Regularity aspects of Leray-Hopf solutions to the 2D Inhomogeneous Navier-Stokes system and applications to weak-strong uniqueness

TL;DR

This paper characterizes when Leray-Hopf solutions to the 2D inhomogeneous Navier-Stokes system become regular over time, using energy inequalities and pressure regularity, and applies this to prove weak-strong uniqueness.

Contribution

It provides a new characterization of solution regularity and a unified framework for weak-strong uniqueness in the 2D inhomogeneous Navier-Stokes equations.

Findings

Leray-Hopf solutions become strong solutions for positive times under certain conditions.

The pressure regularity plays a key role in solution regularity.

A weak-strong uniqueness result is established.

Abstract

We characterize the Leray--Hopf solutions of the 2D inhomogeneous Navier--Stokes system that become strong for positive times. This characterization relies on the strong energy inequality and the regularity properties of the pressure. As an application, we establish a weak-strong uniqueness result and provide a unified framework for several recent advances in the field.