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

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 $L^2\cap {\dot{B}}_{p,1}^{-1+\frac{2}{p}}$ and the initial density ${\rho }_0$ in $L^{\infty }$ and having a positive lower bound, which satisfies $1-{\rho }_0^{-1}\in {\dot{B}}_{\lambda ,2}^{\frac{2}{\lambda }}\cap L^{\infty },$ for $p\in [2,\infty [$ and $\lambda \in [1,\infty [$ with $\frac{1}{2}<\frac{1}{p}+\frac{1}{\lambda }\le 1,$ 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 ${\dot{B}}_{\frac{2}{\epsilon },1}^{\epsilon }$, 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.

36 pages

Cited by in corpus (1)

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