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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

带非阻尼和扩散的非线性发展方程零扩散极

限的边界层

彭红云，阮立志，朱长江

华中师范大学数学与统计学学院，武汉　430079

摘要：文考虑某种带阻尼和扩散的非线性发展方程, 主要研究了当扩散参数 β 趋于 0 时的边界

层效应和收敛率，并给出了边界层厚度的阶数 O (β

) (0 < γ <

). 与文献 [L.Z. Ruan and

C.J. Zhu, Discrete Contin. Dyn. Syst. Ser.A, 32(2012), 331-352] 相比, 本文不再要求参数 ν

和 β 线性相关, 并且收敛率在 W

1,∞

模意义下也得到了提高.

关键词：偏微分方程，非线性发展方程，零扩散极限，边界层，边界层厚度

中图分类号： O175.29

Boundary Layer of Zero Diﬀusion Limit for

Nonlinear Evolution Equations with Damping

and Diﬀusions

PENG Hong-yun , RUAN Li-zhi ,ZHU Chang-jiang

The Hubei Key Laboratory of Mathematical Physics, School of Mathematics and

Statistics, Central China Normal University, Wuhan, 430079,

Abstract: In this paper, we consider an initial-boundary value problem for some nonlinear

evolution equations with damping and diﬀusions on the strip. Our main purpose is to

investigate the boundary layer eﬀect and the convergence rates as the diﬀusion parameter β

goes to zero. It is also shown that the boundary layer thickness is of the order O (β

) with

0 < γ <

. In contrast with [L.Z. Ruan and C.J. Zhu, Discrete Contin. Dyn. Syst. Ser.A,

32(2012), 331-352], the important point in this paper is that the restriction on the linear

relation between the parameters ν and β is removed. In addition, the convergence rates in

1,∞

norm are also improved.

Key words: Nonlinear evolution equations, zero diﬀusion limit, boundary layer,

BL-thickness.

基金项目： the PhD specialized grant of the Ministry of Education of China (20100144110001)

作者简介： PENG Hong-Yun（1984-），male，major research direction：Partial diﬀerential equation. Correspondence

author：RUAN Li-zhi（1976-），male，major research direction：Partial diﬀerential equation. ZHU Chang-jiang（1961-），

male，Professor, major research direction：Partial diﬀerential equation.

- 1 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

0 Introduction

In this paper, we consider the following initial-boundary value problem for the nonlinear

evolution equations with damping and diﬀusions on the strip [0, 1] × [0, ∞)







= −(σ − α)ψ

− σθ

+ αψ

= −(1 − β)θ

+ νψ

+ 2ψ

+ βθ

, 0 < x < 1, t > 0,

(1)

with initial data



, θ



(x, 0) = (ψ

, θ

)(x) 0 ≤ x ≤ 1, (2)

and the boundary conditions



, θ



(0, t) =



, θ



(1, t) = (0, 0), t ≥ 0, (3)

which implies



, θ



(0, t) =



, θ



(1, t) = (0, 0), t ≥ 0, (4)

where we assume the following compatible condition is valid: (ψ

, θ

)(0) = (ψ

, θ

)(1) = (0, 0).

Here α, β, σ and ν are all positive constants satisfying the relation α < σ and 0 < β < 1. The

system (1.1) was originally proposed by Hsieh in [1] to observe the nonlinear interaction between

ellipticity and dissipation. We also refer to [1, 2, 3] for the physical background of (1.1).

Next let us recall some theoretical results on system (1.1), which has been extensively

studied by several authors in diﬀerent contexts, but most of them are concerned with the case

of Cauchy problem, we refer to [4, 5, 6, 7, 8, 9, 10] and references therein.

In case of the initial-boundary value problem, to our knowledge, there was very fewer results

obtained. Based on energy method and pointwise estimates, on the quadrant [0, ∞) × [0, ∞),

the authors in [11] ﬁrstly established the global existence of the solutions to the corresponding

initial-boundary value problem of the nonlinear evolution equations (1.1) with σ = 1 and α = β

and obtained the L

(p ≥ 2) decay rates to diﬀusion waves.

However, all the results mentioned above need to assume that all parameters σ, α, β and

ν are ﬁxed constants. Another interesting problem is the zero diﬀusion limit, i.e., to consider

the limit problem of solution consequences when one or more of parameters vanishes for the

corresponding Cauchy problem or initial-boundary value problem. To our knowledge, very

fewer results have been obtained in this direction, cf. [12, 13].

Firstly, Chen and Zhu in [12] considered the Cauchy problem







= −(1 − α)ψ − θ

+ αψ

= −(1 − α)θ − µψ

+ 2ψθ

+ αθ

(5)

with initial data

(ψ

, θ

)(x, 0) = (ψ

, θ

)(x) → (ψ

, θ

) as x → ±∞ , (6)

- 2 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

where α and µ are positive constants satisfying 0 < α < 1. They proved that the solution

sequences {(ψ

, θ

)} of (5), (6) converge to the solution to the corresponding limit system with

α = 0 as α → 0

Recently, Ruan and Zhu in [13] considered the the following initial-boundary value problem

on the strip [0, 1] × [0, ∞)







= −(σ − α)ψ

− σθ

+ αψ

= −(1 − β)θ

+ µβψ

+ 2ψ

+ βθ

, 0 < x < 1, t > 0,

(7)

with initial data



, θ



(x, 0) = (ψ

, θ

)(x), 0 ≤ x ≤ 1, (8)

and the boundary conditions



, θ



(0, t) =



, θ



(1, t) = (0, 0), t ≥ 0. (9)

They proved that the solution sequences {(ψ

, θ

)} of (7)-(9) converge to the solution to the

corresponding limit system with β = 0 as β → 0

and showed that the boundary layer thickness

is of the order O (β

) with 0 < γ <

The theory of boundary layers has been one of the fundamental and important issues in

ﬂuid dynamics [14] since it was proposed by Prandtl in 1904. For example, there are a number of

papers dedicated to the questions of boundary layers for both incompressible and compressible

Navier-Stokes equations, cf. [15, 16, 17, 18, 19, 20, 21, 22]. 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, [23, 24, 25] and the references cited

therein. The question of boundary layer for other equation was also addressed in a number of

works, we refer to [13, 26].

The main purpose of this paper is to investigate the boundary layer eﬀect and the conver-

gence rates on the initial-boundary value problem (1)-(3) as the diﬀusion parameter β → 0

The limit problem of the vanishing parameter β is formally formulated as the following initial-

boundary value problem







= −(σ − α)ψ

− σθ

+ αψ

= −θ

+ νψ

+ 2ψ

, 0 < x < 1, t > 0,

(10)

with initial data

(ψ

, θ

)(x, 0) = (ψ

(x), θ

(x)), 0 ≤ x ≤ 1, (11)

and the boundary conditions

(0, t) = ψ

(1, t) = 0, t ≥ 0, (12)

- 3 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

which implies







(0, t) = ψ

(1, t) = ψ

(0, t) = ψ

(1, t) = 0,

(σθ

− αψ

)(0, t) = (σθ

− αψ

)(1, t) = 0, t ≥ 0.

(13)

The implicit boundary conditions (13) play an important role in obtaining a priori estimates

on the limit problem (10)-(12). We expect to prove the solution sequences



, θ



of (1)-

(3) converge to the solution (ψ

, θ

) of the limit problem (10)-(12), as the diﬀusion parameter

β → 0

and get the boundary layer thickness (BL-thickness).

For latter presentation, next we state the notation and recall the deﬁnition of BL-thickness

in [16].

Notation: Throughout this paper, we denote positive constant independent of β by C. And

the character “C” may diﬀer in diﬀerent places. L

= L

([0, 1]) and L

∞

= L

∞

([0, 1]) denote

the usual Lebesgue space on [0, 1] with its norms ∥f∥

([0,1])

= ∥f ∥ =





|f(x)|



and

∥f∥

∞

= sup

x∈[0,1]

|f(x)|. H

([0, 1]) denotes the usual l-th order Sobolev space with its norm

∥

([0,1])

= ∥f∥

= (



i=0

∥∂

∥

)

. For simplicity, ∥f(·, t)∥

, ∥f(·, t)∥

∞

and ∥f(·, t)∥

are

denoted by ∥f(t)∥, ∥f (t)∥

∞

and ∥f(t)∥

respectively.

