Stability of Degenerate Stationary Waves for Viscous Gases

作者：Ueda Yoshihiro^*; Nakamura Tohru; Kawashima Shuichi

来源：Archive for Rational Mechanics and Analysis, 2010, 198(3): 735-762.

DOI：10.1007/s00205-010-0369-8

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-(alpha/4) as t -> 8, provided that the initial perturbation is in the weighted space L(alpha)(2) = L(2)(R(+); (1 + x)(alpha) dx). This convergence rate t-(alpha/4) is weaker than the one for the non-degenerate case and requires the restriction alpha < alpha(*)(q), where alpha(*)(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 L(alpha)(2) a for alpha > alpha*(q) with another critical value alpha*(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.

• 出版日期2010-12