Energy regularized models for logarithmic SPDEsJ. Cui, D. Hou and Z. Qiao

# Energy regularized models for logarithmic SPDEs and their numerical
approximations11footnotemark: 1††thanks: The research is partially supported
by the Hong Kong Research Grant Council ECS grant 25302822, start-up funds
(P0039016, P0041274) from Hong Kong Polytechnic University and the CAS AMSS-
PolyU Joint Laboratory of Applied Mathematics. D. Hou’s work is partially
supported by NSFC grant 12001248, NSF of Jiangsu Province grant BK20201020,
and Hong Kong Polytechnic University grant 1-W00D. Z. Qiao’s work is partially
supported by Hong Kong Research Council RFS grant RFS2021-5S03 and GRF grant
15302919, Hong Kong Polytechnic University grant 4-ZZLS, and CAS AMSS-PolyU
Joint Laboratory of Applied Mathematics.

Jianbo Cui22footnotemark: 2 Department of Applied Mathematics, The Hong Kong
Polytechnic University, Hung Hom, Hong Kong. Corresponding author:
<EMAIL_ADDRESS>Dianming Hou33footnotemark: 3 School of Mathematics
and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China.
Current address:Department of Applied Mathematics, The Hong Kong Polytechnic
University, Hung Hom, Hong Kong<EMAIL_ADDRESS>Zhonghua
Qiao44footnotemark: 4 Department of Applied Mathematics, The Hong Kong
Polytechnic University, Hung Hom, Hong Kong<EMAIL_ADDRESS>
###### Abstract

Understanding the properties of the stochastic phase field models is crucial
to model processes in several practical applications, such as soft matters and
phase separation in random environments. To describe such random evolution,
this work proposes and studies two mathematical models and their numerical
approximations for parabolic stochastic partial differential equation (SPDE)
with a logarithmic Flory–Huggins energy potential. These multiscale models are
built based on a regularized energy technique and thus avoid possible
singularities of coefficients. According to the large deviation principle, we
show that the limit of the proposed models with small noise naturally recovers
the classical dynamics in deterministic case. Moreover, when the driving noise
is multiplicative, the Stampacchia maximum principle holds which indicates the
robustness of the proposed model. One of the main advantages of the proposed
models is that they can admit the energy evolution law and asymptotically
preserve the Stampacchia maximum bound of the original problem. To numerically
capture these asymptotic behaviors, we investigate the semi-implicit
discretizations for regularized logrithmic SPDEs. Several numerical results
are presented to verify our theoretical findings.

###### keywords:

parabolic stochastic partial differential equation; logarithmic Flory–Huggins
potential; Stampacchia maximum bound; energy regularization; semi-implicit
discretization

60H15; 65M15; 35R60; 65M12; 60H35;

##  1 Introduction

The phase field model, such as the Allen-Cahn equation [1], has become an
popular tool to model processes involving thin interface layers between almost
homogeneous regions emerged from many scientific, engineering, and industrial
applications. The unknown solution of the model, also called the order
parameter, could represent the normalized density of the involved phase for
phase separation process [9]. For example, the sets of $\\{u=1\\}$ and
$\\{u=-1\\}$ often denotes the pure regions. Nowadays, more attention has been
paid on stochastic phase field models (see, e.g., [3, 22, 11, 32, 43, 44])
since uncertainty and randomness may arise from thermal effects, impurities of
materials, intrinsic instabilities of dynamics, etc.

In this paper, we are mainly interested in the following parabolic SPDE with a
Flory–Huggins logarithmic potential,

(1) $\displaystyle du(t)-\sigma Au(t)dt+F_{\log}^{\prime}(u(t))dt=\epsilon
B(u(t))dW(t).$

Here $A$ is the Laplacian operator defined on a Lipschitz bounded domain
$\mathcal{D}\subset\mathbb{R}^{d},d\leq 3,$ equipped with the homogeneous
Neumann boundary condition or periodic boundary condition, and $\sigma>0$ is
associated with the diffuse interface width. The logarithmic Flory–Huggins
potential $\int_{\mathcal{D}}F_{\log}(u)dx$ is determined by the following
function

$F_{log}(\xi):=(1+\xi)\ln(1+\xi)+(1-\xi)\ln(1-\xi)-c\xi^{2},\;\xi\in(-1,1),\;c\geq
0,$

which has two global minima points in the interior of the physically relevant
domain. The driving noise $W$ is a cylindrical Wiener process, $\epsilon$ is
the noise intensity and $B$ is a suitable Hilbert–Schmidt operator. This kind
of noise is important in soft matters since the elements constituting soft
matter (colloidal particles, polymer molecules, etc.) are doing Brownian
motion, and their behaviour determines the dynamical response of soft matter
(also known as the fluctuation-dissipation theorem in [25]). It is known that
the free energy with the logarithmic potential is often considered to be more
physically realistic than that with a smooth polynomial potential since the
free energy with the logarithmic potential can be derived from the regular or
ideal solution theories [25]. The main difficulties of building mathematical
models for logarithmic SPDEs lie on the singularity of the logarithmic
Flory–Huggins potential and random effect which emerge from the polymer
science community.

In deterministic case ($\epsilon=0$), some strategies have been successfully
used to deal with the logarithmic Flory–Huggins potential, for example,
replacing it by a polynomial of even degree with a strictly positive dominant
coefficient or using the penalization method (see, e.g., [23]). For parabolic
SPDEs of phase field type in the smooth nonlinearity case, there have also
been fruitful theoretical and numerical results [2, 4, 7, 8, 5, 12, 18, 17,
19, 20, 30, 28, 33, 36, 35, 38, 39, 42], just to name a few. However, much
less is known in the stochastic case, especially on the modeling and
simulation of (1). In [24, 31], the authors obtained the well-posedness of a
stochastic Cahn–Hilliard equation with logarithmic potential by imposing
additional reflection measures. A variational approach based on suitable
monotonicity and compactness arguments to study parabolic SPDEs with
logarithmic potential driving by multiplicative noise was proposed (see [40,
41] for the stochastic Cahn–Hilliard equation and [6] for the stochastic
Allen–Cahn equation). To better understand the properties of SPDE with the
Flory–Huggins potential, one useful and important tool is through the
numerical approximation. However, this approach requires a well-defined
mathematical model which is also easy to handle and maintains as much of the
nature and properties of the original system as possible. This is very
challenging in the stochastic case for the soft matter dynamics with the
logarithmic Flory–Huggins potential. First, different from the deterministic
case, one can not expect that the classical maximum principle of parabolic PDE
or the convexity of the energy functional could be used to obtain the point-
wise bound (see, e.g., [15, 14, 16, 26, 27]) due to the random effect. Second,
the existing results either use the polynomial approximation or add the
reflection measure for the drift coefficient of the stochastic dynamics, which
may changes the properties of the original problem, especially on the ideal
solution theory [25] and energy evolution. Last but not least, it is still
unknown how to design a suitable numerical approximation and analyze its
convergence properties for parabolic SPDE with the Flory–Huggins potential.

To overcome the above issues, we will first adopt the energy regularization
approach in [10, 21] to propose a convergent and well-posed multiscale problem
with the regularization parameter $\delta$. In particular, we will prove the
convergence of the regularized problems in two cases, i.e., the small noise
($\epsilon\sim o(1)$) in Case 1 and a special multiplicative noise in Case 2.
We would like mention that Case 1 is inspired by the large deviation principle
of random perturbations of classical dynamical systems [29] and the
observation that the simulation of (1) may escape the maximum bound in
applications. The assumptions in Case 2 is similar to that of [31] where the
Stampacchia maximum principle is expected to hold (see [13] for the
Stampacchia maximum principle). Then applying numerical discretization to the
regularized problem will not suffer from the numerical vacuum issue.
Specifically, we study the stabilized semi-implicit numerical method, where
the spectral Galerkin method is chosen as the space discretization. To analyze
the convergence of the energy regularized numerical method, the key ingredient
of our approach lies on the estimate of the probability that the solution
escapes Stampacchia maximum bound. As a consequence, the error analysis of the
proposed methods is presented.

We would like to remark that this energy regularized technique could directly
provide a well-posed and natural regularized model for (1) which may also
works for fourth order logarithmic SPDE, like the stochastic Cahn–Hilliard
equation with logarithmic potential. Meanwhile, the Stampacchia maximum
principle compensates for the deficiencies of regularization technique that it
may not capture the singular behavior of the solution. The numerical
simulation also indicates the advantages of the proposed energy regularized
models and numerical approximations. For example, it is observed that the
Stampacchia maximum principle is asymptotically preserved by the energy
regularized scheme for the first time. The limit of (1) with small noise could
recover the Allen–Cahn equation with Flory–Huggins potential (see, e.g.,
[14]).

The rest of this paper is organized as follows. In section 2, we present the
main setting and the useful properties of regularized parabolic SPDEs. In
section 3, we propose semi-implicit discretizations for the regularized
models, and study their convergence. Several numerical examples are shown in
section 4 to verify the theoretical findings.

##  2 Regularized parabolic SPDEs with logarithmic potential

This section is devoted to presenting the useful properties of the regularized
SPDEs with logarithmic potential via the energy regularization technique.
Before that, let us first introduce the main setting of this paper.

###  2.1 Setting

Let $\bs\geq 0,T>0$. Assume that there exists an orthonormal basis of $e_{i}$
of $H=L^{2}(\mathcal{D})$ such that $-Ae_{i}=\lambda_{i}e_{i}$ with
$\lambda_{0}=0<\lambda_{1}<\cdots<\lambda_{j}<\lambda_{j+1}\to\infty.$ For
example, $\mathcal{D}$ can be chosen as the cuboid in
$\mathbb{R}^{d},d\in\mathbb{N}^{+}.$ Denote $H^{\bs}:=W^{\bs,2}(\mathcal{D})$
which is equivalent to the fractional Sobolev space
$\mathbb{L}H\oplus(-A)^{\frac{\bs}{2}}$ (see, e.g., [20]). Here $\mathbb{L}H$
is the projection from $H$ to Span$\\{e_{0}\\}$. Let $\mathbb{H}$ and
$\widetilde{H}$ be two Hilbert space. The set of bounded linear operators from
$\HH$ to $\widetilde{H}$ is denoted by $\mathcal{L}(\HH,\widetilde{H})$, and
the set of Hilbert–Schmidt linear operators from $\HH$ to $\widetilde{H}$ is
denoted by $\mathcal{L}_{2}(\HH,\widetilde{H})$, with associated norm denoted
by $\|\cdot\|_{\mathcal{L}_{2}(\HH,\widetilde{H})}$. We also denote by
$\mathcal{L}_{2}(\mathbb{H}):=\mathcal{L}_{2}(\HH,\HH).$ For convenience, we
assume that the initial value $u(0)\in H^{\bs}$ for some $\bs\geq 0$
satisfying $\|u(0)\|_{E}<1$, where $E=\mathcal{C}(\mathcal{D})$. In this
paper, the following two type of noises will be considered. In Case 1, we
assume that $\epsilon\sim o(1)$ and the operator $B$ satisfies that

(2) $\displaystyle\|B(u)-B(v)\|_{\mathcal{L}_{2}(H,H^{\gamma})}$
$\displaystyle\leq C_{\bs}\|u-v\|_{H^{\gamma}},$
$\displaystyle\|B(u)\|_{\mathcal{L}_{2}(H,H^{\gamma})}$ $\displaystyle\leq
C_{\bs}(1+\|u\|_{H^{\gamma}})$

for any $0\leq\gamma\leq\bs$ and some $C_{\bs}\geq 0$. In Case 2, we assume
that $\epsilon\sim O(1)$, and $B$ is defined by

$\displaystyle B(v)e_{i}:=h_{i}(v),\;h_{i}\in
W_{0}^{\bs+1,\infty}[-1,1],\;\sum_{i\in\mathbb{N}}\|h_{i}\|_{W^{\bs+1,\infty}_{0}[-1,1]}^{2}<\infty.$

As a consequence, Case 2 also implies (2) with some $\bs\geq 0$ (see [6]).

