (\sigma _k^2)\right) (F_\varepsilon ^2)''(\hat{u})\right) (\partial _x \hat{u})^2 \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\quad \leqq \delta \sup _{t' \in [0,t]}\left\Vert \partial _x \hat{u}(t') \right\Vert _{{L^2(\mathbb {T})}}^2+ C_{\sigma ,L,T,\delta } \left( 1+|\mathcal {A}(\hat{u} ^{(0)} ) |^2+\left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^{\frac{4}{4-n}}\right) , \end{aligned}$$

(4.18)

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, for any $\delta > 0$.

$(\partial _x \hat{u})^0$-term We first use

$$\begin{aligned} \sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \left( 4 \sigma _k \, \partial _x^4 \sigma _k - \partial _x^4 (\sigma _k^2)\right) F_\varepsilon ^2(\hat{u}) \, \mathrm {d}x \, \mathrm {d}t' {\mathop {\leqq }\limits ^{(2.2)}} C_\sigma \int _0^t \int _{\mathbb {T}} F_\varepsilon ^2(\hat{u}) \, \mathrm {d}x \, \mathrm {d}t' \end{aligned}$$

and

$$\begin{aligned} \int _0^t \int _{\mathbb {T}} F_\varepsilon ^2(\hat{u}) \, \mathrm {d}x \, \mathrm {d}t' {\mathop {\leqq }\limits ^{(3.3)}} C_{L,T} \left( 1 + \int _0^t \int _{\mathbb {T}} \left|\hat{u} \right|^n \mathrm {d}x \, \mathrm {d}t'\right) . \end{aligned}$$

Then, we estimate, using conservation of mass (Remark 3.3) and the Sobolev embedding,

$$\begin{aligned} \int _{\mathbb {T}} \left|\hat{u} \right|^n \mathrm {d}x\leqq & {} C_{L} \left\Vert \hat{u} \right\Vert _{L^\infty (\mathbb {T})}^n \leqq C_{L,n} \left( 1 + \left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})}^n+ |\mathcal {A}(\hat{u} ^{(0)} ) |^n \right) \\\leqq & {} C_{L,n} \left( 1 + \left\Vert \partial _x \hat{u} \right\Vert _{L^2(\mathbb {T})}^{n-2} \Vert \partial ^2_x \hat{u} \Vert _{L^2(\mathbb {T})}^2 + |\mathcal {A}(\hat{u} ^{(0)} ) |^n\right) , \end{aligned}$$

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, where we have also employed that $\partial _x \hat{u}$ has zero average. Integrating in time yields

$$\begin{aligned} \int _0^t \int _{\mathbb {T}} \left|\hat{u} \right|^n \mathrm {d}x \, \mathrm {d}t' \leqq C_{L,T,n} \left( 1 + \sup _{t' \in [0,t]} \left\Vert \partial _x \hat{u}(t') \right\Vert _{L^2(\mathbb {T})}^{n-2} \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^2+|\mathcal {A}(\hat{u} ^{(0)} ) |^n \right) , \end{aligned}$$

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, so that by Young’s inequality it follows that, for any $\delta > 0$,

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}}^2 \int _{\mathbb {T}} \left( 4 \sigma _k \, \partial _x^4 \sigma _k - \partial _x^4 (\sigma _k^2)\right) F_\varepsilon ^2(\hat{u}) \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\quad \leqq \delta \sup _{t' \in [0,t]}\left\Vert \partial _x \hat{u}(t') \right\Vert _{{L^2(\mathbb {T})}}^2+ C_{\sigma ,L,T,\delta } \left( 1+|\mathcal {A}(\hat{u} ^{(0)} ) |^n +\left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^{\frac{4}{4-n}}\right) , \end{aligned}$$

(4.19)

$\mathrm {d}\hat{\mathbb {P}}$-almost surely.

Closing the estimate Inserting all the previous estimates (4.15), (4.17), (4.18), and (4.19) in (4.10) and choosing $\delta$ sufficiently small and an appropriate ${\vartheta }$, we arrive for $n \in \left[ \frac{8}{3},4\right)$ at

$$\begin{aligned}&\left\Vert \partial _x \hat{u}(t) \right\Vert _{L^2(\mathbb {T})}^2 + \int _0^t \int _{\mathbb {T}} F_\varepsilon ^2(\hat{u}) \, (\partial _x^3 \hat{u})^2 \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\quad \leqq 2 \left\Vert \partial _x \hat{u}^{(0)} \right\Vert _{L^2(\mathbb {T})}^2 + 2 \left|M(t) \right| \nonumber \\&\qquad + C_{q,\sigma ,n,L,T,\delta } \left( 1 + |\mathcal {A}(\hat{u} ^{(0)} ) |^n+\sup _{t' \in [0,t]} \left\Vert G_\varepsilon (\hat{u}(t')) \right\Vert _{L^1(\mathbb {T})}^{2} \mathbb {1}_{\left[ \frac{8}{3},3\right] }(n) + \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_t)}^{4q}\right) , \end{aligned}$$

(4.20)

$\mathrm {d}\hat{\mathbb {P}}$-almost surely, where $q \geqq \max \left\{ \frac{1}{4-n},\frac{n-2}{2n-5}\right\}$ and $q > 1$ with a constant $C_{q,\sigma ,n,L,T,\delta } < \infty$. Here,

$$\begin{aligned} M(t) {:}{=}\sum _{ k \in \mathbb {Z}} \int _0^t \gamma _{\hat{u}} \int _{\mathbb {T}} \sigma _k F_\varepsilon (\hat{u}) \, \partial ^3_x \hat{u} \, \mathrm {d}x \, \mathrm {d}\hat{\beta }^k(t') \end{aligned}$$

denotes the martingale in the last line of (4.10). Let us set as usual

$$\begin{aligned} \tau _m = \inf \left\{ t>0 : \int _0^t \int _{\mathbb {T}} F_\varepsilon ^2(\hat{u}) (\partial _x^3 \hat{u} )^2 \, \mathrm {d}x \, \mathrm {d}t' >m \right\} \wedge T \end{aligned}$$

We now discard the second term on the left-hand side of (4.20), we take suprema in time up to $T \wedge \tau _m$, we raise to the power $\frac{p}{2}$, and we take expectation to obtain with help of the Burkholder-Davis-Gundy inequality

$$\begin{aligned}&\hat{\mathbb {E}} \sup _{t \in [0,T]} \left\Vert \partial _x \hat{u}(t\wedge \tau _m) \right\Vert _{L^2(\mathbb {T})}^p\nonumber \\&\quad \leqq C_p \left( \hat{\mathbb {E}} \left\Vert \partial _x \hat{u}^{(0)} \right\Vert _{L^2(\mathbb {T})}^p + \hat{\mathbb {E}} \langle M \rangle _{T \wedge \tau _m} ^{\frac{p}{4}}\right) \nonumber \\&\qquad + C_{p,q,\sigma ,n,L,T} \left( 1 + \hat{\mathbb {E}} \sup _{t \in [0,T]} \left\Vert G_\varepsilon (\hat{u}(t)) \right\Vert _{L^1(\mathbb {T})}^{p} \mathbb {1}_{\left[ \frac{8}{3},3\right] }(n) + \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_T)}^{2pq}\right) ,\nonumber \\ \end{aligned}$$

(4.21)

$\mathrm {d}\hat{\mathbb {P}}$-almost surely. Next, notice that

$$\begin{aligned} \mathbb {E}\langle M \rangle _{T \wedge \tau _m}^{\frac{p}{4}} {\mathop {\leqq }\limits ^{(\hbox {2.2f})}} C_p \, \hat{\mathbb {E}} \left( \int _0^{T \wedge \tau _m} \int _{\mathbb {T}} F_\varepsilon ^2(\hat{u}) \, (\partial _x^3 \hat{u})^2 \, \mathrm {d}x \, \mathrm {d}t \right) ^{\frac{p}{4}}. \end{aligned}$$

(4.22)

Now we go back to (4.20), we discard the first term at the left-hand side and we conclude that

$$\begin{aligned}&\mathbb {E}\left( \int _0^{T \wedge \tau _m} \int _{\mathbb {T}} F_\varepsilon ^2(\hat{u}) \, (\partial _x^3 \hat{u})^2 \, \mathrm {d}x \, \mathrm {d}t \right) ^{\frac{p}{2}} \\&\quad \leqq \ \ C_p \left( \hat{\mathbb {E}} \left\Vert \partial _x \hat{u}^{(0)} \right\Vert _{L^2(\mathbb {T})}^p + \hat{\mathbb {E}} \langle M \rangle _{T \wedge \tau _m}^{\frac{p}{4}}\right) \\&\qquad + C_{p,q,\sigma ,n,L,T} \left( 1 +|\mathcal {A}(\hat{u} ^{(0)} ) |^n+ \hat{\mathbb {E}} \sup _{t \in [0,T]} \left\Vert G_\varepsilon (\hat{u}(t)) \right\Vert _{L^1(\mathbb {T})}^{p} \mathbb {1}_{\left[ \frac{8}{3},3\right] }(n) + \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_T)}^{2pq}\right) \\&\quad {\mathop {\leqq }\limits ^{(4.22)}} C_p \left( \hat{\mathbb {E}} \left\Vert \partial _x \hat{u}^{(0)} \right\Vert _{L^2(\mathbb {T})}^p + \mathbb {E}\left( \int _0^{T \wedge \tau _m} \int _{\mathbb {T}} F_\varepsilon ^2(\hat{u}) \, (\partial _x^3 \hat{u})^2 \, \mathrm {d}x \, \mathrm {d}t \right) ^{\frac{p}{4}}\right) \\&\qquad + C_{p,q,\sigma ,n,L,T} \left( 1 +|\mathcal {A}(\hat{u} ^{(0)} ) |^{\frac{np}{2}}+ \hat{\mathbb {E}} \sup _{t \in [0,T]} \left\Vert G_\varepsilon (\hat{u}(t)) \right\Vert _{L^1(\mathbb {T})}^{p} \mathbb {1}_{\left[ \frac{8}{3},3\right] }(n) + \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_T)}^{2pq}\right) , \end{aligned}$$

$\mathrm {d}\hat{\mathbb {P}}$-almost surely. From this it follows with Young’s inequality that

$$\begin{aligned}&\mathbb {E}\left( \int _0^{T \wedge \tau _m} \int _{\mathbb {T}} F_\varepsilon ^2(\hat{u}) \, (\partial _x^3 \hat{u})^2 \, \mathrm {d}x \, \mathrm {d}t \right) ^{\frac{p}{2}} \\&\quad \leqq C_{p,q,\sigma ,n,L,T} \left( 1 +|\mathcal {A}(\hat{u} ^{(0)} ) |^{\frac{np}{2}}+ \hat{\mathbb {E}} \left\Vert \partial _x \hat{u}^{(0)} \right\Vert _{L^2(\mathbb {T})}^p\right) \\&\qquad + C_{p,q,\sigma ,n,L,T} \left( \hat{\mathbb {E}} \sup _{t \in [0,T]} \left\Vert G_\varepsilon (\hat{u}(t)) \right\Vert _{L^1(\mathbb {T})}^{p} \mathbb {1}_{\left[ \frac{8}{3},3\right] }(n) + \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_T)}^{2pq}\right) , \end{aligned}$$

which, combined with (4.21) and (4.22), yields, by virtue of Fatou’s lemma,

$$\begin{aligned}&\hat{\mathbb {E}} \left[ \sup _{t \in [0,T]} \left\Vert \partial _x \hat{u}(t) \right\Vert _{L^2(\mathbb {T})}^p + \left( \int _0^T \int _{\mathbb {T}} F_\varepsilon ^2(\hat{u}) \, (\partial _x^3 \hat{u})^2 \, \mathrm {d}x \, \mathrm {d}t \right) ^{\frac{p}{2}}\right] \\&\quad \leqq C_{p,q,\sigma ,n,L,T} \left( 1 + |\mathcal {A}(\hat{u} ^{(0)} ) |^{\frac{np}{2}}+\hat{\mathbb {E}} \left\Vert \partial _x \hat{u}^{(0)} \right\Vert _{L^2(\mathbb {T})}^p\right) \\&\qquad + C_{p,q,\sigma ,n,L,T} \left( \hat{\mathbb {E}} \sup _{t \in [0,T]} \left\Vert G_\varepsilon (\hat{u}(t)) \right\Vert _{L^1(\mathbb {T})}^{p} \mathbb {1}_{\left[ \frac{8}{3},3\right] }(n) + \left\Vert \partial _x^2 \hat{u} \right\Vert _{L^2(Q_T)}^{2pq}\right) , \end{aligned}$$

which was stated in (4.9). $\square$

4.3 Passage to the limit to remove the cut off

In this section we consider the SPDE

$$\begin{aligned} \mathrm {d}u_{\varepsilon }= & {} \partial _x\left( -F_\varepsilon ^2(u_{\varepsilon }) \partial _x^3u_{\varepsilon }\right) \mathrm {d}t +\frac{1}{2} \sum _{k \in \mathbb {Z}}\partial _x \left( \sigma _k F_\varepsilon '(u_{\varepsilon }) \partial _x(\sigma _k F_\varepsilon (u_{\varepsilon }))\right) \mathrm {d}t \nonumber \\&+ \sum _{k \in \mathbb {Z}} \left( \partial _x\left( \sigma _k F_\varepsilon (u_{\varepsilon })\right) \right) \mathrm {d}\beta ^k, \end{aligned}$$

(4.23a)

$$\begin{aligned} u_{\varepsilon }(0,\cdot )= & {} u^{(0)}. \end{aligned}$$

(4.23b)

