Inhomogenous Navier--Stokes equations with unbounded density

TL;DR

This paper proves the global existence and uniqueness of solutions to the incompressible Navier-Stokes equations with unbounded density, extending the mathematical understanding of such flows in fluid dynamics.

Contribution

It introduces a novel proof demonstrating existence and uniqueness of solutions with unbounded density, utilizing Desjardins' inequality and a blow-up criterion.

Findings

Established global existence of solutions with unbounded density

Derived time-weighted estimates for velocity regularity

Ensured equivalence of Eulerian and Lagrangian formulations

Abstract

In the current state of the art regarding the Navier--Stokes equations, the existence of unique solutions for incompressible flows in two spatial dimensions is already well-established. Recently, these results have been extended to models with variable density, maintaining positive outcomes for merely bounded densities, even in cases with large vacuum regions. However, the study of incompressible Navier-Stokes equations with unbounded densities remains incomplete. Addressing this gap is the focus of the present paper. Our main result demonstrates the global existence of a unique solution for flows initiated by unbounded density, whose regularity/integrability is characterized within a specific subset of the Yudovich class of unbounded functions. The core of our proof lies in the application of Desjardins' inequality, combined with a blow-up criterion for ordinary differential equations. Furthermore, we derive time-weighted estimates that guarantee the existence of a $C^1$ velocity field and ensure the equivalence of Eulerian and Lagrangian formulations of the equations. Finally, by leveraging results from \cite{DanMu}, we conclude the uniqueness of the solution.