Now we introduce the energy regularized SPDE,

(3) $\displaystyle du^{\delta}(t)-\sigma
Au^{\delta}(t)dt+(F_{\log}^{\delta})^{\prime}(u^{\delta}(t))dt=\epsilon
B(u^{\delta}(t))dW(t),\;u^{\delta}(0)=u(0)$

with a regularized logarithmic potential function

$\displaystyle F_{log}^{\delta}(\xi):=$
$\displaystyle\frac{1}{2}(\xi+1)\ln((\xi+1)^{2}+\delta)+\sqrt{\delta}\arctan(\frac{\xi+1}{\sqrt{\delta}})-\xi$
$\displaystyle-(\frac{1}{2}(\xi-1)\ln((\xi-1)^{2}+\delta)+\sqrt{\delta}\arctan(\frac{\xi-1}{\sqrt{\delta}})-\xi)-c\xi^{2},c\geq
0.$

Here $\delta>0$ is a small parameter. The mild solution $u^{\delta}$ of (3) is
a adapted stochastic process satisfying that for $t\in[0,T],$

$\displaystyle
u^{\delta}(t)=S(t)u(0)-\int_{0}^{t}S(t-s)(F_{\log}^{\delta})^{\prime}(u^{\delta}(s))ds+\epsilon\int_{0}^{t}S(t-s)B(u^{\delta}(s))ds,\;a.s.$

Here $S(\cdot)$ is the semigroup generated by $A$ on $H$. We denote
$F_{\log,2}^{\delta}(\xi):=-c\xi^{2},F_{\log,1}^{\delta}=F_{\log}^{\delta}-F_{\log,2}^{\delta}.$
Furthermore, direct calculations yields that

$\displaystyle(F_{log}^{\delta})^{\prime}(\xi)$
$\displaystyle=\frac{1}{2}\ln((\xi+1)^{2}+\delta)-\frac{1}{2}\ln((\xi-1)^{2}+\delta)-2c\xi,$
$\displaystyle(F_{log}^{\delta})^{\prime\prime}(\xi)$
$\displaystyle=\frac{2(1-\xi^{2})+2\delta}{((1+\xi)^{2}+\delta)((1-\xi)^{2}+\delta)}-2c.$

As we can see, this regularization will let the drift coefficient satisfy the
Lipschitz condition. Thus, similar to [21], the modified energy
$H_{\delta}(v)=\frac{\sigma}{2}\|\nabla
v\|^{2}+\int_{\mathcal{D}}F_{\log}^{\delta}(v(\xi))d\xi$ is well-defined.
However, as a cost, the bound of the first derivative of
$F_{\log}^{\delta}\sim O(\ln(\frac{1}{\delta}))$ and that of higher
derivatives may depend on $\frac{1}{\delta}$ polynomially.

###  2.2 A priori bound of regularized SPDE with logarithmic potential

In this part, we present some useful properties, including the well-posedness
and moment bounds in the energy space, of (3). In what follows, let $\sigma=1$
for the convenience of the presentation. In order to apply the Itô formula
rigorously, one could use the spectral Galerkin approximation of (3) and then
taking limit on the Galerkin projection parameter. For convenience, this
procedure has been omitted in this paper.

######  Proposition 1.

Let Setting 2.1 hold with some $\bs\geq 0$. There exists a unique mild
solution of (3) satisfying for any $p\geq 1$,

$\displaystyle\sup_{t\in[0,T]}\|u^{\delta}(t)\|_{L^{2p}(\Omega;H)}^{2p}\leq
C(p,T,u(0)).$

Let Setting 2.1 hold with some $\bs\geq 1$. In addition assume that
$\epsilon\sim O(\delta^{\frac{1}{2}})$ in Case 1. It holds that

$\displaystyle\sup_{t\in[0,T]}\|H_{\delta}(u^{\delta}(t))\|_{L^{p}(\Omega;\mathbb{R})}\leq
C(p,T,u(0)).$

######  Proof 2.1.

The proof of existence and uniqueness of the mild solution is standard due to
the Lipschitz continuity of $F_{\log}^{\delta}$ and the assumption (2) on $B$
(see, e.g., [34]). It suffices to show the desired a priori bound. By applying
Itô’s formula to $\frac{1}{2}\|u^{\delta}(t)\|^{2}$ and using the integration
by parts, it follows that

$\displaystyle\frac{1}{2}\|u^{\delta}(t)\|^{2}=$
$\displaystyle\frac{1}{2}\|u(0)\|^{2}-\int_{0}^{t}\<\nabla
u^{\delta}(s),\nabla
u^{\delta}(s)\>ds-\int_{0}^{t}\<u^{\delta}(s),(F_{\log}^{\delta})^{\prime}(u^{\delta}(s))\>ds$
$\displaystyle+\frac{\epsilon^{2}}{2}\int_{0}^{t}\|B(u^{\delta}(s))\|_{\mathcal{L}_{2}(H)}^{2}ds+\int_{0}^{t}\epsilon\<u^{\delta}(s),B(u^{\delta}(s))dW(s)\>.$

Notice that $\ln(\frac{(\xi+1)^{2}+\delta}{(\xi-1)^{2}+\delta})$ is monotone
on $(-1,1)$ and that $\ln(\frac{(\xi+1)^{2}+\delta}{(\xi-1)^{2}+\delta})\xi>0$
when $|\xi|\geq 1$. By taking the $p$th moment, using Burkholder’s inequality,
Young’s inequality and Hölder’s inequality, it holds that for $p\geq 1,$

$\displaystyle\tps\|u^{\delta}(t)\|_{L^{2p}(\Omega;H)}^{2p}+\Big{\|}\int_{0}^{t}\|\nabla
u^{\delta}(s)\|^{2}ds\Big{\|}_{L^{p}(\Omega;\mathbb{R})}^{p}+\Big{\|}\int_{0}^{t}\<\ln(\frac{(u^{\delta}(s)+1)^{2}+\delta}{(u^{\delta}(s)-1)^{2}+\delta}),u^{\delta}(s)\>ds\Big{\|}_{L^{p}(\Omega;\mathbb{R})}^{p}$
$\displaystyle\leq
C(p,c,T)(1+\epsilon^{2p})\int_{0}^{t}(1+\|u^{\delta}(s)\|_{L^{2p}(\Omega;H)}^{2p})ds.$

Then Gronwall’s inequality leads to

(4)
$\displaystyle\sup_{s\in[0,T]}\|u^{\delta}(t)\|_{L^{2p}(\Omega;H)}^{2p}+\Big{\|}\int_{0}^{T}\|\nabla
u^{\delta}(s)\|^{2}ds\Big{\|}_{L^{p}(\Omega;\mathbb{R})}^{p}$
$\displaystyle+\Big{\|}\int_{0}^{T}\<\ln(\frac{(u^{\delta}+1)^{2}+\delta}{(u^{\delta}-1)^{2}+\delta}),u^{\delta}(s)\>ds\Big{\|}_{L^{p}(\Omega;\mathbb{R})}^{p}\leq
C(p,c,T)\|u(0)\|_{L^{2p}(\Omega;H)}^{2p}.$

We proceed to estimating the bound of the modified energy in Case 1 and Case
2. By applying the Ito’s formula to $H_{\delta}(u^{\delta}(t))$, we have that

$\displaystyle H_{\delta}(u^{\delta}(t))=$ $\displaystyle
H_{\delta}(u^{\delta}(0))-\int_{0}^{t}\|Au^{\delta}(s)-(F_{\log}^{\delta})^{\prime}(u^{\delta}(s))\|^{2}ds$
$\displaystyle+\int_{0}^{t}\<(F_{log}^{\delta})^{\prime}(u^{\delta}(s)),\epsilon
B(u^{\delta}(s))dW(s)\>+\epsilon\int_{0}^{t}\<\nabla u^{\delta}(s),\nabla
B(u^{\delta}(s))dW(s)\>$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\sum_{j\in\mathbb{N}}\epsilon^{2}\<\nabla(B(u^{\delta}(s))e_{j}),\nabla(B(u^{\delta}(s))e_{j})\>ds$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\epsilon^{2}\sum_{j\in\mathbb{N}}\<(F_{log}^{\delta})^{\prime\prime}(u^{\delta}(s))B(u^{\delta}(s))e_{j},B(u^{\delta}(s))e_{j}\>ds.$

In Case 1, taking $p$th moment yields that

$\displaystyle\|H_{\delta}(u^{\delta}(t))\|_{L^{p}(\Omega;\mathbb{R})}^{p}+\Big{\|}\int_{0}^{t}\|Au^{\delta}(s)-(F_{\log}^{\delta})^{\prime}(u^{\delta}(s))\|^{2}ds\Big{\|}_{L^{p}(\Omega;\mathbb{R})}^{p}$
$\displaystyle\leq
C(p,T)\|H_{\delta}(u^{\delta}(0))\|_{L^{p}(\Omega;\mathbb{R})}^{p}+C(p,T)\epsilon^{2p}\int_{0}^{t}\|\nabla
B(u^{\delta}(s))\|_{L^{2p}(\Omega;{\mathcal{L}_{2}(H)})}^{2p}ds$
$\displaystyle+C(p,T)\int_{0}^{t}\frac{\epsilon^{2p}}{\delta^{p}}\|B(u^{\delta}(s))\|_{L^{2p}(\Omega;{\mathcal{L}_{2}(H)})}^{2p}ds.$

This, together with (2) and (4), implies that for $\epsilon\sim
O(\delta^{\frac{1}{2}}),$ it holds that

$\displaystyle\sup_{t\in[0,T]}\|H_{\delta}(u^{\delta}(t))\|_{L^{p}(\Omega;\mathbb{R})}\leq
C(p,T,u(0)).$

Similarly, in Case 2 using the fact that $h_{i}(\pm 1)=0$ and $h_{i}\in
W_{0}^{\bs+1,\infty}[-1,1]$, it holds that

$\displaystyle\|H_{\delta}(u^{\delta}(t))\|_{L^{p}(\Omega;\mathbb{R})}^{p}+\Big{\|}\int_{0}^{t}\|Au^{\delta}(s)-(F_{\log}^{\delta})^{\prime}(u^{\delta}(s))\|^{2}ds\Big{\|}_{L^{p}(\Omega;\mathbb{R})}^{p}$
$\displaystyle\leq
C(p,T)\|H_{\delta}(u^{\delta}(0))\|_{L^{p}(\Omega;\mathbb{R})}^{p}+C(p,T,u(0))\epsilon^{2p}.$

Combining the above estimates, we complete the proof.

As a consequence of Proposition 1, the averaged evolution law of the modified
energy holds,

(5) $\displaystyle\E[H_{\delta}(u^{\delta}(t))]$
$\displaystyle=\E[H_{\delta}(u^{\delta}(0))]-\int_{0}^{t}\E[\|Au^{\delta}(s)-(F_{\log}^{\delta})^{\prime}(u^{\delta}(s))\|^{2}]ds$
$\displaystyle+\frac{1}{2}\epsilon^{2}\int_{0}^{t}\E[\|\nabla
B(u^{\delta}(s))\|_{\mathcal{L}_{2}(H)}^{2}]ds$
$\displaystyle+\frac{1}{2}\epsilon^{2}\int_{0}^{t}\sum_{j\in\mathbb{N}}\E[\<(F_{log}^{\delta})^{\prime\prime}(u^{\delta}(s))B(u^{\delta}(s))e_{j},B(u^{\delta}(s))e_{j}\>]ds.$

It should be remarked that the energy regularization may be not available for
the SPDE with logarithmic potential driven by the space-time white noise. This
will be studied in the future. The following is devoted to the moment bounds
of the solution in the Sobolev spaces.

######  Proposition 2.

Let Setting 2.1 hold with some $\bs\geq 1$. In addition assume that
$\epsilon\sim O(\delta^{\frac{1}{2}})$ in Case 1. Then it holds that for
$p\geq 1,$

$\displaystyle\sup_{t\in[0,T]}\|u^{\delta}(t)\|_{L^{2p}(\Omega;H^{1})}\leq
C(p,T,u(0)).$

######  Proof 2.2.

Since $\ln\xi\leq C\xi^{\theta}$ if $\xi\geq 1$ and $|\ln\xi|\leq
C\xi^{-\theta}$ if $0<\xi<1$ for $\theta>0$, it holds that

$\displaystyle|\int_{\mathcal{D}}F_{log}^{\delta}(u^{\delta}(t))dx|$
$\displaystyle\leq\Big{|}\int_{\mathcal{D}}\frac{1}{2}(u^{\delta}(t)+1)\ln((u^{\delta}(t)+1)^{2}+\delta)-(\frac{1}{2}(u^{\delta}(t)-1)\ln((u^{\delta}(t)-1)^{2}+\delta)dx\Big{|}$
$\displaystyle+\Big{|}\int_{\mathcal{D}}\sqrt{\delta}\arctan(\frac{u^{\delta}(t)+1}{\sqrt{\delta}})-\sqrt{\delta}\arctan(\frac{u^{\delta}(t)-1}{\sqrt{\delta}}))dx\Big{|}+c\|u^{\delta}(t)\|^{2}.$

The boundedness of $H_{\delta}(u^{\delta}(t))$ and $\|u^{\delta}(t)\|$ in
Proposition 1 implies the desired result.

