American Institute of Mathematical Sciences

Advanced
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
[1]

Ning-An Lai, Yi Zhou. Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1499-1510. doi: 10.3934/cpaa.2018072
[2]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019
[3]

Türker Özsarı, Nermin Yolcu. The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3285-3316. doi: 10.3934/cpaa.2019148
[4]

Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431
[5]

Yacheng Liu, Runzhang Xu. Potential well method for initial boundary value problem of the generalized double dispersion equations. Communications on Pure & Applied Analysis, 2008, 7 (1) : 63-81. doi: 10.3934/cpaa.2008.7.63
[6]

Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319
[7]

Yoshihiro Ueda, Tohru Nakamura, Shuichi Kawashima. Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space. Kinetic & Related Models, 2008, 1 (1) : 49-64. doi: 10.3934/krm.2008.1.49
[8]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[9]

Chérif Amrouche, Yves Raudin. Singular boundary conditions and regularity for the biharmonic problem in the half-space. Communications on Pure & Applied Analysis, 2007, 6 (4) : 957-982. doi: 10.3934/cpaa.2007.6.957
[10]

Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212
[11]

Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709
[12]

Jun Zhou. Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28 (1) : 67-90. doi: 10.3934/era.2020005
[13]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
[14]

Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917
[15]

Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure & Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161
[16]

Wenning Wei. On the Cauchy-Dirichlet problem in a half space for backward SPDEs in weighted Hölder spaces. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5353-5378. doi: 10.3934/dcds.2015.35.5353
[17]

J. Colliander, A. D. Ionescu, C. E. Kenig, Gigliola Staffilani. Weighted low-regularity solutions of the KP-I initial-value problem. Discrete & Continuous Dynamical Systems, 2008, 20 (2) : 219-258. doi: 10.3934/dcds.2008.20.219
[18]

Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465
[19]

Imen Manoubi. Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2837-2863. doi: 10.3934/dcdsb.2014.19.2837
[20]

Maya Bassam, Denis Mercier, Ali Wehbe. Optimal energy decay rate of Rayleigh beam equation with only one boundary control force. Evolution Equations & Control Theory, 2015, 4 (1) : 21-38. doi: 10.3934/eect.2015.4.21

 Impact Factor: 

Tools

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences