Boundary Layer Study for an Ocean Related System with a Small Viscosity Parameter

TL;DR

This paper investigates the behavior of a linearized ocean system with small viscosity, establishing convergence results and boundary layer thickness estimates as viscosity approaches zero.

Contribution

It provides the first rigorous analysis of boundary layer effects and convergence for the linearized primitive equations with small viscosity.

Findings

Proves $L^ abla$ convergence as viscosity tends to zero.

Estimates the boundary layer thickness in the system.

Analyzes the impact of small viscosity on the system's behavior.

Abstract

We study an ocean related system with a small viscosity parameter, which is the linearized version of the modified Primitive Equations. As the parameter goes to zero, a $L^\infty$ convergence result is obtained together with the estimation on the thickness of the boundary layer.

Equations

$$\{\begin{array}{rl}& u_t^{\epsilon }+{\overline{U}}_0{u}_x^{\epsilon }+{\psi }_x^{\epsilon }-2\epsilon u_{xx}^{\epsilon }=f,\\ & {\psi }_t^{\epsilon }+{\overline{U}}_0{\psi }_x^{\epsilon }+{\lambda }^{-2}{u}_x^{\epsilon }=g,\end{array}$$

$$\{\begin{array}{rl}& u^{\epsilon }(0,t)=u^{\epsilon }(L,t)=0,{\psi }^{\epsilon }(0,t)=0,0<t<T,\\ & u^{\epsilon }(x,0)=u_0(x),{\psi }^{\epsilon }(x,0)={\psi }_0(x),0<x<L,\end{array}$$

$$\{\begin{array}{rl}& u_t^0+{\overline{U}}_0{u}_x^0+{\psi }_x^0=f,\\ & {\psi }_t^0+{\overline{U}}_0{\psi }_x^0+{\lambda }^{-2}{u}_x^0=g.\end{array}$$

$$\{\begin{array}{rl}& u^0(0,t)+{\overline{U}}_0{\lambda }^2{\psi }^0(0,t)=0,u^0(L,t)=0,0<t<T,\\ & u^0(x,0)=u_0(x),{\psi }^0(x,0)={\psi }_0(x),0<x<L.\end{array}$$

$$U^{\epsilon }=u^{\epsilon }-u^0-{\theta }_u^{\epsilon },{\Psi }^{\epsilon }={\psi }^{\epsilon }-{\psi }^0-{\theta }_{\psi }^{\epsilon },$$

$$\begin{array}{rl}& \Vert (U^{\epsilon },{\Psi }^{\epsilon }){\Vert }_{L^{\infty }(0,T;L^2(0,L))}+{\epsilon }^{1/2}\Vert (U_x^{\epsilon },{\Psi }_x^{\epsilon }){\Vert }_{L^2((0,L)\times (0,T))}\le C\epsilon ,\\ & \Vert (U^{\epsilon },{\Psi }^{\epsilon }){\Vert }_{L^{\infty }(0,T;L^{\infty }(0,L))}\le C{\epsilon }^{1/2}.\end{array}$$

$$\left(\begin{array}{rl}& {\theta }_u^{\epsilon }(x,t)\\ & {\theta }_{\psi }^{\epsilon }(x,t)\end{array}\right)=e^{-rx/\epsilon }\left(\begin{array}{rl}& A^{\epsilon }(t)\\ & B^{\epsilon }(t)\end{array}\right)+\left(\begin{array}{rl}& C^{\epsilon }(t)\\ & D^{\epsilon }(t)\end{array}\right),$$

$$\left(\begin{array}{rl}& A^{\epsilon }\\ & B^{\epsilon }\\ & C^{\epsilon }\\ & D^{\epsilon }\end{array}\right)=\left(\begin{array}{rl}& -(1-e^{-rL/\epsilon }{)}^{-1}{u}^0(0,t)\\ & {{\overline{U}}_0}^{-1}{\lambda }^{-2}(1-e^{-rL/\epsilon }{)}^{-1}{u}^0(0,t)\\ & e^{-rL/\epsilon }(1-e^{-rL/\epsilon }{)}^{-1}{u}^0(0,t)\\ & -{{\overline{U}}_0}^{-1}{\lambda }^{-2}{e}^{-rL/\epsilon }(1-e^{-rL/\epsilon }{)}^{-1}{u}^0(0,t)\end{array}\right).$$