Let Setting 2.1 hold with $\bs\geq{1+\beta}$, where $\beta\in[0,1)$. Then it
holds that for $p\geq 2,$

$\displaystyle\|u^{\delta}(t)\|_{L^{p}(\Omega;H^{\beta+1})}$
$\displaystyle\leq C(p,T,u(0))|\ln(\delta)|,$
$\displaystyle\|u^{\delta}(t)-u^{\delta}(s)\|_{L^{p}(\Omega;H)}$
$\displaystyle\leq
C(p,T,u(0))\Big{(}|\ln(\delta)|(t-s)^{\frac{\beta+1}{2}}+\epsilon(t-s)^{\frac{1}{2}}\Big{)}.$

######  Proof 2.3.

Applying the property that $\|S(t)(-A)^{\alpha}v\|\leq
C(1+t^{-\alpha})\|v\|_{H^{2\alpha}},$ $\alpha>0$ (see e.g. [34]), we have that

$\displaystyle\|(-A)^{\frac{\beta+1}{2}}u^{\delta}(t)\|$
$\displaystyle\leq\|(-A)^{\frac{\beta+1}{2}}S(t)u(0)\|+\|\int_{0}^{t}(-A)^{\frac{\beta+1}{2}}S(t-s)(F_{\log}^{\delta})^{\prime}(u^{\delta}(s))ds\|$
$\displaystyle+\|\int_{0}^{t}(-A)^{\frac{\beta+1}{2}}S(t-s)\epsilon
B(u^{\delta}(s))dW(s)\|$ $\displaystyle\leq
C\|u(0)\|_{H^{\beta+1}}+C\int_{0}^{t}(t-s)^{-\frac{\beta+1}{2}}(1+|\ln(\delta)|+\|u^{\delta}(s)\|^{2})ds$
$\displaystyle+\|\int_{0}^{t}(-A)^{\frac{\beta+1}{2}}S(t-s)\epsilon
B(u^{\delta}(s))dW(s)\|.$

Therefore, it suffices to bound the stochastic integral term. According to
Burkholder’s inequality, Minkowski inequality and (2), we obtain that for any
$p\geq 2$,

$\displaystyle\Big{\|}\int_{0}^{t}(-A)^{\frac{\beta+1}{2}}S(t-s)\epsilon
B(u^{\delta}(s))dW(s)\Big{\|}_{L^{p}(\Omega;H)}$ $\displaystyle\leq
C\epsilon\Big{(}\int_{0}^{t}\Big{(}\E\Big{[}\Big{(}\sum_{i\in\mathbb{N}}\|S(t-s)(-A)^{\frac{\beta+1}{2}}B(u^{\delta}(s))e_{i}\|^{2}\Big{)}^{\frac{p}{2}}\Big{]}\Big{)}^{\frac{2}{p}}ds\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq
C\epsilon\Big{(}\int_{0}^{t}(t-s)^{-\beta}(\E[\|(-A)^{\frac{1}{2}}B(u^{\delta}(s))\|^{p}_{\mathcal{L}_{2}(H)}])^{\frac{2}{p}}ds\Big{)}^{\frac{1}{2}}\leq
C(T,u(0),p)\epsilon.$

As a consequence, it holds that

$\displaystyle\|(-A)^{\frac{\beta+1}{2}}u^{\delta}(t)\|_{L^{p}(\Omega;H)}$
$\displaystyle\leq
C\|u(0)\|_{H^{\beta+1}}+C(T,u(0),p)(|\ln(\delta)|+\epsilon).$

Using the property that $\|(S(t)-I)v\|\leq Ct^{\alpha}\|v\|_{H^{2\alpha}}$
with $\alpha\in[0,1]$ (see e.g. [34]), Burkholder’s inequality and Proposition
1, it follows that for $0\leq s\leq t\leq T,$

$\displaystyle\|u^{\delta}(t)-u^{\delta}(s)\|_{L^{p}(\Omega;H)}$
$\displaystyle\leq$
$\displaystyle\Big{\|}\int_{s}^{t}S(t-r)(F_{\log}^{\delta})^{\prime}(u^{\delta}(r))dr\Big{\|}_{L^{p}(\Omega;H)}$
$\displaystyle+\Big{\|}\int_{0}^{s}(S(t-r)-S(s-r))(F_{\log}^{\delta})^{\prime}(u^{\delta}(r))dr\Big{\|}_{L^{p}(\Omega;H)}$
$\displaystyle+\Big{\|}\int_{s}^{t}S(t-r)\epsilon
B(u^{\delta}(r))dW(r)\Big{\|}_{L^{p}(\Omega;H)}$
$\displaystyle+\Big{\|}\int_{0}^{s}(S(t-r)-S(s-r))\epsilon
B(u^{\delta}(r))dW(r)\Big{\|}_{L^{p}(\Omega;H)}$ $\displaystyle\leq$
$\displaystyle
C(T,u(0),p)|\ln(\delta)|((t-s)+(t-s)^{\frac{\beta+1}{2}})+C(T,u(0),p)\epsilon(t-s)^{\frac{1}{2}}.$

Combining the above estimates, we complete the proof.

###  2.3 Limit of regularized SPDE with logarithmic potential

Now, we are in a position to present the strong convergence of the regularized
models. More precisely, we prove that in Case 1 ($\epsilon\to 0,\delta\to 0$),
the limit equation of (3) is

(6) $\displaystyle
du_{det}(t)-Au_{det}(t)dt+(F_{\log})^{\prime}(u_{det}(t))dt=0,\;$

while in Case 2 ($\epsilon\sim O(1),\delta\to 0$), the limit equation is

(7) $\displaystyle du(t)-Au(t)dt+(F_{\log})^{\prime}(u(t))dt=\epsilon
B(u(t))dW(t).$

The well-posedness of (7) has been shown in [6]. Our strategy is constructing
a sequence of solutions $u^{\delta}$ and showing its convergence (see, e.g.,
[21]). The key step of our approach lies on the upper bound estimate of the
probability that the solution escapes the Stampacchia maximum bound, i.e.,
$u^{\delta}(t,x)\notin[-1,1]$.

To this end, we first show the tail estimate of $u^{\delta},$ which indicates
that $(1-u^{\delta})^{-}$ and $(u^{\delta}+1)^{-}$ are small in the sense of
$L^{2}(\mathcal{O}),$ by setting up the Itô formula of some suitable
functionals as in [13]. We define the upper bound functional
$\int_{\mathcal{D}}|f^{1}(v(\xi))|^{2}d\xi:=\int_{\mathcal{D}}|(1-v(\xi))^{-}|^{2}d\xi$
and the lower bound functional
$\int_{\mathcal{D}}|f^{2}(v(\xi))|^{2}d\xi:=\int_{\mathcal{D}}|(v(\xi)+1)^{-}|^{2}d\xi$.
Let $\kappa$ be a small positive parameter. Define the regularization
approximations of $f^{j}$ by $f_{\kappa}^{j},j=1,2$ such that
$f_{\kappa}^{j}\in\mathcal{C}^{2}(\mathbb{R})$ satisfying

(8) $\displaystyle
f_{\kappa}^{j}(\xi)=0,\;\xi\in[-1,1],\;f_{\kappa}^{j}(\xi)=0\;\text{otherwise},$
$\displaystyle|f_{\kappa}^{j}(\xi)|\leq
c_{1}(1+|\xi|),\;|(f_{\kappa}^{j})^{\prime}(\xi)|\leq c_{1},$
$\displaystyle|(f_{\kappa}^{j})^{\prime\prime}(\xi)|\leq
c_{1},|f^{j}_{\kappa}(\xi)(f_{\kappa}^{j})^{\prime}(\xi)|\leq c_{1}.$

Here $c_{1}$ is some positive constant. We in addition suppose that

$\displaystyle
f^{1}_{\kappa}(\xi)=\xi-1,\;\text{if}\;1-\xi<-\kappa,\;f^{1}_{\kappa}(\xi)=0,\;\text{if}\;1-\xi\geq
0.$

and

$\displaystyle
f^{2}_{\kappa}(\xi)=-\xi-1,\;\text{if}\;\xi+1<-\kappa,\;f^{2}_{\kappa}(\xi)=0,\;\text{if}\;\xi+1\geq
0.$

It can be seen that the set of $f^{j}_{\kappa}$ is not empty by using a
$C^{2}$ interpolation (see, e.g., [13]). Below is one concrete example.

(9) $f_{\kappa}^{1}(\xi)=\left\\{\begin{aligned}
&\xi-1&\;\text{if}\;1-\xi<-\kappa,\\\
&-\frac{3}{\kappa^{4}}(1-\xi)^{5}-\frac{8}{\kappa^{3}}(1-\xi)^{4}-\frac{6}{\kappa^{2}}(1-\xi)^{3}&\;\text{if}\;-\kappa\leq
1-\xi<0,\\\ &0&\;\text{if}\;1-\xi\geq 0,\end{aligned}\right.$ (10)
$f_{\kappa}^{2}(\xi)=\left\\{\begin{aligned}
&-(\xi+1)&\;\text{if}\;\xi+1<-\kappa,\\\
&-\frac{3}{\kappa^{4}}(\xi+1)^{5}-\frac{8}{\kappa^{3}}(\xi+1)^{4}-\frac{6}{\kappa^{2}}(\xi+1)^{3}&\;\text{if}\;-\kappa\leq\xi+1<0,\\\
&0&\;\text{if}\;\xi+1\geq 0.\end{aligned}\right.$

Let Setting 2.1 hold with some $\bs>\frac{d}{2}$. Then the following tail
estimates holds,

$\displaystyle\mathbb{P}(\sup_{s\in[0,T]}|u^{\delta}(s)|_{E}>1+\delta_{0})\to
0,\;\text{as}\;\epsilon\sim o(\delta_{0})\to 0$

in Case 1, and

$\displaystyle\mathbb{P}(\sup_{s\in[0,T]}|u^{\delta}(s)|_{E}>1)=0.$

in Case 2.

######  Proof 2.4.

Since the derivative of $F_{\log}^{\delta}$ is finite, one can show that
$u^{\delta}\in E,a.s.$ by a standard procedure (see, e.g., [7, 18]). However,
its $E$-norm may be not uniformly bounded with respect to $\delta$ by this
argument. Instead, we will consider the evolution of
$\|f^{1}_{\kappa}(u(t))\|^{2}$ and $\|f^{2}_{\kappa}(u(t))\|^{2}$, and then
take limits on $\kappa.$ Let us illustrate the procedures to estimate
$\|f^{1}(u(t))\|^{2}$ since that of the case $\|f^{2}(u(t))\|^{2}$ is similar.

Applying the Itô formula to $\frac{1}{2}\|f^{1}_{\kappa}(u(t))\|^{2}$, we
obtain

$\displaystyle\frac{1}{2}\|f^{1}_{\kappa}(u^{\delta}(t))\|^{2}$
$\displaystyle~{}=\frac{1}{2}\|f^{1}_{\kappa}(u(0))\|^{2}+\int_{0}^{t}\<(f^{1}_{\kappa})^{\prime}(u^{\delta})f^{1}_{\kappa}(u^{\delta}),Au^{\delta}(s)-(F_{log}^{\delta})^{\prime}(u^{\delta}(s))\>ds$
$\displaystyle~{}~{}+\int_{0}^{t}\epsilon\<(f^{1}_{\kappa})^{\prime}(u^{\delta}(s))f^{1}_{\kappa}(u^{\delta}(s)),B(u^{\delta}(s))dW(s)\>$
$\displaystyle~{}~{}+\int_{0}^{t}\frac{1}{2}\epsilon^{2}\sum_{j\in\mathbb{N}}\<(f^{1}_{\kappa})^{\prime\prime}(u^{\delta}(s))f_{\kappa}^{1}(u^{\delta}(s))+|(f^{1}_{\kappa})^{\prime}(u^{\delta}(s))|^{2})B(u^{\delta}(s))e_{j},B(u^{\delta}(s))e_{j}\>ds.$

Taking $\kappa\to 0$, thanks to Propositions 1, (8) and (2), we have that

$\displaystyle\frac{1}{2}\|f^{1}(u^{\delta}(t))\|^{2}$
$\displaystyle~{}=\frac{1}{2}\|f^{1}(u(0))\|^{2}-\int_{0}^{t}\<\mathbb{I}_{\\{u^{\delta}(s)>1\\}}(u^{\delta}(s)-1),(F_{log}^{\delta})^{\prime}(u^{\delta}(s))\>ds$
$\displaystyle~{}~{}-\int_{0}^{t}\|\mathbb{I}_{\\{u^{\delta}(s)>1\\}}\nabla
u^{\delta}(s)\|^{2}ds+\int_{0}^{t}\epsilon\<\mathbb{I}_{\\{u^{\delta}(s)>1\\}}(u^{\delta}(s)-1),B(u^{\delta}(s))dW(s)\>$
$\displaystyle~{}~{}+\int_{0}^{t}\frac{1}{2}\epsilon^{2}\sum_{j\in\mathbb{N}}\<\mathbb{I}_{\\{u^{\delta}(s)>1\\}}B(u^{\delta}(s))e_{j},B(u^{\delta}(s))e_{j}\>ds.$

