Abstract
This paper is concerned with the asymptotic behavior toward the rarefaction wave of the solution of a one-dimensional barotropic model system for compressible viscous gas. We assume that the initial data tend to constant states atx=±∞, respectively, and the Riemann problem for the corresponding hyperbolic system admits a weak continuous rarefaction wave. If the adiabatic constant γ satisfies 1≦γ≦2, then the solution is proved to tend to the rarefaction wave ast→∞ under no smallness conditions of both the difference of asymptotic values atx=±∞ and the initial data. The proof is given by an elementaryL 2-energy method.
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Hattori, Y., Nishihara, K.: A note on the stability of the rarefaction wave of the Burgers equation. Jpn. J. Appl. Math.8, 85–96 (1991)
Il'in, A. M., Oleinik, O. A.: Asymptotic behavior of the solutions of the Cauchy problem for certain quasilinear equations for large time (Russian). Mat. Sb.51, 191–216 (1960)
Kanel', Ya.: On a model system of equations of one-dimensional gas motion (Russian). Differencial'nya Uravnenija4, 374–380 (1968)
Kawashima, S., Matsumura, A.: Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion. Commun. Math. Phys.101, 97–127 (1985)
Kawashima, S., Matsumura, A., Nishihara, K.: Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas. Proc. Jpn. Acad.62, 249–252 (1986)
Lax, P. D.: Hyperbolic systems of conservation laws. II. Commun. Pure Appl. Math.10, 537–566 (1957)
Liu, T.-P.: Nonlinear stability of shock waves for viscous conservation laws. Memoirs AMS328, 1–108 (1985)
Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ.20, 67–104 (1980)
Matsumura, A., Nishida, T.: Periodic solutions in the half-space for a one-dimensional model of visco-elasticity (to appear)
Matsumura, A., Nishihara, K.: On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas. Jpn. J. Appl. Math.2, 17–25 (1985)
Matsumura, A., Nishihara, K.: Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas. Jpn. J. Appl. Math.3, 1–13 (1986)
Nishihara, K.: A note on the stability of travelling wave solutions of Burger's equation. Jpn. J. Appl. Math.2, 27–35 (1985)
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Communicated by S.-T. Yau
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Matsumura, A., Nishihara, K. Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas. Commun.Math. Phys. 144, 325–335 (1992). https://doi.org/10.1007/BF02101095
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DOI: https://doi.org/10.1007/BF02101095