$$\{\begin{array}{rl}& {\overline{U}}_0{\theta }_{ux}^{\epsilon }+{\theta }_{\psi x}^{\epsilon }-2\epsilon {\theta }_{uxx}^{\epsilon }=0,{\overline{U}}_0{\theta }_{\psi x}^{\epsilon }+{\lambda }^{-2}{\theta }_{ux}^{\epsilon }=0,\\ & {\theta }_u^{\epsilon }(0,t)=-u^0(0,t),{\theta }_{\psi }^{\epsilon }(0,t)=-{\psi }^0(0,t),{\theta }_u^{\epsilon }(L,t)=0.\end{array}$$

$$\{\begin{array}{rl}& \Vert ({\theta }_u^{\epsilon },{\theta }_{\psi }^{\epsilon },{\theta }_{ut}^{\epsilon },{\theta }_{\psi t}^{\epsilon }){\Vert }_{L^{\infty }(0,T;L^2(0,L))}\le C{\epsilon }^{1/2},\\ & \Vert ({\theta }_{utx}^{\epsilon },{\theta }_{\psi tx}^{\epsilon }){\Vert }_{L^{\infty }(0,T;L^2(0,L))}\le C,\\ & \Vert x({\theta }_{ut}^{\epsilon },{\theta }_{\psi t}^{\epsilon },\epsilon {\theta }_{utx},\epsilon {\theta }_{\psi tx}){\Vert }_{L^{\infty }(0,T;L^2(0,L))}\le C{\epsilon }^{3/2}.\end{array}$$

$$\{\begin{array}{rl}& u_t+{\overline{U}}_0{u}_x+{\psi }_x-2\epsilon u_{xx}=2\epsilon u_{xx}^0-{\theta }_{ut}^{\epsilon },\\ & {\psi }_t+{\overline{U}}_0{\psi }_x+{\lambda }^{-2}{u}_x=-{\theta }_{\psi t}^{\epsilon },\end{array}$$

$$\{\begin{array}{rl}& u(0,t)=u(L,t)=0,\psi (0,t)=0,0<t<T,\\ & u(x,0)=0,\psi (x,0)=0,0<x<L.\end{array}$$

$$\begin{array}{rl}& (u-\lambda \psi {)}_t+{\overline{U}}_0(u-\lambda \psi {)}_x-{\lambda }^{-1}(u-\lambda \psi {)}_x-2\epsilon u_{xx}=2\epsilon u_{xx}^0-{\theta }_{ut}^{\epsilon }+\lambda {\theta }_{\psi t}^{\epsilon }.\end{array}$$

$$\begin{array}{r}2\epsilon {\lambda }^2{\psi }_{xt}+2\epsilon {\overline{U}}_0{\lambda }^2{\psi }_{xx}+2\epsilon u_{xx}=-2\epsilon {\lambda }^2{\theta }_{\psi tx}^{\epsilon }.\end{array}$$

$$\begin{array}{rl}& W_t+{\overline{U}}_0{W}_x-{\lambda }^{-1}(u-\lambda \psi {)}_x=2\epsilon u_{xx}^0-{\theta }_{ut}^{\epsilon }+\lambda {\theta }_{\psi t}^{\epsilon }-2\epsilon {\lambda }^2{\theta }_{\psi tx}^{\epsilon }=:2\epsilon u_{xx}^0+F^{\epsilon }.\end{array}$$