Taking supreme over $t\in[0,t_{1}]$ for $t_{1}\leq T$, then taking $p$th
moment and using Burkerhold’s inequality, we obtain that

$\displaystyle\E[\sup_{t\in[0,t_{1}]}\|f^{1}(u^{\delta}(t))\|^{2p}]+\E\Big{[}\Big{(}\int_{0}^{t_{1}}\mathbb{I}_{\\{u^{\delta}(s)>1\\}}\|\nabla
u^{\delta}(s)\|^{2}ds\Big{)}^{p}\Big{]}$
$\displaystyle+\E\Big{[}\Big{(}\int_{0}^{t_{1}}\mathbb{I}_{\\{u^{\delta}(s)>1\\}}\<(u^{\delta}(s)-1),(F_{log,1}^{\delta})^{\prime}(u^{\delta}(s))\>ds\Big{)}^{p}\Big{]}$
$\displaystyle\leq
C(p)\E[\|f^{1}(u^{\delta}(0))\|^{2p}]+C(p,T)\epsilon^{2p}\int_{0}^{t_{1}}(1+\E[\|B(u^{\delta}(s)))\mathbb{I}_{\\{u^{\delta}(s)>1\\}}\|_{\mathcal{L}_{2}(H)}^{2p}])ds.$

As a consequence, it holds that in Case 1,

(11) $\displaystyle\E[\sup_{t\in[0,T]}\|f^{1}(u^{\delta}(t))\|^{2p}]\leq
C(T,p)(\|f^{1}(u(0))\|^{2p}+\epsilon^{2p}\|u(0)\|^{2p})$

and that in Case 2,

(12) $\displaystyle\E[\sup_{t\in[0,T]}\|f^{1}(u^{\delta}(t))\|^{2p}]\leq
C(T,p)(\|f^{1}(u(0))\|^{2p}).$

By repeating the above procedures to $f^{2}$ and using the fact that
$f^{j}(u(0))=0,j=1,2$, it is not hard to see that in Case 2,

$\mathbb{P}(\sup_{s\in[0,T]}|u^{\delta}(s)|_{E}>1)=0.$

Let $\delta_{0}>0$ be a small number. Denote
$\mathcal{A}_{s}=\\{x\in\mathcal{D}|u^{\delta}(s)>1+\delta_{0}\\}$,
$\mathcal{B}_{s}=\\{x\in\mathcal{D}|u^{\delta}(s)<-1-\delta_{0}\\}.$ In Case
1, notice that for any $s\in[0,T]$,

(13)
$\displaystyle|\mathcal{A}_{s}|+|\mathcal{B}_{s}|\leq\frac{\|f^{1}(u^{\delta}(s))\|^{2}+\|f^{2}(u^{\delta}(s))\|}{\delta_{0}^{2}}\leq\eta_{s}\frac{\epsilon^{2}}{\delta_{0}^{2}},$

where $\eta_{s}$ is a positive stochastic process with any finite $p$th
moment. As a consequence, the Lebesgue measures
$|\mathcal{A}_{s}|,|\mathcal{B}_{s}|\to 0,a.s.$ as $\epsilon\sim
o(\delta_{0})\to 0,$ which completes the proof.

Next, we provide the strong convergence analysis of (3).

Let Setting 2.1 hold with some $\bs>\frac{d}{2}$, $p\geq 1$. It holds that in
Case 1,

$\displaystyle\E[\|u^{\delta}(t)-u_{\det}(t)\|^{2p}]\leq
C(\delta^{2}+\epsilon^{2}+(1+\ln^{2}(\delta))\frac{\epsilon^{2}}{\delta_{0}^{2}})^{p},$

where $\delta_{0}\sim O(\delta)$ is the small parameter in Lemma 2.3, and that
in Case 2,

$\displaystyle\E[\|u^{\delta}(t)-u(t)\|^{2p}]\leq C\delta^{2p},$

######  Proof 2.5.

We only present the details of $p=1$ for simplicity. We first show the
convergence in Case 1. Denote by $u_{det}(t)$ the exact solution of the
deterministic Allen-Cahn equation with $F_{log}$, i.e.,

$\displaystyle du_{det}(t)-Au_{det}(t)dt+F^{\prime}_{log}(u_{det}(t))dt=0,\;$

which satisfies the maximum bound principle

$\displaystyle|u_{\det}(t,x)|<1,\;\text{if}\;|u_{\det}(0,x)|<1,$

and $u(t)\in H^{\bs}(\mathcal{D})$ if $u(0)\in H^{\bs}(\mathcal{D})$.
Considering the difference between $u^{\delta}$ and $u$, taking expectation
and applying Burkholder’s inequality , we have that

$\displaystyle\frac{1}{2}\E[\|u^{\delta}(t)-u_{det}(t)\|^{2}]+\int_{0}^{t}\E[\|\nabla(u^{\delta}(s)-u_{det}(s))\|^{2}]ds$
$\displaystyle=-\int_{0}^{t}\E[\<(F_{\log}^{\delta})^{\prime}(u^{\delta}(s))-(F_{\log})^{\prime}(u_{det}(s)),u^{\delta}(s)-u_{det}(s)\>]ds$
$\displaystyle+\int_{0}^{t}\frac{\epsilon^{2}}{2}\E[\|B(u^{\delta}(s))\|_{\mathcal{L}_{2}(H)}^{2}]ds+\epsilon\E[\int_{0}^{t}\<u_{det}(s)-u^{\delta}(s),B(u^{\delta}(s))dW_{s}\>]$
$\displaystyle\leq-\int_{0}^{t}\E[\int_{\mathcal{A}_{s}\cup\mathcal{B}_{s}}(F_{\log}^{\delta})^{\prime}(u^{\delta}(s))-(F_{\log})^{\prime}(u_{det}(s)))(u^{\delta}(s)-u_{det}(s))dx]ds$
$\displaystyle-\int_{0}^{t}\E[\int_{\mathcal{A}_{s}^{c}\cap\mathcal{B}_{s}^{c}}((F_{\log}^{\delta})^{\prime}(u^{\delta}(s))-(F_{\log})^{\prime}(u_{det}(s)))(u^{\delta}(s)-u_{det}(s))dx]ds$
$\displaystyle+\int_{0}^{t}\frac{\epsilon^{2}}{2}\E[\|B(u^{\delta}(s))\|_{\mathcal{L}_{2}(H)}^{2}]ds=:II_{1}+II_{2}+II_{3}.$

Since $II_{3}\sim O(\epsilon^{2}),$ it suffices to estimate $II_{1}$-$II_{2}.$
Thanks to Proposition 1 and the proof of Lemma 2.3, applying Hölder’s
inequality, it holds that

$\displaystyle|II_{1}|$ $\displaystyle\leq
C\int_{0}^{t}\E[\|u^{\delta}(s)-u_{det}(s)\|^{2}]ds$
$\displaystyle+\int_{0}^{t}\E\Big{[}\|(u^{\delta}(s)-u_{det}(s))\|\sqrt{\int_{\mathcal{A}_{s}\cup\mathcal{B}_{s}}|(F_{\log}^{\delta})^{\prime}(u^{\delta}(s))-(F_{\log})^{\prime}(u_{det}(s)))|^{2}dx}\Big{]}ds$
$\displaystyle\leq
C\int_{0}^{t}\E[\|u^{\delta}(s)-u_{det}(s)\|^{2}]ds+C(T)(1+\ln^{2}(\delta))\frac{\epsilon^{2}}{\delta_{0}^{2}}.$

Thanks to the fact that

$\displaystyle(F_{\log}^{\delta})^{\prime}(\xi_{1})-(F_{\log}^{\delta})^{\prime}(\xi_{2})$
$\displaystyle=\frac{2(1-\chi^{2})+2\delta}{((1+\chi)^{2}+\delta)((1-\chi)^{2}+\delta)}(\xi_{1}-\xi_{2}),$

where $\chi=\theta\xi_{1}+(1-\theta)\xi_{2}$ for some $\theta\in(0,1)$, it
follows that if $|\chi|\in[1,1+\delta_{0}]$, then

$\displaystyle|2(1-\chi^{2})|\leq 2((1+\delta_{0})^{2}-1)\leq
4\delta_{0}+2\delta_{0}^{2}.$

In the following, we take $\delta_{0}\sim O(\delta).$ Notice that

$\displaystyle II_{2}$ $\displaystyle\leq
C\int_{0}^{t}\E[\|u^{\delta}(s)-u_{det}(s)\|^{2}]ds$
$\displaystyle-\int_{0}^{t}\E[\int_{\mathcal{A}_{s}^{c}\cap\mathcal{B}_{s}^{c}}((F_{\log,1}^{\delta})^{\prime}(u^{\delta}(s))-(F_{\log,1}^{\delta})^{\prime}(u_{det}(s)))(u^{\delta}(s)-u_{det}(s))dx]ds$
$\displaystyle-\int_{0}^{t}\E[\int_{\mathcal{A}_{s}^{c}\cap\mathcal{B}_{s}^{c}}((F_{\log,1}^{\delta})^{\prime}(u_{det}(s))-((F_{\log,1})^{\prime}(u_{det}(s)))(u^{\delta}(s)-u_{det}(s))dx]ds,$

where $(F_{\log,1})^{\prime}(\xi):=\ln(1+\xi)-\ln(1-\xi).$ Therefore, we have
that

$\displaystyle II_{2}$ $\displaystyle\leq
C\int_{0}^{t}\E[\|u^{\delta}(s)-u_{det}(s)\|^{2}]ds+C\delta^{2}.$

Combining the above estimates with Gronwall’s inequality yields that

$\displaystyle\E[\|u^{\delta}(t)-u_{det}(t)\|^{2}]\leq
C(\delta^{2}+\epsilon^{2}+(1+\ln^{2}(\delta))\frac{\epsilon^{2}}{\delta_{0}^{2}}).$

Similar steps as in Case 1, together with Lemma 2.3, yield the desired result
in Case 2.

To end this section, we show the random effect on the behaviors of energy
thanks to Proposition 2.2 and Theorem 2.3.

######  Proposition 3.

Let Setting 2.1 hold with some $\bs=1+\beta>\frac{d}{2}$, $\beta\in(0,1)$ and
$\delta_{0}\sim O(\delta)$. Then it holds that in Case 1, for $\epsilon\sim
o(\delta_{0}),$

(14) $\displaystyle\E[|H(u^{\delta}(t))-H(u_{\det}(t))|]$ $\displaystyle\leq
C(\delta+\epsilon+|\ln(\delta)|\frac{\epsilon}{\delta_{0}})^{\frac{(\bs-1)}{\bs}}|\ln(\delta)|,$

and that in Case 2,

(15) $\displaystyle\E[|H(u^{\delta}(t))-H(u(t))|]$ $\displaystyle\leq
C\delta^{\frac{(\bs-1)}{\bs}}|\ln(\delta)|.$

######  Proof 2.6.

Using the Sobolev interpolation inequality

$\|(-A)^{\frac{1}{2}}v\|\leq
C\|v\|^{\frac{\bs-1}{\bs}}\|(-A)^{\frac{\bs}{2}}v\|^{\frac{1}{\bs}},$

and applying Proposition 2.2 and Theorem 2.3, it follows that in Case 1,

$\displaystyle\|\nabla(u^{\delta}(t)-u_{\det}(t))\|_{L^{2}(\Omega;H)}$
$\displaystyle\leq
C(\delta+\epsilon+|\ln(\delta)|\frac{\epsilon}{\delta_{0}})^{\frac{(\bs-1)}{\bs}}|\ln(\delta)|^{\frac{1}{\bs}}.$

and that in Case 2,

$\displaystyle\|\nabla(u^{\delta}(t)-u(t))\|_{L^{2}(\Omega;H)}$
$\displaystyle\leq
C\delta^{\frac{(\bs-1)}{\bs}}|\ln(\delta)|^{\frac{1}{\bs}}.$

On the other hand, by Proposition 2.3, Hölder’s and Young’s inequalities, we
have that for $v(t)=u(t)$ or $u_{det}(t),$

$\displaystyle\E[\|F_{\log}^{\delta}(u^{\delta}(t))-F_{\log}(v(t))\|_{L^{1}}]$
$\displaystyle\leq\E[\|F_{\log}^{\delta}(v(t))-F_{\log}(v(t))\|_{L^{1}}]+\E[\|F_{\log}^{\delta}(u^{\delta}(t))-F_{\log}^{\delta}(v(t))\|_{L^{1}}]$
$\displaystyle\leq
C(\delta+\E[\|v(t)\|^{2}])+\E[\|F_{\log}^{\delta}(u^{\delta}(t))-F_{\log}^{\delta}(v(t))\|_{L^{1}}]$
$\displaystyle\leq
C(\delta+(1+\ln(\delta))\sqrt{\E[\|v(t)-u^{\delta}(t)\|^{2}]}).$

Thus, we get that in Case 1,

$\displaystyle\E[\|F_{\log}^{\delta}(u^{\delta}(t))-F_{\log}(u_{det}(t))\|_{L^{1}}]\leq
C(\delta+(1+|\ln(\delta)|)(\epsilon+\delta+|\ln(\delta)|\frac{\epsilon}{\delta_{0}}))$

and that in Case 2,

$\displaystyle\E[\|F_{\log}^{\delta}(u^{\delta}(t))-F_{\log}(u(t))\|_{L^{1}}]\leq
C(\delta+(1+|\ln(\delta)|)\delta).$

Combining the above estimates, we complete the proof.

