An estimate on the thickness of boundary layer for nonlinear evolution equations with damping and diffusion

Abstract

The main purpose of this paper is to estimate the thickness of boundary layer for nonlinear evolution equations with damping and diffusion as the diffusion parameter β goes to zero. We prove that the thickness of layer is of the order $O(\beta^{\gamma})$ with $0<\gamma<1$, thus improving the corresponding result in (Ruan and Zhu in Discrete Contin. Dyn. Syst. 32(1) 331-352, 2012) where $0<\gamma<1/2$ is obtained.

1 Introduction

In this paper, we consider the nonlinear evolution equations with damping and diffusion:

$$ \textstyle\begin{cases} \psi_{t}^{\beta}=-( \sigma-\alpha)\psi^{\beta}-\sigma\theta _{x}^{\beta}+ \alpha\psi^{\beta}_{xx}, \\ \theta_{t}^{\beta}=-(1-\beta)\theta^{\beta}+\mu\beta \psi^{\beta}_{x}+2\psi^{\beta}\theta_{x}^{\beta}+ \beta\theta^{\beta}_{xx},\quad 0< x< 1, t>0, \end{cases} $$

(1.1)

with the initial-boundary conditions

$$ \begin{aligned} &\bigl(\psi^{\beta}, \theta^{\beta}\bigr) (x,0)=(\psi_{0},\theta_{0}) (x),\quad 0 \leq x\leq1, \\ &\bigl(\psi^{\beta},\theta^{\beta}\bigr) (1,t)=\bigl( \psi^{\beta},\theta^{\beta}\bigr) (0,t)=(0,0),\quad t \geq0, \end{aligned} $$

(1.2)

where $\sigma, \alpha, \beta$, and μ are positive constants with $\alpha<\sigma$ and $0<\beta<1$. The corresponding problem of zero diffusion limit as $\beta\rightarrow0$ is the following:

$$ \textstyle\begin{cases} \psi_{t}^{0}=-( \sigma-\alpha)\psi^{0}-\sigma\theta_{x}^{0}+\alpha \psi^{0}_{xx}, \\ \theta_{t}^{0}=-\theta^{0}+2\psi^{0} \theta_{x}^{0},\quad 0< x< 1, t>0, \end{cases} $$

(1.3)

with the initial-boundary conditions

$$ \begin{aligned} &\bigl(\psi^{0}, \theta^{0}\bigr) (x,0)=(\psi_{0},\theta_{0}) (x),\quad 0 \leq x\leq1, \\ & \psi^{0}(1,t)=\psi^{0}(0,t)=0,\quad t \geq0. \end{aligned} $$

(1.4)

The system (1.1) was originally proposed by Hsieh in [2] to observe the nonlinear interaction between ellipticity and dissipation. In [3], Hsieh et al. established a link between this interaction and chaos. We also refer to [4, 5] for the physical background of (1.1). Some similar problems were studied in [6, 7] and the references therein.

Our main purpose is to estimate the thickness of boundary layer for problem (1.1)-(1.2) as $\beta\rightarrow0$. Before stating the main result, we first recall the concept of BL-thickness in the sprit of [8].

Definition 1.1

A function $\delta(\beta)$ is called a BL-thickness for problem (1.1)-(1.2) with vanishing diffusion if $\delta(\beta)\downarrow0$ as $\beta\downarrow0$, and

$$ \begin{aligned} &\lim_{\beta\rightarrow0}\bigl\Vert \psi^{\beta}-\psi^{0}\bigr\Vert _{L^{\infty}(0,T;L^{\infty}[0,1])}=0, \\ &\lim_{\beta\rightarrow0}\bigl\Vert \theta^{\beta}- \theta^{0}\bigr\Vert _{L^{\infty}(0,T;L^{\infty}[\delta,1-\delta])}=0, \\ &\mathop{\inf\lim} _{\beta\rightarrow0}\bigl\Vert \theta^{\beta}- \theta^{0}\bigr\Vert _{L^{\infty}(0,T;L^{\infty}[0,1])}>0, \end{aligned} $$

for any $T>0$, where $(\psi^{\beta}, \theta^{\beta})$ (resp. $(\psi^{0}, \theta^{0})$) is the solution for problem (1.1)-(1.2) (resp. problem (1.3)-(1.4)).