The definition of a weak solution $\{ (\check{\Omega }, \check{\mathcal {F}},\check{\mathbb {F}},\check{\mathbb {P}}), \ (\check{\beta }_k)_{k \in \mathbb {Z}},\ \check{u}^{(0)}, \check{u} \}$ of equation (4.23) is covered by Definition 3.2 taking for $g_R$ the function $g_\infty \equiv 1.$

Proposition 4.7

Suppose that $n \in \left[ \frac{8}{3},4\right)$, $\mathfrak {p} \geqq n+2$, $\varepsilon \in (0,1)$, and $q > 1$ satisfying $q \geqq \max \left\{ \frac{1}{4-n},\frac{n-2}{2n-5}\right\}$. Suppose that $u^{(0)} \in L^{\mathfrak {p}}\left( \Omega ,\mathcal {F}_0,\mathbb {P};H^1(\mathbb {T})\right)$ such that

$$\begin{aligned} \mathcal {K}(u^{(0)}, \mathfrak {p}, q, \varepsilon ){:}{=} \mathbb {E}\left|\mathcal {A}(u^{(0)}) \right|^{2\mathfrak {p}q} + \mathbb {E}\left\Vert G_\varepsilon (u^{(0)}) \right\Vert _{L^1(\mathbb {T})}^{\mathfrak {p}q}+ \mathbb {E}\left\Vert \partial _x u^{(0)} \right\Vert ^{\mathfrak {p}} < \infty . \end{aligned}$$

Then there exists a weak solution $\{ (\check{\Omega }, \check{\mathcal {F}},\check{\mathbb {F}},\check{\mathbb {P}}), \ (\check{\beta }_k)_{k \in \mathbb {Z}},\ \check{u}^{(0)}_\varepsilon , \check{u}_\varepsilon \}$ to (4.23) in the sense of Definition 3.2, satisfying

$$\begin{aligned}&\check{\mathbb {E}} \left[ \sup _{t \in [0,T]} \left\Vert \partial _x \check{u}_\varepsilon (t) \right\Vert _{L^2(\mathbb {T})}^{\mathfrak {p}} + \left\Vert F_\varepsilon (\check{u}_\varepsilon ) \partial _x^3 \check{u}_\varepsilon \right\Vert _{L^2(Q_T)}^{\mathfrak {p}} + \sup _{t \in [0,T]} \left\Vert G_\varepsilon (\check{u}_\varepsilon (t)) \right\Vert _{L^1(\mathbb {T})}^{\mathfrak {p}q}\right. \nonumber \\&\qquad \qquad \left. + \left\Vert \partial _x^2\check{u}_\varepsilon \right\Vert _{L^2(Q_T)}^{2\mathfrak {p}q}\right] \nonumber \\&\quad \leqq C \left( 1 + \mathcal {K}(u^{(0)}, \mathfrak {p}, q, \varepsilon )\right) , \end{aligned}$$

(4.24)

where $C < \infty$ is a constant depending only on $\mathfrak {p}$, q, $\sigma = (\sigma _k)_{k \in \mathbb {Z}}$, n, L, and T.

Proof

Let $\hat{u}_{\varepsilon , R}$ be a weak solution of (3.8a3.8b). By Lemmata 4.3 and 4.6 we have

$$\begin{aligned}&\hat{\mathbb {E}} \left[ \sup _{t \in [0,T]} \left\Vert \partial _x \hat{u}_{\varepsilon , R}(t) \right\Vert _{L^2(\mathbb {T})}^{\mathfrak {p}} + \left\Vert F_\varepsilon ( \hat{u}_{\varepsilon , R}) \partial _x^3 \hat{u}_{\varepsilon , R} \right\Vert _{L^2(Q_T)}^{\mathfrak {p}}\right] \\&\qquad + \hat{\mathbb {E}}\left[ \sup _{t \in [0,T]} \left\Vert G_\varepsilon ( \hat{u}_{\varepsilon , R}(t)) \right\Vert _{L^1(\mathbb {T})}^{\mathfrak {p}q} + \left\Vert \partial _x^2 \hat{u}_{\varepsilon , R} \right\Vert _{L^2(Q_T)}^{2\mathfrak {p}q}\right] \\&\quad \leqq C_{p,q,\sigma ,n,L,T} \left( 1 + \mathcal {K}(u^{(0)}, \mathfrak {p}, q, \varepsilon )\right) , \end{aligned}$$

and notice that the constant does not depend on R. From this estimate, the construction of a weak solution $\{ (\check{\Omega }, \check{\mathcal {F}},\check{\mathbb {F}},\check{\mathbb {P}}), \ (\check{\beta }_k)_{k \in \mathbb {Z}},\ \check{u}^{(0)}, \check{u} \}$ is very similar to the construction in Proposition 3.4 (in fact, easier) and is left to the reader. Estimate (4.24) follows from the above estimate and Fatou’s lemma. $\square$

Remark 4.8

The right-hand side of (4.24) can be formulated independently of $\varepsilon$, just noting the inequality

$$\begin{aligned}&G_\varepsilon (r) {\mathop {=}\limits ^{(4.1)}} \int _r^\infty \int _{r'}^\infty \frac{1}{F_\varepsilon ^2(r'')} \, \mathrm {d}r'' \, \mathrm {d}r' {\mathop {\leqq }\limits ^{(3.3)}} \int _r^\infty \int _{r'}^\infty \frac{1}{F_0^2(r'')} \, \mathrm {d}r'' \, \mathrm {d}r' {\mathop {=}\limits ^{(4.1)}} G_0(r)\\&\text{ and } \text{ choosing }~\varepsilon =0~\text{ in }~\mathcal K(u^{(0)}, p,q,\varepsilon ).{\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad \square } \end{aligned}$$

5 The Degenerate Limit

In order to prove Theorem 2.2, we first prove additional regularity in time in order to obtain $\mathrm {d}\tilde{\mathbb {P}}$-almost surely uniform convergence in the limit $\varepsilon \searrow 0$ using a version of Prokhorov’s theorem (cf. [35, Theorem 2]) and a compactness argument. Subsequently, we prove that (2.6) is recovered in this limit by employing the energy-entropy estimate, Proposition 4.7 . The proof is concluded by showing that the weak formulation (2.5) is valid, which follows by applying [33, Proposition A.1] to characterize the martingale.

For $\varepsilon \in \{ n^{-1} \}_{n=1}^\infty$, we denote by $\{ (\check{\Omega }_\varepsilon , \check{\mathcal {F}}_\varepsilon ,\check{\mathbb {F}}_\varepsilon ,\check{\mathbb {P}}_\varepsilon ), \ (\check{\beta }^k_\varepsilon )_{k \in \mathbb {Z}},\ \check{u}^{(0)}_\varepsilon , \check{u}_\varepsilon \}$ the weak solution of (4.23a4.23b) constructed in Proposition 4.7. In order to drop the $\varepsilon$-dependence from the probability space we will be considering $( (\check{\beta }^k_\varepsilon )_{k \in \mathbb {Z}},\ \check{u}^{(0)}_\varepsilon , \check{u}_\varepsilon )$ on a common probability space given by

$$\begin{aligned} (\check{\Omega }, \check{\mathcal {F}},\check{\mathbb {F}},\check{\mathbb {P}}){:}{=} \prod _{\varepsilon } (\check{\Omega }_\varepsilon , \check{\mathcal {F}}_\varepsilon ,\check{\mathbb {F}}_\varepsilon ,\check{\mathbb {P}}_\varepsilon ). \end{aligned}$$

5.1 Compactness

The reasoning of this section uses techniques of [17] and of [21, §4].

Lemma 5.1

(Regularity in time). Suppose that $T\in (0,\infty )$, $\varepsilon \in (0,1]$, $n\in [8/3,4)$, ${p}>1$, $q > 1$ satisfying $q \geqq \max \left\{ \frac{1}{4-n},\frac{(n-2)}{2n-5}\right\}$, and

$$\begin{aligned} u^{(0)}\in L^{(n+2){p}}\left( \Omega ,\mathbb {F}_{0},\mathbb {P};H^{1}(\mathbb {T})\right) \end{aligned}$$

such that $u^{(0)}\geqq 0$ $\mathrm {d}\mathbb {P}$-almost surely, $\mathbb {E}\left|{\mathcal {A}(u^{(0)})} \right|^{2(n+2){p}q}<\infty$, and$\mathbb {E}\left\Vert G_{0}\left( u^{(0)}\right) \right\Vert _{L^{1}(\mathbb {T})}^{(n+2){p}q}<\infty$. Then, the weak solutions $\check{u}_{\varepsilon }$ constructed in Proposition 4.7 satisfy for any ${p'}\in [1,2{p})$,

$$\begin{aligned} \check{u}_ \varepsilon \in L^{{p'}}\left( \check{\Omega },\check{\mathcal {F}},\check{\mathbb {P}};C^{\frac{1}{4}}\left( [0,T];{L^2(\mathbb {T})}\right) \right) \end{aligned}$$

(5.1a)

with

$$\begin{aligned}&\left\Vert \check{u}_\varepsilon \right\Vert _{L^{{p'}}(\check{\Omega };C^{\frac{1}{4}}([0,T];{L^2(\mathbb {T})}))}\nonumber \\&\quad \leqq C\,\left[ \mathbb {E}\left( 1+\left|{\mathcal {A}(u^{(0)})} \right|^{2(n+2){p}q}+\left\Vert G_{0}(u^{(0)}) \right\Vert _{L^{1}(\mathbb {T})}^{(n+2){p}q}+\left\Vert \partial _{x}u^{(0)} \right\Vert _{L^{2}(\mathbb {T})}^{(n+2){p}}\right) \right] ^{\frac{1}{2{p}}}, \end{aligned}$$

(5.1b)

where C is a constant depending only on ${p},{p'},q,(\sigma _k)_{k\in \mathbb {Z}},L,$ and T.

Proof

Starting from the weak formulation (cf. Definition 3.2 (ii))

$$\begin{aligned}&\left( \check{u}_\varepsilon (t_{2})-\check{u}_\varepsilon (t_{1}),\varphi \right) _{{L^2(\mathbb {T})}}+\int _{t_{1}}^{t_{2}}{\int _{\mathbb {T}}}{F^2_\varepsilon }(\check{u}_\varepsilon )\partial _{x}^{3}\check{u}_\varepsilon \partial _{x}\varphi \ \mathrm {d}x \, \mathrm {d}t\nonumber \\&\quad +\tfrac{1}{2}\sum _{k\in \mathbb {Z}}\int _{t_{1}}^{t_{2}}\sigma _{k}{F'_\varepsilon }(\check{u}_\varepsilon )\partial _{x}\left( {\sigma _k}{F_\varepsilon }(\check{u}_\varepsilon )\right) \partial _{x}\varphi \, \mathrm {d}x \, \mathrm {d}t = \sum _{k\in \mathbb {Z}}\int _{t_{1}}^{t_{2}}{\int _{\mathbb {T}}}\partial _{x}\left( {\sigma _k}{F_\varepsilon }(\check{u}_\varepsilon )\right) \varphi \, \mathrm {d}x \, \mathrm {d}\check{\beta }^{k}_\varepsilon \end{aligned}$$

(5.2)

for all $\varphi \in H^{1}(\mathbb {T})$, $t_{1},t_{2}\in [0,T]$ with $t_{1}\leqq t_{2}$ and $\mathrm {d}\check{\mathbb {P}}$-almost surely, we obtain, by an approximation argument based on the separability of $H^1(\mathbb {T})$ that the $\hat{\mathbb {P}}$-zero set can be chosen independently of $\varphi$, that is, $\mathrm {d}\check{\mathbb {P}}$-almost surely,

$$\begin{aligned}&\left( \check{u}_\varepsilon (t_{2})-\check{u}_\varepsilon (t_{1}),\varphi \right) _{{L^2(\mathbb {T})}}+\int _{t_{1}}^{t_{2}}{\int _{\mathbb {T}}}{F^2_\varepsilon }(\check{u}_\varepsilon )\partial _{x}^{3}\check{u}_\varepsilon \partial _{x}\varphi \, \mathrm {d}x \, \mathrm {d}t\\&\quad +\tfrac{1}{2}\sum _{k\in \mathbb {Z}}\int _{t_{1}}^{t_{2}}\sigma _{k}{F'_\varepsilon }(\check{u}_\varepsilon )\partial _{x}\left( {\sigma _k}{F_\varepsilon }(\check{u}_\varepsilon )\right) \partial _{x}\varphi \, \mathrm {d}x \, \mathrm {d}t\leqq \sup _{\left\Vert \psi \right\Vert _{{L^2(\mathbb {T})}}\leqq 1}\left|(I_{\varepsilon }(t_{2})-I_{\varepsilon }(t_{1}),\psi ) \right| \end{aligned}$$

for all $\varphi \in H^{1}(\mathbb {T})$ with $\left\Vert \varphi \right\Vert _{{L^2(\mathbb {T})}}=1.$ Here, we have used the abbreviation

$$\begin{aligned} I_{\varepsilon }(t){:}{=}\sum _{k\in \mathbb {Z}}\int _{0}^{t}\partial _{x}\left( {\sigma _k}{F_\varepsilon }(\check{u}_\varepsilon )\right) \mathrm {d}\check{\beta }^{k}. \end{aligned}$$

(5.3)

Following the lines of the proof of Lemma 4.10 in [17], the choice $\varphi {:}{=}\frac{\check{u}_\varepsilon (t_{2})-\check{u}_\varepsilon (t_{1})}{\left\Vert \check{u}_\varepsilon (t_{2})-\check{u}_\varepsilon (t_{1}) \right\Vert _{{L^2(\mathbb {T})}}}$ and Young’s inequality imply that there is a finite constant C independent of $\varepsilon >0$ such that, $\mathrm {d}\hat{\mathbb {P}}$-almost surely,

$$\begin{aligned}&\left\Vert \check{u}_\varepsilon (t_{2})-\check{u}_\varepsilon (t_{1}) \right\Vert _{{L^2(\mathbb {T})}}^2 \nonumber \\&\quad \leqq C\Bigg (\left|\int _{t_{1}}^{t_{2}}{\int _{\mathbb {T}}}{F^2_\varepsilon }(\check{u}_\varepsilon )\partial _{x}^{3}\check{u}_\varepsilon \partial _{x}\left( \check{u}_\varepsilon (t_{2})-\check{u}_\varepsilon (t_{1})\right) \mathrm {d}x \, \mathrm {d}t \right| \nonumber \\&\qquad +\tfrac{1}{2}\left|\sum _{k\in \mathbb {Z}}\int _{t_{1}}^{t_{2}}{\int _{\mathbb {T}}}{\sigma _k}{F'_\varepsilon }(\check{u}_\varepsilon )\partial _{x}\left( \check{u}_\varepsilon (t_{2})-\check{u}_\varepsilon (t_{1})\right) \partial _{x}\left( {\sigma _k}{F_\varepsilon }(\check{u}_\varepsilon )\right) \mathrm {d}x \, \mathrm {d}t \right|\Bigg ) \nonumber \\&\qquad +\left\Vert I_{\varepsilon }(t_{2})-I_{\varepsilon }(t_{1}) \right\Vert _{{L^2(\mathbb {T})}}^2 {=}{:} R_{1}^2(t_1,t_2)+R_{2}^2(t_1,t_2)+R_{3}^2(t_1,t_2). \end{aligned}$$

(5.4)

By [46, Theorem 3.2 (vi)], for all $0<\sigma <1/2$, $1\leqq {p'}< 2{p}$,

$$\begin{aligned}&\left\Vert I_{\varepsilon } \right\Vert _{L^{{p'}}(\check{\Omega };C^{\frac{1}{2}-\sigma }\left( [0,T];L^{2}(\mathbb {T})\right) )}\\&\quad \,\,\, \leqq C_{T,{p'},{p},\sigma }\Vert (\partial _{x}\left( {\sigma _k}{F_\varepsilon }(\check{u}_\varepsilon )\right) )_{k\in \mathbb {Z}}\Vert _{L^{2{p}}(\Omega ;L^{\infty }(0,T;L_{2}(\ell ^2(\mathbb {Z});L^2(\mathbb {T}))))}\\&\quad \,\,\, = C\,\left[ \check{\mathbb {E}}\ \sup _{t\in [0,T]}\left( \sum _{k\in \mathbb {Z}}\Vert \partial _{x}(\sigma _{k}F_{\varepsilon } (\check{u}_\varepsilon ))\Vert _{L^{2}(\mathbb {T})}^{2}\right) ^{{p}}\right] ^{\frac{1}{2{p}}}\\&\quad \,\,\, \leqq C\,\left[ \check{\mathbb {E}}\ \sup _{t\in [0,T]}\left( \sum _{k\in \mathbb {Z}}\Vert \partial _{x}\sigma _{k}\Vert _{L^{\infty }(\mathbb {T})}^{2}\Vert F_{\varepsilon }(\check{u}_\varepsilon )\Vert _{L^{2}(\mathbb {T})}^{2}\right) ^{{p}}\right] ^{\frac{1}{2{p}}}\\&\qquad \,\, + C\,\left[ \check{\mathbb {E}}\ \sup _{t\in [0,T]}\left( \sum _{k\in \mathbb {Z}}\Vert \sigma _{k}\Vert _{L^{2}(\mathbb {T})}^{2}\Vert F_{\varepsilon }'(\check{u}_\varepsilon )\Vert _{L^{\infty }(\mathbb {T})}^{2}\Vert \partial _{x}\check{u}_\varepsilon \Vert _{L^{2}(\mathbb {T})}^{2}\right) ^{{p}}\right] ^{\frac{1}{2{p}}}\\&\quad {\mathop {\leqq }\limits ^{(\hbox {2.2e})}} C\,\left[ \check{\mathbb {E}}\ \sup _{t\in [0,T]}\left( \Vert F_{\varepsilon }(\check{u}_\varepsilon )\Vert _{L^{2}(\mathbb {T})}^{2{p}}+\Vert F_{\varepsilon }'(\check{u}_\varepsilon )\Vert _{L^{\infty }(\mathbb {T})}^{2{p}}\Vert \partial _{x}\check{u}_\varepsilon \Vert _{L^{2}(\mathbb {T})}^{2{p}}\right) \right] ^{\frac{1}{2{p}}}. \end{aligned}$$

We then use that

$$\begin{aligned} \Vert F_{\varepsilon }(\check{u}_\varepsilon )\Vert _{L^{2}(\mathbb {T})}^{2{p}}=\left( \int _{\mathbb {T}}F_{\varepsilon }^{2}(\check{u}_\varepsilon )\right) ^{{p}}&{\mathop {\leqq }\limits ^{(3.3)}}C\left( \left|\mathcal {A}(\check{u}^{(0)}_\varepsilon ) \right|^{n}+\Vert \partial _{x}\check{u}_\varepsilon \Vert _{L^{2}}^{n}+1\right) {}^{{p}}\\&\,\, \leqq C\left( 1+\left|\mathcal {A}(\check{u}^{(0)}_\varepsilon ) \right|^{{p}n}+\Vert \partial _{x}\check{u}_\varepsilon \Vert _{L^{2}}^{{p}n}\right) \end{aligned}$$

and

$$\begin{aligned} \Vert F_{\varepsilon }'(\check{u}_\varepsilon )\Vert _{L^{\infty }(\mathbb {T})}^{2{p}}\Vert \partial _{x}\check{u}_\varepsilon \Vert _{L^{2}(\mathbb {T})}^{2{p}}&{\mathop {\leqq }\limits ^{(3.3)}} \left( 1+\Vert \check{u}_\varepsilon \Vert _{L^{\infty }(\mathbb {T})}\right) ^{2\left( \frac{n}{2}-1\right) {p}}\Vert \partial _{x}\check{u}_\varepsilon \Vert _{L^{2}(\mathbb {T})}^{2{p}}\\&\,\,\leqq C\left( 1+\left|\mathcal {A}(\check{u}^{(0)}_\varepsilon ) \right|^{{p}n}+\Vert \partial _{x}\check{u}_\varepsilon \Vert _{L^{2}}^{{p}n}\right) \end{aligned}$$

to get for $R_3$ the estimate

$$\begin{aligned} \check{\mathbb {E}}\sup _{t_1,t_2\in [0,T]}\left( \frac{R_3(t_1,t_2)}{|t_2-t_1|^{(1/2-\sigma )}}\right) ^{p'}\leqq C\,\left[ \check{\mathbb {E}}\ \sup _{t\in [0,T]}\left( 1+\left|\mathcal {A}(\check{u}^{(0)}_\varepsilon ) \right|^{{p}n}+\Vert \partial _{x}\check{u}_\varepsilon (t)\Vert _{L^{2}}^{{p}n}\right) \right] ^{\frac{{p'}}{2{p}}}, \end{aligned}$$

(5.5)

valid for any $\sigma \in \left( 0,\frac{1}{2}\right)$. For $R_{1}$, we estimate, using ${F_\varepsilon }(r) {\mathop {\leqq }\limits ^{(3.3)}} (r^{2}+\varepsilon ^{2})^{\frac{n}{4}}$,

$$\begin{aligned}&\left|R_{1}(t_1,t_2) \right| \nonumber \\&\quad \leqq C\Bigg (\int _{t_{1}}^{t_{2}}{\int _{\mathbb {T}}}{F^2_\varepsilon }(\check{u}_\varepsilon ) \left( \partial _{x}(\check{u}_\varepsilon (t_{2})-\check{u}_\varepsilon (t_{1}))\right) ^{2} \mathrm {d}x \, \mathrm {d}t\Bigg )^{\frac{1}{4}}\nonumber \\&\qquad \times \Bigg ({\int _{t_1}^{t_2}}{\int _{\mathbb {T}}}{F^2_\varepsilon }(\check{u}_\varepsilon ) \, (\partial _{x}^{3}\check{u}_\varepsilon )^{2} \, \mathrm {d}x \, \mathrm {d}t\Bigg )^{\frac{1}{4}}\nonumber \\&\quad \leqq C |t_{2}-t_{1}|^{\frac{1}{4}}\left( 1+|\mathcal {A}(\check{u}_\varepsilon ^{(0)})|^{\frac{n+2}{4}}+\sup _{t\in [0,T]}\left\Vert \partial _{x}\check{u}_\varepsilon \right\Vert _{{L^2(\mathbb {T})}}^{\frac{n+2}{4}}\right) \nonumber \\&\qquad \times \left( \int _{0}^{T}{\int _{\mathbb {T}}}{F^2_\varepsilon }(\check{u}_\varepsilon ) \, (\partial _{x}^{3}\check{u}_\varepsilon )^{2} \, \mathrm {d}x \, \mathrm {d}t\right) ^{\frac{1}{4}} \nonumber \\&\quad \leqq C |t_{2}-t_{1}|^{\frac{1}{4}} \nonumber \\&\qquad \times \left( 1+|\mathcal {A}(\check{u}_\varepsilon ^{(0)})|^{\frac{n+2}{2}}+\sup _{t\in [0,T]}\left\Vert \partial _{x}\check{u}_\varepsilon \right\Vert _{{L^2(\mathbb {T})}}^{\frac{n+2}{2}}+\left( \int _{0}^{T}{\int _{\mathbb {T}}}{F^2_\varepsilon }(\check{u}_\varepsilon ) \, (\partial _{x}^{3}\check{u}_\varepsilon )^{2} \, \mathrm {d}x \, \mathrm {d}t\right) ^{\frac{1}{2}}\right) . \end{aligned}$$

(5.6)

Hence,

$$\begin{aligned}&\check{\mathbb {E}}\left( \sup _{t_1,t_2\in [0,T]}\frac{|R_{1}(t_1,t_2)|}{|t_2-t_1|^{\frac{1}{4}}}\right) ^{{p'}} \nonumber \\&\quad \leqq C \, \check{\mathbb {E}}\left( 1+|\mathcal {A}(\check{u}_\varepsilon ^{(0)})|^{\frac{(n+2){p'}}{2}}+\sup _{t\in [0,T]}\left\Vert \partial _{x}\check{u}_\varepsilon \right\Vert _{{L^2(\mathbb {T})}}^{\frac{(n+2){p'}}{2}}+\left( \int _{0}^{T}{\int _{\mathbb {T}}}{F^2_\varepsilon }(\check{u}_\varepsilon ) \, (\partial _{x}^{3}\check{u}_\varepsilon )^{2} \, \mathrm {d}x \, \mathrm {d}t\right) ^{\frac{{p'}}{2}}\right) . \end{aligned}$$

(5.7)

The term $R_{2}^2$ is split as follows:

$$\begin{aligned} 2R_{2}^2(t_1,t_2)&=\sum _{k\in \mathbb {Z}}{\int _{t_1}^{t_2}}{\int _{\mathbb {T}}}\sigma _{k}^{2} {F'_\varepsilon }(\check{u}_\varepsilon )^{2}\partial _{x}\left( \check{u}_\varepsilon (t_{2})-\check{u}_\varepsilon (t_{1})\right) \partial _{x}\check{u}_\varepsilon (t) \, \mathrm {d}x \, \mathrm {d}t \nonumber \\&\quad +\sum _{k\in \mathbb {Z}} {\int _{t_1}^{t_2}}{\int _{\mathbb {T}}}\sigma _{k}(\partial _x\sigma _{k}) {F'_\varepsilon }(\check{u}_\varepsilon ){F_\varepsilon }(\check{u}_\varepsilon )\partial _{x}\left( \check{u}_\varepsilon (t_{2})-\check{u}_\varepsilon (t_{1})\right) \mathrm {d}x \, \mathrm {d}t \nonumber \\&{=}{:}R_{21}^2+R_{22}^2. \end{aligned}$$

(5.8)

For $R_{21}$, we estimate

$$\begin{aligned} |R_{21}|&{\mathop {\leqq }\limits ^{(3.3)}} C\left( \sum _{k\in \mathbb {Z}} \left\Vert \sigma _{k} \right\Vert _{{{L^\infty (\mathbb {T})}}}^{2} {\int _{t_1}^{t_2}}\sup _{t\in [0,T]}\left( 1+\left\Vert \check{u}_\varepsilon (t) \right\Vert _{{{L^\infty (\mathbb {T})}}}^{n-2}\right) \sup _{t\in [0,T]}\left\Vert \partial _{x}\check{u}_\varepsilon (t) \right\Vert _{{L^2(\mathbb {T})}}^{2}\right) ^{\frac{1}{2}} \nonumber \\&{\mathop {\leqq }\limits ^{(\hbox {2.2e})}} C T^{\frac{1}{4}}\left|t_{2}-t_{1} \right|^{\frac{1}{4}} \left( 1+\left|{\mathcal {A}(u^{(0)})} \right|^{\frac{n}{2}}+\sup _{t\in [0,T]}\left\Vert \partial _{x}\check{u}_\varepsilon (t) \right\Vert _{{L^2(\mathbb {T})}}^{\frac{n}{2}}\right) . \end{aligned}$$

(5.9)

