TY - JOUR

T1 - Stationary waves to viscous heat-conductive gases in half-space

T2 - Existence, stability and convergence rate

AU - Kawashima, Shuichi

AU - Nakamura, Tohru

AU - Nishibata, Shinya

AU - Zhu, Peicheng

N1 - Funding Information:
T.N. was supported in part by Grant-in-Aid for Young Scientists (B) 21740100 of the Ministry of Education, Culture, Sports, Science and Technology. P.Z. was supported in part by JSPS postdoctoral fellowship under P99217.

PY - 2010/12

Y1 - 2010/12

N2 - The main concern of this paper is to study large-time behavior of solutions to an ideal polytropic model of compressible viscous gases in one-dimensional half-space. We consider an outflow problem and obtain a convergence rate of solutions toward a corresponding stationary solution. Here the existence of the stationary solution is proved under a smallness condition on the boundary data with the aid of center manifold theory. We also show the time asymptotic stability of the stationary solution under smallness assumptions on the boundary data and the initial perturbation in the Sobolev space, by employing an energy method. Moreover, the convergence rate of the solution toward the stationary solution is obtained, provided that the initial perturbation belongs to the weighted Sobolev space. The proof is based on deriving a priori estimates by using a time and space weighted energy method.

AB - The main concern of this paper is to study large-time behavior of solutions to an ideal polytropic model of compressible viscous gases in one-dimensional half-space. We consider an outflow problem and obtain a convergence rate of solutions toward a corresponding stationary solution. Here the existence of the stationary solution is proved under a smallness condition on the boundary data with the aid of center manifold theory. We also show the time asymptotic stability of the stationary solution under smallness assumptions on the boundary data and the initial perturbation in the Sobolev space, by employing an energy method. Moreover, the convergence rate of the solution toward the stationary solution is obtained, provided that the initial perturbation belongs to the weighted Sobolev space. The proof is based on deriving a priori estimates by using a time and space weighted energy method.

KW - Compressible Navier-Stokes equation

KW - Eulerian coordinate

KW - boundary layer solution

KW - ideal polytropic model

KW - outflow problem

KW - weighted energy method

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U2 - 10.1142/S0218202510004908

DO - 10.1142/S0218202510004908

M3 - Article

AN - SCOPUS:78651230839

VL - 20

SP - 2201

EP - 2235

JO - Mathematical Models and Methods in Applied Sciences

JF - Mathematical Models and Methods in Applied Sciences

SN - 0218-2025

IS - 12

ER -