Stability of Degenerate Stationary Waves for Viscous Gases |
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Authors: | Yoshihiro Ueda Tohru Nakamura Shuichi Kawashima |
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Affiliation: | 1. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan 2. Faculty of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan
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Abstract: | This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous gases in the half space. We discuss the following two cases: (1) viscous conservation laws and (2) damped wave equations with nonlinear convection. In each case, we prove that the solution converges to the corresponding degenerate stationary wave at the rate t −α/4 as t → ∞, provided that the initial perturbation is in the weighted space L2a=L2(mathbb R+; (1+x)a dx){L^2_alpha=L^2({mathbb R}_+;,(1+x)^alpha dx)} . This convergence rate t −α/4 is weaker than the one for the non-degenerate case and requires the restriction α < α*(q), where α*(q) is the critical value depending only on the degeneracy exponent q. Such a restriction is reasonable because the corresponding linearized operator for viscous conservation laws cannot be dissipative in L2a{L^2_alpha} for α > α*(q) with another critical value α*(q). Our stability analysis is based on the space–time weighted energy method in which the spatial weight is chosen as a function of the degenerate stationary wave. |
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