$L^{p}$-solutions of the Navier-Stokes equation with fractional Brownian noise

TL;DR

This paper investigates the existence and uniqueness of mild solutions in L^p spaces for the Navier-Stokes equations affected by fractional Brownian noise on bounded domains in two and three dimensions.

Contribution

It establishes local existence and uniqueness results for L^p solutions of Navier-Stokes equations with fractional Brownian noise, extending previous stochastic fluid dynamics research.

Findings

Proves local existence of solutions for p > d.

Establishes uniqueness of solutions in the specified setting.

Analyzes the impact of fractional Brownian noise on solution behavior.

Abstract

We study the Navier-Stokes equations on a smooth bounded domain $D\subset \mathbb R^d$ ($d=2$ or 3), under the effect of an additive fractional Brownian noise. We show local existence and uniqueness of a mild $L^p$-solution for $p>d$.