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

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.