Regularity criteria and Liouville Theorem for 3D inhomogeneous Navier-Stokes flows with vacuum

TL;DR

This paper establishes regularity criteria and Liouville theorems for 3D inhomogeneous Navier-Stokes flows with vacuum, using Lorentz space conditions to understand solution behavior.

Contribution

It introduces new regularity criteria and Liouville theorems for inhomogeneous Navier-Stokes equations with vacuum in Lorentz spaces.

Findings

Derived regularity criteria under Lorentz space conditions.

Proved Liouville type theorems for smooth solutions.

Enhanced understanding of vacuum flow behavior in 3D Navier-Stokes equations.

Abstract

In this paper, we investigate the 3D inhomogeneous Navier-Stokes flows with vacuum, and obtain regularity criteria and Liouville type theorems in the Lorentz space if a smooth solution $(\rho, \mathbf{u})$ satisfies suitable conditions.