# The Euler equations in planar nonsmooth convex domains

@article{Bardos2013TheEE, title={The Euler equations in planar nonsmooth convex domains}, author={Claude W. Bardos and Francesco Di Plinio and Roger Temam}, journal={Journal of Mathematical Analysis and Applications}, year={2013}, volume={407}, pages={69-89} }

Abstract As a model problem for the barotropic mode of the primitive equations of the oceans and atmosphere, we consider the Euler system on a bounded convex planar domain Ω , endowed with non-penetrating boundary conditions. For 4 3 ≤ p ≤ 2 , and initial and forcing data with L p ( Ω ) vorticity we show the existence of a weak solution, enriching and extending the results of Taylor (2000) [32] . In the physical case of a rectangular domain Ω = [ 0 , L 1 ] × [ 0 , L 2 ] , a similar result holds…

## 18 Citations

Uniqueness for the 2-D Euler equations on domains with corners

- Mathematics
- 2013

For a large class of non smooth bounded domains, existence of a global weak solution of the 2D Euler equations, with bounded vorticity, was established by G\'erard-Varet and Lacave. In the case of…

Uniqueness for Two-Dimensional Incompressible Ideal Flow on Singular Domains

- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2015

We prove uniqueness of the weak solution of the Euler equations for compactly supported, single signed, and bounded initial vorticity in simply connected planar domains with corners forming angles…

Euler Equations on General Planar Domains

- MathematicsAnnals of PDE
- 2021

We obtain a general sufficient condition on the geometry of possibly singular planar domains that guarantees global uniqueness for any weak solution to the Euler equations on them whose vorticity is…

MODELING THE LID DRIVEN FLOW : THEORY AND COMPUTATION

- 2017

Abstract. Motivated by the study of the corner singularities in the so-called cavity flow, we establish in the first part of this article, the existence and uniqueness of solutions in L(Ω) for the…

The Euler Equations in Planar Domains with Corners

- MathematicsArchive for Rational Mechanics and Analysis
- 2019

When the velocity field is not a priori known to be globally almost Lipschitz, global uniqueness of solutions to the two-dimensional Euler equations has been established only in some special cases,…

Uniqueness of the 2D Euler equation on a corner domain with non-constant vorticity around the corner

- Physics, Mathematics
- 2020

We consider the 2D incompressible Euler equation on a corner domain $\Omega$ with angle $\nu\pi$ with $\frac{1}{2}<\nu<1$. We prove that if the initial vorticity $\omega_0 \in L^{1}(\Omega)\cap…

Very weak solutions of the Stokes problem in a convex polygon

- Mathematics
- 2015

Motivated by the study of the corner singularities in the so-called cavity flow, we establish in this article, the existence and uniqueness of solutions in $L^2(\Omega)^2$ for the Stokes problem in a…

Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion

- MathematicsPhysica D: Nonlinear Phenomena
- 2018

Abstract We study the global well-posedness and stability/instability of perturbations near a special type of hydrostatic equilibrium associated with the 2D Boussinesq equations without buoyancy…

The Two Dimensional Euler Equations on Singular Exterior Domains

- Mathematics
- 2015

This paper is a follow-up of Gérard-Varet and Lacave (Arch Ration Mech Anal 209(1):131–170, 2013), on the existence of global weak solutions to the two dimensional Euler equations in singular…

The 2D Euler–Boussinesq Equations in Planar Polygonal Domains with Yudovich’s Type Data

- Mathematics
- 2014

We address the well-posedness of the 2D (Euler)–Boussinesq equations with zero viscosity and positive diffusivity in the polygonal-like domains with Yudovich’s type data, which gives a positive…

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