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:
[1]

K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. U.S.A., 68 (1971), 1686-1688. doi: 10.1073/pnas.68.8.1686.

[2]

S. K. Godunov, An interesting class of quasi-linear systems, Dokl. Akad. Nauk SSSR, 139 (1961), 521-523.

[3]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto University, 1984.

[4]

S. Kawashima and K. Kurata, Hardy type inequality and application to the stability of degenerate stationary waves, J. Funct. Anal., 257 (2009), 1-19. doi: 10.1016/j.jfa.2009.04.003.

[5]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.

[6]

S. KawashimaT. NakamuraS. Nishibata and P. Zhu, Stationary waves to viscous heat-conductive gases in half-space: Existence, stability and convergence rate, Math. Models Methods Appl. Sci., 20 (2010), 2201-2235. doi: 10.1142/S0218202510004908.

[7]

S. KawashimaS. Nishibata and P. Zhu, Asymptotic stability of the stationary solution to the compressible {N}avier-{S}tokes equations in the half space, Comm. Math. Phys., 240 (2003), 483-500. doi: 10.1007/s00220-003-0909-2.

[8]

S. Kawashima and Y. Shizuta, On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws, Tohoku Math. J.(2), 40 (1988), 449-464. doi: 10.2748/tmj/1178227986.

[9]

A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys., 165 (1994), 83-96. doi: 10.1007/BF02099739.

[10]

T. Nakamura, Degenerate boundary layers for a system of viscous conservation laws, Anal. Appl. (Singap.), 14 (2016), 75-99. doi: 10.1142/S0219530515400047.

[11]

T. Nakamura and S. Nishibata, Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas, J. Hyperbolic Differ. Equ., 8 (2011), 651-670. doi: 10.1142/S0219891611002524.

[12]

T. Nakamura and S. Nishibata, Existence and asymptotic stability of stationary waves for symmetric hyperbolic-parabolic systems in half-line, Math. Models Methods Appl. Sci., 27 (2017), 2071-2110. doi: 10.1142/S0218202517500397.

[13]

T. NakamuraS. Nishibata and T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line, J. Differential Equations, 241 (2007), 94-111. doi: 10.1016/j.jde.2007.06.016.

[14]

M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac., 41 (1998), 107-132.

[15]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275. doi: 10.14492/hokmj/1381757663.

[16]

Y. UedaT. Nakamura and S. Kawashima, Stability of degenerate stationary waves for viscous gases, Arch. Ration. Mech. Anal., 198 (2010), 735-762. doi: 10.1007/s00205-010-0369-8.

[17]

T. UmedaS. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457. doi: 10.1007/BF03167068.

show all references

References:
[1]

K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. U.S.A., 68 (1971), 1686-1688. doi: 10.1073/pnas.68.8.1686.

[2]

S. K. Godunov, An interesting class of quasi-linear systems, Dokl. Akad. Nauk SSSR, 139 (1961), 521-523.

[3]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto University, 1984.

[4]

S. Kawashima and K. Kurata, Hardy type inequality and application to the stability of degenerate stationary waves, J. Funct. Anal., 257 (2009), 1-19. doi: 10.1016/j.jfa.2009.04.003.

[5]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.

[6]

S. KawashimaT. NakamuraS. Nishibata and P. Zhu, Stationary waves to viscous heat-conductive gases in half-space: Existence, stability and convergence rate, Math. Models Methods Appl. Sci., 20 (2010), 2201-2235. doi: 10.1142/S0218202510004908.

[7]

S. KawashimaS. Nishibata and P. Zhu, Asymptotic stability of the stationary solution to the compressible {N}avier-{S}tokes equations in the half space, Comm. Math. Phys., 240 (2003), 483-500. doi: 10.1007/s00220-003-0909-2.

[8]

S. Kawashima and Y. Shizuta, On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws, Tohoku Math. J.(2), 40 (1988), 449-464. doi: 10.2748/tmj/1178227986.

[9]

A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys., 165 (1994), 83-96. doi: 10.1007/BF02099739.

[10]

T. Nakamura, Degenerate boundary layers for a system of viscous conservation laws, Anal. Appl. (Singap.), 14 (2016), 75-99. doi: 10.1142/S0219530515400047.

[11]

T. Nakamura and S. Nishibata, Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas, J. Hyperbolic Differ. Equ., 8 (2011), 651-670. doi: 10.1142/S0219891611002524.

[12]

T. Nakamura and S. Nishibata, Existence and asymptotic stability of stationary waves for symmetric hyperbolic-parabolic systems in half-line, Math. Models Methods Appl. Sci., 27 (2017), 2071-2110. doi: 10.1142/S0218202517500397.

[13]

T. NakamuraS. Nishibata and T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line, J. Differential Equations, 241 (2007), 94-111. doi: 10.1016/j.jde.2007.06.016.

[14]

M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac., 41 (1998), 107-132.

[15]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275. doi: 10.14492/hokmj/1381757663.

[16]

Y. UedaT. Nakamura and S. Kawashima, Stability of degenerate stationary waves for viscous gases, Arch. Ration. Mech. Anal., 198 (2010), 735-762. doi: 10.1007/s00205-010-0369-8.

[17]

T. UmedaS. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457. doi: 10.1007/BF03167068.

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