##  3 Numerical approximation

There have been a lot of numerical results on the discretization for SPDEs
with Lipschitz nonlinearities. But less attention has been paid on parabolic
SPDEs with logarithmic potentials. Thanks to the regularized models in section
2, we are able to do numerical analysis for (1). Although the implicit
discretization is a suitable choice of (1), it often requires to solve
nonlinear algebraic equations with randomness. It is still highly desirable to
develop stable numerical approximations for (1) which could be solved
explicitly to save computational costs. However, due to the singularity of the
drift coefficient, it is not easy to achieve this goal.

In this section, we propose a semi-implicit scheme for the considered model.
We present a detailed analysis for the stabilized full discretization. Similar
arguments are also applicable for the classical implicit full discretizations.
In the spatial direction, we take the spectral Galerkin method for simplicity.

###  3.1 Semi-implicit scheme with stabilization term

In order to propose a stable semi-implicit scheme, we add a small
stabilization term into the numerical discretization. For convenience, let us
use the following stabilized scheme with a parameter $\alpha\geq 0$,

(16) $\displaystyle u_{j+1}^{N}=u_{j}^{N}+(A+2c)\tau
u_{j+1}^{N}-(F^{\delta}_{\log,1})^{\prime}(u_{j}^{N})\tau-\alpha(u_{j+1}^{N}-u_{j}^{N})\tau+\epsilon
P^{N}B(u_{j}^{N})\delta W_{j}.$

which uses the backward Euler discretization for the linear unbounded part and
the forward Euler discretization for the potential and stochastic terms. Here
$u_{0}^{N}=P^{N}u(0)$, $S_{\tau}=(I-A\tau)^{-1},$ $\delta
W_{j}=W(t_{j+1})-W(t_{j}),t_{j}=j\tau$, $j\leq J$, $J\tau=T$, and $N$ is the
parameter of the spectral Galerkin method, i.e., the dimension of the
projection on to the span of first $N+1$ eigenspace. Note that the above full
discretization of (1) can be rewritten as

(17) $\displaystyle
u^{N}_{j+1}=S_{\tau}\Big{(}u^{N}_{j}-P^{N}(F_{\log,1}^{\delta})^{\prime}(u^{N}_{j})\tau-(\alpha-2c)\tau
u_{j+1}^{N}+\alpha u_{j}^{N}\tau+P^{N}\epsilon B(u_{j}^{N})\delta
W_{j}\Big{)}.$

The key steps to derive the convergence of stable numerical scheme lies on the
regularity estimates in time and space.

Let Setting 2.1 hold with some $\bs\geq 0$ and $p\geq 1$. There exists
$C:=C(p,T,u(0))$ such that

$\displaystyle\sup_{N\in\mathbb{N}^{+}}\sup_{j\leq
J}\|u_{j}^{N}\|_{L^{2p}(\Omega;H)}\leq
C(1+|\ln(\delta)|\tau^{\frac{1}{2}}).\;$

######  Proof 3.1.

For convenience, we only give the proof for $p=1$. By taking $H$-inner product
on the both sides of (16) with $u_{j+1}^{N}$, and using integration by parts
and (17), one can obtain that

$\displaystyle\frac{1}{2}(1+\alpha\tau)\|u_{j+1}^{N}\|^{2}+\|\nabla
u_{j+1}^{N}\|^{2}\tau$
$\displaystyle\leq\frac{1}{2}(1+\alpha\tau)\|u_{j}^{N}\|^{2}+2c\|u_{j+1}^{N}\|^{2}\tau-\<(F^{\delta}_{\log,1})^{\prime}(u_{j}^{N}),u_{j+1}^{N}\>\tau+\epsilon\<B(u_{j}^{N})\delta
W_{j},u_{j+1}^{N}\>.$

Using Hölder’s and Young’ inequality, as well as the property that
$\|(S_{\tau}-I)v\|\leq C\tau^{\alpha}\|v\|_{H^{2\alpha}},$ $\alpha\in[0,1],$
it follows that

$\displaystyle\frac{1}{2}(1+\alpha\tau)\|u_{j+1}^{N}\|^{2}+\|\nabla
u_{j+1}^{N}\|^{2}\tau$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}(1+\alpha\tau)\|u_{j}^{N}\|^{2}+2c\|u_{j+1}^{N}\|^{2}\tau-\<(F^{\delta}_{\log,1})^{\prime}(u_{j}^{N}),u_{j}^{N}\>\tau-$
$\displaystyle\<(F^{\delta}_{\log,1})^{\prime}(u_{j}^{N}),(S_{\tau}-I)u_{j}^{N}\>\tau-\<(F^{\delta}_{\log,1})^{\prime}(u_{j}^{N}),-S_{\tau}P^{N}(F^{\delta}_{\log,1})^{\prime}(u_{j}^{N})\tau$
$\displaystyle-S_{\tau}(\alpha-2c)u_{j+1}^{N}+\alpha
u_{j}^{N}\tau\>\tau-\<(F^{\delta}_{\log,1})^{\prime}(u_{j}^{N}),S_{\tau}\epsilon
B(u_{j}^{N})\delta W_{j}\>\tau$ $\displaystyle+\epsilon\<B(u_{j}^{N})\delta
W_{j},S_{\tau}u_{j}^{N}\>+\epsilon\<B(u_{j}^{N})\delta
W_{j},-S_{\tau}(F_{\log,1}^{\delta})^{\prime}(u_{j}^{N})\tau$
$\displaystyle-(\alpha-2c)\tau u_{j+1}^{N}+\alpha
u_{j}^{N}\tau\>+\epsilon^{2}\<B(u_{j}^{N})\delta
W_{j},S_{\tau}B(u_{j}^{N})\delta W_{j}\>$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}(1+\alpha\tau)\|u_{j}^{N}\|^{2}+2c\|u_{j+1}^{N}\|^{2}\tau+C|\ln(\delta)|^{2}\tau^{2}+C(\|u_{j+1}^{N}\|^{2}+\|u_{j}^{N}\|^{2})\tau^{2}$
$\displaystyle-\<(F^{\delta}_{\log,1})^{\prime}(u_{j}^{N}),S_{\tau}\epsilon
B(u_{j}^{N})\delta W_{j}\>\tau+\epsilon\<B(u_{j}^{N})\delta
W_{j},S_{\tau}u_{j}^{N}\>$ $\displaystyle+\epsilon\<B(u_{j}^{N})\delta
W_{j},-S_{\tau}(F_{\log,1}^{\delta})^{\prime}(u_{j}^{N})\tau-(\alpha-2c)\tau
u_{j+1}^{N}+\alpha u_{j}^{N}\tau\>$
$\displaystyle+\epsilon^{2}\<B(u_{j}^{N})\delta
W_{j},S_{\tau}B(u_{j}^{N})\delta W_{j}\>.$

Taking expectation and using Gronwall’s inequality, it follows that

(18) $\displaystyle\E[\|u_{j+1}^{N}\|^{2}]+\E[\sum_{i=0}^{j}\|\nabla
u_{j+1}^{N}\|^{2}]\tau$ $\displaystyle\leq
C(T)\|u_{0}^{N}\|^{2}(1+\ln^{2}(\delta)\tau+\epsilon^{2}).$

Now, we are in a position to present its moment bounds and regularity
estimates in Sobolev norms.

Let Setting 2.1 hold with some $\bs=1+\beta$, $\beta\in[0,1)$ and $p\geq 2.$
It holds that

$\displaystyle\sup_{N\in\mathbb{N}^{+}}\sup_{j\leq
J-1}\|u_{j+1}^{N}-u_{j}^{N}\|_{L^{p}(\Omega;H)}$ $\displaystyle\leq
C(T,u(0),p)|\ln(\delta)|(\tau^{\frac{\beta+1}{2}}+\epsilon\tau^{\frac{1}{2}}),\;$
$\displaystyle\sup_{N\in\mathbb{N}^{+}}\sup_{j\leq
J}\|u_{j+1}^{N}\|_{L^{p}(\Omega;H^{\bs})}$ $\displaystyle\leq
C(T,u(0),p,\beta)(\epsilon+|\ln(\delta)|).$

######  Proof 3.2.

We first show the regularity estimate in space direction. Using the fact that

$\|S_{\tau}^{k}(-A)^{\alpha}v\|\leq
C(1+(k\tau)^{-\alpha})\|v\|,\alpha\in(0,1),k\in\mathbb{N}^{+},$

and Burkholder’s inequality, we have that for $\beta\in(0,1),$

$\displaystyle\|(-A)^{\frac{\beta+1}{2}}u_{j+1}^{N}\|_{L^{p}(\Omega;H)}$
$\displaystyle=\|S_{\tau}^{j+1}(-A)^{\frac{\beta+1}{2}}u_{0}^{N}\|+\|P^{N}\sum_{i=0}^{j}(-A)^{\frac{\beta+1}{2}}S_{\tau}^{j+1-i}(F_{\log,1}^{\delta})^{\prime}(u_{i}^{N})\|_{L^{p}(\Omega;H)}\tau$
$\displaystyle+\|P^{N}\sum_{i=0}^{j}(-A)^{\frac{\beta+1}{2}}S_{\tau}^{j+1-i}(\alpha-2c)u_{j+1}^{N}+\alpha
u_{j}^{N}\|_{L^{p}(\Omega;H)}\tau$
$\displaystyle+\|P^{N}\sum_{i=0}^{j}(-A)^{\frac{\beta+1}{2}}S_{\tau}^{j+1-i}\epsilon
B(u_{i}^{N})\delta W_{j}\|_{L^{p}(\Omega;H)}$ $\displaystyle\leq
C\|u(0)\|_{H^{\frac{\beta+1}{2}}}+C\sum_{i=0}^{j}(t_{j+1}-t_{i})^{-\frac{\beta+1}{2}}\tau(1+|\ln(\delta)|)$
$\displaystyle+C\Big{(}\sum_{i=0}^{j}(t_{j+1}-t_{i})^{-\beta}\tau(\E[\|(-A)^{\frac{1}{2}}B(u_{i}^{N})\|_{\mathcal{L}_{2}(H)}^{p}])^{\frac{2}{p}}\Big{)}^{\frac{1}{2}}\epsilon$
$\displaystyle\leq C(T,u(0),p,\beta)(\epsilon+|\ln(\delta)|).$

Next we prove the discrete time regularity estimate. By Burkholder’s
inequality, Lemma 3.1, and the property that $\|(S_{\tau}-I)v\|\leq
C\tau^{\alpha}\|v\|_{H^{2\alpha}},\alpha\in[0,1],$ we obtain

$\displaystyle\|u_{j+1}^{N}-u_{j}^{N}\|_{L^{p}(\Omega;H)}$
$\displaystyle\leq\|(S_{\tau}-I)u_{j}^{N}\|_{L^{p}(\Omega;H)}+\tau\|(F_{\log,1}^{\delta})^{\prime}(u_{j}^{N})\|_{L^{p}(\Omega;H)}+\epsilon\|B(u_{j}^{N})\delta
W_{j}\|_{L^{p}(\Omega;H)}$
$\displaystyle+C(\|u_{j}^{N}\|_{L^{p}(\Omega;H)}+\|u_{j+1}^{N}\|_{L^{p}(\Omega;H)})\tau$
$\displaystyle\leq
C\tau^{\frac{1}{2}}\|u_{j}^{N}\|_{L^{p}(\Omega;H^{1})}+C\tau|\ln(\delta)|+C\epsilon\|B(u_{j})\|_{L^{p}(\Omega;\mathcal{L}_{2}(H))}\tau^{\frac{1}{2}}$
$\displaystyle\leq
C(T,u(0),p)|\ln(\delta)|(\tau^{\frac{\beta+1}{2}}+\epsilon\tau^{\frac{1}{2}}).$

###  3.2 Strong convergence analysis

Now we present its convergence result of the considered scheme under
$L^{2p}(\Omega;H)$ with $p=1$. Same convergence rate result still holds for
other $p>1.$

Let Setting 2.1 hold with some $\bs=1+\beta>\frac{d}{2},\beta\in[0,1)$. It
holds that for $\delta_{0}\sim O(\delta),$

$\displaystyle\sup_{j\leq
J}\|u^{\delta}(t_{j})-u_{j}^{N}\|_{L^{2}(\Omega;H)}^{2}\leq$ $\displaystyle
C(T,u(0))|\ln(\delta)|^{2}\Big{(}\frac{\lambda_{N}^{-{\beta+1}}}{\delta^{2}}+\frac{\|f^{1}(u_{0}^{N})\|^{2}+\|f^{2}(u_{0}^{N})\|^{2}}{\delta_{0}^{2}}$
$\displaystyle+\frac{\epsilon^{2}}{\delta_{0}^{2}}+\frac{\tau^{\beta+1}}{\delta_{0}^{2}\delta^{2}}+\frac{\tau^{2}}{\delta_{0}^{2}\delta^{2}}+\frac{\tau\epsilon^{2}}{\delta_{0}^{2}\delta^{2}}\Big{)}$

in Case 1, and

$\displaystyle\sup_{j\leq
J}\|u^{\delta}(t_{j})-u_{j}^{N}\|_{L^{2}(\Omega;H)}^{2}$ $\displaystyle\leq
C(T,u(0))|\ln(\delta)|^{2}\Big{(}\frac{\lambda_{N}^{-{\beta+1}}}{\delta^{2}}+\frac{\|f^{1}(u_{0}^{N})\|^{2}+\|f^{2}(u_{0}^{N})\|^{2}}{\delta_{0}^{2}}+\frac{\tau^{\beta+1}}{\delta_{0}^{2}\delta^{2}}+\frac{\tau^{2}}{\delta_{0}^{2}\delta^{2}}\Big{)}$

in Case 2.