Finally,

$$\begin{aligned} \left|R_{22} \right|&\leqq C\left( \sum _{k\in \mathbb {Z}} \left\Vert \sigma _k \right\Vert _{{{L^\infty (\mathbb {T})}}} \left\Vert \partial _x \sigma _k \right\Vert _{{{L^\infty (\mathbb {T})}}} {\int _{t_1}^{t_2}}{\int _{\mathbb {T}}}\left|{F'_\varepsilon }(\check{u}_\varepsilon ){F_\varepsilon }(\check{u}_\varepsilon )\partial _{x}\left( \check{u}_\varepsilon (t_{2})-\check{u}_\varepsilon (t_{1})\right) \right| \mathrm {d}x \, \mathrm {d}t\right) ^{\frac{1}{2}}\nonumber \\&{\mathop {\leqq }\limits ^{(\hbox {2.2e}), (3.3)}} C T^{\frac{1}{4}} \left|t_{2}-t_{1} \right|^{\frac{1}{4}} \left( 1+\left|\mathcal {A}(\check{u}_\varepsilon ^{(0)}) \right|^{\frac{n}{2}}+\sup _{t\in [0,T]}\left\Vert \partial _{x}\check{u}_\varepsilon (t) \right\Vert _{{L^2(\mathbb {T})}}^{\frac{n}{2}}\right) . \end{aligned}$$

(5.10)

Hence, we get

$$\begin{aligned} \check{\mathbb {E}}\left( \sup _{t_1,t_2\in [0,T]}\frac{|R_{2}(t_1,t_2)|}{|t_2-t_1|^{\frac{1}{4}}}\right) ^{{p'}} \leqq C\check{\mathbb {E}}\left( 1+\left|\mathcal {A}(\check{u}_\varepsilon ^{(0)}) \right|^{\frac{n{p'}}{2}}+\sup _{t\in [0,T]}\left\Vert \partial _{x}\check{u}_\varepsilon (t) \right\Vert _{{L^2(\mathbb {T})}}^{\frac{n{p'}}{2}}\right) . \end{aligned}$$

(5.11)

Altogether, combining (5.5), (5.7), (5.11) and choosing $\sigma = \frac{1}{4}$, we get

$$\begin{aligned}&\Vert \check{u}_\varepsilon \Vert _{L^{{p'}}(\check{\Omega };C^{\frac{1}{4}} ([0,T];{L^2(\mathbb {T})}))}^{{p'}} \\&\quad \leqq C\,\left[ \check{\mathbb {E}}\ \sup _{t\in [0,T]}\left( 1+\left|{\mathcal {A}(u^{(0)})} \right|^{{p}n}+\Vert \partial _{x}\check{u}_{\varepsilon }(t)\Vert _{L^{2}}^{{p}n}\right) \right] ^{\frac{{p'}}{2{p}}}\\&\qquad +C\left[ \check{\mathbb {E}}\left( 1+\left|\mathcal {A}(\check{u}_\varepsilon ^{(0)}) \right|^{(n+2){p}}+\sup _{t\in [0,T]}\left\Vert \partial _{x}\check{u}_\varepsilon \right\Vert _{{L^2(\mathbb {T})}}^{(n+2){p}}\right) \right] ^{\frac{{p'}}{2{p}}} \\&\qquad +C\left[ \check{\mathbb {E}}\left( \int _{0}^{T}{\int _{\mathbb {T}}}{F^2_\varepsilon }(\check{u}_\varepsilon ) \, (\partial _{x}^{3}\check{u}_\varepsilon )^{2} \, \mathrm {d}x \, \mathrm {d}t\right) ^{{p}}\right] ^{\frac{{p'}}{2{p}}} \\&\qquad +C\left[ \check{\mathbb {E}}\left( 1+\left|\mathcal {A}(\check{u}_\varepsilon ^{(0)}) \right|^{n{p}}+\sup _{t\in [0,T]}\left\Vert \partial _{x}\check{u}_\varepsilon (t) \right\Vert _{{L^2(\mathbb {T})}}^{n{p}}\right) \right] ^{\frac{{p'}}{2{p}}}\\&\quad \leqq C\left[ \check{\mathbb {E}}\left( 1+\left|\mathcal {A}(\check{u}_\varepsilon ^{(0)}) \right|^{(n+2){p}}+\sup _{t\in [0,T]}\left\Vert \partial _{x}\check{u}_\varepsilon \right\Vert _{{L^2(\mathbb {T})}}^{(n+2){p}}\right) \right] ^{\frac{{p'}}{2{p}}}\\&\qquad + C\left[ \check{\mathbb {E}}\left( \int _{0}^{T}{\int _{\mathbb {T}}}{F^2_\varepsilon }(\check{u}_\varepsilon ) \, (\partial _{x}^{3}\check{u}_\varepsilon )^{2} \, \mathrm {d}x \, \mathrm {d}t\right) ^{{p}}\right] ^{\frac{{p'}}{2{p}}}\\&\, {\mathop {\leqq }\limits ^{(4.24)}} C\left[ 1+\mathbb {E}\left|{\mathcal {A}(u^{(0)})} \right|^{2(n+2){p}q}+\mathbb {E}\left\Vert G_{0}(u^{(0)}) \right\Vert _{L^{1}(\mathbb {T})}^{(n+2){p}q}+\mathbb {E}\left\Vert \partial _{x}u^{(0)} \right\Vert _{L^{2}(\mathbb {T})}^{(n+2){p}}\right] ^{\frac{{p'}}{2{p}}}, \end{aligned}$$

which gives the desired estimate (5.1b). Note that we have used Proposition 4.7 to get an estimate in terms of the initial data. $\square$

By interpolation, we get the following result on Hölder regularity with respect to space and time (for details, see [17, Lemma 4.11, p. 437]). Note that in Corollary 5.2, we control the moment of order $p'$ of $\left\Vert \check{u}_\varepsilon \right\Vert _{{C^{\frac{1}{8}, \frac{1}{2}}}(Q_T)}$ for any $p'\in [1,2p)$ while in [17] only estimates for second moments have been provided. This is due to (5.1a) as the analogous estimate in [17] has been formulated only for $p'=2.$

Corollary 5.2

(Hölder-continuity) Under the assumptions of Lemma 5.1, the solutions $\check{u}_\varepsilon$ constructed in Proposition 4.7 are space-time Hölder-continuous, $\mathrm {d}\check{\mathbb {P}}$-almost surely. In particular, there is a finite constant C independent of $\varepsilon >0$ such that

$$\begin{aligned} \check{\mathbb {E}}\left[ \left\Vert \check{u}_\varepsilon \right\Vert ^{p'}_{{C^{\frac{1}{8}, \frac{1}{2}}}(Q_T)}\right] \leqq C \end{aligned}$$

(5.12)

for any $p'\in [1,2p)$.

In the next proposition we will consider the space of all functions $u : Q_T \rightarrow \mathbb {R}$ such that $u \in {C^{\frac{\gamma }{4},\gamma }}(Q_T)$ for all $\gamma \in (0, 1/2)$, that is,

$$\begin{aligned} {C^{\frac{1}{8} -, \frac{1}{2} -}}(Q_T){:}{=}\bigcap \limits _{\gamma \in (0, 1/2)} {C^{\frac{\gamma }{4},\gamma }}(Q_T). \end{aligned}$$

We endow ${C^{\frac{1}{8} -, \frac{1}{2} -}}(Q_T)$ with the topology generated by the metric

$$\begin{aligned} d(u, v) {:}{=} \sum _{n=1}^\infty 2^{-n} \big ( \Vert u -v \Vert _{C^{\frac{\gamma _n}{4}, \gamma _n,}(Q_T)} \wedge 1 \big ), \end{aligned}$$

where $\gamma _n= \frac{1}{2} - \frac{1}{2^{n+1}}$.

Remark 5.3

Despite the fact that for each $\gamma < \frac{1}{2}$, ${C^{\frac{\gamma }{4},\gamma }}(Q_T)$ is not separable, the space ${C^{\frac{1}{8} -, \frac{1}{2} -}}(Q_T)$ is separable: If $u \in C^{ \frac{1}{8}-, \frac{1}{2}-}$ there exists $u_n \in C^\infty (Q_T)$ such that, for all $\gamma < 1/2$,

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert u_n-u\Vert _{{C^{\frac{\gamma }{4},\gamma }}(Q_T)}=0. \end{aligned}$$

Moreover, there exists a countable set $\mathcal {D} \subset C^\infty (Q_T)$ such that for all $v \in C^\infty (Q_T)$ and all $\varepsilon >0$, there exists $\tilde{v} \in \mathcal {D}$ such that $\Vert \tilde{v}- v \Vert _{C^1(Q_T)} \leqq \varepsilon$. It follows that $\mathcal {D}$ is dense in ${C^{\frac{1}{8} -, \frac{1}{2} -}}(Q_T)$.

In addition, it is complete, since ${C^{\frac{\gamma _n}{4}, \gamma _n}(Q_T)}$ is complete for each n. Therefore it is a Polish space.

Finally, since every bounded sequence in ${C^{\frac{1}{8}, \frac{1}{2}}}(Q_T)$ has a subsequence that converges in ${C^{\frac{\gamma }{4},\gamma }}(Q_T)$, for all $\gamma < \frac{1}{2}$, it follows that the embedding $C^{\frac{1}{8}, \frac{1}{2}}(Q_T) \subset {C^{\frac{1}{8} -, \frac{1}{2} -}}(Q_T)$ is compact.

Proposition 5.4

(Point-wise convergence) Let $T \in (0,\infty )$, $n \in \left[ \frac{8}{3},4\right)$, $\varepsilon \in (0,1]$, $p> 1$, $q > 1$ satisfying $q \geqq \max \left\{ \frac{1}{4-n},\frac{n-2}{2n-5}\right\}$. Suppose that

$$\begin{aligned} u^{(0)} \in L^{(n+2)p}\left( \Omega ,\mathcal {F}_0,\mathbb {P};H^1(\mathbb {T})\right) \end{aligned}$$

such that $u^{(0)} \geqq 0$, $\mathrm {d}\mathbb {P}$-almost surely, $\mathbb {E}\left|{\mathcal {A}(u^{(0)})} \right|^{2(n+2)pq} < \infty$, and$\mathbb {E}\left\Vert G_0\left( u^{(0)}\right) \right\Vert _{L^1(\mathbb {T})}^{(n+2)pq} < \infty$. Let $\check{u}_\varepsilon$ be the weak solution constructed in Proposition 4.7. Further define $\check{J} _\varepsilon {:}{=} F_\varepsilon (\check{u}_\varepsilon ) \, \partial _x^3 \check{u}_\varepsilon$ (pseudo-flux density) and$\check{W}_\varepsilon {:}{=} {\sum _{k\in \mathbb {Z}}}\sigma _k \check{\beta }^k_\varepsilon$. Then, up to taking a subsequence of $u_\varepsilon$, on the probability space $([0,1], \mathcal {B}([0,1]), \lambda _{[0,1]})$, there exist random variables

$$\tilde{u}, \tilde{u}_\varepsilon : [0,1] \rightarrow C^{\frac{1}{2}-, \frac{1}{8}- }(Q_T)\mathrm{,}$$

$$\tilde{J}, \tilde{J}_\varepsilon : [0,1] \rightarrow L^2(Q_T)\mathrm{,}$$

$$\tilde{W}, \tilde{W}_\varepsilon : [0,1] \rightarrow C\left( [0,T];H^2(\mathbb {T})\right) \mathrm{,}$$

with

$$\begin{aligned} (\tilde{u}_\varepsilon , \tilde{J}_\varepsilon , \tilde{W}_\varepsilon ) \sim (\check{u}_\varepsilon , \check{J}_\varepsilon , \check{W}_\varepsilon ), \end{aligned}$$

(5.13)

such that

$$\begin{aligned} \tilde{u}_\varepsilon (\omega )&\rightarrow \tilde{u}(\omega ) \quad \text{ as } \quad \varepsilon \searrow 0 \quad \text{ in } \quad {C^{\frac{1}{8}-, \frac{1}{2}- }}(Q_T), \end{aligned}$$

(5.14a)

$$\begin{aligned} \tilde{J}_\varepsilon (\omega )&\rightharpoonup \tilde{J}(\omega ) \quad \text{ as } \quad \varepsilon \searrow 0 \quad \text{ in } \quad L^2\left( Q_T\right) , \end{aligned}$$

(5.14b)

$$\begin{aligned} \tilde{W}_\varepsilon (\omega )&\rightarrow \tilde{W}(\omega ) \,\, \text{ as } \quad \varepsilon \searrow 0 \quad \text{ in } \quad C\left( [0,T];H^2(\mathbb {T})\right) \end{aligned}$$

(5.14c)

for every $\omega \in [0,1]$. It holds that

$$\begin{aligned} \tilde{u}\in L^{p'}(\tilde{\Omega }, \tilde{\mathbb {F}}, \tilde{\mathbb {P}}; C^{\gamma , \frac{\gamma }{4}}(Q_T)),\quad \text{ for } \text{ all } \quad \gamma \in \left( 0,\tfrac{1}{2}\right) \quad \text{ and } \quad p'\in \left[ 1,2p\right) . \end{aligned}$$

(5.15)

Proof

It suffices to show the tightness of the laws $\mu _{\check{u}_\varepsilon }$, $\mu _{\check{J}_\varepsilon }$, and $\mu _{\check{W}_\varepsilon }$ corresponding to the families $\check{u}_\varepsilon$, $\check{J}_\varepsilon$, and $\check{W}_\varepsilon$. The proposition then follows by applying [35, Theorem 2].

Tightness of the law $\mu _{\check{W}}$ follows because $\mu _{\check{W}}$ is a Radon measure in the Polish space $C\left( [0,T];H^2(\mathbb {T})\right)$ implying regularity from interior and thus tightness.

Tightness for $\mu _{\check{u}_\varepsilon }$ as a family of measures on ${C^{\frac{1}{8}-, \frac{1}{2}- }}(Q_T)$ is a direct consequence of Corollary 5.2, in particular estimate (5.12) (see also Remark 5.3) .

