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 Hölder spaces. Local well-posedness allows us to derive the one-dimensional system on 𝕊1 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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BibTeX
@misc{suleiman2026wellposedness,
title = {{A well-posedness theory for the 2D inhomogeneous incompressible Euler equations and a class of global-in-time solutions}},
author = {Ibrahim Suleiman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30096},
eprint = {2610.30096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30096},
}
RIS
TY - UNPB
TI - A well-posedness theory for the 2D inhomogeneous incompressible Euler equations and a class of global-in-time solutions
AU - Ibrahim Suleiman
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30096
UR - https://arxiv.org/abs/2610.30096
AB - 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 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.
ER -