On strongly Petrovskii's parabolic SPDEs in arbitrary dimension and the stochastic Cahn-Hilliard equation

Local and global well-posedness of SPDE with generalized coercivity conditions

In this paper we establish local and global existence and uniqueness of solutions for general nonlinear evolution equations with coefficients satisfying some local monotonicity and generalized coercivity conditions. An analogous result is obtained for stochastic evolution equations in Hilbert space with general additive noise. As applications, the main results are applied to obtain simpler proofs in known cases as the stochastic 3D Navier-Stokes equation, the tamed 3D Navier-Stokes equation and the Cahn-Hilliard equation, but also to get new results for stochastic surface growth PDE and stocha…

一类具有完全局部单调系数的SPDE的Wong-Zakai逼近及其应用

Wong-Zakai approximation for a class of SPDEs with fully local monotone coefficients and its application

本文针对具有完全局部单调系数的随机偏微分方程，通过Faedo-Galerkin方法等建立Wong-Zakai逼近，并应用于确定解的拓扑支集，涵盖多种流体模型。

An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs

We show uniqueness in law for the critical SPDE $$ dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,\;\; X_0 =x \in H, $$ where $A$ $ : dom(A) \subset H \to H$ is a negative definite self-adjoint operator on a separable Hilbert space $H$ having $A^{-1}$ of trace class and $W$ is a cylindrical Wiener process on $H$. Here $F: H \to H $ can be continuous with at most linear growth (some functions $F$ which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers' equations and for three-dimensional stochastic Cahn-Hilliard type equations …

Small time asymptotics for a class of stochastic partial differential equations with fully monotone coefficients forced by multiplicative Gaussian noise

The main goal of this article is to study the effect of small, highly nonlinear, unbounded drifts (small time large deviation principle (LDP) based on exponential equivalence arguments) for a class of stochastic partial differential equations (SPDEs) with fully monotone coefficients driven by multiplicative Gaussian noise. The small time LDP obtained in this paper is applicable for various quasi-linear and semilinear SPDEs such as porous medium equations, Cahn-Hilliard equation, 2D Navier-Stokes equations, convection-diffusion equation, 2D liquid crystal model, power law fluids, Ladyzhenskaya …

Correction to "An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs"

We show uniqueness in law for the critical SPDE \begin{eqnarray} \label{qq1} dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,\;\; X_0 =x \in H, \end{eqnarray} where $A$ $ : \text{dom}(A) \subset H \to H$ is a negative definite self-adjoint operator on a separable Hilbert space $H$ having $A^{-1}$ of trace class and $W$ is a cylindrical Wiener process on $H$. Here $F: H \to H $ can be locally Hölder continuous with at most linear growth (some functions $F$ which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers equations and for…

非线性SPDE与极大正则性：扩展综述

Nonlinear SPDEs and Maximal Regularity: An Extended Survey

本文综述了基于极大正则性和临界空间理论的随机演化方程适定性研究成果，应用于多种非线性SPDE，并提出了新的爆破准则与开放问题。