By Markov’s inequality we have for any $R \in (0,\infty )$, using conservation of mass (cf. Remark 3.3) and the Sobolev embedding theorem,

$$\begin{aligned}&\check{\mathbb {P}}\left\{ \left\Vert J_\varepsilon \right\Vert _{L^2(Q_T)} > R\right\} \\&\quad \leqq \frac{1}{R}\check{\mathbb {E}} \left\Vert \check{J}_\varepsilon \right\Vert _{L^2(Q_T)} \\&\quad \leqq \frac{C}{R} \, \check{\mathbb {E}} \left[ 1 + \left\Vert F_\varepsilon (\check{u}_\varepsilon ) \partial _x^3 \check{u}_\varepsilon \right\Vert _{L^2(Q_T)}^{n+2}\right] \\&\quad {\mathop {\leqq }\limits ^{(4.24)}} \frac{C}{R} \, \mathbb {E}\left[ 1 + \left|{\mathcal {A}(u^{(0)})} \right|^{2(n+2)q} + \left\Vert G_0(u^{(0)}) \right\Vert _{L^1(\mathbb {T})}^{(n+2)q} + \left\Vert \partial _x u^{(0)} \right\Vert _{L^2(\mathbb {T})}^{n+2}\right] \rightarrow 0 \quad \end{aligned}$$

$\text{ as } R \rightarrow \infty$, where we have used Proposition 4.7.

Finally, (5.15) holds by virtue of (5.14a) and Corollary 5.2. This finishes the proof. $\square$

For what follows, we define $\tilde{\mathbb {F}} = (\tilde{\mathcal {F}}_t)_{t \in [0,T]}$ as the augmented filtration of $(\tilde{\mathcal {F}}_t')_{t \in [0,T]}$, where

$$\begin{aligned} \tilde{\mathcal {F}}_t' {:}{=} \sigma \big (\tilde{u}(t'), \tilde{W}(t') : 0 \leqq t' \big ). \end{aligned}$$

(5.16)

We further define

$$\begin{aligned} \tilde{\beta }^k_\varepsilon (t) {:}{=} \frac{ \big (\sigma _k,\tilde{W}_\varepsilon (t) \big )_{H^2(\mathbb {T})}}{ \left\Vert \sigma _k \right\Vert _{H^2(\mathbb {T})}^2} \quad \text{ and } \quad \tilde{\beta }^k(t) {:}{=} \frac{\big ( \sigma _k,\tilde{W}(t) \big ) _{H^2(\mathbb {T})}}{ \left\Vert \sigma _k \right\Vert _{H^2(\mathbb {T})}^2}. \end{aligned}$$

(5.17)

Then, we work in the filtered probability space

$$\begin{aligned} \big (\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {F}},\tilde{\mathbb {P}}\big ) {:}{=} \big ([0,1],\mathcal {B}([0,1]), (\tilde{\mathcal {F}}_t )_{t \in [0,T]},\lambda _{[0,1]}\big ). \end{aligned}$$

Lemma 5.5

The $\tilde{\beta }^k$ are mutually independent, standard, real-valued $\big (\tilde{\mathcal {F}}_t\big )$-Wiener processes.

Proof

The reasoning is quite standard and contained in detail for instance in [21, Proposition 5.3] or [10, Proof of Proposition 5.4] or [17, Lemma 5.7]. $\square$

Proposition 5.6

(Weak convergence, a-priori estimate, non-negativity, continuity) Let $T \in (0,\infty )$, $n \in \left[ \frac{8}{3},4\right)$, $\varepsilon \in (0,1]$, $p>1$, $q > 1$ satisfying $q \geqq \max \left\{ \frac{1}{4-n},\frac{n-2}{2n-5}\right\}$. Suppose that

$$\begin{aligned} u^{(0)} \in L^{(n+2)p}\left( \Omega ,\mathcal {F}_0,\mathbb {P};H^1(\mathbb {T})\right) \end{aligned}$$

such that $u^{(0)} \geqq 0$, $\mathrm {d}\mathbb {P}$-almost surely, $\mathbb {E}\left|{\mathcal {A}(u^{(0)})} \right|^{2(n+2)pq} < \infty$, and $\mathbb {E}\left\Vert G_0\left( u^{(0)}\right) \right\Vert _{L^1(\mathbb {T})}^{(n+2)pq} < \infty$. With the notation of Proposition 5.4, up to taking subsequences, it holds that

$$\begin{aligned} \partial _x \tilde{u}_\varepsilon \overset{*}{\rightharpoonup } \partial _x \tilde{u} \quad&\text{ in } \quad L^{(n+2)p}\big (\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^\infty (0,T;L^2(\mathbb {T}))\big ),\\ \partial _x^2 \tilde{u}_\varepsilon \overset{*}{\rightharpoonup } \partial _x^2 \tilde{u} \quad&\text{ in } \quad L^{2(n+2)pq}\big (\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(Q_T)\big ),\\ \tilde{J}_\varepsilon \overset{*}{\rightharpoonup } \tilde{J} \quad&\text{ in } \quad L^{(n+2)p}\big (\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(Q_T)\big ) \end{aligned}$$

as $\varepsilon \searrow 0$, and the energy-entropy estimate

$$\begin{aligned}&\tilde{\mathbb {E}} \left[ \sup _{t \in [0,T]} \left\Vert \partial _x \tilde{u}(t) \right\Vert _{L^2(\mathbb {T})}^{(n+2)p} + \sup _{t \in [0,T]} \left\Vert G_0\left( \tilde{u}(t)\right) \right\Vert _{L^1(\mathbb {T})}^{(n+2)pq} + \left\Vert \partial _x^2 \tilde{u} \right\Vert _{L^2(Q_T)}^{2(n+2)pq}+\Vert \tilde{J}\Vert _{L^2(Q_T)}^{(n+2)p}\right] \nonumber \\&\quad \leqq C \, \mathbb {E}\left[ 1 + \left|{\mathcal {A}(u^{(0)})} \right|^{2(n+2)pq} + \left\Vert G_0(u^{(0)}) \right\Vert _{L^1(\mathbb {T})}^{(n+2)pq} + \left\Vert \partial _x u^{(0)} \right\Vert _{L^2(\mathbb {T})}^{(n+2)p}\right] , \end{aligned}$$

(5.18)

is satisfied, with a constant $C < \infty$ depending only on $p,q,\sigma =(\sigma _k)_{k\in \mathbb {Z}}, n,L,$ and T. It holds $\tilde{u} \geqq 0$ and $\left|\{\tilde{u} = 0\} \right| = 0$, $\mathrm {d}\mathbb {P}$-almost surely. Furthermore, $\tilde{u}$ is a continous $H^1_\mathrm {w}(\mathbb {T})$-valued process.

Proof

From Proposition 5.4 we derive that, up to taking subsequences, we have $\tilde{u}_\varepsilon (\omega ) \rightarrow \tilde{u}(\omega )$ in $C^{\gamma , \frac{\gamma }{4}}\left( Q_T\right)$ for $\gamma < \frac{1}{2}$, and $\tilde{J}_\varepsilon (\omega ) \rightharpoonup \tilde{J}(\omega )$ in $L^2\left( Q_T\right)$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely. From (4.24) of Proposition 4.7 we can conclude by compactness that, up to taking subsequences once more, we have

$$\begin{aligned} \partial _x \tilde{u}_\varepsilon&\overset{*}{\rightharpoonup } \tilde{u}_1 \quad \text{ in } \quad L^{(n+2)p}\big (\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^\infty (0,T;L^2(\mathbb {T}))\big ), \\ \partial _x^2 \tilde{u}_\varepsilon&\overset{*}{\rightharpoonup } \tilde{u}_2 \quad \text{ in } \quad L^{2(n+2)pq}\big (\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(Q_T)\big ), \\ \tilde{J}_\varepsilon&\overset{*}{\rightharpoonup } \tilde{J}_1 \quad \text{ in } \quad L^{(n+2)p}\big (\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(Q_T)\big ) \end{aligned}$$

as $\varepsilon \searrow 0$. From (5.12) of Corollary 5.2 we have that $\tilde{\mathbb {E}} \left\Vert \tilde{u}_\varepsilon \right\Vert _{C^{\gamma ,\frac{\gamma }{4}}(Q_T)}^{p'}$ for $p' \in [1,2p)$ is uniformly bounded in $\varepsilon$. Hence, we obtain with Vitali’s convergence theorem $\tilde{\mathbb {E}} \left\Vert \tilde{u}_\varepsilon - \tilde{u} \right\Vert _{C^{\gamma ,\frac{\gamma }{4}}(Q_T)} \rightarrow 0$ as $\varepsilon \searrow 0$. Thus, we have for $j \in \{1,2\}$ and $\tilde{\phi } \in L^\infty \big (\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};C^2(Q_T)\big )$,

$$\begin{aligned} \tilde{\mathbb {E}} \left\langle \partial _x^j \tilde{u}_\varepsilon - \tilde{u}_j,\tilde{\phi }\right\rangle _{L^1\left( 0,T;L^1(\mathbb {T})\right) \times L^\infty \left( 0,T;L^\infty (\mathbb {T})\right) } \rightarrow 0 \quad \text{ as } \quad \varepsilon \searrow 0 \end{aligned}$$

by weak-$*$-convergence while by the Cauchy-Schwarz inequality

$$\begin{aligned}&\left|\tilde{\mathbb {E}} \left\langle \tilde{u}_\varepsilon - \tilde{u},\partial _x^j \tilde{\phi }\right\rangle _{L^1\left( 0,T;L^1(\mathbb {T})\right) \times L^\infty \left( 0,T;L^\infty (\mathbb {T})\right) } \right| \\&\quad \leqq \tilde{\mathbb {E}} \left\Vert \tilde{u}_\varepsilon - \tilde{u} \right\Vert _{C(Q_T)} \left\Vert \partial _x^j \tilde{\phi } \right\Vert _{L^\infty (\tilde{\Omega }\times Q_T)} \rightarrow 0 \quad \text{ as } \quad \varepsilon \searrow 0. \end{aligned}$$

This implies $\tilde{u}_j = \partial _x^j \tilde{u}$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely.

Since $\tilde{J}_\varepsilon (\omega ) \rightharpoonup \tilde{J}(\omega )$ in $L^2\left( Q_T\right)$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely, and

$$\tilde{J}_\varepsilon \in L^{(n+2)p}\big (\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(Q_T)\big )$$

is uniformly bounded, we have

$$\begin{aligned} \big (\tilde{J}_\varepsilon ,\phi \big )_{L^2(Q_T)} \rightarrow \big (\tilde{J},\phi \big )_{L^2(Q_T)} \end{aligned}$$

strongly in $L^{(n+2)p'}\big (\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}}\big )$ for any $p'<p$ and $\phi \in L^2\left( Q_T\right)$. Since also $\big (\tilde{J}_\varepsilon ,\phi \big )_{L^2(Q_T)} \rightharpoonup \big (\tilde{J}_1,\phi \big )_{L^2(Q_T)}$ weakly in $L^{(n+2)p'}\big (\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}}\big )$ we have $\tilde{J}_1 = \tilde{J}$.

Estimate (5.18) follows from (4.24) of Proposition 4.7 by weak lower-semicontinuity of the appearing norms and Fatou’s lemma.

The fact that $\tilde{u} \geqq 0$ and $\left|\{\tilde{u} = 0\} \right| = 0$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely, is a consequence of $\tilde{u} \in C(Q_T)$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely, $\tilde{u}(0,\cdot ) \geqq 0$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely, and finiteness of $\tilde{\mathbb {E}} \sup _{t \in [0,T]} \left\Vert G_0\left( \tilde{u}\right) \right\Vert _{L^1(\mathbb {T})}^{(n+2)pq}$ because of (5.18).

The fact that $\tilde{u}$ is an $H^1_\mathrm {w}(\mathbb {T})$-valued process follows at once from $\tilde{u} \in L^\infty \left( 0,T;H^1(\mathbb {T})\right) \cap C^{\gamma , \frac{\gamma }{4}}(Q_T)$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely. $\square$

Proposition 5.7

Assume the conditions of Proposition 5.6 and

$$u^{(0)} \in L^{2np}\left( \Omega ,\mathcal {F}_0,\mathbb {P};H^1(\mathbb {T})\right) .$$

Then, the distributional derivative $\partial _{x}^{3}\tilde{u}$ fulfills $\partial _{x}^{3}\tilde{u}\in L_{\mathrm {loc}}^{2}\left( \left\{ \tilde{u}>0\right\} \right)$ and we can identify $\tilde{J}_{\varepsilon }=F_{\varepsilon }\left( \tilde{u}_{\varepsilon }\right) (\partial _{x}^{3}\tilde{u}_{\varepsilon })$ and $\tilde{J}=\mathbb {1}_{\{\tilde{u}>0\}}\tilde{u}^{\frac{n}{2}}(\partial _{x}^{3}\tilde{u})$. In particular, the following energy-dissipation estimate holds

$$\begin{aligned}&\tilde{\mathbb {E}} \Big [\Vert \mathbb {1}_{\{\tilde{u}>0\}}\tilde{u}^{\frac{n}{2}}(\partial _{x}^{3}\tilde{u})\Vert _{L^2(Q_T)}^{(n+2)p}\Big ] \nonumber \\&\quad \leqq C \, \mathbb {E}\left[ 1 + \left\Vert \partial _x u^{(0)} \right\Vert _{L^2(\mathbb {T})}^{(n+2)p} + \left|{\mathcal {A}(u^{(0)})} \right|^{2(n+2)pq} + \left\Vert G_0(u^{(0)}) \right\Vert _{L^1(\mathbb {T})}^{(n+2)pq}\right] . \end{aligned}$$

(5.19)

Proof

The proof follows the lines of the proof of [21, Proposition 5.6], with the additional complication of taking care of the approximation $F_\varepsilon (r)$ of the square root of the mobility $F(r)=\left|r \right|^{\frac{n}{2}}$.

