Critical strong Feller regularity for Markov solutions to the Navier-Stokes equations

TL;DR

This paper proves that Markov solutions to the 3D Navier-Stokes equations with Gaussian noise possess the strong Feller property within a critical functional space, enhancing understanding of their regularity and probabilistic behavior.

Contribution

It establishes the strong Feller property for Markov solutions to the 3D Navier-Stokes equations at a critical regularity level, which was previously unknown.

Findings

Markov solutions exhibit strong Feller property in the critical topology

The result applies to solutions driven by Gaussian noise

Advances the understanding of regularity for stochastic Navier-Stokes equations

Abstract

The main purpose of this paper is to show that Markov solutions to the 3D Navier--Stokes equations driven by Gaussian noise have the strong Feller property up to the critical topology given by the domain of the Stokes operator to the power one-fourth.