On the global existence and uniqueness of solution to 2-D inhomogeneous incompressible Navier-Stokes equations in critical spaces

TL;DR

This paper proves the global existence and uniqueness of solutions for 2-D inhomogeneous incompressible Navier-Stokes equations with initial data in critical spaces, extending previous results and relaxing some assumptions.

Contribution

It establishes new conditions for global existence and uniqueness of solutions in critical spaces, improving upon prior work by weakening initial density assumptions.

Findings

Global solutions exist under specified initial data conditions.

Uniqueness holds for p=2 and with additional density assumptions for p>2.

Results extend and improve previous work on inhomogeneous Navier-Stokes equations.

Abstract

In this paper, we establish the global existence and uniqueness of solution to $2$-D inhomogeneous incompressible Navier-Stokes equations \eqref{1.2} with initial data in the critical spaces. Precisely, under the assumption that the initial velocity $u_0$ in $L^2 \cap\dot B^{-1+\frac{2}{p}}_{p,1}$ and the initial density $\rho_0$ in $L^\infty$ and having a positive lower bound, which satisfies $1-\rho_0^{-1}\in \dot B^{\frac{2}{\lambda}}_{\lambda,2}\cap L^\infty,$ for $p\in[2,\infty[$ and $\lambda\in [1,\infty[$ with $\frac{1}{2}<\frac{1}{p}+\frac{1}{\lambda}\leq1,$ the system \eqref{1.2} has a global solution. The solution is unique if $p=2.$ With additional assumptions on the initial density in case $p>2,$ we can also prove the uniqueness of such solution. In particular, this result improves the previous work in \cite{AG2021} where $u_{0}$ belongs to $\dot{B}_{2,1}^{0}$ and $\rho_0^{-1}-1$ belongs to $\dot{ B}_{\frac{2}{\varepsilon},1}^{\varepsilon}$, and we also remove the assumption that the initial density is close enough to a positive constant in \cite{DW2023} yet with additional regularities on the initial density here.