Since the laws of $(\check{J}_{\varepsilon }, \check{u}_\varepsilon )$ and $(\tilde{J}_{\varepsilon }, \tilde{u}_\varepsilon )$ coincide (cf. Proposition 5.4), it holds for $\phi \in C^{\infty }(Q_{T})$,

$$\begin{aligned} 0=\mathbb {E}\left|\left( \check{J}_{\varepsilon }-F_{\varepsilon }(\check{u}_\varepsilon )\,\partial _{x}^{3}\check{u}_\varepsilon ,\phi \right) _{L^{2}(Q_{T})} \right|=\tilde{\mathbb {E}}\left|\left( \tilde{J}_{\varepsilon }-F_{\varepsilon }\left( \tilde{u}_{\varepsilon }\right) \partial _{x}^{3}\tilde{u}_{\varepsilon },\phi \right) _{L^{2}(Q_{T})} \right|, \end{aligned}$$

which implies $\tilde{J}_{\varepsilon }=F_{\varepsilon }\left( \tilde{u}_{\varepsilon }\right) \partial _{x}^{3}\tilde{u}_{\varepsilon }$.

Because of estimate (4.24) of Proposition 4.7, $\tilde{J}_{\varepsilon }=F_{\varepsilon }(\tilde{u}_{\varepsilon })\,\partial _{x}^{3}\tilde{u}_{\varepsilon }$, and (3.3), it holds for fixed $r>0$,

$$\begin{aligned}&\tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}(\partial _{x}^{3}\tilde{u}_{\varepsilon })^{2}\,\mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}<\frac{r}{2}\right\} \cap \left\{ \tilde{u}>r\right\} }\,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\quad \leqq \frac{2^n}{r^n}\tilde{\mathbb {E}}\int _{0}^{T}\int _{\left\{ \tilde{u}_{\varepsilon }(t)>\frac{r}{2}\right\} }F_{\varepsilon }^2(\tilde{u}_{\varepsilon })\,(\partial _{x}^{3}\tilde{u}_{\varepsilon })^{2}\,\mathrm {d}x\,\mathrm {d}t\leqq C(r,u_{0}), \end{aligned}$$

(5.20)

where $C(r,u_{0})<\infty$ is independent of $\varepsilon$. Hence, by taking a subsequence again denoted by $\tilde{u}_{\varepsilon }$, it holds that for some $\eta ^r \in L^{2}( \tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{2}(Q_{T}))$,

$$\begin{aligned} \partial _{x}^{3}\tilde{u}_{\varepsilon }\,\mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}<\frac{r}{2}\right\} \cap \left\{ \tilde{u}>r\right\} }\rightharpoonup \tilde{\eta }^r \,\mathbb {1}_{\left\{ \tilde{u}>r\right\} }\quad \text{ as }\quad \varepsilon \searrow 0\quad \end{aligned}$$

(5.21)

$\text{ in }~ L^{2}(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{2}(Q_{T})).$

We next show that $\tilde{\eta }^r=\partial _{x}^{3}\tilde{u}$ on $\left\{ \tilde{u}> r\right\}$ for any $r > 0$, that is, for almost every $(\omega ,t) \in \Omega \times [0,T]$ and all $\tilde{\varphi } \in C_\mathrm {c}^\infty \left( \{\tilde{u}(\omega ,t) > r\}\right)$ we have

$$\begin{aligned} \int _\mathbb {T}\tilde{\eta }^r \, \tilde{\varphi } \, \mathrm {d}x = - \int _\mathbb {T}\tilde{u} \, \partial _x^3\tilde{\varphi } \, \mathrm {d}x. \end{aligned}$$

It is enough to show that for almost every $\omega \in \Omega$ and all $\tilde{\varphi } \in C_\mathrm {c}^\infty \left( \{\tilde{u}(\omega ) > r\}\right)$ we have

$$\begin{aligned} \int _0^T \int _\mathbb {T}\tilde{\eta }^r \, \tilde{\varphi } \, \mathrm {d}x \, \mathrm {d}t = - \int _0^T \int _\mathbb {T}\tilde{u} \, \partial _x^3\tilde{\varphi } \, \mathrm {d}x \, \mathrm {d}t. \end{aligned}$$

For every $\omega \in \Omega$ and $N \in \mathbb {N}$ let $\tilde{\chi }_N \in C^\infty (\mathbb {R}^2)$ such that $\tilde{\chi }_N(t,x) = 0$ for $(t,x) \in \{\tilde{u}(\omega ) > r\}^\mathrm {c}$ and

$$\begin{aligned} \tilde{\chi }_N(t,x) = {\left\{ \begin{array}{ll} 1 &{} \text{ if } \mathrm {dist}\left( (t,x),\partial \{\tilde{u}(\omega )> r\}\right) \geqq \frac{1}{N} \\ 0 &{} \text{ if } \mathrm {dist}\left( (t,x),\partial \{\tilde{u}(\omega ) > r\}\right) \leqq \frac{1}{N+1} \end{array}\right. } \end{aligned}$$

for all $(t,x) \in \{\tilde{u}(\omega ) > r\}$. Then it is enough to show that for almost every $\omega \in \Omega$, all $\tilde{\varphi } \in C_\mathrm {c}^\infty \left( \mathbb {R}^2\right)$, and all $N \in \mathbb {N}$ it holds

$$\begin{aligned} \int _0^T \int _\mathbb {T}\tilde{\eta }^r \, \tilde{\varphi } \, \tilde{\chi }_N \, \mathrm {d}x \, \mathrm {d}t = - \int _0^T \int _\mathbb {T}\tilde{u} \, \partial _x^3 (\tilde{\varphi } \, \tilde{\chi }_N) \, \mathrm {d}x \, \mathrm {d}t. \end{aligned}$$

Observe that for every $\varphi \in C^\infty _\mathrm {c}(\{\tilde{u}(\omega )>r\})$, we have

$$\begin{aligned} \mathrm {dist}(\mathrm {supp}\ \varphi , \partial \{\tilde{u}(\omega )>r\})>0. \end{aligned}$$

Hence, for N sufficiently large,

$$\begin{aligned} \tilde{\chi }_N \tilde{\varphi }=\tilde{\varphi }\quad \text {for} \quad \tilde{\varphi }\in C^\infty _\mathrm {c}(\{\tilde{u}(\omega )>r\}). \end{aligned}$$

The equality above in particular holds if for all $\tilde{\varphi } \in C_\mathrm {c}^\infty \left( \mathbb {R}^2\right)$, $N \in \mathbb {N}$, and $\tilde{\theta } \in L^\infty (\tilde{\Omega },\tilde{\mathbb {F}}, \tilde{\mathbb {P}})$ we have

$$\begin{aligned} \tilde{\mathbb {E}} \int _0^T \int _\mathbb {T}\tilde{\eta }^r \, \tilde{\varphi } \, \tilde{\chi }_N \, \tilde{\theta } \, \mathrm {d}x \, \mathrm {d}t = - \tilde{\mathbb {E}} \int _0^T \int _\mathbb {T}\tilde{u} \, \partial _x^3(\tilde{\varphi } \, \tilde{\chi }_N \, \tilde{\theta }) \, \mathrm {d}x \, \mathrm {d}t. \end{aligned}$$

Since

$$\tilde{\varphi } \, \tilde{\chi }_N \, \tilde{\theta } \in L^\infty (\tilde{\Omega },\tilde{\mathbb {F}}, \tilde{\mathbb {P}};C^3(Q_T)),$$

it suffices to prove that for $\tilde{\zeta }\in L^{\infty }(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};C^{3}(Q_{T}))$ such that ${\mathrm {supp}}_{(t,x)\in Q_{T}}\tilde{\zeta }\Subset \left\{ \tilde{u}>r\right\}$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely, we have

$$\begin{aligned} \tilde{\mathbb {E}} \int _0^T \int _\mathbb {T}\tilde{\eta }^r \, \tilde{\zeta } \, \mathrm {d}x \, \mathrm {d}t = - \tilde{\mathbb {E}} \int _0^T \int _\mathbb {T}\tilde{u} \, \partial _x^3\tilde{\zeta } \, \mathrm {d}x \, \mathrm {d}t. \end{aligned}$$

(5.22)

Therefore, take $\tilde{\zeta } \in L^{\infty }(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};C^{3}(Q_{T}))$ such that ${\mathrm {supp}}_{(t,x)\in Q_{T}}\tilde{\zeta }\Subset \left\{ \tilde{u}>r\right\}$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely. Then, integration by parts gives

$$\begin{aligned}&\tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}(\partial _{x}^{3}\tilde{u}_{\varepsilon })\,\mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}<\frac{r}{2}\right\} \cap \left\{ \tilde{u}>r\right\} }\,\tilde{\zeta }\,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\quad = \tilde{\mathbb {E}}\left[ \mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}<\frac{r}{2}\right\} }\int _{0}^{T}\int _{\mathbb {T}}(\partial _{x}^{3}\tilde{u}_{\varepsilon }) \, \tilde{\zeta } \,\mathrm {d}x\,\mathrm {d}t\right] \nonumber \\&\quad = -\tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\tilde{u}_{\varepsilon } \, \mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}<\frac{r}{2}\right\} } \, \partial _{x}^{3}\tilde{\zeta } \,\mathrm {d}x\,\mathrm {d}t\nonumber \\&\quad \rightarrow -\tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\tilde{u} \, \partial _{x}^{3}\tilde{\zeta } \,\mathrm {d}x\,\mathrm {d}t\quad \text{ as }\quad \varepsilon \searrow 0 \end{aligned}$$

(5.23)

for any $r>0$, where in the last line we have applied Vitali’s convergence theorem. Indeed, by Proposition 5.4 it holds that

$$\begin{aligned} \tilde{u}_{\varepsilon } \, \mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}<\frac{r}{2}\right\} } \,(\partial _{x}^{3}\tilde{\zeta })\rightarrow \tilde{u}\, \partial _{x}^{3}\tilde{\zeta } \quad \text{ as }\quad \varepsilon \searrow 0, \end{aligned}$$

$\mathrm {d}\tilde{\mathbb {P}}\otimes \mathrm {d}t\otimes \mathrm {d}x$-almost everywhere, and for some $p'>1$, we have that

$$\begin{aligned}&\tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\left|\tilde{u}_{\varepsilon } \, \mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}<\frac{r}{2}\right\} } \, (\partial _{x}^{3}\tilde{\zeta }) \right|^{p'}\mathrm {d}x\,\mathrm {d}t\\&\quad \leqq \Vert \partial _{x}^{3}\tilde{\zeta }\Vert _{{L^{\infty }\left( \tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^\infty (Q_{T})\right) }}^{p'} \left( \tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}(\tilde{u}_{\varepsilon })^{p'} \, \mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}<\frac{r}{2}\right\} } \, \mathrm {d}x \, \mathrm {d}t\right) \\&\quad \leqq C(u_0), \end{aligned}$$

for some constant $C(u_0)$ independent of $\varepsilon ,r$, where we have used the $\varepsilon$-uniform bound (4.24) of Proposition 4.7. Therefore, by (5.21) and (5.23), we get (5.22), which in turn implies that

$$\begin{aligned} \tilde{\eta }^r=\partial _{x}^{3}\tilde{u} \end{aligned}$$

(5.24)

on $\left\{ \tilde{u}> r\right\}$ for every $r > 0$.

Now, take $\tilde{\phi }\in L^{\infty }(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{\infty }(Q_{T}))$, $r>0$, $\varepsilon \in (0,1]$, and separate according to

$$\begin{aligned} I(\varepsilon ){:}{=}\tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\tilde{J}_{\varepsilon }\,\tilde{\phi }\,\mathrm {d}x\,\mathrm {d}t=I_{1}(r,\varepsilon )+I_{2}(r,\varepsilon ), \end{aligned}$$

(5.25)

where because of $\tilde{u}\geqq 0$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely (cf. Proposition 5.6), we may choose

$$\begin{aligned} I_{1}(r,\varepsilon )&{:}{=}\tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\tilde{J}_{\varepsilon }\,\mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}<\frac{r}{2}\right\} \cap \left\{ \tilde{u}>r\right\} }\,\tilde{\phi }\,\mathrm {d}x\,\mathrm {d}t,\\ I_{2}(r,\varepsilon )&{:}{=}\tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\tilde{J}_{\varepsilon }\,\mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}\geqq \frac{r}{2}\right\} \cup \left\{ 0<\tilde{u}\leqq r\right\} }\,\tilde{\phi }\,\mathrm {d}x\,\mathrm {d}t. \end{aligned}$$

Then, we separate according to

$$\begin{aligned} I_{1}(r,\varepsilon )-\tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\tilde{u}^{\frac{n}{2}}\,(\partial _{x}^{3}\tilde{u})\,\mathbb {1}_{\left\{ \tilde{u}>r\right\} }\,\mathrm {d}x\,\mathrm {d}t=I_{11}(r,\varepsilon )+I_{12}(r,\varepsilon ), \end{aligned}$$