The theory of boundary layers is one of the most fundamental and important issues in fluid dynamics (cf. [9, 10]) since the seminal work by Prandtl in 1904. There are a number of papers dedicated to the questions of boundary layers for the Navier-Stokes equations; see for instance [8, 11–17] and the references therein. Moreover, the boundary layer problem also arises in the theory of hyperbolic systems when parabolic equations with small viscosity are applied as perturbations; see for instance [18–23] and the references therein.

Recently, Ruan and Zhu [1], Theorem 1.3, discussed the existence and zero diffusion limit for problem (1.1)-(1.2), and proved that the thickness of boundary layer is of the order $O(\beta^{\gamma})$ with $0<\gamma<1/2$ if $\frac{(\sigma+\mu\beta)^{2}}{4(1-\beta)}< \alpha<\sigma$ and if the initial data satisfy $\psi_{0}\in H^{2}([0,1]), \theta_{0}\in H^{3}([0,1]), (\psi_{0},\theta_{0})(1)=(\psi_{0},\theta_{0})(0)=(0, 0)$, and $\Vert (\psi_{0},\theta _{0})\Vert _{2}$ is sufficiently small. Here $H^{l}([0, 1])$ denotes the usual lth order Sobolev space with the norm $\Vert f\Vert _{l}= (\sum_{i=0}^{l}\int_{0}^{1} \vert \partial_{x}^{i} f\vert ^{2}\,dx )^{1/2}$. In the present paper, we improve the result by extending the range of γ to $(0, 1)$. Our main result can be stated as follows.

Theorem 1.2

Let $0<\beta<1$ and $\frac{(\sigma+\mu\beta)^{2}}{4(1-\beta)}< \alpha<\sigma$. Assume that the initial data satisfy $\psi_{0}\in H^{2}([0,1]), \theta_{0}\in H^{3}([0,1]), (\psi_{0},\theta_{0})(1)=(\psi_{0},\theta_{0})(0)=(0, 0)$, and $\Vert (\psi_{0},\theta_{0})\Vert _{2}$ is sufficiently small. Then any function $\delta(\beta)$ satisfying $\delta(\beta) \downarrow 0$ and $\frac{\beta}{\delta(\beta)} \rightarrow0$ as $\beta\downarrow 0$ is a BL-thickness such that

$$ \begin{aligned} \bigl\Vert \theta^{\beta}- \theta^{0}\bigr\Vert _{L^{\infty}(0,T;L^{\infty}[\delta,1-\delta])} \leq C \sqrt{\frac{\beta}{\delta}},\quad \forall\delta\in(0, 1/2), \end{aligned} $$

(1.5)

where $T>0$, and C is a positive constant independent of β and δ.

The proof of Theorem 1.2 will be given in the next section.

2 Proof of Theorem 1.2

To prove Theorem 1.2, we need the following result, which can be found in [1], Lemmas 2.2, 2.4, 2.5 and 3.1.

Lemma 2.1

Let the assumptions of Theorem 1.2 hold. Then there exists a positive constant independent of β such that

$$\begin{aligned} & \int_{0}^{1} \bigl[\bigl(\psi^{\beta}_{x} \bigr)^{2}+\bigl(\theta^{\beta}_{x} \bigr)^{2}+\bigl(\psi^{\beta}_{xx} \bigr)^{2}+\beta\bigl(\theta^{\beta}_{xx} \bigr)^{2} \bigr]\,dx \leq C, \end{aligned}$$