$$\begin{array}{rl}& \frac{1}{2}{\int }_0^L{W}^{2}dx+\frac{{\overline{U}}_0}{2}{\int }_0^t{W}^2{|}_{x=L}ds+2\epsilon {\lambda }^2{\int }_0^t{\int }_0^L{\psi }_x^{2}dxds\\ & =\frac{1}{2}{\int }_0^L{W}^2{|}_{t=0}dx+\frac{{\overline{U}}_0}{2}{\int }_0^t{W}^2{|}_{x=0}ds+\frac{1}{2\lambda }{\int }_0^t(u-\lambda \psi {)}^2{|}_{x=0}^{x=L}ds\\ & +2\epsilon \lambda {\int }_0^t{\int }_0^L{u}_x{\psi }_{x}dxds+{\int }_0^t{\int }_0^L(2\epsilon u_{xx}^0+F^{\epsilon })Wdxds\\ & =\frac{\lambda }{2}{\int }_0^t{\psi }^2{|}_{x=L}ds+2{\epsilon }^2{\overline{U}}_0{\lambda }^4{\int }_0^t{\psi }_x^2{|}_{x=0}ds+2\epsilon \lambda {\int }_0^t{\int }_0^L{u}_x{\psi }_{x}dxds\\ & +{\int }_0^t{\int }_0^{L}2\epsilon u_{xx}^{0}Wdxds+{\int }_0^t{\int }_0^L{F}^{\epsilon }(u-\lambda \psi )dxds+2\epsilon {\lambda }^2{\int }_0^t{\int }_0^L{F}^{\epsilon }{\psi }_{x}dxds\\ & =:\frac{\lambda }{2}{\int }_0^t{\psi }^2{|}_{x=L}ds+\sum _{i=1}^5{E}_i.\end{array}$$

$$\begin{array}{r}{\int }_0^L\frac{u^2}{x^2}dx\le 4{\int }_0^L{u}_x^{2}dx,{\int }_0^L\frac{{\psi }^2}{x^2}dx\le 4{\int }_0^L{\psi }_x^{2}dx.\end{array}$$

$$\begin{array}{r}\Vert F^{\epsilon }{\Vert }_{L^{\infty }(0,T;L^2(0,L))}^2\le C\epsilon ,\Vert x{F}^{\epsilon }{\Vert }_{L^{\infty }(0,T;L^2(0,L))}^2\le C{\epsilon }^3.\end{array}$$

$$\begin{array}{rl}{E}_4\le & \frac{C}{\epsilon }\Vert x{F}^{\epsilon }{\Vert }_{L^2((0,T)\times (0,L))}^2+\frac{\epsilon }{32}{\int }_0^t{\int }_0^L\frac{(u-\lambda \psi {)}^2}{x^2}dxds\\ \le & C{\epsilon }^2+\frac{\epsilon }{4}{\int }_0^t{\int }_0^L(u_x^2+{\lambda }^2{\psi }_x^2)dxds,\end{array}$$

$$\begin{array}{rl}{E}_5\le & C{\epsilon }^2+\frac{\epsilon {\lambda }^2}{4}{\int }_0^t{\int }_0^L{\psi }_x^{2}dxds.\end{array}$$

$$\begin{array}{rl}{E}_1\le & C{\epsilon }^2+\frac{4{\epsilon }^2}{{\overline{U}}_0}{\int }_0^t{u}_x^2(0,s)ds.\end{array}$$

$$\begin{array}{rl}{E}_1\le & C{\epsilon }^2+\frac{8{\epsilon }^2}{{\overline{U}}_0}{\int }_0^t{\int }_0^L|u_x{u}_{xx}|dxds\\ \le & C{\epsilon }^2+\frac{64\epsilon }{{\overline{U}}_0^2{\lambda }^2}{\int }_0^t{\int }_0^L{u}_x^{2}dxds+\frac{{\epsilon }^3{\lambda }^2}{4}{\int }_0^t{\int }_0^L{u}_{xx}^{2}dxds.\end{array}$$

