$L^{p}$ Mild Solution to Stochastic Incompressible inhomogeneous Navier-Stokes Equations

TL;DR

This paper establishes the existence and uniqueness of global $L^{p}$ mild solutions for stochastic inhomogeneous incompressible Navier-Stokes equations on a torus, introducing a novel semigroup approach with time- and space-dependent generators.

Contribution

It introduces a new iteration scheme and semigroup framework for stochastic Navier-Stokes equations with inhomogeneous density, extending solution existence results.

Findings

Proved local existence and uniqueness of mild solutions.

Established global existence using Zorn's lemma.

Developed a semigroup approach with generators depending on both time and space.

Abstract

In this paper, we establish the global $L^{p}$ mild solution of inhomogeneous incompressible Navier-Stokes equations in the torus $\mathbb{T}^{N}$ with $N<p<6$, $ 1 \leqslant N \leqslant 3$, driven by the Wiener Process. We introduce a new iteration scheme coupled the density $\rho$ and the velocity $\mathbf{u}$ to linearize the system, which defines a semigroup. Notably, unlike semigroups dependent solely on $x$, the generators of this semigroup depend on both time $t$ and space $x$. After demonstrating the properties of this time- and space-dependent semigroup, we prove the local existence and uniqueness of mild solution, employing the semigroup theory and Banach's fixed point theorem. Finally, we show the global existence of mild solutions by Zorn's lemma. Moreover, for the stochastic case, we need to use the operator splitting method to do some estimates separately.