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ON THE STABILITY OF CONTACT DISCONTINUITY FOR CAUCHY PROBLEM OF ...
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一般初值的可压NAVIER-STOKES方程组Cauchy问题接触间断解的稳定性,郑婷婷,赵俊宁,本文研究了具有一般初值的一维可压Navier-Stokes方程Cauchy问题解的稳定性。在某种条件下,证明了当$tightarrowinfty$ 时, 解的渐进极限是粘性�
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˖ڍመڙጲ
http://www.paper.edu.cn
一般初值的可压 NAVIER-STOKES 方程组
Cauchy 问题接触间断解的稳定性
郑婷婷
1
,赵俊宁
2
1
福建农林大学计算机与信息学院,福州 350002
2
厦门大学数学科学学院,厦门 361005
摘要:本文研究了具有一般初值的一维可压 Navier-Stokes 方程 Cauchy 问题解的稳定性。在
某种条件下,证明了当 t → ∞ 时,解的渐进极限是粘性接触间断波。同时,证明了快速扩散
方程解的衰减速率。本文的证明方法基于基本的能量方法和快速反应扩散方程的渐进行为。
关键词:可压 Navier-Stokes 方程组,全局弱解,大时间渐进性
中图分类号: 0175
ON THE STABILITY OF CONTACT
DISCONTINUITY FOR CAUCHY PROBLEM
OF COMPRESS NAVIER-STOKES
EQUATIONS WITH GENERAL INITIAL
DATA
ZHENG Ting-Ting
1
, ZHAO Jun-Ning
2
1
School of Computer and Information Science, Fujian Agriculture and Forestry University,
Fuzhou 350002, P. R. China
2
School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China
Abstract: In this paper, we study the stability of solutions of the Cauchy problem for 1-D
compressible Narvier-Stokes equations with general initial data. The asymptotic limit of
solution is found, under some conditions. The results in this paper imply the case that the
limit function of solution as t → ∞ is a viscous contact wave, which approximates the contact
discontinuity on any finite-time interval as the heat conduction coefficients toward zero. As a
by-product, the decay rates of solution for the fast diffusion equations are also obtained. The
proofs are based on the elementary energy method and the study of asymptotic behavior of
基金项目: This work was supported by a grant from the Ph.D. Programs Foundation of Ministry of Education of China
(200803840016)
作者简介: Correspondence author:ZHAO Jun-Ning(1945-),male,Professor,major research direction:Paritial dif-
ferential equations. ZHENG Ting-Ting(1980-),female,Assistant professor,major research direction:Paritial differential
equations.
- 1 -
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solution to the fast diffusion equation.
Key words: compressible Navier-Stokes equations, global weak solution, large-time
asymptotic behavior
0 Introduction
We study the asymptotic behavior of the solutions for the Cauchy problem to the one-
dimensional compressible Navier-Stokes equations in Lagrangian coordinates:
v
t
− u
x
= 0,
u
t
+
Rθ
v
x
= µ
u
x
v
x
,
R
γ−1
θ
t
+ R
θ
v
u
x
= κ
θ
x
v
x
+ µ
u
2
x
v
(1)
with initial conditions
(v, u, θ)|
t=0
= (v
0
, u
0
, θ
0
) → (v
±
, 0, θ
±
) as x → ±∞. (2)
Here x ∈ R
1
, t > 0, v(x, t) > 0, u(x, t), θ(x, t) > 0 are the specific volume, velocity, temperature,
while µ > 0, κ > 0 denote the viscosity and heat conduction coefficients respectively, v
±
, θ
±
are
positive constants.
In the ideal fluids, i.e., κ = µ = 0, (1) become the well-known compressible Euler system:
v
t
− u
x
= 0,
u
t
+
Rθ
v
x
= 0 ,
R
γ−1
θ
t
+ R
θ
v
u
x
= 0 .
