主题：Global weak solutions to the isentropic compressible Navier-Stokes equations with vacuum and unbounded density in a half-plane under Dirichlet boundary conditions 半平面内Dirichlet边界条件下含真空及无界密度的等熵可压缩Navier-Stokes方程的整体弱解

主讲人：西南大学数学与统计学院 钟新 教授

主持人：西南财经大学数学学院 王永富

时间：2026年6月20日16:30-17:30

地点：西南财经大学光华校区崇文楼405会议室

主办单位：数学学院 科研处

主讲人简介：

钟新，西南大学数学与统计学院教授，博士生导师，学院和学部学术委员会委员。主要研究领域为可压缩Navier–Stokes 方程组及相关模型解的定性理论，以独立作者或通讯作者在Math. Ann.、J. Math. Pures Appl.、Comm. Partial Differential Equations、Indiana Univ. Math. J.、J. Nonlinear Sci.、Calc. Var. Partial Differential Equations等期刊上发表多篇学术论文，主持国家自然科学基金项目3项、省部级科研项目7项。曾先后入选重庆市巴渝学者计划青年学者、重庆英才计划青年拔尖人才、重庆青年科技创新先锋人物、重庆市科学传播专家，获得重庆市自然科学奖三等奖(独立完成人)、萧文灿数学研究杰出教师奖。

内容提要：

We establish the global existence of a class of weak solutions to the isentropic compressible Navier-Stokes equations in a half-plane with Dirichlet boundary conditions, allowing for vacuum both in the interior and at infinity, under a suitably small initial total energy. The solutions constructed here admit unbounded densities and lie in an intermediate regularity regime between the finite-energy weak solutions of Lions-Feireisl and the framework of Hoff. This result generalizes previous works of Hoff (Comm. Pure Appl. Math. 55 (2002), pp. 1365-1407) and Perepelitsa (Arch. Ration. Mech. Anal. 212 (2014), pp. 709-726) concerning discontinuous solutions by allowing vacuum states and unbounded density. Our analysis relies on the Green function method and new estimates involving the specific structure of the equations and the geometry of the half-plane. To the best of our knowledge, this is the first result concerning global weak solutions within Hoff's framework on an unbounded domain that simultaneously accommodates Dirichlet boundary conditions and far-field vacuum. The intermediate-regularity class developed here may be viewed as a natural extension of Hoff's theory, precisely tailored to overcome the two corresponding obstructions: the lack of global space-time control of the effective viscous flux arising from far-field vacuum and the absence of boundary-induced regularity gains in the no-slip setting. 讲座主要围绕在初始总能量充分小的条件下，证明了半平面内具有Dirichlet边界条件的等熵可压缩Navier-Stokes方程的一类弱解的整体存在性，该解允许在区域内部及无穷远处存在真空展开。此处构造的解容许无界密度，且其正则性介于Lions-Feireisl的有限能量弱解与Hoff的理论框架之间。通过允许真空态和无界密度，这一结果推广了Hoff (Comm. Pure Appl. Math. 55 (2002), pp. 1365-1407) 和 Perepelitsa (Arch. Ration. Mech. Anal. 212 (2014), pp. 709-726) 关于间断解的先前工作。我们的分析依赖于格林函数方法，以及结合方程特定结构与半平面几何特征的新估计。据我们所知，这是无界域上Hoff框架内关于整体弱解的首个结果，该结果同时兼容了Dirichlet边界条件与远场真空。此处发展的中间正则性类可视为Hoff理论的自然推广，其专门旨在克服两个相应的障碍：远场真空导致的有效粘性通量缺乏全局时空控制，以及无滑移条件下缺乏边界诱导的正则性提升。