American Institute of Mathematical Sciences

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2003, 2003(Special): 469-476. doi: 10.3934/proc.2003.2003.469

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

Shuichi Kawashima 1, , Shinya Nishibata 2, and  Masataka Nishikawa 3,

1. 

Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, Japan
2. 

Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan
3. 

Department of Mathematics Sciences, School of Science and Engineering, Waseda University, Tokyo 169-8555, Japan

Received  September 2002 Published  April 2003

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: decay rate., Planar wave, half space, weighted energy method, initial boundary value problem.
Mathematics Subject Classification: Primary 35L65, 35L67, 76L0.
Citation: Shuichi Kawashima, Shinya Nishibata, Masataka Nishikawa. Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane. Conference Publications, 2003, 2003 (Special) : 469-476. doi: 10.3934/proc.2003.2003.469
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