$$\begin{array}{rl}& {\int }_0^t{\int }_0^L(4{\epsilon }^2{u}_{xx}^2+u_t^2)dxds+2\epsilon {\int }_0^L{u}_x^{2}dx\\ & ={\int }_0^t{\int }_0^L(u_t-2\epsilon u_{xx}{)}^{2}dxds\\ & ={\int }_0^t{\int }_0^L(2\epsilon u_{xx}^0-{\theta }_{ut}^{\epsilon }-{\overline{U}}_0{u}_x-{\psi }_x{)}^{2}dxds\\ & \le C\epsilon +4{\overline{U}}_0^2{\int }_0^t{\int }_0^L{u}_x^{2}dxds+4{\int }_0^t{\int }_0^L{\psi }_x^{2}dxds,\end{array}$$

$$\begin{array}{rl}& {\epsilon }^2{\int }_0^t{\int }_0^L{u}_{xx}^{2}dxds\le C\epsilon +{\overline{U}}_0^2{\int }_0^t{\int }_0^L{u}_x^{2}dxds+{\int }_0^t{\int }_0^L{\psi }_x^{2}dxds.\end{array}$$

$$\begin{split}E_{1}\leq&C\varepsilon^{2}+\Big{(}\frac{64}{\overline{U}_{0}^{2}\lambda^{2}}+\frac{\overline{U}_{0}^{2}\lambda^{2}}{4}\Big{)}\varepsilon\int_{0}^{t}\int_{0}^{L}u_{x}^{2}dxds+\frac{\varepsilon\lambda^{2}}{4}\int_{0}^{t}\int_{0}^{L}\psi_{x}^{2}dxds.\\ \end{split}$$

$$\begin{array}{rl}{E}_2+E_3\le & C{\epsilon }^2+\frac{1}{2}{\int }_0^t{\int }_0^L{W}^{2}dxds+4\epsilon {\int }_0^t{\int }_0^L{u}_x^{2}dxds+\frac{\epsilon {\lambda }^2}{4}{\int }_0^t{\int }_0^L{\psi }_x^{2}dxds.\end{array}$$

$$\begin{array}{rl}& \frac{1}{2}{\int }_0^L{W}^{2}dx+\epsilon {\lambda }^2{\int }_0^t{\int }_0^L{\psi }_x^{2}dxds\\ & \le C{\epsilon }^2+\frac{\lambda }{2}{\int }_0^t{\psi }^2{|}_{x=L}ds+C_0\epsilon {\int }_0^t{\int }_0^L{u}_x^{2}dxds+\frac{1}{2}{\int }_0^t{\int }_0^L{W}^{2}dxds,\end{array}$$

$$\begin{array}{rl}& {\int }_0^L(u^2+{\lambda }^2{\psi }^2)dx+{\overline{U}}_0{\lambda }^2{\int }_0^t{\psi }^2{|}_{x=L}ds+4\epsilon {\int }_0^t{\int }_0^L{u}_x^{2}dxds\\ & =2{\int }_0^t{\int }_0^L\left(2\epsilon u_{xx}^{0}u-{\theta }_{ut}^{\epsilon }u-{\lambda }^2{\theta }_{\psi t}^{\epsilon }\psi \right)dxds\\ & \le C{\epsilon }^2+C{\int }_0^t{\int }_0^L{u}^{2}dxds+2\epsilon {\int }_0^t{\int }_0^L{u}_x^{2}dxds+\frac{\epsilon {\lambda }^2}{2(C_0+1/(2\lambda {\overline{U}}_0))}{\int }_0^t{\int }_0^L{\psi }_x^{2}dxds.\end{array}$$

$$\begin{array}{rl}& {\int }_0^L(u^2+{\lambda }^2{\psi }^2)dx+{\overline{U}}_0{\lambda }^2{\int }_0^t{\psi }^2{|}_{x=L}ds+2\epsilon {\int }_0^t{\int }_0^L{u}_x^{2}dxds\\ & \le C{\epsilon }^2+C{\int }_0^t{\int }_0^L{u}^{2}dxds+\frac{\epsilon {\lambda }^2}{2(C_0+1/(2\lambda {\overline{U}}_0))}{\int }_0^t{\int }_0^L{\psi }_x^{2}dxds.\end{array}$$

$$\begin{array}{rl}& \frac{1}{2}{\int }_0^L{W}^{2}dx+C_0{\int }_0^L(u^2+{\lambda }^2{\psi }^2)dx+\epsilon {\int }_0^t{\int }_0^L\left(\frac{{\lambda }^2}{2}{\psi }_x^2+C_0{u}_x^2\right)dxds\\ & \le C{\epsilon }^2+C{\int }_0^t{\int }_0^L(u^2+W^2)dxds.\end{array}$$

Full text

Keywords. Primitive Equations, $L^{\infty}$ convergence, boundary layer.

