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

Ibrahim Suleiman

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arXiv · 2610.30096

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

Ibrahim Suleiman

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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 Hölder 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.

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Ibrahim Suleiman. 2026-10-21. A well-posedness theory for the 2D inhomogeneous incompressible Euler equations and a class of global-in-time solutions. https://arxiv.org/abs/2610.30096

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