(2.1)

$$\begin{aligned} & \int_{0}^{1} \bigl[\bigl(\psi^{0}_{x} \bigr)^{2}+\bigl(\theta^{0}_{x}\bigr)^{2} +\bigl(\theta^{0}_{xx}\bigr)^{2}+\bigl(\psi ^{0}_{xx}\bigr)^{2} \bigr]\,dx \leq C \end{aligned}$$

(2.2)

and

$$ \int_{0}^{1} \bigl[ \bigl(\psi^{\beta}- \psi^{0}\bigr)^{2}+\bigl(\theta^{\beta}- \theta^{0}\bigr)^{2} \bigr]\,dx + \int_{0}^{T} \int _{0}^{1} \bigl(\psi^{\beta}- \psi^{0}\bigr)_{x}^{2} \,dx\,dt \leq C\beta. $$

(2.3)

Proof of Theorem 1.2

It suffices to prove (1.5). Set

$$u^{\beta}=\psi^{\beta}-\psi^{0}, \qquad v^{\beta}= \theta^{\beta}-\theta^{0}. $$

Then it follows from the equation of $\theta^{\beta}$ that

$$ v^{\beta}_{t}=-(1- \beta)v^{\beta}+2\psi^{\beta}v^{\beta}_{x}+2u^{\beta}\theta^{0}_{x} +\beta v^{\beta}_{xx}+\beta \bigl(\mu\psi^{\beta}_{x}+\theta^{0}+ \theta^{0}_{xx}\bigr). $$

Differentiating the equation, we see that $z:=v^{\beta}_{x}$ satisfies

$$ z_{t}=-(1-\beta)z+2\bigl( \psi^{\beta}z\bigr)_{x}+2\bigl(u^{\beta}\theta^{0}_{x}\bigr)_{x}+\beta z_{xx}+ \beta\bigl(\mu\psi^{\beta}_{xx}+\theta^{0}_{x}+ \theta^{0}_{xxx}\bigr). $$

(2.4)

Denote $\varphi_{\varepsilon}$ for $\varepsilon\in(0, 1)$ and $\xi_{\delta}$ for $\delta\in(0, 1/2)$ by

$$ \varphi_{\varepsilon}(s)=\sqrt{s^{2}+ \varepsilon^{2}},\qquad \xi_{\delta}(x)= \textstyle\begin{cases}x,&0\leq x\leq\delta, \\ \delta,& \delta\leq x\leq1-\delta, \\ 1-x,&1-\delta\leq x \leq1. \end{cases} $$

It is easy to check that $\varphi_{\varepsilon}$ satisfies

$$ \textstyle\begin{cases} \vert s\vert \leq \vert \varphi_{\varepsilon}(s)\vert \leq \vert s\vert +1, \\ \vert \varphi_{\varepsilon}'(s)\vert \leq1,\quad 0\leq s\varphi_{\varepsilon}'(s)\leq \varphi_{\varepsilon}(s), \\ \varphi_{\varepsilon}''(s)\geq0,\quad s^{2} \varphi_{\varepsilon}''(s) \geq 0, \end{cases} $$

and $\xi_{\delta}$ satisfies

$$0\leq \xi_{\delta}\leq\delta,\qquad \xi_{\delta}(1)=\xi_{\delta}(0)=0. $$

Multiplying (2.4) by $\varphi_{\varepsilon}'(z)\xi_{\delta}$ and integrating it over $(0, 1)\times(0, t)$, we have

$$\begin{aligned} & \int_{0}^{1}\varphi_{\varepsilon}(z) \xi_{\delta}\,dx- \varepsilon \int_{0}^{1}\xi _{\delta}\,dx \\ & \quad =-(1-\beta) \int_{0}^{t} \int_{0}^{1} z \varphi_{\varepsilon}'(z) \xi_{\delta}\,dx\,d\tau+2 \int_{0}^{t} \int_{0}^{1}\bigl(\psi^{\beta}z \bigr)_{x}\varphi_{\varepsilon}'(z)\xi _{\delta}\, dx\,d\tau \\ &\qquad{} +2 \int_{0}^{t} \int_{0}^{1} \bigl(u^{\beta}\theta^{0}_{x}\bigr)_{x}\varphi_{\varepsilon}'(z) \xi _{\delta}\,dx\,d\tau +\beta \int_{0}^{t} \int_{0}^{1}z_{xx}\varphi_{\varepsilon}'(z) \xi_{\delta}\, dx\,d\tau \\ &\qquad{} +\beta \int_{0}^{t} \int_{0}^{1} \varphi_{\varepsilon}'(z) \xi_{\delta}\bigl(\mu\psi^{\beta}_{xx}+ \theta^{0}_{x}+\theta^{0}_{xxx}\bigr) \,dx \,d\tau=: \sum_{i=1}^{5}E_{j}. \end{aligned}$$

(2.5)

Next we estimate $E_{i} (i=1,2,3,4,5)$. From $0\leq s\varphi_{\varepsilon}'(s)\leq\varphi_{\varepsilon}(s)$, we have

$$ \begin{aligned} E_{1}\leq \int_{0}^{t} \int_{0}^{1} \varphi_{\varepsilon}(z) \xi_{\delta}\,dx\,d\tau. \end{aligned} $$

(2.6)

To estimate $E_{2}$, we note, using integration by parts,

$$\begin{aligned} E_{2}={}&2 \int_{0}^{t} \int_{0}^{1}\psi^{\beta}_{x}z \varphi_{\varepsilon}'(z)\xi_{\delta}\,dx\,d\tau+2 \int_{0}^{t} \int_{0}^{1}\psi^{\beta}z_{x} \varphi_{\varepsilon}'(z)\xi _{\delta}\,dx\,d\tau \\ ={}&2 \int_{0}^{t} \int_{0}^{1}\psi^{\beta}_{x}z \varphi_{\varepsilon}'(z)\xi_{\delta}\,dx\,d\tau - 2 \int_{0}^{t} \int_{0}^{1} \varphi_{\varepsilon}(z) \psi^{\beta}_{x}\xi_{\delta}\,dx\,d\tau \\ &{} - 2 \int_{0}^{t} \int_{0}^{1} \varphi_{\varepsilon}(z) \psi^{\beta}\xi_{\delta}' \,dx\,d\tau \\ =:{}&E_{2}^{1}+E_{2}^{2}+E_{2}^{3}. \end{aligned}$$

(2.7)

By (2.1) and the embedding $W^{1,1}[0, 1]\hookrightarrow L^{\infty}[0, 1]$, we have

$$ \bigl\vert \psi^{\beta}_{x}(x,t) \bigr\vert \leq \int_{0}^{1} \bigl\vert \psi^{\beta}_{x} \bigr\vert \,dx+ \int_{0}^{1}\bigl\vert \psi ^{\beta}_{xx} \bigr\vert \,dx\leq C, $$

(2.8)

where C denotes the generic positive constant independent of $\beta, \delta$, and ε, so

$$ \begin{aligned} &\bigl\vert \psi^{\beta}(x,t) \bigr\vert \leq \int_{0}^{x} \bigl\vert \psi^{\beta}_{y}(y,t) \bigr\vert \,dy\leq Cx\leq C\xi _{\delta}(x), \quad\forall x\in[0, \delta], \\ &\bigl\vert \psi^{\beta}(x,t)\bigr\vert \leq \int_{x}^{1} \bigl\vert \psi^{\beta}_{y}(y,t) \bigr\vert \,dy\leq C(1-x)\leq C\xi_{\delta}(x), \quad \forall x\in[1-\delta,1]. \end{aligned} $$

(2.9)