2010 MSC. 35L50; 47D06; 76D10; 76N20; 86A05.

1 Introduction

We consider the ocean related system with a small parameter $\varepsilon$ in the space-time domain $(0,L)\times(0,T)$:

[TABLE]

with the boundary and initial conditions:

[TABLE]

where $0<\varepsilon\ll 1,{\overline{U}_{0}}>0$ and $\lambda>0$ are constants with ${\overline{U}_{0}}<\lambda^{-1}$. This system was derived from the Primitive Equations (PEs) of the ocean with mild viscosity thanks to a modal decomposition in the vertical direction, see [1]( cf. [3, 2]).

The PEs are one of the fundamental models for geophysical flows and they are used to describe oceanic and atmospheric dynamics, see [4, 5]. In the presence of viscosity, the study of the PEs through analytical means was started by Lions, Temam and Wang in [6, 7], and some recent advances for the PEs have been obtained, see [8] and the survey [9]. In the absence of viscosity, it is known that the PEs are not well-posed for any set of local boundary conditions, see [10]. To overcome this difficulty, a modified PEs (i.e. $\delta$-PEs) was proposed in [1]. System (1.1) is the linearized version of the modified PEs.

Formally setting $\varepsilon=0$ in (1.1), one obtains the following system in the space-time domain $(0,L)\times(0,T)$:

[TABLE]

Like in [3, 2], the boundary and initial conditions are chosen as follows:

[TABLE]

In [3], the existence and uniqueness of solutions of both problem (1.1)-(1.2) and problem (1.3)-(1.4) was proved for some initial data. As numerically shown in [2], some boundary layers appear at the boundary $x=0$ when $\varepsilon$ goes to zero. Actually, it was shown in [3, Theorem 1.3] that $U^{\varepsilon}$ and $\Psi^{\varepsilon}$ are $O(\varepsilon^{1/2})$ in $L^{\infty}(0,T;L^{2}(0,L))$, where

[TABLE]

where $\theta_{u}^{\varepsilon}$ and $\theta_{\psi}^{\varepsilon}$ are the boundary layer correctors and are the same as in Section 2. This implies that $(u^{\varepsilon},\psi^{\varepsilon})$ converges to $(u^{0},\psi^{0})$ in $L^{\infty}(0,T;L^{2}(0,L))$ as $\varepsilon\rightarrow 0^{+}$. Recently, the analysis of the boundary layers for the linearized viscous PEs has been presented in [11] for 2D and in [12, 13, 14] for 3D.

In the present paper, we study the $L^{\infty}$ convergence of $(U^{\varepsilon},\Psi^{\varepsilon})$ as $\varepsilon\rightarrow 0^{+}$. Our main result is as follows.

Theorem 1.1**.**

Assume that $(u^{\varepsilon},\psi^{\varepsilon})\in\mathcal{K}$ and $(u^{0},\psi^{0})\in\mathcal{K}^{0}$ are solutions of problem (1.1)-(1.2) and problem (1.3)-(1.4), respectively, where $\mathcal{K}$ and $\mathcal{K}^{0}$ are the same as in Section 3. Then there exists some constant $C$ independent of $\varepsilon$ such that

[TABLE]

From Theorem 1.1 and the definitions of $\theta_{u}^{\varepsilon}$ and $\theta_{\psi}^{\varepsilon}$ (see Section 2), we immediately obtain

