Asymptotic stability of the rarefaction wave of a onedimensional model system for compressible viscous gas with boundary
 Tao Pan,
 Hongxia Liu,
 Kenji Nishihara
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
This paper is concerned with asymptotic behavior of solutions of a onedimensional barotropic flow governed byv _{t} −u _{x} = 0,u _{t} +p(v) _{x} = μ(u _{x}/v)_{ x } onR _{+} ^{1} with boundary. The initial data of (v,u) have constant states (v _{+},u_{+}) at +∞ and the boundary condition atx = 0 is given only on the velocityu, say u_{−}. By virtue of the boundary effect the solution is expected to behave as outgoing wave. Therefore, whenu _{−} <u _{+},v _{−} is determined as (u _{+},u _{+}) ∈R _{2}(v _{−},u _{−}), 2rarefaction curve for the corresponding hyperbolic system, which admits the 2rarefaction wave (v ^{r},u^{r})(x/t) connecting two constant states (v _{−},u _{−}) and (v _{+},u _{+}). Our assertion is that the solution of the original system tends to the restriction of (v ^{r},u^{r})(x/t) toR _{+} ^{1} as t → ∞ provided that both the initial perturbations and ¦(v _{+} −v _{−},u _{+}u _{t}) are small. The result is given by an elementaryL ^{2} energy method.
 Y. Hattori and K. Nishihara, A note on the stability of rarefaction wave of Burgers equation. Japan J. Indust. Appl. Math.,8 (1991), 85–96. CrossRef
 A.M. Ilin and O.A. Oleinik, Asymptotic behavior of the solution of the Cauchy problem for certain quasilinear equations for large time (Russian). Mat. Sb.,51 (1960), 191–216.
 Ya. I. Kanel’, On a model system of equations of onedimensional gas motion (Russian). Differencial’nye Uravnenija,4 (1968), 374–380.
 S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for onedimensional gas motion. Comm. Math. Phys.,101 (1985), 97–127. CrossRef
 P.D. Lax, Hyperbolic systems of conservation laws II. Comm. Pure. Appl. Math.,10 (1957), 537–566. CrossRef
 J.T.P. Liu, Nonlinear stability of shock waves for viscous conservation laws. Memoirs AMS,328 (1985), 1–108.
 T.P. Liu, A. Matsumura and K. Nishihara, Behavior of solutions for the Burgers equations with boundary corresponding to rarefaction waves. SIAM J. Math. Anal.,29(2) (1998), 293–308. CrossRef
 T.P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect. J. Differential Equations,133 (1997), 296–320. CrossRef
 T.P. Liu and S.H. Yu, Propagation of a stationary shock layer in the presence of a boundary. Arch. Rat. Mech. Anal.,139 (1997), 57–82. CrossRef
 A. Matsumura and M. Mei, Convergence to travelling fronts of solutions of thepsystem with viscosity in the presence of a boundary. Arch. Rat. Mech. Anal.,146 (1999), 1–22. CrossRef
 A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heatconductive gases. J. Math. Kyoto Univ.,20 (1980), 67–104.
 A. Matsumura and K. Nishihara, On the stability of travelling wave solutions of a onedimensional model system for compressible viscous gas. Japan J. Appl. Math.,2 (1985), 17–25.
 A. Matsumura and K. Nishihara, Asymptotic toward the rarefaction waves of the solutions of a onedimensional model system for compressible viscous gas. Japan J. Appl. Math.,3 (1986), 1–13. CrossRef
 A. Matsumura and K. Nishihara, Global stability of the rarefaction wave of the solutions of a onedimensional model system for compressible viscous gas. Comm. Math. Phys.,144 (1992), 325–335. CrossRef
 A. Matsumura and K. Nishihara, Asymptotic stability of the traveling waves for scalar viscous conservation laws with nonconvex nonlinearity. Comm. Math. Phys.,165 (1994), 83–96. CrossRef
 K. Nishihara, A note on the stability of traveling wave solution of Burgers’ equation. Japan J. Appl. Math.,2 (1985), 27–35. CrossRef
 A.I. Vol’pert and S.I. Hujaev, On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSRSB.,16 (1972), 517–544. CrossRef
 Title
 Asymptotic stability of the rarefaction wave of a onedimensional model system for compressible viscous gas with boundary
 Journal

Japan Journal of Industrial and Applied Mathematics
Volume 16, Issue 3 , pp 431441
 Cover Date
 19991001
 DOI
 10.1007/BF03167367
 Print ISSN
 09167005
 Online ISSN
 1868937X
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 asymptotic behavior
 rarefaction wave
 compressible viscous gas
 boundary
 Authors

 Tao Pan ^{(1)}
 Hongxia Liu ^{(2)}
 Kenji Nishihara ^{(3)}
 Author Affiliations

 1. Department of Mathematics, Guangxi University, 530004, Nanning, P.R. China
 2. Department of Mathematics, Jinan University, 510632, Guangzhou, P.R. China
 3. School of Political Science and Economics, Waseda University, 16950, Tokyo, Japan