Global existence and exponential decay of strong solutions for the inhomogeneous incompressible Navier-Stokes equations with vacuum

TL;DR

This paper proves the global existence, uniqueness, and exponential decay over time of strong solutions to the inhomogeneous incompressible Navier-Stokes equations with fractional dissipation, even with vacuum initial conditions.

Contribution

It establishes the existence and decay properties of strong solutions for the inhomogeneous Navier-Stokes equations with vacuum, extending previous results to fractional Laplacian dissipation.

Findings

Global strong solutions exist and are unique for large initial data.

Solutions exhibit exponential decay over time.

Initial vacuum conditions are permitted, including compact support.

Abstract

The inhomogeneous incompressible Navier-Stokes equations with fractional Laplacian dissipations in the multi-dimensional whole space are considered. The existence and uniqueness of global strong solution with vacuum are established for large initial data. The exponential decay-in-time of the strong solution is also obtained, which is different from the homogeneous case. The initial density may have vacuum and even compact support.