Geometric blow-up criteria for the non-homogeneous incompressible Euler equations in 2-D

TL;DR

This paper introduces a new geometric criterion for predicting blow-up in 2-D incompressible Euler equations with variable density, focusing on the gradient of velocity along a specific density-dependent direction.

Contribution

It establishes a novel geometric blow-up criterion based on the control of the velocity gradient along a density-dependent direction, applicable to both subcritical and critical regularity frameworks.

Findings

Blow-up can be predicted by controlling ∇u along a specific density-dependent direction.

The approach recovers global well-posedness for constant density cases.

The method applies to both subcritical and critical regularity settings.

Abstract

This paper concerns the study of the incompressible Euler equations with variable density, in the case of space dimension $d=2$. Contrarily to their homogeneous (constant density) counterpart, those equations are not known to be well-posed globally in time. A classical blow-up/continuation criterion for smooth solutions relies on the control of the Lipschitz norm of the velocity field $u$. Here we show that, for establishing blow-up or continuation of solutions, it is enough to determine a control of $\nabla u$ only along the direction $X=\nabla^\perp\rho$, where $\rho$ represents the density of the fluid. Our results deal with both the subcritical regularity and critical regularity frameworks. They rely on a novel approach to study regularity of solutions for the density-dependent incompressible Euler equations. Besides, they allow to recover the global well-posedness for $\rho\equiv {\rm cst}$ as a particular case.