######  Proof 3.3.

We only present the proof of Case 1 since that of Case 2 is similar. Notice
that $u^{\delta}(t_{j+1})=(I-P^{N})u^{\delta}(t_{j+1})+P^{N}u(t_{j+1}).$ The
error bound of $(I-P^{N})u^{\delta}(t_{j+1})$ is standard due to the fact that

(19) $\displaystyle\|(I-P^{N})v\|\leq
C\lambda_{N}^{-\frac{\bs}{2}}\|v\|_{H^{\bs}}.$

It suffices to derive an iterative formula for
$Er_{j+1}:=P^{N}u^{\delta}(t_{j+1})-u^{N}_{j+1}$. To this end, we consider
$\E[\<Er_{j+1}-Er_{j},Er_{j+1}\>]$ and obtain that

$\displaystyle\frac{1}{2}\E[\|Er_{j+1}\|^{2}]$
$\displaystyle\leq\frac{1}{2}\E[\|Er_{j}\|^{2}]-\int_{t_{j}}^{t_{j+1}}\E[\<\nabla(P^{N}u^{\delta}(s)-u^{N}_{j+1}),\nabla
Er_{j+1}\>]ds$
$\displaystyle-\int_{t_{j}}^{t_{j+1}}\E[\<(F_{\log,1}^{\delta})^{\prime}(u^{\delta}(s))-(F_{\log,1}^{\delta})^{\prime}(u_{j}^{N}),Er_{j+1}\>]ds$
$\displaystyle+\E[\int_{t_{j}}^{t_{j+1}}\epsilon\<Er_{j+1}-Er_{j},(B(u^{\delta}(s))-B(u_{j}^{N}))dW(s))\>]$
$\displaystyle+\E[\int_{t_{j}}^{t_{j+1}}\<-2cu^{\delta}(s)+2cu_{j+1}^{N}-\alpha(u_{j+1}^{N}-u_{j}^{N}),Er_{j+1}\>ds]$
$\displaystyle=:\frac{1}{2}\E[\|Er_{j}\|^{2}]+III_{1}+III_{2}+III_{3}+III_{4}.$

By using Young’s inequality and the mild formulation of $u^{\delta}(s)$, we
get that for small $\kappa<1$,

$\displaystyle III_{1}$
$\displaystyle\leq-(1-\kappa)\int_{t_{j}}^{t_{j+1}}\E[\|\nabla
Er_{j+1}\|^{2}]ds$
$\displaystyle+C(\kappa)\int_{t_{j}}^{t_{j+1}}\E\Big{[}\Big{\|}(-A)^{\frac{1}{2}}\int_{0}^{s}(S(t_{j+1}-r)-S(s-r))(F_{\log}^{\delta})^{\prime}(u^{\delta}(r))dr\Big{\|}^{2}\Big{]}ds$
$\displaystyle+C(\kappa)\int_{t_{j}}^{t_{j+1}}\E\Big{[}\Big{\|}(-A)^{\frac{1}{2}}\int_{s}^{t_{j+1}}(S(t_{j+1}-r)(F_{\log}^{\delta})^{\prime}(u^{\delta}(r))dr\Big{\|}^{2}\Big{]}ds$
$\displaystyle+C(\kappa)\int_{t_{j}}^{t_{j+1}}\E\Big{[}\Big{\|}(-A)^{\frac{1}{2}}\int_{0}^{s}(S(t_{j+1}-r)-S(s-r))\epsilon
B(u^{\delta}(r))dW(r)\Big{\|}^{2}\Big{]}ds$
$\displaystyle+C(\kappa)\int_{t_{j}}^{t_{j+1}}\E\Big{[}\Big{\|}(-A)^{\frac{1}{2}}\int_{s}^{t_{j+1}}(S(t_{j+1}-r)\epsilon
B(u^{\delta}(r))dW(r)\Big{\|}^{2}\Big{]}ds.$

By Proposition 2 and the property that $\|A^{-\alpha}(S(t)-I)v\|\leq
Ct^{\alpha}\|v\|$, $\alpha\in(0,1)$, we obtain that

$\displaystyle III_{1}$
$\displaystyle\leq-(1-\kappa)\int_{t_{j}}^{t_{j+1}}\E[\|\nabla
Er_{j+1}\|^{2}]ds+C(\kappa,u(0),T)\tau^{2}(\epsilon^{2}+\ln^{2}(\delta)).$

Similarly, one can obtain that

$\displaystyle III_{4}$ $\displaystyle\leq
CEr_{j+1}^{2}\tau+C|\ln(\delta)|^{2}(\tau^{\frac{\beta+1}{2}}+\epsilon\tau^{\frac{1}{2}}+\lambda_{N}^{-\frac{\beta+1}{2}})^{2}.$

For the term $III_{2}$, by Young’s inequality, (19), (13) and Lemma 2.2, it
can be seen that

$\displaystyle III_{2}\leq$ $\displaystyle
C(\kappa)\E[\|Er_{j+1}\|^{2}]\tau+C(\kappa,T,u(0))\tau|\ln(\delta)|^{2}(\frac{\epsilon^{2}}{\delta^{2}}+\frac{\lambda_{N}^{-\beta-1}}{\delta^{2}})$
$\displaystyle-\int_{t_{j}}^{t_{j+1}}\E[\int_{D_{j+1}\cup\mathcal{E}_{j+1}}((F_{\log,1}^{\delta})^{\prime}(P^{N}u^{\delta}(t_{j}))-(F_{\log,1}^{\delta})^{\prime}(u_{j}^{N}))Er_{j+1}dx]ds$
$\displaystyle-\int_{t_{j}}^{t_{j+1}}\E[\int_{D_{j+1}^{c}\cap\mathcal{E}_{j+1}^{c}}((F_{\log,1}^{\delta})^{\prime}(P^{N}u^{\delta}(t_{j}))-(F_{\log,1}^{\delta})^{\prime}(u_{j}^{N}))Er_{j+1}dx]ds$
$\displaystyle\leq$ $\displaystyle
C(\kappa)\E[\|Er_{j+1}\|^{2}]\tau+C(\kappa,T,u(0))\tau|\ln(\delta)|^{2}(\frac{\epsilon^{2}}{\delta^{2}}+\frac{\lambda_{N}^{-\beta-1}}{\delta^{2}})$
$\displaystyle+\Big{|}\int_{t_{j}}^{t_{j+1}}\E[\int_{\mathcal{E}_{j+1}}((F_{\log,1}^{\delta})^{\prime}(P^{N}u^{\delta}(t_{j}))-(F_{\log,1}^{\delta})^{\prime}(u_{j}^{N}))Er_{j+1}dx]ds\Big{|},$

where $D_{j+1}=\\{x\in\mathcal{D}||u^{\delta}(t_{j+1})|\geq 1+\delta_{0}\\},$
$\mathcal{E}_{j+1}=\\{x\in\mathcal{D}||u_{j+1}^{N}|\geq 1+\delta_{0}\\}.$ Note
that the term
$C(\kappa,T,u(0))\tau|\ln(\delta)|^{2}\frac{\epsilon^{2}}{\delta^{2}}$
disappears in Case 2 thanks to Lemma 2.3.

Next we show a priori bound on $\mathcal{E}_{j+1}$. To this end, we construct
a continuous interpolation of the considered numerical scheme via the flow of
a local continuous problem defined on $[t_{j},t_{j+1}],$

$\displaystyle\widetilde{u}(t)=$
$\displaystyle\widetilde{u}(t_{j})+P^{N}AS_{\tau}\widetilde{u}(t_{j})(t-t_{j})-P^{N}S_{\tau}(F_{\log,1}^{\delta})^{\prime}(u_{j}^{N})(t-t_{j})$
$\displaystyle+2cS_{\tau}P^{N}u_{j+1}^{N}(t-t_{j})-\alpha
S_{\tau}P^{N}(u_{j+1}^{N}-u_{j}^{N})(t-t_{j})$
$\displaystyle+P^{N}S_{\tau}\epsilon B(\widetilde{u}(t_{j}))(W(t)-W(t_{j}))$

with $\widetilde{u}(t_{j})=u_{j}^{N}.$ Similar to the proof of Lemma 3.1, one
can obtain the regularity estimate of $\widetilde{u}$, i.e., for
$\beta\in(0,1),$ $s\leq t,$ $p\geq 2,$

(20) $\displaystyle\|\widetilde{u}(t)-\widetilde{u}(s)\|_{L^{p}(\Omega;H)}$
$\displaystyle\leq
C(T,u(0),p)(|\ln(\delta)|+\epsilon)|\max(t-s,\tau)|^{\min(\frac{\beta+1}{2},\frac{1}{2})},$
(21) $\displaystyle\|\widetilde{u}(t)\|_{L^{p}(\Omega;H^{1+\beta})}$
$\displaystyle\leq C(T,u(0),p,\beta)(\epsilon+|\ln(\delta)|).$

Denote $[s]$ the largest integer $j$ such that $j\tau\leq s$, $s\in[0,T]$.
Following the steps in the proof of Lemma 2.3, it follows that

$\displaystyle\frac{1}{2}\|f^{1}(\widetilde{u}(t))\|^{2}$ $\displaystyle\leq$
$\displaystyle\tps\frac{1}{2}\|f^{1}(\widetilde{u}(0))\|^{2}-\int_{0}^{t}\<\mathbb{I}_{\widetilde{u}(s)>1}\nabla\widetilde{u}(s),\mathbb{I}_{\widetilde{u}(s)>1}\nabla
S_{\tau}\widetilde{u}([s]\tau)\>ds$
$\displaystyle\tps-\int_{0}^{t}\<\widetilde{u}(s)-1,\mathbb{I}_{\widetilde{u}(s)>1}P^{N}S_{\tau}(F_{\log,1}^{\delta})^{\prime}(\widetilde{u}([s]\tau)\>ds$
$\displaystyle\tps+\int_{0}^{t}\epsilon\<\mathbb{I}_{\widetilde{u}(s)>1}(\widetilde{u}(s)-1),P^{N}S_{\tau}B(\widetilde{u}[s]\tau)dW(s)\>+\frac{\epsilon^{2}}{2}\int_{0}^{t}\mathbb{\|}\mathbb{I}_{\widetilde{u}(s)>1}B(\widetilde{u}([s]\tau))\|^{2}_{\mathcal{L}_{2}(H)}ds$
$\displaystyle\tps+\int_{0}^{t}\<\widetilde{u}(s)-1,\mathbb{I}_{\widetilde{u}(s)>1}P^{N}S_{\tau}(2c\widetilde{u}([s]\tau+\tau)-\alpha(\widetilde{u}([s]\tau+\tau)-\widetilde{u}([s]\tau))\>ds.$

Using the property of $(F_{\log,1}^{\delta})^{\prime}$, Burkholder’s
inequality and (20)-(21), we get