Corollary 1.1**.**

Under the assumptions of Theorem 1.1, we have $\lim_{\varepsilon\rightarrow 0^{+}}\|(u^{\varepsilon}-u^{0},\psi^{\varepsilon}-\psi^{0})\|_{L^{\infty}(0,T;L^{\infty}(\delta(\varepsilon),L))}=0$ for any nonnegative function $\delta(\varepsilon)$ satisfying $\delta(\varepsilon)\rightarrow 0$ and ${\delta(\varepsilon)}/{\varepsilon}\rightarrow+\infty$ as $\varepsilon\rightarrow 0^{+}$, and $\mathop{\underline{\lim}}_{\varepsilon\rightarrow 0^{+}}\|(u^{\varepsilon}-u^{0},\psi^{\varepsilon}-\psi^{0})\|_{L^{\infty}(0,T;L^{\infty}(0,\delta(\varepsilon)))}>0$ for any nonnegative function $\delta(\varepsilon)$ satisfying $\delta(\varepsilon)\rightarrow 0$ and ${\delta(\varepsilon)}/{\varepsilon}\rightarrow c~{}(\hbox{nonnegative constant})$ as $\varepsilon\rightarrow 0^{+}$ whenever $(u^{0}(0,t),\psi^{0}(0,t))\not\equiv(0,0)$ in $(0,T)$. This implies that the boundary layer thickness is of the order $O(\varepsilon)$.

Theorem 1.1 will be proved in Section 2. The main difficulty in the proof is that system (1.1) is an incompletely parabolic perturbation of a hyperbolic system. To overcome the difficulty, a key observation is to obtain the equation (2.9). With this, one can deduce the required estimates, for example, (2.25). In Section 3, some remarks on the regularities of $(u^{\varepsilon},\psi^{\varepsilon})$ and $(u^{0},\psi^{0})$ will be presented. The regularity conditions will be used to show the equation (2.8).

2 Proof of Theorem 1.1

Let $(u^{0},\psi^{0})\in\mathcal{K}^{0}$ be a solution of problem (1.3)-(1.4). As in [3], $\theta_{u}^{\varepsilon}$ and $\theta_{\psi}^{\varepsilon}$ are defined as follows:

[TABLE]

where $r=\frac{1}{2{\overline{U}_{0}}\lambda^{2}}-\frac{{\overline{U}_{0}}}{2}>0$, and

[TABLE]

Due to $u^{0}(0,t)=-\overline{U}_{0}\lambda^{2}\psi^{0}(0,t)$, it is easy to verify that $\theta_{u}^{\varepsilon}$ and $\theta_{\psi}^{\varepsilon}$ satisfy

[TABLE]

From now on, we use $C$ to denote a positive generic constant independent of $\varepsilon$.

Lemma 2.1**.**

Assume that $(u^{0},\psi^{0})\in\mathcal{K}^{0}$ is a solution of problem (1.3)-(1.4). Then

[TABLE]

Proof.

Due to $u^{0}(L,t)=0$, $u_{t}^{0}(0,t)=-\int_{0}^{L}u^{0}_{xt}(x,t)dx$. Thus, $u_{t}^{0}(0,t)\in L^{\infty}(0,T)$ since $u^{0}_{xt}\in L^{\infty}(0,T;L^{2}(0,L))$. This together with $u^{0}(0,t)=-\overline{U}_{0}\lambda^{2}\psi^{0}(0,t)$ gives $\psi_{t}^{0}(0,t)\in L^{\infty}(0,T)$. Using the inequality $e^{-x}\leq 6/x^{3}~{}(\forall x>0)$, we have $e^{-2rL/\varepsilon}\leq C\varepsilon^{3}$. Noticing $\|xe^{-rx/\varepsilon}\|_{L^{2}(0,L)}\leq C\varepsilon^{3/2}$, we deduce that $\|x\theta_{ut}\|_{L^{\infty}(0,T;L^{2}(0,L))}\leq C\varepsilon^{3/2}.$ The other estimates of (2.4) can be deduced similarly. The proof is completed. ∎

Proof of Theorem 1.1 For simplicity, write $u=U^{\varepsilon}$ and $\psi=\Psi^{\varepsilon}$. It follows from (1.1), (1.3) and (2.3) that $u$ and $\psi$ satisfy