(3)
which is one of the most important nonlinear hyperbolic systems of conservation laws. It is well
known that the basic waves for the system (3) are dilation invariant solutions: shock waves,
rarefaction waves, contact discontinuities, and the linear combinations of these waves, called
Riemann solutions, which govern both local and large-time asymptotic behavior of general
solutions of Euler system (3)). Since the system (3) is an idealization when the dissipative
effects are neglected, it is of great importance to study the large-time asymptotic behavior
of solutions of the corresponding viscous system, such as (1), toward the viscous versions
of these basic waves. Up to now, some deeper results have been obtained on the asymptotic
stability toward nonlinear waves, viscous shock profiles and viscous rarefaction, for quite general
perturbation of the Navier-Stokes system (1) and general systems of viscous strictly hyperbolic
conservation laws(see [1],[2]-[13]). However the problem of stability of contact discontinuities
is more subtle. To study the stability toward contact waves for systems of viscous conservation
- 2 -
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laws, the concept of metastability of a weak contact discontinuity is introduced in [14], i.e.,
although a contact discontinuity is not an asymptotic attract for viscous system, there is a
viscous wave, which approximates the contact discontinuity on any finite time interval and is
the asymptotic limit of the solution of the viscous system. The problem of metastability of
contact discontinuity for the Navier-Stokes system (1) has already been well studied. For a
free-boundary value problem of (1) with a particle path as free boundary, the stability of a
viscous contact waves is proved in the supnorm (see [15, 16]). For the Cauchy problem of (1),
the stability of a viscous contact discontinuity is proved in [17], see also [18, 19].
The main purpose of this paper is to generalize the result in [18], we will give a class of
initial data, which ensures the contact discontinuity for the compressible Euler system (3) are
metastable waves patterns for the Navier-Stokes system (1), i.e., we will show that for a class
of initial value of the Cauchy problem (1), it is possible to construct a viscous contact wave
for the Navier-Sokes system, which is smooth, solves (1) asymptotically, and approximates the
given contact discontinuity on any finite-time interval as κ → 0.
We now construct the viscous contact wave for the Navier-Sokes system (1). First, consider
the corresponding Euler equations (3) with the Riemann initial data
(v, u, θ)(x, 0) = (v
−
, 0, θ
−
) if x < 0,
(v, u, θ)(x, 0) = (v
+
, 0, θ
+
) if x > 0.
(4)
Riemann problem (3)-(4) admits a contact discontinuity
(V , U, Θ) =
(v
−
, 0, θ
−
), x < 0,
(v
+
, 0, θ
+
), x > 0;
(5)
provided that
p
−
=
Rθ
−
v
−
= p
+
=
Rθ
+
v
+
.
In the next, as that in [15, 16, 17] we conjecture that the asymptotic limit (V, U, Θ) of (1) is as
follows
P (V, Θ) = R
Θ
V
= p
+
, U(x, t) =
κ(γ − 1)Θ
x
γRΘ
. (6)
and Θ is the solution of the following Cauchy problem
Θ
t
= a(ln Θ)
xx
, a =
κp
+
(γ−1)
γR
2
> 0,
Θ(x, 0) = θ
0
.
(7)
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(V, U, Θ) in (6) satisfies
R
Θ
V
= p
+
,
V
t
= U
x
,
U
t
+ p(V, Θ)
x
= µ
U
x
V
x
+ F,
R
γ−1
Θ
t
+ R
Θ
V
U
x
= κ
Θ
x
V
x
+ µ
U
2
x
V
+ G,
(8)
where
G = −µ
U
2
x
V
= O((ln Θ)
2
xx
), F (x, t) =
κ(γ − 1)
γR
(ln Θ)
xt
− µ
(ln Θ)
xx
V
x
. (9)
Denote
φ(x, t) = v(x, t) − V (x, t),
ψ(x, t) = u(x, t) − U(x, t),
ζ(x, t) = θ(x, t) − Θ(x, t). (10)
Combining (8) and (1), we can get
φ
t
= ψ
x
,
ψ
t
− (
RΘ
vV
φ)
x
+ (
Rζ
v
)
x
= −µ(
U
x
vV
φ)
x
+ µ(
ψ
x
v
)
x
− F,
R
γ−1
ζ
t
+
Rθ
v
(ψ
x
+ U
x
) −
RΘ
V
U
x
= k(
ζ
x
v
)
x
− k(
Θ
x
φ
vV
)
x
+ µ(
u
x
2
v
−
U
x
2
V
) − G,
(11)
and
(φ
0
, ψ
0
, ζ
0
) = (v
0
− V
0
, u
0
− U
0
, 0).
