# American Institute of Mathematical Sciences

August  2018, 11(4): 757-793. doi: 10.3934/krm.2018031

## Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space

 1 Department of Applied Mathematics, Kumamoto University, Kumamoto 860-8555, Japan 2 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan

Received  September 2017 Revised  December 2017 Published  April 2018

Fund Project: The first author's work was supported in part by Grant-in-Aid for Scientific Research (C) 16K05237 of Japan Society for the Promotion of Science.

In the present paper, we study a system of viscous conservation laws, which is rewritten to a symmetric hyperbolic-parabolic system, in one-dimensional half space. For this system, we derive a convergence rate of the solutions towards the corresponding stationary solution with/without the stability condition. The essential ingredient in the proof is to obtain the a priori estimate in the weighted Sobolev space. In the case that all characteristic speeds are negative, we show the solution converges to the stationary solution exponentially if an initial perturbation belongs to the exponential weighted Sobolev space. The algebraic convergence is also obtained in the similar way. In the case that one characteristic speed is zero and the other characteristic speeds are negative, we show the algebraic convergence of solution provided that the initial perturbation belongs to the algebraic weighted Sobolev space. The Hardy type inequality with the best possible constant plays an essential role in deriving the optimal upper bound of the convergence rate. Since these results hold without the stability condition, they immediately mean the asymptotic stability of the stationary solution even though the stability condition does not hold.

Citation: Tohru Nakamura, Shinya Nishibata, Naoto Usami. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinetic & Related Models, 2018, 11 (4) : 757-793. doi: 10.3934/krm.2018031
##### References:

show all references

##### References:
 [1] O. Guès, G. Métivier, M. Williams, K. Zumbrun. Boundary layer and long time stability for multi-D viscous shocks. Discrete & Continuous Dynamical Systems, 2004, 11 (1) : 131-160. doi: 10.3934/dcds.2004.11.131 [2] Tong Yang, Huijiang Zhao. Asymptotics toward strong rarefaction waves for $2\times 2$ systems of viscous conservation laws. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 251-282. doi: 10.3934/dcds.2005.12.251 [3] Weishi Liu. Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 871-884. doi: 10.3934/dcds.2004.10.871 [4] 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 [5] Zefu Feng, Changjiang Zhu. Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3069-3097. doi: 10.3934/dcds.2019127 [6] Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic & Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409 [7] Jan Březina, Eduard Feireisl, Antonín Novotný. On convergence to equilibria of flows of compressible viscous fluids under in/out–flux boundary conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3615-3627. doi: 10.3934/dcds.2021009 [8] Xin Liu. Compressible viscous flows in a symmetric domain with complete slip boundary: The nonlinear stability of uniformly rotating states with small angular velocities. Communications on Pure & Applied Analysis, 2019, 18 (2) : 751-794. doi: 10.3934/cpaa.2019037 [9] Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345 [10] Tong Tang, Hongjun Gao. Local strong solutions to the compressible viscous magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1617-1633. doi: 10.3934/dcdsb.2016014 [11] Boling Guo, Guangwu Wang. Existence of the solution for the viscous bipolar quantum hydrodynamic model. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3183-3210. doi: 10.3934/dcds.2017136 [12] Christophe Prieur. Control of systems of conservation laws with boundary errors. Networks & Heterogeneous Media, 2009, 4 (2) : 393-407. doi: 10.3934/nhm.2009.4.393 [13] Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717 [14] Cristian A. Coclici, Jörg Heiermann, Gh. Moroşanu, W. L. Wendland. Asymptotic analysis of a two--dimensional coupled problem for compressible viscous flows. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 137-163. doi: 10.3934/dcds.2004.10.137 [15] Zhi-Ying Sun, Lan Huang, Xin-Guang Yang. Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry. Electronic Research Archive, 2020, 28 (2) : 861-878. doi: 10.3934/era.2020045 [16] Chiu-Ya Lan, Chi-Kun Lin. Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach. Discrete & Continuous Dynamical Systems, 2004, 11 (1) : 161-188. doi: 10.3934/dcds.2004.11.161 [17] Xiaoping Zhai, Hailong Ye. On global large energy solutions to the viscous shallow water equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4277-4293. doi: 10.3934/dcdsb.2020097 [18] Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden, Adele Manes. Energy stability for thermo-viscous fluids with a fading memory heat flux. Evolution Equations & Control Theory, 2015, 4 (3) : 265-279. doi: 10.3934/eect.2015.4.265 [19] Takayuki Kubo, Yoshihiro Shibata, Kohei Soga. On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3741-3774. doi: 10.3934/dcds.2016.36.3741 [20] Jaemin Shin, Yongho Choi, Junseok Kim. An unconditionally stable numerical method for the viscous Cahn--Hilliard equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1737-1747. doi: 10.3934/dcdsb.2014.19.1737

2020 Impact Factor: 1.432