Global Well-posedness of Strong Solutions to the Cauchy Problem of 2D Nonhomogeneous Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum

TL;DR

This paper proves the global existence and behavior of strong solutions to 2D nonhomogeneous Navier-Stokes equations with density-dependent viscosity, allowing for vacuum states and large initial data.

Contribution

It establishes the global well-posedness of strong solutions without smallness assumptions on initial data, even with vacuum and large initial densities.

Findings

Global existence of strong solutions with vacuum

Key estimates for density gradients without smallness constraints

Asymptotic behavior of velocity and pressure gradients

Abstract

This paper is concerned with the Cauchy problem for the modified two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity. By fully using the structure of the system, we can obtain the key estimates of $\|\nabla \rho\|_{L_t^\infty L_x^q},q>2$ without any smallness asuumption on the initial data, and thus establish the global existence of the strong solutions with the far-field density being either vacuum or nonvacuum. Notably, the initial data can be arbitrarily large and the initial density is allowed to vanish. Furthermore, the large-time asymptotic behavior of the gradients of the velocity and the pressure is also established.