Abstract
The asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics. 1) The solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet −γ (for someγ>0) ast→∞, provided that the initial disturbance is small and of integral zero, and in addition decays at an algebraic rate for |x|→∞. 2) The traveling wave solution with Nishida and Smoller's condition of the system of a viscous heat-conductive ideal gas is asymptotically stable, provided the initial disturbance is small and of integral zero. 3) The traveling wave solution with weak shock profile of the Broadwell model system of the Boltzmann equation is asymptotically stable, provided the initial disturbance is small and its hydrodynamical moments are of integral zero. Each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one. The property of integral zero of the initial disturbance plays a crucial role in this procedure.
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References
Caflisch, R. E.: Navier-Stokes and Boltzmann shock profiles for a model of gas dynamics. Commun Pure Appl. Math.32, 521–554 (1979)
Gilbarg, D.: The existence and limit behavior of the one-dimensional shock layer. Am. J. Math.73, 256–274 (1951)
Goodman, J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws (to appear)
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. Shornik51, 191–216 (1960)
Jones, C.: Stability of the traveling wave solution of the FitzHugh-Nagumo system (to appear)
Matsumura, A., Nishihara, K.: On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas, to appear in Jpn. J. Appl. Math
Nishida, T., Smoller, J. A.: Solutions in the large for some nonlinear hyperbolic conservation laws. Commun. Pure Appl. Math.26, 183–200 (1983)
Nishihara, K.: A note on the stability of traveling wave solutions of the Burgers' equation, to appear in Jpn. J. Appl. Math
Vol'pert, A. I., Hudjaev, S. I.: On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSR Sbornik16, 517–544 (1972)
Kawashima, S., Nishida, T.: Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases. J. Math. Kyoto Univ.21, 825–837 (1981)
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)
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Communicated by L. Nirenberg
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Kawashima, S., Matsumura, A. Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Commun.Math. Phys. 101, 97–127 (1985). https://doi.org/10.1007/BF01212358
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DOI: https://doi.org/10.1007/BF01212358