[TABLE]

with the boundary and initial conditions

[TABLE]

From (2.5), we obtain

[TABLE]

Differentiating (2.5)2 in $x$ and multiplying the resulting equation by $2\varepsilon\lambda^{2}$, we have

[TABLE]

Denote $W=u-\lambda\psi+2\varepsilon\lambda^{2}\psi_{x}$. Adding (2.7) and (2.8) yields

[TABLE]

Multiplying (2.9) by $W$, integrating over $(0,L)\times(0,t)$ and using (2.6), we have

[TABLE]

Using the Hardy inequality (cf. [15]) and noticing (2.6), we obtain

[TABLE]

By Lemma 2.1, we have

[TABLE]

Applying the Young inequality, (2.12) and (2.11), we deduce that

[TABLE]

and

[TABLE]

Using (2.5)2, we have $\psi_{x}|_{x=0}=-{\overline{U}_{0}}^{-1}\lambda^{-2}u_{x}|_{x=0}-{\overline{U}_{0}}^{-1}\theta^{\varepsilon}_{\psi t}|_{x=0}$. Noticing $\psi_{t}^{0}(0,t)\in L^{\infty}(0,T)$ (see the proof of Lemma 2.1), we have

[TABLE]

Since $u(0,t)=u(L,t)=0$, there exists some $\xi=\xi_{t}\in(0,L)$ such that $u_{x}(\xi,t)=0$, thus, $u_{x}^{2}(0,t)=-\int_{0}^{\xi}(u_{x}^{2})_{x}dx.$ Substituting it into (2.15) and using the Young inequality, we obtain

[TABLE]

Using $u^{0}_{xx}\in L^{\infty}(0,T;L^{2}(0,L))$ and Lemma 2.1, we derive from (2.5)1 that

[TABLE]

consequently,

[TABLE]

Substituting (2.18) into (2.16) yields

[TABLE]

By the Young inequality, we have

[TABLE]

Plugging (2.13), (2.14), (2.19) and (2.20) into (2.10), we obtain

[TABLE]

where $C_{0}=\frac{17}{4}+\frac{64}{\overline{U}_{0}^{2}\lambda^{2}}+\frac{\overline{U}_{0}^{2}\lambda^{2}}{4}$.

On the other hand, multiplying (2.5)1 and (2.5)2 by $2u$ and $2\lambda^{2}\psi$ respectively, integrating over $(0,L)\times(0,t)$, using (2.6) and (2.11), and performing a similar argument to (2.13), we obtain

[TABLE]

Hence,

[TABLE]

Multiplying (2.23) by $C_{0}+1/(2\lambda{\overline{U}_{0}})$ and adding the resulting equations to (2.21), we have

[TABLE]

Then, the Gronwall inequality gives

[TABLE]

Recalling (2.17) and $W=u-\lambda\psi+2\varepsilon\lambda^{2}\psi_{x}$ and using (2.24), we deduce that

[TABLE]

Thanks to $u(0,t)=0$, we have $u^{2}(x,t)=\int_{0}^{x}(u^{2})_{x}dx$. Then, using the Hölder inequality, (2.24) and (2.25), we obtain

[TABLE]

thus, $\|U^{\varepsilon}\|_{L^{\infty}(0,T;L^{\infty}(0,L))}\leq C\varepsilon^{1/2}.$ Similarly, $\|\Psi^{\varepsilon}\|_{L^{\infty}(0,T;L^{\infty}(0,L))}\leq C\varepsilon^{1/2}.$ The proof of Theorem 1.1 is completed.