$\displaystyle\E[\sup_{t\in[0,T]}\|f^{1}(\widetilde{u}(t))\|^{2p}]$
$\displaystyle\leq$ $\displaystyle
C(T,u(0),p)\Big{(}\|f^{1}(\widetilde{u}(0))\|^{2p}+|\ln(\delta)|^{2p}(\epsilon^{2p}+(\frac{{\tau^{\frac{\beta+1}{2}}}}{\delta}+\frac{\tau}{\delta}+\frac{\epsilon\tau^{\frac{1}{2}}}{\delta})^{2p})\Big{)}.$

Similar results hold for
$\E[\sup_{t\in[0,T]}\|f^{2}(\widetilde{u}(t))\|^{2p}].$ Thus, we conclude that
there exists a positive stochastic process $\eta_{2}(s)$ with any finite
$p$th-moment such that

$\displaystyle\mathcal{E}_{j+1}\leq\eta_{2}(s)\Big{(}\frac{\|f^{1}(\widetilde{u}(0))\|^{2}+\|f^{2}(\widetilde{u}(0))\|^{2}}{\delta_{0}^{2}}+\frac{|\ln(\delta)|^{2}}{\delta_{0}^{2}}({\epsilon^{2}}+\frac{\tau^{\beta+1}}{\delta^{2}}+\frac{\tau^{2}}{\delta^{2}}+\frac{\tau\epsilon^{2}}{\delta^{2}})\Big{)},\;a.s.$

Substituting the above inequality into the estimate of $III_{2}$, we have that

$\displaystyle III_{2}\leq$ $\displaystyle
C(\kappa)\E[\|Er_{j+1}\|^{2}]\tau+C(\kappa,T,u(0))\tau|\ln(\delta)|^{2}(\frac{\epsilon^{2}}{\delta^{2}}+\frac{\lambda_{N}^{-\beta-1}}{\delta^{2}})+C(T,u(0),p)$
$\displaystyle\Big{(}\frac{\|f^{1}(\widetilde{u}(0))\|^{2}+\|f^{2}(\widetilde{u}(0))\|^{2}}{\delta_{0}^{2}}+\frac{|\ln(\delta)|^{2}}{\delta_{0}^{2}}({\epsilon^{2}}+\frac{\tau^{\beta+1}}{\delta^{2}}+\frac{\tau^{2}}{\delta^{2}}+\frac{\tau\epsilon^{2}}{\delta^{2}})\Big{)}.$

Thanks to the properties of the conditional expectation and stochastic
integral, it follows that

$III_{3}=III_{31}+III_{32}+III_{33}+III_{34}+III_{35},$

where

$\displaystyle III_{31}:=$
$\displaystyle-\E\Big{[}\int_{t_{j}}^{t_{j+1}}\epsilon\<\int_{t_{j}}^{t_{j+1}}S(t_{j+1}-r)P^{N}(F_{\log}^{\delta})^{\prime}(u^{\delta}(r))dr,$
$\displaystyle(B(u^{\delta}(s))-B(u^{\delta}(t_{j})))dW(s)\>\Big{]},$
$\displaystyle III_{32}:=$
$\displaystyle-\E\Big{[}\int_{t_{j}}^{t_{j+1}}\epsilon\<\int_{t_{j}}^{t_{j+1}}S(t_{j+1}-r)P^{N}(F_{\log}^{\delta})^{\prime}(u^{\delta}(r))dr,$
$\displaystyle P^{N}(B(u^{\delta}(t_{j}))-B(u_{j}^{N}))dW(s)\>\Big{]},$
$\displaystyle III_{33}:=$
$\displaystyle\E\Big{[}\int_{t_{j}}^{t_{j+1}}\epsilon\<S_{\tau}(\alpha
u_{j+1}^{N})\tau,P^{N}(B(u^{\delta}(s))-B(u_{j}^{N}))dW(s)\>\Big{]},$
$\displaystyle III_{34}:=$
$\displaystyle\E\Big{[}\int_{t_{j}}^{t_{j+1}}\epsilon\<S_{\tau}(\alpha
u_{j+1}^{N})\tau,P^{N}(B(u^{\delta}(t_{j}))-B(u_{j}^{N}))dW(s)\>\Big{]},$

and $III_{35}$ is defined as

$\displaystyle III_{35}:=$
$\displaystyle\E\Big{[}\int_{t_{j}}^{t_{j+1}}\epsilon^{2}\<\int_{t_{j}}^{t_{j+1}}(S(t_{j+1}-r)B(u^{\delta}(r))-S_{\tau}B(u_{j}^{N}))dW(r),P^{N}(B(u^{\delta}(s))$
$\displaystyle-B(u_{j}^{N}))dW(s)\>\Big{]}.$

Similar to the estimates of $III_{1}$-$III_{2}$, by using (19) and Proposition
2, one can show that

$\displaystyle III_{31}+III_{32}+III_{33}+III_{34}$ $\displaystyle\leq$
$\displaystyle
C\epsilon\tau|\ln(\delta)|(\tau|\ln(\delta)|+\epsilon\tau)+C\epsilon\lambda_{N}^{-1-\beta}|\ln(\delta)|\tau+C\|Er_{j}\|^{2}\tau.$

Next, we deal with $III_{35}.$ Using the property that $\|(S_{\tau}-I)v\|\leq
C\|v\|_{H^{\alpha}}\tau^{\frac{\alpha}{2}},\alpha\in[0,2]$, Burkholder’s
inequality and Proposition 2, we have that

$\displaystyle III_{35}$
$\displaystyle\leq\epsilon^{2}\E[\int_{t_{j}}^{t_{j+1}}\<\int_{t_{j}}^{t_{j+1}}B(u^{\delta}(r))-B(u^{\delta}(t_{j}))dW(r),P^{N}(B(u^{\delta}(s))-B(u^{\delta}(t_{j})))dW(s)\>]$
$\displaystyle+\epsilon^{2}\E[\int_{t_{j}}^{t_{j+1}}\<\int_{t_{j}}^{t_{j+1}}B(u^{\delta}(r))-B(u^{\delta}(t_{j}))dW(r),P^{N}(B(u^{\delta}(t_{j}))-B(u^{N}_{j}))dW(s)\>]$
$\displaystyle+\epsilon^{2}\E[\int_{t_{j}}^{t_{j+1}}\<\int_{t_{j}}^{t_{j+1}}B(u^{\delta}(t_{j}))-B(u_{j}^{N})dW(r),P^{N}(B(u^{\delta}(s))-B(u^{\delta}(t_{j})))dW(s)\>]$
$\displaystyle+\epsilon^{2}\E[\int_{t_{j}}^{t_{j+1}}\<\int_{t_{j}}^{t_{j+1}}B(u^{\delta}(t_{j}))-B(u_{j}^{N})dW(r),P^{N}(B(u^{\delta}(t_{j}))-B(u_{j}^{N}))dW(s)\>]$
$\displaystyle+\epsilon^{2}\E[\int_{t_{j}}^{t_{j+1}}\<\int_{t_{j}}^{t_{j+1}}((S(t_{j+1}-r)-I)B(u^{\delta}(r))-(S_{\tau}-I)B(u_{j}^{N}))dW(r),$
$\displaystyle\quad P^{N}(B(u^{\delta}(s))-B(u^{\delta}(t_{j})))dW(s)\>]$
$\displaystyle+\epsilon^{2}\E[\int_{t_{j}}^{t_{j+1}}\<\int_{t_{j}}^{t_{j+1}}((S(t_{j+1}-r)-I)B(u^{\delta}(r))-(S_{\tau}-I)B(u_{j}^{N}))dW(r),$
$\displaystyle\quad P^{N}(B(u^{\delta}(t_{j}))-B(u_{j}^{N}))dW(s)\>]$
$\displaystyle\leq
C\epsilon^{2}(\tau^{2}|\ln(\delta)|^{2}+\epsilon^{2}\tau^{2}+\lambda_{N}^{-1-\beta}|\ln(\delta)|\tau)+C\|Er_{j}\|^{2}\tau.$

Summarizing the above estimates, Gronwall’s inequality yields the desired
result.

Thanks to Theorem 3.2, we are in a position to present the convergence of the
energy functional as in Proposition 3.

######  Proposition 4.

Let Setting 2.1 hold with some $\bs=1+\beta>\frac{d}{2}$, $\beta\in[0,1)$. It
holds that in Case 1, for a small $\delta_{0}\sim O(\delta)$,

$\displaystyle\E[|H_{\delta}(u^{\delta}(t_{j}))-H_{\delta}(u_{j}^{N})|]$
$\displaystyle\leq
C\Big{(}\frac{\lambda_{N}^{-\frac{\beta+1}{2}}}{\delta}+\frac{\|f^{1}(u_{0}^{N})\|+\|f^{2}(u_{0}^{N})\|}{\delta_{0}}+\frac{|\ln(\delta)|}{\delta_{0}}({\epsilon}+\frac{\tau^{\frac{\beta+1}{2}}}{\delta}+\frac{\tau}{\delta}+\frac{\tau^{\frac{1}{2}}\epsilon}{\delta})\Big{)}^{\frac{(\bs-1)}{\bs}}|\ln(\delta)|,$

and that in Case 2,

$\displaystyle\E[|H_{\delta}(u^{\delta}(t_{j}))-H_{\delta}(u_{j}^{N})|]$
$\displaystyle\leq
C\Big{(}\frac{\lambda_{N}^{-\frac{\beta+1}{2}}}{\delta}+\frac{\|f^{1}(u_{0}^{N})\|+\|f^{2}(u_{0}^{N})\|}{\delta_{0}}+\frac{|\ln(\delta)|}{\delta_{0}}(\frac{\tau^{\frac{\beta+1}{2}}}{\delta}+\frac{\tau}{\delta})\Big{)}^{\frac{(\bs-1)}{\bs}}|\ln(\delta)|.$

######  Proof 3.4.

Following the steps as in the proof of Proposition 3, we have that

$\displaystyle\|\nabla(u^{\delta}(t_{j})-u_{j}^{N})\|_{L^{2}(\Omega;H)}$
$\displaystyle\leq
C\|(u^{\delta}(t_{j})-u_{j}^{N})\|_{L^{2}(\Omega;H)}^{\frac{\bs-1}{\bs}}|\ln(\delta)|^{\frac{1}{\bs}},$

and that

$\displaystyle\E[\|F_{\log}^{\delta}(u^{\delta}(t_{j}))-F_{\log}(u_{j}^{N})\|_{L^{1}}]$
$\displaystyle\leq C(1+|\ln(\delta)|\sqrt{\E[\|u(t)-u^{\delta}(t)\|^{2}]}).$

Combining the above estimates with Theorem 3.2 and (19), we complete the
proof.

The above result also implies that the proposed scheme is stable in the energy
space. Thanks to Theorem 3.2 and Theorem 2.3, one can obtain the overall
strong error estimate.

######  Corollary 5.

Let Setting 2.1 hold with some $\bs=1+\beta>\frac{d}{2},\beta\in[0,1)$. It
holds that for $\delta_{0}\sim O(\delta),$

$\displaystyle\sup_{j\leq
J}\|u_{det}(t_{j})-u_{j}^{N}\|_{L^{2}(\Omega;H)}^{2}$ $\displaystyle\leq
C|\ln(\delta)|^{2}\Big{(}\frac{\lambda_{N}^{-{\beta+1}}}{\delta^{2}}+\frac{\|f^{1}(u_{0}^{N})\|^{2}+\|f^{2}(u_{0}^{N})\|^{2}}{\delta_{0}^{2}}+\frac{\epsilon^{2}}{\delta_{0}^{2}}+\frac{\tau^{\beta+1}}{\delta_{0}^{2}\delta^{2}}+\frac{\tau^{2}}{\delta_{0}^{2}\delta^{2}}+\frac{\tau\epsilon^{2}}{\delta_{0}^{2}\delta^{2}}\Big{)}$
$\displaystyle+C(\delta^{2}+\epsilon^{2})$

in Case 1, and

$\displaystyle\sup_{j\leq J}\|u(t_{j})-u_{j}^{N}\|_{L^{2}(\Omega;H)}^{2}$
$\displaystyle\leq
C|\ln(\delta)|^{2}\Big{(}\frac{\lambda_{N}^{-{\beta+1}}}{\delta^{2}}+\frac{\|f^{1}(u_{0}^{N})\|^{2}+\|f^{2}(u_{0}^{N})\|^{2}}{\delta_{0}^{2}}+\frac{\tau^{\beta+1}}{\delta_{0}^{2}\delta^{2}}+\frac{\tau^{2}}{\delta_{0}^{2}\delta^{2}}\Big{)}+C\delta^{2}$

in Case 2.

