报告题目：Stochastic Euler Equations with Pseudo-differential Noise: Continuous and Discontinuous Perturbations in Compressible and Incompressible Flows

报告人：唐昊教授，天津大学

报告时间：2026年7月18日（周六）15:00-17:00

报告地点：20-200

报告摘要：We study stochastic Euler equations in both compressible and incompressible regimes, on the whole space and on the torus, driven by genuinely mixed multiplicative noise: continuous Stratonovich/Itô components and a discontinuous Marcus component. The Stratonovich and Marcus noise amplitudes are (nonlocal) pseudo-differential operators that include the classical transport operator as a special case. Within this setting, we develop a local-in-time theory of classical solutions for both regimes, establishing existence, uniqueness, and a blow-up criterion. The presence of discontinuous pseudo-differential Marcus noise necessitates new analytical tools, which we develop to control the delicate interaction between jump discontinuities and nonlocal operators.

For the compressible barotropic case, we establish a transformation principle that generalizes the classical Makino transform beyond the standard polytropic γ-law. This extension accommodates a broad class of physically relevant equations of state, including piecewise-definedγ-laws, (piecewise-defined) Chaplygin-type laws, and the pressure law for white dwarf stars, many of which have remained unexplored in the stochastic compressible setting even under purely Itô-type forcing.

For the incompressible damped case, we identify a hierarchy of damping–noise regimes that successively guarantee global-in-time existence, uniform-in-time bounds, and exponential decay. To study the long-time statistical behavior, we establish a novel abstract existence criterion for invariant probability measures tailored to Markov semigroups satisfying a restricted Feller property under mismatched metrics. By explicitly circumventing the requirement of Feller continuity within a single topology, this framework provides a robust extension of the classical Krylov–Bogoliubov theory. Utilizing this criterion, we construct invariant probability measures for a broad class of singular stochastic evolution systems in Hilbert spaces, notably encompassing the stochastic damped Euler equations. As a principal application, we provide what appears to be the first positive answer to Shirikyan’s open problem regarding the existence, uniqueness, and mixing of invariant measures for the damped Euler equations on T^2. In fact, our approach goes beyond the original problem, resolving a substantially generalized version of the problem on both T^d and R^d across all spatial dimensions d≥2, under genuinely mixed multiplicative noise.

报告人简介：唐昊，天津大学教授。于2018年10月获得香港城市大学博士学位，随后于2019年至2024年间，先后在德国斯图加特大学（受洪堡基金资助）与挪威奥斯陆大学从事博士后研究工作。其主要研究方向为随机偏微分方程及其相关领域。近年来，以独立作者或通讯作者身份在《J. Lond. Math. Soc.》《J. Funct. Anal.》《SIAM J. Math. Anal.》《Ann. Inst. Henri Poincaré Probab. Stat.》《Commun. Contemp. Math.》《J. Differential Equations》《Stochastic Process. Appl.》等国际学术期刊上发表多篇研究论文。