发表机构
The University of Kansas; The University of Texas at Austin; Institute of Applied Mathematics, AMSS, CAS; University of Chinese Academy of Sciences（堪萨斯大学; 德克萨斯大学奥斯汀分校; 中国科学院数学与系统科学研究院应用数学研究所; 中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对二维可压缩欧拉方程平面激波解的非唯一性，通过引入密度依赖黏性的纳维-斯托克斯方程全局弱解框架，建立首个消失黏性极限选择准则，证明近平面初始数据下激波解的唯一性与稳定性。
AI 中文摘要
众所周知，如Chiodaroli、De Lellis和Kreml [Comm. Pure Appl. Math. (2015)]所示，具有平面激波初始数据的可压缩欧拉方程可能允许无穷多个熵弱解。本文针对$\u211d \times \u2124$中具有密度依赖黏性的二维(2D)可压缩纳维-斯托克斯方程，引入了一个合适的全局时间弱解概念。在此解框架内，我们建立了二维可压缩欧拉方程物理相关平面激波解的第一个消失黏性极限选择准则。更精确地说，我们证明了在从近平面初始数据出发的纳维-斯托克斯解的无黏极限类中，二维欧拉平面激波解的唯一性和稳定性，而激波方向扰动可以任意大。从Bresch、Noble和Vila [ESAIM Proc. Surveys (2017)]在$\u2124^2$中二维可压缩纳维-斯托克斯方程的相对$\u03ba$-熵出发，我们首先建立了$\u211d \times \u2124$中二维可压缩纳维-斯托克斯方程弱解与$\u211d$中平均初始数据的一维(1D)强解之间的相对$\frac{1}{2}$-熵。然后，通过成功弥合相对$\frac{1}{2}$-熵中原始纳维-斯托克斯系统的二维欧拉表述与Kang和Vasseur [Invent. Math. (2021)]在拉格朗日坐标中欧拉方程激波解的一维消失黏性极限之间的基本结构差距，我们实现了所需的消失黏性极限选择准则。本文开发的框架可应用于$\u211d \times \u2124^2$中的三维极限选择准则，以及更复杂的平面黎曼解和相互作用波型。
英文摘要
It is well known that the compressible Euler equations with planar shock initial data may admit infinitely many entropy weak solutions, as shown by Chiodaroli, De Lellis and Kreml [Comm. Pure Appl. Math. (2015)]. In this paper, we introduce a suitable notion of global-in-time weak solutions to the two-dimensional (2D) compressible Navier-Stokes equations with density-dependent viscosities in $\mathbb{R} \times \mathbb{T}$. Within this solution framework, we establish the first vanishing viscosity limit selection criterion for the physically relevant planar shock solution to the 2D compressible Euler equations. More precisely, we prove the uniqueness and stability of the 2D Euler planar shock solution within a class of inviscid limits of Navier-Stokes solutions emanating from near-planar initial data while the shock direction perturbation can be arbitrarily large. Starting from the relative $κ$-entropy for 2D compressible Navier-Stokes equations in $\mathbb{T}^2$ by Bresch, Noble and Vila [ESAIM Proc. Surveys (2017)], we first establish the relative $\frac12$-entropy between 2D weak solution of compressible Navier-Stokes equations in $\mathbb{R} \times \mathbb{T}$ and one-dimensional (1D) strong solution for averaged initial data in $\mathbb{R}$. Then we achieve the desired vanishing viscosity limit selection criterion by successfully bridging the fundamental structural gap between 2D Eulerian formulation of the original Navier-Stokes system in the relative $\frac12$-entropy and 1D vanishing viscosity limit to a shock solution of Euler equations in Lagrangian coordinates by Kang and Vasseur [Invent. Math. (2021)]. The framework developed herein can be applied to the three-dimensional limit selection criterion in $\mathbb{R} \times \mathbb{T}^2$, as well as to more complex planar Riemann solutions and interacting wave patterns.
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