Deﬁnition 1.1. A function δ(β) is called a BL-thickness for the problem (1)-(3) with vanishing

diﬀusion β, if δ(β) ↓ 0 as β ↓ 0, and

lim

β→0



− ψ



∞

(0,T ;L

∞

[0,1])

= 0, (14)

lim

β→0



− θ



∞

(0,T ;L

∞

[δ,1−δ])

= 0 , (15)

lim inf

β→0



− θ



∞

(0,T ;L

∞

[0,1])

> 0, (16)

where 0 < δ = δ(β) < 1, and (ψ

, θ

) (rep. (ψ

, θ

)) is the solution to the problem (1)-(3)

(resp. to the limit problem (10)-(12)).

Clearly, this deﬁnition does not determine the BL-thickness uniquely, since any function

∗

(β) satisfying the inequality δ

∗

(β) ≥ δ(β) for small β is also a BL-thickness if δ(β) is a

BL-thickness.

Our ﬁrst result shows that the initial-boundary value problem (1)-(3) admits a unique

global smooth solution



, θ



Theorem 1.1. Assume that the initial data satisfy the conditions: (ψ

, θ

) ∈ H

, (ψ

, θ

)(0) =

(ψ

, θ

)(1) = (0, 0) and ∥ψ

∥

+ ∥θ

∥

is suﬃciently small. The parameters σ, α and β satisfy

the relation

(σ +ν)

4(1−β)

< α < σ and 1 − β −

σν

> 0.

- 4 -

˖ڍመ᝶஠ڙጲ

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Then there exists a unique solution



, θ



to the initial-boundary value problem (1)-(3)

satisfying

∈ L

∞

(0, T ; H

)



(0, T ; H

) , ψ

∈ L

∞

(0, T ; H

)



(0, T ; H

) ,

∈ L

∞

(0, T ; H

)



(0, T ; H

) , θ

∈ L

∞

(0, T ; H

)



(0, T ; H

) ,

where the norms are all uniform in β. There also exists a positive constant C independent of

β, such that



xxx



∞

(0,T ;L

)

+ β





xxx



(0,T ;L

)



xxt



(0,T ;L

)



≤ C.

Our second result shows that the initial-boundary value problem (10)-(12) admits a unique

global smooth solution (ψ

, θ

). In addition, the regularity on the solutions is also improved if

the initial data is more regular, which will play important role in proving Theorem 1.3 later.

Theorem 1.2. Under the conditions of Theorem 1.1, we assume the initial data ψ

, θ

satisfy

further regularity (ψ

, θ

) ∈ H

and ∥θ

∥

+ ∥ψ

∥

is suﬃciently small. Then there exists a

unique solution (ψ

, θ

) to the limit problem (10)-(12) satisfying

∈ L

∞

(0, T ; H

)



(0, T ; H

) , ψ

∈ L

∞

(0, T ; H

)



(0, T ; H

) ,

∈ L

∞

(0, T ; H

)



(0, T ; H

) , θ

∈ L

∞

(0, T ; H

)



(0, T ; H

) .

Another interesting problem is to give the convergence rates and boundary layer thickness.

Theorem 1.3. Under the conditions of Theorem 1.2. Then any function δ(β) satisfying the

conditions δ(β) → 0 and β

/δ(β) → 0 as β → 0

, is a BL-thickness such that



− ψ



∞

(0,T ;L

∞

[0,1])

≤ Cβ





− ψ





∞

(0,T ;L

∞

[δ,1−δ])

≤ Cδ

−1



− θ



∞

(0,T ;L

∞

[δ,1−δ])

≤ Cδ

−1





− θ





∞

(0,T ;L

∞

[δ,1−δ])

≤ Cδ

−1

lim inf

β→0



− θ



∞

(0,T ;L

∞

[0,1])

> 0.

Consequently,

lim

β→0



− ψ



∞

(0,T ;L

∞

[0,1])

= 0,

lim

β→0



− θ



1,∞

(0,T ;L

∞

[δ,1−δ])

= 0.

Before concluding this section, we outline the novelty and the main ideas we used in

this paper. As is well-known that it is great diﬃcult to obtain the estimates on higher order

derivatives of solutions since the boundary eﬀect appears. Generally speaking, the presence of

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