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

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 $\mathrm{\parallel }\mathrm{\nabla }\rho {\mathrm{\parallel }}_{L_t^{\mathrm{\infty }}{L}_x^q},q>2$ 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.