By $0\leq s\varphi_{\varepsilon}'(s)\leq\varphi_{\varepsilon}(s)$ and (2.8), we obtain

$$ \begin{aligned} E_{2}^{1}+E_{2}^{2} \leq C \int_{0}^{t} \int_{0}^{1} \varphi_{\varepsilon}(z) \xi_{\delta}\,dx\,d\tau. \end{aligned} $$

(2.10)

By the definition of $\xi_{\delta}$ and (2.9), we have

$$ \begin{aligned} E_{2}^{3} &= - 2 \int_{0}^{t} \int_{0}^{\delta}\varphi_{\varepsilon}(z) \psi^{\beta}\,dx\,d\tau+ 2 \int_{0}^{t} \int_{1-\delta}^{1} \varphi_{\varepsilon}(z) \psi^{\beta}\,dx\,d\tau \\ &\leq C \int_{0}^{t} \int_{0}^{\delta}\varphi_{\varepsilon}(z) \xi_{\delta}\,dx\,d\tau+ C \int_{0}^{t} \int_{1-\delta}^{1} \varphi_{\varepsilon}(z) \xi_{\delta}\,dx\,d\tau \\ &\leq C \int_{0}^{t} \int_{0}^{1} \varphi_{\varepsilon}(z) \xi_{\delta}\,dx\,d\tau. \end{aligned} $$

(2.11)

Thus

$$ \begin{aligned} E_{2} \leq C \int_{0}^{t} \int_{0}^{1} \varphi_{\varepsilon}(z) \xi_{\delta}\,dx\,d\tau. \end{aligned} $$

(2.12)

Using integration by parts and noticing $\varphi_{\varepsilon}''\geq 0$ and $\vert \varphi_{\varepsilon}'\vert \leq1$, we have

$$ \begin{aligned} E_{4}={}&-\beta \int_{0}^{t} \int_{0}^{1} z_{x}^{2} \varphi_{\varepsilon}''(z)\xi_{\delta}\,dx\,d\tau- \beta \int_{0}^{t} \int_{0}^{1} z_{x} \varphi_{\varepsilon}'(z) \xi_{\delta}' \,dx\,d\tau \\ \leq{}&- \beta \int_{0}^{t} \int_{0}^{1} z_{x} \varphi_{\varepsilon}'(z) \xi_{\delta}' \,dx\,d\tau \\ ={}&-\beta \int_{0}^{t} \int_{0}^{\delta}z_{x} \varphi_{\varepsilon}'(z) \,dx\,d\tau+\beta \int_{0}^{t} \int_{1-\delta}^{1} z_{x} \varphi_{\varepsilon}'(z) \,dx\,d\tau \\ \leq{}&\beta \biggl( \int_{0}^{t} \int_{0}^{\delta} \vert z_{x}\vert \,dx\,d \tau+ \int_{0}^{t} \int _{1-\delta}^{1} \vert z_{x}\vert \,dx\,d \tau \biggr), \end{aligned} $$

and, by Hölder’s inequality and (2.1), we obtain

$$\begin{aligned} E_{4} \leq{}& C\beta \delta^{1/2} \biggl[ \biggl( \int_{0}^{t} \int_{0}^{\delta} \vert z_{x}\vert ^{2}\,dx\,d\tau \biggr)^{1/2}+ \biggl( \int_{0}^{t} \int_{1-\delta}^{1} \vert z_{x}\vert ^{2}\,dx\,d\tau \biggr)^{1/2} \biggr] \\ \leq{}& C\beta^{1/2} \delta^{1/2}. \end{aligned}$$

(2.13)