3 Remarks on Regularity of Solutions

By the Hille-Yosida theorem, the authors in [3] proved the following results on existence and uniqueness of both problem (1.1)-(1.2) and problem (1.3)-(1.4):

(i)  If $(f,g)\in L^{1}(0,T;H)$, $(u_{0},\psi_{0})\in D(A)$, and $(f,g)$ is continuous in $H$ at $t=0$, then for every $\varepsilon>0$ problem (1.1)-(1.2) admits a unique solution $(u^{\varepsilon},\psi^{\varepsilon})$ in ${\mathcal{F}}=C([0,T];H)\cap L^{\infty}(0,T;D(A))$ with $(u^{\varepsilon}_{t},\psi^{\varepsilon}_{t})\in L^{\infty}(0,T;H)$, where $H=L^{2}(0,L)\times L^{2}(0,L)$ and $D(A)=\{(u,\psi)\in H|u_{x},\psi_{x},u_{xx}\in L^{2}(0,L),u(0)=\psi(0)=u(L)=0\}.$

(ii)  If $(f,g)\in L^{1}(0,T;H)$ and $(u_{0},\psi_{0})\in D(A^{0})$, then problem (1.1)-(1.2) admits a unique solution $(u^{0},\psi^{0})$ in ${\cal F}^{0}=C([0,T];H)\cap L^{\infty}(0,T;D(A^{0}))$ with $(u^{0}_{t},\psi^{0}_{t})\in L^{\infty}(0,T;H)$, where $D(A^{0})=\{(u,\psi)\in H|u_{x},\psi_{x}\in L^{2}(0,L),u(0)+{\overline{U}_{0}}\lambda^{2}\psi(0)=0,u(L)=0\}.$

To get $(u^{\varepsilon},\psi^{\varepsilon})\in\mathcal{K}$, where

[TABLE]

some additional conditions on $f,g,u_{0},\psi_{0}$ must be imposed, for example, the following conditions: $(f_{t},g_{t})\in L^{1}(0,T;H)$, $(u_{0},\psi_{0})\in D(A)\cap(H^{4}(0,L)\times H^{3}(0,L))$, $(f_{t},g_{t})$ is continuous in $H$ at $t=0$, and

[TABLE]

Indeed, we observe by differentiating the equations in (1.1) with respect to $t$ that $(u^{\varepsilon}_{t},\psi^{\varepsilon}_{t})$ satisfies (1.1) with $(f_{t},g_{t})$ instead of $(f,g)$ and the initial condition $(u^{\varepsilon}_{t}|_{t=0},\psi^{\varepsilon}_{t}|_{t=0})=(f(x,0)-{\overline{U}_{0}}u_{0x}-\psi_{0x}+2\varepsilon u_{0xx},~{}~{}g(x,0)-{\overline{U}_{0}}\psi_{0x}-\lambda^{-2}u_{0x}).$ From the above assumptions, we find that $(u^{\varepsilon}_{t}|_{t=0},\psi^{\varepsilon}_{t}|_{t=0})\in D(A)$. Thus, by a similar argument to (i), one has $(u^{\varepsilon}_{t},\psi^{\varepsilon}_{t})\in C([0,T];H)\cap L^{\infty}(0,T;D(A))$ and then, one deduces from (1.1)2 that $\psi^{\varepsilon}_{x}\in L^{\infty}(0,T;H^{1}(0,L))$ if $g_{x}\in L^{\infty}(0,T;L^{2}(0,L))$. Consequently, $(u^{\varepsilon},\psi^{\varepsilon})\in\mathcal{K}$.

To get $(u^{0},\psi^{0})\in\mathcal{K}^{0}$, where

[TABLE]

we observe that under (3.1), $f,g,u_{0},\psi_{0}$ satisfy the compatibility conditions (1.72) of [3]. Consequently, if in addition we assume that $(f_{x},g_{x})\in L^{\infty}(0,T;H)$ and $u_{0xx},\psi_{0xx},f_{t}|_{t=0},g_{t}|_{t=0}\in L^{2}(0,L)$, then $(u^{0},\psi^{0})\in\mathcal{K}^{0}$, see [3, Remark 1.4] for the detail.

Acknowledgments

The research was supported by the NSFC (11571062, 11571380), the Program for Liaoning Innovative Talents in University (LR2016004), Guangzhou Science and Technology Program (201607010144) and the Fundamental Research Fund for the Central Universities (DMU).