Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane

Pages: 469 - 476, Issue Special, July 2003

 Abstract        Full Text (183.9K)              

Shuichi Kawashima - Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, Japan (email)
Shinya Nishibata - Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan (email)
Masataka Nishikawa - Department of Mathematics Sciences, School of Science and Engineering, Waseda University, Tokyo 169-8555, Japan (email)

Abstract: We investigate the asymptotic stability of a stationary solution to an initial boundary value problem for a 2-dimensional viscous conservation law in half plane. Precisely, we show that under suitable boundary and spatial asymptotic conditions, a solution converges to the corresponding stationary solution as time tends to infinity. The proof is based on an a priori estimate in the $H^2$-Sobolev space, which is obtained by a standard energy method. In this computation, we utilize the Poincaré type inequality. In addition, we obtain a convergence rate under the assumption that the initial data converges to a spatial asymptotic state algebraically fast. This result is obtained by a weighted energy estimate.

Keywords:  Planar wave, weighted energy method, initial boundary value problem, half space, decay rate.
Mathematics Subject Classification:  Primary 35L65, 35L67, 76L05.

Received: September 2002; Published: April 2003.