A well-posedness theory for the 2D inhomogeneous incompressible Euler equations and a class of global-in-time solutions

Abstract

We consider the Cauchy problem for the 2D inhomogeneous incompressible Euler equations. We show that the Cauchy problem is locally well-posed in a class of $m$-fold symmetric functions at critical regularity and allowing the velocity field to grow linearly at spatial infinity. This was a problem posed by P.L. Lions in the homogeneous, not necessarily symmetric case. A key ingredient is a variable-coefficient elliptic theory developed using quasiconformal mappings and Schauder estimates in certain scale-invariant H\"{o}lder spaces. Local well-posedness allows us to derive the one-dimensional system on $\mathbb{S}^1$ satisfied by solutions with $0$-homogeneous vorticity and density. For this system, we can obtain global well-posedness for a class of initial data away from equilibrium. We are unaware of any previous nonperturbative class of time-dependent solutions to the two-dimensional inhomogeneous Euler equations for which global well-posedness has been established. Finally, for some signed global solutions, we show a dichotomy between density homogenization and averaged relaxation in infinite time.