(5.26)

where

$$\begin{aligned} I_{11}(r,\varepsilon )&{:}{=}\tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\left( F_{\varepsilon }\left( \tilde{u}_{\varepsilon }\right) -\tilde{u}^{\frac{n}{2}}\right) (\partial _{x}^{3}\tilde{u}_{\varepsilon })\,\mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}<\frac{r}{2}\right\} \cap \left\{ \tilde{u}>r\right\} }\,\tilde{\phi }\,\mathrm {d}x\,\mathrm {d}t,\\ I_{12}(r,\varepsilon )&{:}{=}\tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\left( \partial _{x}^{3}\tilde{u}_{\varepsilon }-\partial _{x}^{3}\tilde{u}\right) \mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}<\frac{r}{2}\right\} \cap \left\{ \tilde{u}>r\right\} }\,\tilde{u}^{\frac{n}{2}}\,\tilde{\phi }\,\mathrm {d}x\,\mathrm {d}t, \end{aligned}$$

where we have used $\tilde{J}_{\varepsilon }=F_{\varepsilon }\left( \tilde{u}_{\varepsilon }\right) \partial _{x}^{3}\tilde{u}_{\varepsilon }$. For the first integral, we note that

$$\begin{aligned} I_{11}(r,\varepsilon )&\leqq \left( \tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\left( \frac{F_{\varepsilon }\left( \tilde{u}_{\varepsilon }\right) -\tilde{u}^{\frac{n}{2}}}{F_{\varepsilon }\left( \tilde{u}_{\varepsilon }\right) }\right) ^{2}\mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}<\frac{r}{2}\right\} \cap \left\{ \tilde{u}>r\right\} }\,\mathrm {d}x\,\mathrm {d}t\right) ^{\frac{1}{2}}\nonumber \\&\times \left( \tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}F_{\varepsilon }^{2}\left( \tilde{u}_{\varepsilon }\right) (\partial _{x}^{3}\tilde{u}_{\varepsilon })^{2}\,\mathrm {d}x\,\mathrm {d}t\right) \left\Vert \tilde{\phi } \right\Vert _{L^{\infty }\left( \tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{\infty }(Q_{T})\right) }\nonumber \\&{\mathop {\leqq }\limits ^{(3.3), (4.24)}} \frac{C}{F_{\varepsilon }\left( \frac{r}{2}\right) }\left( \tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\left( F_{\varepsilon }\left( \tilde{u}_{\varepsilon }\right) -\tilde{u}^{\frac{n}{2}}\right) ^{2}\,\mathrm {d}x\,\mathrm {d}t\right) ^{\frac{1}{2}}\left\Vert \tilde{\phi } \right\Vert _{L^{\infty }\left( \tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{\infty }(Q_{T})\right) },\nonumber \\&\rightarrow 0\quad \text{ as }\quad \varepsilon \searrow 0, \end{aligned}$$

(5.27)

where $C<\infty$ is independent of r, $\varepsilon$, and $\tilde{\phi }$, and we used Vitali’s convergence theorem and Proposition 4.7.

Because of

$$\begin{aligned} \tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\left( \tilde{u}^{\frac{n}{2}}\,\tilde{\phi }\right) ^{2}\mathrm {d}x\,\mathrm {d}t&\leqq C \left( 1 + \tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\tilde{u}^{n+2} \, \mathrm {d}x\,\mathrm {d}t\right) \left\Vert \tilde{\phi } \right\Vert _{L^{\infty }\left( \tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{\infty }(Q_{T})\right) }^{2} \\&{\mathop {\leqq }\limits ^{(5.18)}} C(u^{(0)}) \left\Vert \tilde{\phi } \right\Vert _{L^{\infty }\left( \tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{\infty }(Q_{T})\right) }^{2} < \infty , \end{aligned}$$

where the Sobolev embedding, mass conservation (Remark 3.3), and Proposition 5.6 have been applied, we have $\tilde{u}^{\frac{n}{2}} \, \tilde{\phi }\in L^{2}\left( \tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{2}(Q_{T})\right)$ and by the weak convergence stated in (5.21), combined with (5.24), it follows that $I_{12}(r,\varepsilon )\rightarrow 0$ as $\varepsilon \searrow 0$, which in conjunction with (5.26) and (5.27) gives

$$\begin{aligned} I_{1}(r,\varepsilon )\rightarrow \tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\tilde{u}^{\frac{n}{2}} \, (\partial _{x}^{3}\tilde{u})\,\mathbb {1}_{\left\{ \tilde{u}>r\right\} }\,\tilde{\phi }\,\mathrm {d}x\,\mathrm {d}t\quad \text{ as }\quad \varepsilon \searrow 0. \end{aligned}$$

(5.28)

The integral $I_{2}(r,\varepsilon )$ in (5.25) can be estimated as

$$\begin{aligned} \left|I_{2}(r,\varepsilon ) \right| \leqq&C\left( \tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}F_{\varepsilon }^{2}(\tilde{u}_{\varepsilon })\left( \partial _{x}^{3}\tilde{u}_{\varepsilon }\right) ^{2}\mathrm {d}x\,\mathrm {d}t\right) \left\Vert \tilde{\phi } \right\Vert _{L^{\infty }\left( \tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{\infty }(Q_{T})\right) }\\&\times \left( \tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}} \mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}\geqq \frac{r}{2}\right\} \cup \left\{ 0<\tilde{u}\leqq r\right\} }\,\mathrm {d}x\,\mathrm {d}t\right) ^{\frac{1}{2}}\\ {\mathop {\leqq }\limits ^{(4.24)}}&C(u^{(0)}) \, \left\Vert \tilde{\phi } \right\Vert _{L^{\infty }\left( \tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{\infty }(Q_{T})\right) }\\&\times \left( \tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}} \mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}\geqq \frac{r}{2}\right\} \cup \left\{ 0<\tilde{u}\leqq r\right\} } \,\mathrm {d}x\,\mathrm {d}t\right) ^{\frac{1}{2}}, \end{aligned}$$

where $C, C(u^{(0)}) < \infty$ are independent of r, $\varepsilon$, and $\tilde{\phi }$, and Proposition 4.7 has been used. Then, we note that

$$\begin{aligned} \mathbb {1}_{\left\{ \left\Vert \tilde{u}_{\varepsilon }-\tilde{u} \right\Vert _{L^{\infty }(Q_{T})}\geqq \frac{r}{2}\right\} \cup \left\{ 0<\tilde{u}\leqq r\right\} } \rightarrow \mathbb {1}_{\left\{ 0<\tilde{u}\leqq r\right\} }\quad \text{ as }\quad \varepsilon \searrow 0, \end{aligned}$$

$\mathrm {d}\tilde{\mathbb {P}}\otimes \mathrm {d}t\otimes \mathrm {d}x$-almost everywhere, due to Proposition 5.4. Therefore, by bounded convergence it follows that

$$\begin{aligned} \limsup _{\varepsilon \searrow 0}I_{2}(r,\varepsilon )&\leqq C\,\left\Vert \tilde{\phi } \right\Vert _{L^{\infty }\left( \tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{\infty }(Q_{T})\right) }\times \left( \tilde{\mathbb {E}} \left|\left\{ 0 < \tilde{u} \leqq r\right\} \right| \right) ^{\frac{1}{2}}, \end{aligned}$$

which, in combination with (5.25) and (5.28), leads to

$$\begin{aligned}&\limsup _{\varepsilon \searrow 0}\left| I(\varepsilon )-\tilde{\mathbb {E}}\int _{0}^{T}\int _{\mathbb {T}}\tilde{u}^{n}\,(\partial _{x}^{3}\tilde{u})\,\mathbb {1}_{\left\{ \tilde{u}>r\right\} }\,\tilde{\phi }\,\mathrm {d}x\,\mathrm {d}t\right| \\&\quad \leqq C \left( \tilde{\mathbb {E}} \left|\left\{ 0 < \tilde{u} \leqq r\right\} \right|\right) ^{\frac{1}{2}} \left\Vert \tilde{\phi } \right\Vert _{L^{\infty }\left( \tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{\infty }(Q_{T})\right) }, \end{aligned}$$

where $C < \infty$ is independent of r. In the limit $r \searrow 0$, we infer by monotone convergence $\limsup _{r \searrow 0} \tilde{\mathbb {E}} \left|\left\{ 0 < \tilde{u} \leqq r\right\} \right| = 0$, which finishes the proof. $\square$

5.2 Recovering the SPDE

In this section, we give the proof of the main result Theorem 2.2:

Proof of Theorem 2.2

We first note that the results of §5.1 can be applied due to the assumptions of Theorem 2.2, with p in §5.1 replaced by $\frac{p}{n+2}$ from the statement of Theorem 2.2.

We will show that $\{ (\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {F}},\tilde{\mathbb {P}}), \ (\tilde{\beta }_k)_{k \in \mathbb {Z}},\ \tilde{u}(0), \tilde{u} \}$ is a solution of (2.4). The fact that $\tilde{u}(0)$ has the same distribution as $u^{(0)}$ follows from Proposition 5.4 and (5.13) therein. By Proposition 5.6, $\tilde{u}$ is an $\tilde{\mathbb {F}}$-adapted continuous $H^1_\mathrm {w}(\mathbb {T})$-valued process, so that in particular $\tilde{u}(0)$ is $\tilde{\mathcal {F}}_0$-measurable. The fact that the $\tilde{\beta }^k$ are independent real-valued standard $\tilde{\mathbb {F}}$-Wiener processes is the content of Lemma 5.5. The Hölder regularity stated in (2.7) is a consequence of (5.15) of Proposition 5.4. Moreover, (i) and (ii) from Definition 2.1 follow from (5.18) and (5.19). Hence, we only have to show (iii).

Denote by $\tilde{u}_\varepsilon$ the sequence of Proposition 5.4 and notice that by (5.13) we have that $\tilde{u}_\varepsilon$ satisfies (4.23), that is, for $\varphi \in C^\infty (\mathbb {T})$ we have, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely, that

$$\begin{aligned} \left( \tilde{u}_\varepsilon (t), \varphi \right) _{L^2(\mathbb {T})}= & {} \left( \tilde{u}_\varepsilon (0), \varphi \right) _{L^2(\mathbb {T})} + \int _0^t \left( F^2_\varepsilon \left( \tilde{u}_\varepsilon (t')\right) \partial ^3_x \tilde{u}_\varepsilon (t'), \partial _x\varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t' \nonumber \\&- \frac{1}{2} {\sum _{k\in \mathbb {Z}}}\int _0^t \left( \sigma _k F_\varepsilon '\left( \tilde{u}_\varepsilon (t')\right) \partial _x \left( \sigma _k F_\varepsilon \left( \tilde{u}_\varepsilon (t')\right) \right) , \partial _x \varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t' \nonumber \\&- {\sum _{k\in \mathbb {Z}}}\int _0^t \left( \sigma _k F_\varepsilon \left( \tilde{u}_\varepsilon (t')\right) , \partial _x\varphi \right) _{L^2(\mathbb {T})} \mathrm {d}\tilde{\beta }^k_\varepsilon (t'), \end{aligned}$$

(5.29)

for all $t \in [0,T]$. We claim that for all $t \in [0,T]$,

$$\begin{aligned}&\left( \tilde{u}_\varepsilon (t), \varphi \right) _{L^2(\mathbb {T})} \rightarrow \left( \tilde{u}(t), \varphi \right) _{L^2(\mathbb {T})}, \end{aligned}$$

(5.30a)

$$\begin{aligned}&\int _0^t {\int _{\mathbb {T}}}F^2_\varepsilon \left( \tilde{u}_\varepsilon (t')\right) \left( \partial ^3_x \tilde{u}_\varepsilon (t')\right) \partial _x\varphi \, \mathrm {d}x \, \mathrm {d}t' \nonumber \\&\quad \rightarrow \int _0^t \int _{\left\{ \tilde{u}(t')>0\right\} } \left( \tilde{u}(t')\right) ^n \left( \partial ^3_x \tilde{u}(t')\right) \partial _x\varphi \, \mathrm {d}x \, \mathrm {d}t', \end{aligned}$$

(5.30b)

and

$$\begin{aligned}&\frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t \left( \sigma _k F_\varepsilon '\left( \tilde{u}_\varepsilon (t')\right) \partial _x \left( \sigma _k F_\varepsilon \left( \tilde{u}_\varepsilon (t')\right) \right) , \partial _x \varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t' \nonumber \\&\quad \rightarrow \frac{1}{2} \sum _{k \in \mathbb {Z}} \int _0^t \left( \sigma _k F_0'\left( \tilde{u}(t')\right) \partial _x \left( \sigma _k F_0\left( \tilde{u}(t')\right) \right) , \partial _x \varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t', \end{aligned}$$

(5.30c)

as $\varepsilon \rightarrow 0$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely.

Argument for (5.30a) Since by Proposition 5.4 it holds that $\left\Vert \tilde{u}_\varepsilon - \tilde{u} \right\Vert _{C(Q_T)} \rightarrow 0$ as $\varepsilon \searrow 0$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely, it follows that

$$\begin{aligned}&\sup _{t \leqq T} \left|\left( \tilde{u}_\varepsilon (t), \varphi \right) _{L^2(\mathbb {T})} - \left( \tilde{u}(t), \varphi \right) _{L^2(\mathbb {T})} \right| \leqq \left\Vert \tilde{u}_\varepsilon - \tilde{u} \right\Vert _{C(Q_T)} \left\Vert \varphi \right\Vert _{L^1(\mathbb {T})} \rightarrow 0 \end{aligned}$$

as $\varepsilon \searrow 0, \mathrm {d}\tilde{\mathbb {P}}$-almost surely.