(V, U, Θ)(x, 0) = (
R
p
+
θ
0
, a
θ
0x
θ
0
, θ
0
).
The main results of this paper are as follows:
Theorem 1. Let θ
0
−θ
±
∈ L
2
(R
±
), ln θ
0x
∈ L
2
(R)∩L
1
(R) and ∥ln θ
0x
∥
L
2
< δ
0
, for some small
positive constant δ
0
. Then there exists a constant η
0
> 0, such that if ∥v
0
−V
0
, u
0
−U
0
∥
H
1
≤ η
0
,
(1)-(2) has a global solution satisfying
(φ, ζ)(x, t) ∈ C(0, ∞; H
1
0
(R)), ψ(x, t) ∈ C(0, ∞; H
1
(R))
(ψ
x
, ζ
x
) ∈ L
2
(0, ∞; H
1
(R)), φ
x
∈ L
2
(0, ∞; L
2
(R))
and
sup
x∈R
|(φ, ψ, ζ)| → 0, as t → ∞.
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To show that (1)-(2) has a viscous contact discontinuity wave independent on the initial
data of (1), we consider the following diffusion problem
Θ
t
= a(ln Θ)
xx
, a =
κp
+
(γ − 1)
γR
2
> 0, Θ(±∞, t) = θ
±
. (12)
(12)has a unique self similar solution Θ
1
= Θ
1
(ξ), ξ =
x
√
1+t
(see [18] or [17]). Set
V
1
=
RΘ
1
p
+
, U
1
=
κ(γ − 1)Θ
1x
γRΘ
1
.
From Theorem 0.1, we can deduce the following corollary.
Corollary 1. Let
θ
0
− θ
±
∈ L
2
(R
±
), ln θ
0x
∈ L
2
(R) ∩ L
1
(R), ∥ln θ
0x
∥
L
2
< δ
0
,
for some small positive constant δ
0
. Then there exists a constant η
0
> 0, such that if ∥v
0
−
V
0
, u
0
− U
0
∥
H
1
≤ η
0
, (1)-(2) has a global solution satisfying
(φ, ζ)(x, t) ∈ C(0, ∞; H
1
0
(R)), ψ(x, t) ∈ C(0, ∞; H
1
(R))
(ψ
x
, ζ
x
) ∈ L
2
(0, ∞; H
1
(R)), φ
x
∈ L
2
(0, ∞; L
2
(R))
and
sup
x∈R
|(v(x, t) − V
1
(x, t), u(x, t) − U
1
(x, t), θ(x, t) − Θ
1
(x, t))| → 0, as t → ∞.
Proof. According to Theorem 1, we need only to prove
sup
x∈R
|(V (x, t) − V
1
(x, t), U(x, t) − U
1
(x, t), Θ(x, t) − Θ
1
(x, t))| → 0, as t → ∞,
where V, U, Θ are the functions in (6). From Remark 1 in next section, we have
∥Θ
x
∥
2
2
, ∥Θ
1x
∥
2
2
≤ Ct
−
1
2
, |Θ
x
|, |Θ
1x
| ≤ Ct
−
1
2
.
Hence
sup
x∈R
|Θ(x, t) − Θ
1
(x, t)|
2
≤ sup
x∈R
|
x
−∞
∂
∂x
(Θ(s, t) − Θ
1
(s, t))
2
ds|
≤ 2∥Θ(·, t) − Θ
1
(·, t)∥
2
∥Θ
x
(·, t) − Θ
1x
(·, t)∥
2
≤ Ct
−
1
4
→ 0 as t → ∞,
sup
x∈R
|V (x, t) − V
1
(x, t)|
2
=
R
2
p
2
+
sup
x∈R
|Θ(x, t) − Θ
1
(x, t)|
2
→ 0 as t → ∞,
sup
x∈R
|U(x, t) − U
1
(x, t)|
2
= sup
x∈R
(
κ(γ − 1)Θ
1x
γRΘ
1
−
κ(γ − 1)Θ
x
γRΘ
)
2
≤ Ct
−1
→ 0.
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