It should be noticed that the derived convergence rate may be not optimal
since it relies on $\frac{1}{\delta}$. The upper bound of
$\|f^{1}(u_{0}^{N})\|$ and $\|f^{2}(u_{0}^{N})\|$ could be estimated as
follows. Let us illustrate the estimate of $\|f^{1}(u_{0}^{N})\|$ since that
of $\|f^{2}(u_{0}^{N})\|$ is similar. By Hölder’s inequality, Chebyshev’s
inequality, the Sobolev embedding theorem $L^{4}(\mathcal{D})\longrightarrow
H^{1}$ for $d\leq 3$ and (19), it holds that

$\displaystyle\|f^{1}(u_{0}^{N})\|$
$\displaystyle\leq|\\{x\in\mathcal{D}||u_{0}^{N}|>1\\}|^{\frac{1}{2}}\|1-u_{0}^{N}\|_{L^{4}(\mathcal{D})}$
$\displaystyle\leq
C(u(0))|\\{x\in\mathcal{D}||u_{0}^{N}-u(0)|>c_{0}\\}|^{\frac{1}{2}}$
$\displaystyle\leq C(u(0))\frac{\|u_{0}^{N}-u(0)\|}{c_{0}}\leq
C(u(0),c_{0})\lambda_{N}^{-\frac{\bs}{2}},$

where $c_{0}=1-|u(0)|_{E}>0$. In particular, if $\bs>\frac{d}{2}$, one can
obtain an improved bound,

$\displaystyle\|f^{1}(u_{0}^{N})\|$
$\displaystyle\leq|\\{x\in\mathcal{D}||u_{0}^{N}|>1\\}|\|1-u_{0}^{N}\|_{E}$
$\displaystyle\leq C(u(0),c_{0})\lambda_{N}^{-\bs}.$

One can slightly modify the condition of $u(0)$ and consider the random
initial data.

To end this part, we show that the proposed scheme nearly preserve the
Stampacchhia maximum principle via a strong convergence result in
$L^{p}(\Omega;E).$ Thanks to the Gagliardo–Nirenberg inequality, one can see
that the following asymptotic maximum principle holds. Due to Lemma 3.1 and
Proposition 4, from the Gagliardo–Nirenberg inequality, it holds that

$\displaystyle\|u^{\delta}(t_{j+1})-u_{j+1}^{N}\|_{L^{p}(\Omega;E)}$
$\displaystyle\leq
C\|u^{\delta}(t_{j+1})-u_{j+1}^{N}\|_{L^{p}(\Omega;H)}^{\theta}(\|u^{\delta}(t_{j+1})\|_{L^{p}(\Omega;H^{{\beta+1}})}^{1-\theta}+\|u_{j+1}^{N}\|_{L^{p}(\Omega;H^{{\beta+1}})}^{1-\theta}),$

where $\theta=\frac{d}{2(\beta+1)}\in(0,1).$ This implies that $d<2\beta+2$
with $\beta\in(0,1).$ Applying Theorem 3.2, we get

$\displaystyle\|u^{\delta}(t_{j+1})-u_{j+1}^{N}\|_{L^{p}(\Omega;E)}\to 0$

as
$|\ln(\delta)|(\frac{\lambda_{N}^{-\frac{\beta+1}{2}}}{\delta}+\frac{\|f^{1}(u_{0}^{N})\|+\|f^{2}(u_{0}^{N})\|}{\delta_{0}}+\frac{\tau^{\frac{\beta+1}{2}}}{\delta_{0}\delta}+\frac{\tau}{\delta_{0}\delta}+\frac{\tau^{\frac{1}{2}}}{\delta_{0}\delta})\sim
o(1)$. Up to a subsequence, we conclude that in Case 1,

$\displaystyle\lim_{(\epsilon,\delta,\tau,N)\to(0,0,0,+\infty)}\mathbb{P}(|u_{j+1}^{N}|<1)=1,$

and in Case 2,

$\displaystyle\lim_{(\delta,\tau,N)\to(0,0,+\infty)}\mathbb{P}(|u_{j+1}^{N}|<1)=1.$

##  4 Numerical experiments

In this section, we present several numerical experiments to validate our
theoretical findings for (1) and (3). For simplicity, in what follows, the
computational domains $\mathcal{D}$ are set to be $(0,2\pi)\times(0,2\pi)$ for
the periodic boundary condition and $(-1,1)\times(-1,1)$ for the homogeneous
Neumann boundary condition, respectively. The operator $B(u)$ satisfies

$\displaystyle B(u)e_{l,k}=B(u)\sqrt{q_{k,l}}e_{l,k},$

where $q_{k,l}=\frac{1}{1+k^{2}+l^{2}}$, $e_{l,k}=\exp\big{(}i(kx+ly)\big{)}$
for the periodic boundary condition and $e_{l,k}=\phi_{k}(x)\phi_{l}(y)$ for
the homogeneous Neumann boundary condition. Here $\phi_{k}(x)$ is defined by

$\phi_{k}(x):=L_{k}(x)-\frac{k(k+1)}{(k+2)(k+3)}L_{k+2}(x)$

with $L_{k}(x)$ being the Legendre polynomial with degree $k$, satisfying the
homogeneous Neumann boundary condition. For the evaluation of the noise term,
we have used the Algorithm 10.6 reported in [37]. Moreover, we always set
$c=3/2$ and choose the stabilizing parameter as $\alpha=2$ for the stabilized
scheme (16) unless specified otherwise.

Figure 1: The evolutions in time of the supremum norm (left) and the energy
(right) of the simulated solution with $B(u)=1$ (top) and $B(u)=\sin^{2}(\pi
u)$ (bottom).

###  4.1 Comparison for two type of noises:

In this test, two type of noises $B(u)=1$ (in Case 1) and $B(u)=\sin^{2}(\pi
u)$ (in Case 2) are considered for the model (3) with
$\sigma=1,\delta=1e-18,\epsilon=1$, and the initial data $u(0)=0.4[\cos(\pi
x)\sin(\pi y)+\sin(\pi x)\cos(2\pi y)]+0.2.$ The operator $A$ is subject to
the homogeneous Neumann boundary condition. The simulation is performed by the
stabilized scheme (16) with time step size $\tau=1e-3$, in which the Legendre
spectral method with $128\times 128$ modes is used for the spatial
discretization. In Figure 1, we plot the evolutions in time of the supremum
norm and the energy of the numerical solutions for both two types of noises up
to $T=300.$ It is observed that the numerical solution and the corresponding
energy will blow up at a finite time (about $t=10$) for the case of $B(u)=1$,
which indicates the ill-posedness of the model with $\epsilon=1$ and $B(u)=1$.
As shown in the last line of Figure 1, the regularized model (3) with
$B(u)=\sin^{2}(\pi u)$ is a more reasonable model, in which the Stampacchia
maximum bound and the energy evolution law are almost preserved. Therefore, in
what follows, we will focus our attention on Case 2 to test the performance of
proposed numerical schemes. For simplicity, we always choose
$B(u)=\sin^{2}(\pi u)$ in the rest of this section.

###  4.2 Test of the convergence accuracy

For simplicity, we only numerically study the convergence order of (16) for
(3) under the homogeneous Neumann boundary condition. Both the diffuse
interface width $\sigma$ and the noise density $\epsilon$ are set to be $1$,
and the initial data considered in this test is given by

$u(0)=0.4[\cos(\pi x)\sin(\pi y)+\sin(\pi x)\cos(2\pi y)]+0.2.$

The Legendre spectral method with $128\times 128$ modes is used for the
spatial discretization. Since there is no exact solution available, we will
use the numerical solution by stabilized semi-implicit scheme (16) with a
small enough time step size $\tau=1e-6$ as a reference solution. The averaged
$L^{2}$ errors over 1000 realizations as functions of the time step sizes are
plotted in log-log scale for several different tested $\delta$ in Figure 2. It
is observed that the stabilized semi-implicit scheme (16) achieves half order
in time with several different tested $\delta=0.1,1e-4$ and $1e-18$.

(a) $\delta=0.1$

(b) $\delta=1e-4$

(c) $\delta=1e-18$

Figure 2: The error behavior with respect to the time step size for the
stabilized semi-implicit scheme $B(u)=\sin^{2}(\pi u)$ at $T=0.1$.

###  4.3 Test on the effect of $\delta$ on the regularized model

In this test, we investigate the influence of the regularized parameter
$\delta$ on the model (3). The considered problem is subject to the
homogeneous Neumann boundary condition with following initial condition:

$u(0)=0.5\cos(\pi x)\cos(\pi y)+0.3.$

The simulation up to $T=300$ is performed by the stabilized semi-implicit
scheme (16) with $\tau=0.01$ and $128\times 128$ Legendre-type basis modes. In
Figure 3, the evolutions of the computed averaged energy over 500 realizations
are plotted for the model with several different regularized parameters, that
is $\delta=1e-2,1e-4,1e-6,1e-8,1e-12,$ and $1e-18$. It is observed that the
computed average energies converge to a limited one as $\delta\rightarrow 0.$

(a) energy evolution in time

(b) zoom in at $[0,10]$

Figure 3: The averaged energy evolution in time of (3) with $\sigma=0.1$,
$\epsilon=1e-4$ and several different values of
$\delta=1e-2,1e-4,1e-6,1e-8,1e-12,1e-18$.

(a)

(b)

(c)

Figure 4: The energy evolution in time of (3) with $\delta=1e-18$,
$\sigma=1e-2$ and several different values of $\epsilon=1e-n$ with
$n=2,4,6,8,10,12,14,16,18$.

###  4.4 Coarsening dynamics

The coarsening dynamics of the regularized model (3) and the deterministic
model are numerically investigated in this part. The related parameters in (3)
are set to be $\sigma=0.01,\delta=1e-18$, and the periodic boundary condition
is considered with the initial data $u(0)=0.01\cos(\pi x)\sin(\pi y)$.

Figure 5: Snapshots of the phase function $u$ produced by scheme (16) with
$\epsilon=1e-6$(the first column), $\epsilon=1e-8$(the second column),
$\epsilon=1e-10$(the third column) for (3) and the stabilized ETDRK2 scheme in
[27] (the last column) for the corresponding deterministic model at the times
$t=5$, $20$, $40$, $60$, $100$, $200$ and $500$ from top to bottom.

The simulations of coarsening dynamics for the model (3) and the deterministic
model are performed by the stabilized semi-implicit scheme (16) and the
second-order stabilized exponential time-differencing Runge–Kutta (ETDRK2)
scheme proposed in [27], respectively. This time step size used for the
testing schemes is set to be $\tau=1e-3$. For the spatial discretization, the
Fourier spectral method with $256\times 256$ modes is used for the model (3),
and the central finite difference method with uniform spatial grid spacing
$h=2\pi/256$ is applied to the deterministic model. Figure 4 shows a
comparison on the energy evolution between (3) with several different noise
densities $\epsilon$ and the deterministic model. It can be seen that the
energy produced by stabilized semi-implicit scheme (16) converges to the one
of the determined model computed by the stabilized ETDRK2 scheme as
$\epsilon\rightarrow 0$. Moreover, it is shown in Figure 4 (c) that the
computed energies for the regularized model (3) are consistent with the
determined ones within the range of time-stepping scheme error about $\tau$,
if the noise density $\epsilon\leq 1e-10$ for the tested $\epsilon$. This is
also shown in columns 3 and 4 of Figure 5 showing several almost consistent
snapshots of $u$ for the case of $\epsilon=1e-10$ and the deterministic
problem along the coarsening process, respectively. Another notable
observation in Figure 4 is that the computed energies are consistent with each
other for all tested cases at early time (about $t=12$), and the ones of the
tested cases $\epsilon\geq 1e-6$ exhibit completely different behaviors at the
later time, which leads a totally different phase transition process seeing
the first column of Figure 5. For the tested cases of $\epsilon\leq 1e-8$, the
computed energies along the time are almost similar behaviors except the case
of $\epsilon=1e-8$ with $t\in(60,100)$, as shown in Figure 4 (b). Furthermore,
it also leads to different phase transition behaviors for the case of
$\epsilon=1e-8$ at the later time comparing with the tested case
$\epsilon=1e-10$ and the deterministic problem, as shown in the last four rows
of Figure 5 column 2-4.

##  5 Conclusions

In this paper, we propose and study two multiscale models for a parabolic SPDE
with a Flory–Huggins logarithmic potential which emerges from the soft matter
and phase separation. The key tools are the energy regularized technique for
the Flory–Huggins logarithmic potential and the Stampacchia maximum principle
for studying the possible singularity of the solution. Then we show the
stability and strong convergence of a stabilized scheme for the considered
logarithmic SPDE. Following this work, many open problems deserve further
investigation. For example, it is unknown how to establish the optimal strong
and weak convergence rate of the energy regularization model and energy
regularized numerical approximations. What can we expect by extending the
energy-regularization technique to stochastic Cahn–Hillard equation with a
Flory–Huggins logarithmic potential? These problems are very crucial to
improve accuracy of numerical simulation and to design high-order convergent
schemes for logarithmic SPDEs. We plan to investigate them in the future.

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