Yudovich theory under geometric regularity for density-dependent incompressible fluids
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912456499003392 |
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| author | Fanelli, Francesco |
| author_facet | Fanelli, Francesco |
| contents | 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 $X:=\nabla^\perpρ$, where $ρ$ is the fluid density, low regularity solutions \textsl{à 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23365 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Yudovich theory under geometric regularity for density-dependent incompressible fluids Fanelli, Francesco Analysis of PDEs 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 $X:=\nabla^\perpρ$, where $ρ$ is the fluid density, low regularity solutions \textsl{à 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. |
| title | Yudovich theory under geometric regularity for density-dependent incompressible fluids |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.23365 |