Stability of Degenerate Stationary Waves for Viscous Gases
Stability of Degenerate Stationary Waves for Viscous Gases
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DOI：
10.1007/s00205-010-0369-8
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发表时间：
2010-09
影响因子：
2.5
通讯作者：
Yoshihiro Ueda;Tohru Nakamura;S. Kawashima
中科院分区：
文献类型：
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作者：
Yoshihiro Ueda;Tohru Nakamura;S. Kawashima
本文研究半空间中粘性气体退化驻波的渐近稳定性。我们讨论了以下两种情形：（1）粘性守恒律和（2）带非线性对流项的阻尼波动方程。在每种情况下，我们证明了解在速率−α/4ast→ ∞收敛到相应的退化驻波，只要初始扰动在加权空间中。这个收敛率−α/4比非退化情形的收敛率弱，并且需要限制α < α*（q），其中α*（q）是仅依赖于退化指数q的临界值。这种限制是合理的，因为当α > α*（q）具有另一个临界值α*（q）时，相应的粘性守恒律线性化算子不可能是耗散的。我们的稳定性分析是基于时空加权能量法，其中的空间权重被选择为退化驻波的函数。
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 ratet−α/4ast→ ∞, provided that the initial perturbation is in the weighted space. This convergence ratet−α/4is 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 exponentq. Such a restriction is reasonable because the corresponding linearized operator for viscous conservation laws cannot be dissipative infor 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.