Argument for (5.30b) By Proposition 5.7, we can identify

$$\begin{aligned} \int _0^t {\int _{\mathbb {T}}}F^2_\varepsilon \left( \tilde{u}_\varepsilon (t')\right) \left( \partial ^3_x \tilde{u}_\varepsilon (t')\right) \partial _x\varphi \, \mathrm {d}x \, \mathrm {d}t' = \int _0^t \int _{\mathbb {T}} F_\varepsilon \left( \tilde{u}_\varepsilon (t')\right) \tilde{J}_\varepsilon (t') \, \partial _x\varphi \, \mathrm {d}x \, \mathrm {d}t' \end{aligned}$$

and

$$\begin{aligned} \int _0^t \int _{\left\{ \tilde{u}(t')>0\right\} } \left( \tilde{u}(t')\right) ^n \left( \partial ^3_x \tilde{u}(t')\right) \partial _x\varphi \, \mathrm {d}x \, \mathrm {d}t' = \int _0^t \int _{\mathbb {T}} \left( \tilde{u}(t')\right) ^{\frac{n}{2}} \tilde{J}(t') \, \partial _x\varphi \, \mathrm {d}x \, \mathrm {d}t', \end{aligned}$$

so that the limit in (5.30b) follows from (3.3), $\tilde{u}_\varepsilon \rightarrow \tilde{u}$ in $C(Q_T)$, and $\tilde{J}_\varepsilon \rightharpoonup \tilde{J}$ in $L^2(Q_T)$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely.

Argument for (5.30c) Applying the chain rule and integration by parts yields

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _0^t \left( \sigma _k F_\varepsilon '\left( \tilde{u}_\varepsilon (t')\right) \partial _x \left( \sigma _k F_\varepsilon \left( \tilde{u}_\varepsilon (t')\right) \right) , \partial _x \varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t' \\&\quad =-\sum _{k \in \mathbb {Z}} \int _0^t \left( \int _1^{\tilde{u}_\varepsilon (t')}(F'_\varepsilon )^2(r) \, \mathrm {d}r, \partial _x(\sigma _k^2 \partial _x \varphi ) \right) _{L^2(\mathbb {T})} \mathrm {d}t' \nonumber \\&\qquad +\frac{1}{2}\sum _{k \in \mathbb {Z}} \int _0^t \left( F_\varepsilon '\left( \tilde{u}_\varepsilon (t')\right) F_\varepsilon \left( \tilde{u}_\varepsilon (t')\right) , (\partial _x\sigma _k^2)\partial _x \varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t'. \end{aligned}$$

Since by Proposition 5.4 we have $\left\Vert \tilde{u}_\varepsilon - \tilde{u} \right\Vert _{C(Q_T)} \rightarrow 0$ as $\varepsilon \searrow 0$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely, and by reversing the application of the chain rule and integration by parts, we obtain

$$\begin{aligned}&\sum _{k \in \mathbb {Z}} \int _0^t \left( \sigma _k F_\varepsilon '\left( \tilde{u}_\varepsilon (t')\right) \partial _x \left( \sigma _k F_\varepsilon \left( \tilde{u}_\varepsilon (t')\right) \right) , \partial _x \varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t' \\&\quad \rightarrow {-\sum _{k \in \mathbb {Z}} \int _0^t \left( \int _1^{\tilde{u}(t')}(F'_0)^2(r) \, \mathrm {d}r, \partial _x(\sigma _k^2 \partial _x \varphi ) \right) _{L^2(\mathbb {T})} \mathrm {d}t'} \\&\qquad +{\frac{1}{2}\sum _{k \in \mathbb {Z}} \int _0^t \left( F_0'\left( \tilde{u}(t')\right) F_0 \left( \tilde{u}(t')\right) , (\partial _x\sigma _k^2)\partial _x \varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t'} \nonumber \\&\quad ={\sum _{k \in \mathbb {Z}} \int _0^t \left( \sigma _k F_0'\left( \tilde{u}(t')\right) \partial _x \left( \sigma _k F_0 \left( \tilde{u}(t')\right) \right) , \partial _x \varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t'} \end{aligned}$$

uniformly in $t \in [0,T]$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely. We note that the application of the chain rule is justified due to the $\mathrm {d}\tilde{\mathbb {P}}$-almost surely boundedness of $u_\varepsilon$ and $\tilde{u}$ and the local Lipschitz continuity of the occurring nonlinearities.

Hence, we have showed (5.30). It follows that for all $t \in [0,T]$,

$$\begin{aligned} M_{\varepsilon ,\varphi }(t){:}{=} - \sum _{k \in \mathbb {Z}} \int _0^t \left( \sigma _k F_\varepsilon \left( \tilde{u}_\varepsilon (t')\right) , \partial _x\varphi \right) _{L^2(\mathbb {T})} \mathrm {d}\beta ^k_\varepsilon (t') \rightarrow M_ \varphi (t) \end{aligned}$$

(5.31)

as $\varepsilon \searrow 0$, $\mathrm {d}\tilde{\mathbb {P}}$-almost surely, for some process $M_\varphi (t)$. Since the limits in (5.30) are continuous processes, so is $M_\varphi$. By virtue of Proposition 4.7, we can choose $\kappa \in (1,2)$ such that for any $\varepsilon \in (0,1)$

$$\begin{aligned}&\tilde{\mathbb {E}} \sup _{t \in [0,T]} | M_{\varepsilon , \varphi }(t)|^{2 \kappa } \\&\quad \,\,\,\leqq C \ \tilde{\mathbb {E}} \left( \sum _{k \in \mathbb {Z}} \int _0^T \left( \sigma _k F_\varepsilon \left( \tilde{u}_\varepsilon (t)\right) , \partial _x\varphi \right) _{L^2(\mathbb {T})}^2\mathrm {d}t \right) ^\kappa \\&\quad \,\,\,\leqq C \, T^\kappa \left( \sum _{k \in \mathbb {Z}} \left\Vert \sigma _k \right\Vert _{L^2(\mathbb {T})}^2\right) ^\kappa \tilde{\mathbb {E}} \sup _{t \in [0,T]} \left\Vert F_\varepsilon \left( \tilde{u}(t)\right) \right\Vert _{L^\infty (\mathbb {T})}^{2\kappa } \left\Vert \partial _x\varphi \right\Vert _{L^2(\mathbb {T})}^{2\kappa } \\&{\mathop {\leqq }\limits ^{(\hbox {2.2e}), (3.3), (4.24)}} C \left( 1 + \tilde{\mathbb {E}} \left|{\mathcal {A}(\tilde{u}^{(0)})} \right|^{\kappa n} + \tilde{\mathbb {E}} \sup _{t \in [0,T]} \left\Vert \partial _x \tilde{u}(t) \right\Vert _{L^2(\mathbb {T})}^{\kappa n}\right) \\&\quad {\mathop {\leqq }\limits ^{(4.24)}} C\left( u^{(0)}\right) , \end{aligned}$$

where $C\left( u^{(0)}\right) < \infty$ is independent of $\varepsilon$ and where Remark 3.3 (mass conservation), the Sobolev embedding theorem, and Poincaré’s inequality have been applied. In particular, we have

$$\begin{aligned}&\sup _{\varepsilon \in (0,1)} \tilde{\mathbb {E}} \!\!\sup _{t \in [0,T]} \!| M_{\varepsilon , \varphi }(t)|^{2 \kappa } \!+\!\!\! \sup _{\varepsilon \in (0,1)} \tilde{\mathbb {E}} \left( \sum _{k \in \mathbb {Z}} \!\int _0^T \left( \sigma _k F_\varepsilon \left( \tilde{u}_\varepsilon (t)\right) , \partial _x\varphi \right) _{L^2(\mathbb {T})}^2\mathrm {d}t\! \right) ^\kappa \!\!<\! \infty , \end{aligned}$$

(5.32)

which implies by Fatou’s lemma that

$$\begin{aligned} \tilde{\mathbb {E}}|M_\varphi (t)|^{2\kappa } < \infty . \end{aligned}$$

In order to complete the proof, we only have to show that

$$\begin{aligned} M_\varphi (t) = - \sum _{k \in \mathbb {Z}} \int _0^t \left( \sigma _k F_0 \left( \tilde{u}(t')\right) , \partial _x\varphi \right) _{L^2(\mathbb {T})} \mathrm {d}\beta ^k(t'). \end{aligned}$$

(5.33)

For this, it suffices by virtue of [33, Proposition A.1] to verify that for $0 \leqq t' \leqq t \leqq T$ and $k \in \mathbb {Z}$, we have

$$\begin{aligned}&\tilde{\mathbb {E}}\left[ M_\varphi (t) - M_\varphi (t') \bigg | \tilde{\mathcal {F}}_{t'}\right] = 0, \end{aligned}$$

(5.34)

$$\begin{aligned}&\tilde{\mathbb {E}}\left[ \left( M_\varphi (t)\right) ^2 - \left( M_\varphi (t')\right) ^2 - \sum _{k \in \mathbb {Z}} \int _{t'}^t \left( \sigma _k F_0 \left( \tilde{u}(t'',\cdot )\right) , \partial _x\varphi \right) _{L^2(\mathbb {T})}^2 \mathrm {d}t'' \bigg | \tilde{\mathcal {F}}_{t'}\right] = 0, \end{aligned}$$

(5.35)

$$\begin{aligned}&\tilde{\mathbb {E}}\left[ \tilde{\beta }^k(t) \, M_\varphi (t) - \tilde{\beta }^k(t') \, M_\varphi (t') + \int _{t'}^t \left( \sigma _k F_0 \left( \tilde{u}(t'',\cdot )\right) , \partial _x\varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t'' \bigg | \tilde{\mathcal {F}}_{t'}\right] = 0. \end{aligned}$$

(5.36)

Notice that $M_{\varepsilon ,\varphi }$ as defined in (5.31) is a square-integrable $\tilde{\mathcal {F}}_{\varepsilon ,t}'$-martingale, where $\tilde{\mathcal {F}}_{\varepsilon ,t}' {:}{=} \sigma \left( \tilde{u}_\varepsilon (t'), \tilde{W}_\varepsilon (t') : 0 \leqq t' \leqq t\right)$. Hence, we infer that for $0 \leqq t' \leqq t \leqq T$ and any

$$\begin{aligned} \Phi \in C\left( C(Q_{t'}) \times C\left( [0,t'];H^2(\mathbb {T})\right) ;[0,1]\right) \end{aligned}$$

it holds that

$$\begin{aligned}&\tilde{\mathbb {E}}\left[ \left( \tilde{M}_{\varepsilon ,\varphi }(t) - \tilde{M}_{\varepsilon ,\varphi }(t')\right) {\tilde{\Phi }_{\varepsilon }}(t')\right] = 0, \\&\tilde{\mathbb {E}}\left[ \left( \left( \tilde{M}_{\varepsilon ,\varphi }(t)\right) ^2 - \left( \tilde{M}_{\varepsilon ,\varphi }(t')\right) ^2 - \sum _{k \in \mathbb {Z}} \int _{t'}^t \left( \sigma _k F_\varepsilon \left( \tilde{u}_\varepsilon (t'',\cdot )\right) , \partial _x\varphi \right) _{L^2(\mathbb {T})}^2 \mathrm {d}t''\right) {\tilde{\Phi }_{\varepsilon }}(t')\right] = 0, \\&\tilde{\mathbb {E}}\left[ \left( \tilde{\beta }^k_\varepsilon (t) \, \tilde{M}_{\varepsilon ,\varphi }(t) - \tilde{\beta }^k_\varepsilon (t') \, \tilde{M}_{\varepsilon ,\varphi }(t') + \int _{t'}^t \left( \sigma _k F_\varepsilon \left( \tilde{u}_\varepsilon (t'',\cdot )\right) , \partial _x\varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t''\right) {\tilde{\Phi }_{\varepsilon }}(t')\right] = 0, \end{aligned}$$

where

$$\begin{aligned} {\tilde{\Phi }_{\varepsilon }}(t') {:}{=} \Phi \left( \tilde{u}_\varepsilon |_{[0,t'] }, \tilde{W}_\varepsilon |_{[0,t'] }\right) . \end{aligned}$$

By the convergence stated in (5.14a) and (5.14c) of Proposition 5.4 and (5.31), combined with the uniform integrability of all the terms appearing in the expectations above, which in turn follows from (5.32), we conclude that

$$\begin{aligned}&\tilde{\mathbb {E}}\left[ \left( M_{\varphi }(t) - M_{\varphi }(t')\right) \tilde{\Phi }(t')\right] = 0, \\&\tilde{\mathbb {E}}\left[ \left( \left( M_{\varphi }(t)\right) ^2 - \left( M_{\varphi }(t')\right) ^2 - \sum _{k \in \mathbb {Z}} \int _{t'}^t \left( \sigma _k F_0 \left( u(t'',\cdot )\right) , \partial _x\varphi \right) _{L^2(\mathbb {T})}^2 \mathrm {d}t''\right) {\tilde{\Phi }}(t')\right] = 0, \\&\tilde{\mathbb {E}}\left[ \left( \tilde{\beta }^k(t) \, M_{\varphi }(t) - \tilde{\beta }^k(t') \, M_{\varphi }(t') + \int _{t'}^t \left( \sigma _k F_0 \left( \tilde{u}(t'',\cdot )\right) , \partial _x\varphi \right) _{L^2(\mathbb {T})} \mathrm {d}t''\right) {\tilde{\Phi }}(t')\right] = 0, \end{aligned}$$

where

$$\begin{aligned} {\tilde{\Phi }}(t') {:}{=} \Phi \left( \tilde{u}|_{[0,t'] }, \tilde{W} |_{[0,t'] }\right) . \end{aligned}$$

Since $\Phi$ was arbitrary, we conclude that (5.34), (5.35), and (5.36) are valid with $\tilde{\mathcal {F}}_{t'}$ replaced by $\tilde{\mathcal {F}}_{t'}'$. The passage from $\tilde{\mathcal {F}}_{t'}'$ to $\tilde{\mathcal {F}}_{t'}$ follows by a standard continuity argument employing Vitali’s convergence theorem. This finishes the proof. $\square$