Global well-posedness and exponential decay to the Cauchy problem of nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum in $\mathbb{R}^2$

TL;DR

This paper proves the global existence, uniqueness, and exponential decay of strong solutions to the 2D nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum, improving previous results and removing extra initial conditions.

Contribution

It establishes global well-posedness and decay rates for solutions with vacuum in the whole space, extending prior work by relaxing initial data restrictions.

Findings

Global existence of strong solutions under smallness condition on initial data

Exponential decay rates of solutions over time

No additional initial compatibility conditions needed despite vacuum presence

Abstract

We study global well-posedness of strong solutions for the nonhomogeneous Navier-Stokes equations with density-dependent viscosity and initial density allowing vanish in $\mathbb{R}^2$. Applying a logarithmic interpolation inequality and delicate energy estimates, we show the global existence of a unique strong solution provided that $\|\nabla\mu(\rho_0)\|_{L^q}$ is suitably small, which improves the previous result of Huang and Wang [SIAM J. Math. Anal. 46, 1771--1788 (2014)] to the whole space case. Moreover, we also derive exponential decay rates of the solution. In particular, there is no need to require additional initial compatibility condition despite the presence of vacuum.