Global wellposedness to incompressible inhomogeneous fluid system with bounded density and non-Lipschitz velocity

TL;DR

This paper establishes the global existence and uniqueness of weak solutions for the incompressible inhomogeneous Navier-Stokes equations with bounded, non-Lipschitz velocity fields, using critical Besov space initial data and heat kernel regularity.

Contribution

It proves the first global well-posedness results for such systems with non-Lipschitz velocities in critical Besov spaces, extending previous theories.

Findings

Global weak solutions exist under smallness conditions.

Uniqueness holds with additional initial regularity.

Classical heat kernel regularity is crucial for proofs.

Abstract

In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier-Stokes equations with initial data in critical Besov spaces, which satisfies a non-linear smallness condition. The regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non-Lipschitz velocity field. Furthermore, with additional regularity assumption on the initial velocity or on the initial density, we can also prove the uniqueness of such solution. We should mention that the classical maximal regularity theorem for the heat kernel plays an essential role in this context.