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（天津大学）
邀请人&主持人：黄辉
报告时间与地点：6月16日 19:30 pm， 腾讯会议：116-393-809
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/Itˆo 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.
We establish a transformation principle for the compressible barotropic case that generalizes the Makino transform beyond the polytropic setting and covers a broad class of physically relevant pressure laws outside the standard polytropic γ -law. This class includes piecewise-defined γ -laws, (piecewise-defined) Chaplygin-type laws, the pressure law for white dwarf stars, and other astrophysically motivated regimes. Most of these equations of state have not been analyzed in the stochastic compressible setting, even under purely Itˆo-type forcing.
For the incompressible damped case, we specify a hierarchy of damping-noise conditions of increasing strength that yield global-in-time existence, uniform-in-time bounds, and exponential decay estimates, respectively. Moreover, we develop a criterion for the existence of invariant probability measures for a general Markov process accommodating mismatched topologies, which extends the classical Krylov–Bogoliubov approach. This abstract criterion allows us to prove the existence and uniqueness of invariant probability measures for a broad class of singular stochastic evolution systems in Hilbert spaces, including the stochastic damped Euler equations. Notably, this application gives what appears to be the first positive answer to Shirikyan’s open problem concerning the invariant measure and mixing of the damped Euler equations on T^2 under genuinely mixed multiplicative noise. Furthermore, our framework goes beyond the original formulation of the problem: it resolves a substantially strengthened version in every dimension d ≥ 2, on both T^d and R^d .
报告人介绍：唐昊现任天津大学教授。他于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。
