Smooth Solutions of Non-linear Stochastic Partial Differential Equations

TL;DR

This paper investigates the regularity and smoothness of solutions to various nonlinear stochastic partial differential equations, providing a unified framework and applying it to key models like stochastic Burgers, Ginzburg-Landau, and Navier-Stokes equations.

Contribution

It develops a general approach to establish the existence of smooth solutions for nonlinear SPDEs within Hilbert scales, covering several important equations.

Findings

Existence of smooth solutions for stochastic Burgers and Ginzburg-Landau equations.

Existence of smooth solutions for 2D and 3D stochastic Navier-Stokes equations.

Unified framework for regularity analysis of nonlinear SPDEs.

Abstract

In this paper, we study the regularities of solutions of nonlinear stochastic partial differential equations in the framework of Hilbert scales. Then we apply our general result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau's equations on the real line, stochastic 2D Navier-Stokes equations in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their respectively smooth solutions.