Yudovich theory under geometric regularity for density-dependent incompressible fluids

TL;DR

This paper extends Yudovich's low-regularity solution theory to density-dependent incompressible Euler equations in 2D, using geometric control of the velocity derivative along a specific vector field.

Contribution

It introduces a new approach to construct and prove uniqueness of low-regularity solutions for density-dependent fluids based on geometric quantities.

Findings

Existence of Yudovich-type solutions under geometric control.

Convergence of smooth approximations to these solutions.

Enhanced uniqueness results with less initial regularity.

Abstract

This paper focuses on the study of the density-dependent incompressible Euler equations in space dimension $d=2$, for low regularity (\textsl{i.e.} non-Lipschitz) initial data satisfying assumptions in spirit of the celebrated Yudovich theory for the classical homogeneous Euler equations. We show that, under an \textsl{a priori} control of a non-linear geometric quantity, namely the directional derivative $\partial_Xu$ of the fluid velocity $u$ along the vector field $X:=\nabla^\perp\rho$, where $\rho$ is the fluid density, low regularity solutions \textsl{\`a la Yudovich} can be constructed also in the non-homogeneous setting. More precisely, we prove the following facts: (i) \emph{stability}: given a sequence of smooth approximate solutions enjoying a uniform control on the above mentioned geometric quantity, then (up to an extraction) that sequence converges to a Yudovich-type solution of the density-dependent incompressible Euler system; \\ (ii) \emph{uniqueness}: there exists at most one Yudovich-type solution of the density-dependent incompressible Euler equations such that $\partial_Xu$ remains finite; besides, this statement improves previous uniqueness results for regular solutions, inasmuch as it requires less smoothness on the initial data.