By $\vert \varphi_{\varepsilon}'\vert \leq1, 0\leq\xi_{\delta}\leq\delta$, Hölder’s inequality, (2.2), and (2.3), we have

$$\begin{aligned} E_{3}={}& 2 \int_{0}^{t} \int_{0}^{1}u^{\beta}_{x} \theta^{0}_{x}\varphi_{\varepsilon}'(z)\xi _{\delta}\,dx\,d\tau+ 2 \int_{0}^{t} \int_{0}^{1}u^{\beta}\theta^{0}_{xx} \varphi _{\varepsilon}'(z)\xi_{\delta}\,dx\,d\tau \\ \leq{}& C\delta \biggl( \int_{0}^{t} \int_{0}^{1}\bigl(u^{\beta}_{x} \bigr)^{2} \,dx\,d\tau \biggr)^{1/2} \biggl( \int_{0}^{t} \int_{0}^{1}\bigl(\theta^{0}_{x} \bigr)^{2} \,dx\,d\tau \biggr)^{1/2} \\ &{}+C\delta \biggl( \int_{0}^{t} \int_{0}^{1}\bigl(u^{\beta}\bigr)^{2} \,dx\,d\tau \biggr)^{1/2} \biggl( \int_{0}^{t} \int_{0}^{1}\bigl(\theta^{0}_{xx} \bigr)^{2} \,dx\,d\tau \biggr)^{1/2} \\ \leq{}&C\delta\beta^{1/2}. \end{aligned}$$

(2.14)

Finally, we estimate $E_{5}$. By $\vert \varphi_{\varepsilon}'\vert \leq1, 0\leq \xi_{\delta}\leq\delta$, and Lemma 2.1, we have

$$\begin{aligned} E_{5}&\leq C \beta\delta \int_{0}^{t} \int_{0}^{1} \bigl(\bigl\vert \psi^{\beta}_{xx} \bigr\vert +\bigl\vert \theta ^{0}_{x}\bigr\vert +\bigl\vert \theta^{0}_{xxx}\bigr\vert \bigr) \,dx\,d\tau \\ &\leq C \beta\delta. \end{aligned}$$

(2.15)

Combining (2.6), (2.12)-(2.15) with (2.5) and noticing

$$\varepsilon \int_{0}^{1} \xi_{\delta}\,dx\leq \varepsilon \delta, $$

we obtain

$$ \begin{aligned} \int_{0}^{1} \varphi_{\varepsilon}(z) \xi_{\delta}\,dx \leq C \int_{0}^{t} \int_{0}^{1} \varphi_{\varepsilon}(z) \xi_{\delta}\, dx\,d\tau+ \varepsilon\delta+C\beta^{1/2} \delta^{1/2}, \end{aligned} $$

so an application of Gronwall’s inequality leads to

$$ \begin{aligned} \int_{0}^{1} \varphi_{\varepsilon}(z) \xi_{\delta}\,dx \leq C \bigl( \varepsilon \delta+\beta^{1/2} \delta^{1/2} \bigr). \end{aligned} $$

From this and the definition of $\xi_{\delta}$ and $\vert z\vert \leq \varphi_{\varepsilon}(z)$, we obtain

$$ \begin{aligned} \int_{\delta}^{1-\delta} \vert z\vert \,dx \leq C \biggl( \varepsilon+\sqrt{\frac {\beta}{\delta}} \biggr). \end{aligned} $$

Letting $\varepsilon\rightarrow0$ yields

$$ \begin{aligned} \int_{\delta}^{1-\delta} \vert z\vert \,dx \leq C \sqrt{ \frac{\beta}{\delta}}. \end{aligned} $$

(2.16)

From (2.3), (2.16), and the embedding $W^{1,1}([\delta, 1-\delta])\hookrightarrow L^{\infty}([\delta, 1-\delta])$ it follows that

$$ \begin{aligned} \bigl\Vert \bigl(\theta^{\beta}- \theta^{0}\bigr) (\cdot, t)\bigr\Vert _{L^{\infty}[\delta,1-\delta]} \leq{} & \int_{0}^{1}\bigl\vert \theta^{\beta}- \theta^{0}\bigr\vert \,dx+ \int_{\delta}^{1-\delta} \vert z\vert \,dx \\ \leq{}& C \sqrt{\frac{\beta}{\delta}}. \end{aligned} $$

Thus (1.5) is proved, and the proof is complete. □