无粘性热传导理想气体系统的一般黎曼解的时间渐近稳定性

中文摘要 AI 辅助

本文研究一维无粘性热传导理想气体系统的一般黎曼解的时间渐近稳定性，其中一般黎曼解由激波、接触间断和稀疏波组成。我们证明，当时间趋于无穷时，该无粘性热传导理想气体系统的解会以随时间变化的平移一致收敛于由稀疏波、粘性接触波和粘性激波构成的复合波。受Kang-Vasseur-Wang近期工作[Arch. Ration. Mech. Anal. 249: 42 (2025)]的启发，我们克服了仅对单个变量耗散的部分耗散双曲-抛物系统中激波与稀疏波共存带来的困难。更值得注意的是，速度耗散的缺失在处理与密度和速度相关的项时产生了新的、内在的困难。为解决这一问题，我们利用控制方程的精确结构和激波的额外性质。此外，我们利用该无粘性系统的波结构，对密度和速度分别进行时空估计。

英文摘要

This paper is concerned with the time-asymptotic stability of the generic Riemann solution for the one-dimensional system of heat-conductive ideal gas without viscosity, where the generic Riemann solution consists of a shock, a contact discontinuity, and a rarefaction wave. We prove that, as time tends to infinity, the solution of the non-viscous and heat-conductive ideal gas system converges uniformly to a composite wave composed of rarefaction wave, viscous contact wave, and viscous shock wave with a time-dependent shift. Motivated by the recent work of Kang-Vasseur-Wang [Arch. Ration. Mech. Anal. 249: 42 (2025)], we overcome the difficulties arising from the concurrence of shock and rarefaction waves for the partially dissipative hyperbolic-parabolic system with dissipation acting only on a single variable. More notably, the absence of velocity dissipation gives rise to new and intrinsic difficulties when handling the terms associated with the density and velocity. To resolve this, we exploit the precise structure of the governing equations and the additional properties of shock waves. Furthermore, we utilize the wave structure of the system without viscosity and perform separate space-time estimates for the density and velocity.

发表机构

• Beijing Normal University（北京师范大学）
• University of Notre Dame（圣母大学）

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