Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems
Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems
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DOI：
10.1007/s00205-003-0293-2
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发表时间：
2004-04-01
影响因子：
2.5
通讯作者：
Zumbrun, K
中科院分区：
文献类型：
--
作者：
Mascia, C;Zumbrun, K
本文在强谱稳定性的必要条件下，即，对于包含可压缩气体动力学和磁流体动力学（MHD）的一类对称双曲-抛物系统，建立了大振幅Lax型激波剖面的非线性L（1）布尔ANDH（3）->L（p）轨道稳定性，2小于或等于无顶小于或等于无穷大，具有急剧的衰减率。一个稳定的点谱的线性化运营商的波，横截性的配置文件，双曲稳定性的相关的理想冲击。特别地，这与[50]的谱稳定性结果一起，给出了当γ>1时，伽马律气体等熵Navier-Stokes方程任意大振幅激波剖面的非线性稳定性：这是真实的系统激波剖面的第一个完整大振幅稳定性结果（即，部分）粘度。在[53，54]中，通过结合“Kawashima型”能量估计和逐点绿色函数界，对Kawashima类的一般系统建立了相应的小振幅结果，其中小振幅假设仅用于关闭能量估计。在这里，在双曲线特征速度（相对于冲击）不仅非零而且具有共同符号的温和附加假设下，我们通过使用Goodman型加权范数[25，26]来关闭估计，该范数旨在控制关键双曲线模式中的估计。
We establish nonlinear L(1)boolean ANDH(3)-->L(p) orbital stability, 2less than or equal topless than or equal toinfinity, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier-Stokes equations for a gamma-law gas as gamma-->1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of ''Kawashima-type'' energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes.