Euler equations with non-homogeneous Navier slip boundary condition

TL;DR

This paper studies the Euler equations with non-homogeneous Navier slip boundary conditions in 2D, establishing solvability for solutions with bounded vorticity by analyzing the vanishing viscosity limit from Navier-Stokes equations.

Contribution

It introduces a framework for solving Euler equations with non-homogeneous slip boundary conditions and proves solvability in the class of solutions with bounded vorticity.

Findings

Established solvability of Euler equations with non-homogeneous boundary conditions.

Connected the problem to the vanishing viscosity limit of Navier-Stokes equations.

Provided conditions for solutions with bounded vorticity.

Abstract

We consider the flow of an { ideal} fluid in a 2D-bounded domain, admitting flows through the boundary of this domain. The flow is described by Euler equations with \textit{non-homogeneous } Navier slip boundary conditions. These conditions can be written in the form $\mathbf{v}\cdot \mathsf{n}=a,$ $2D(\mathbf{v})\mathsf{n}\cdot \mathsf{s}+\alpha \mathbf{v}% \cdot {\mathsf{s}}=b,$ where the tensor $D(\mathbf{v})$ is the rate-of-strain of the fluid's velocity $\mathbf{v}$ and\ $\mathsf{(n},% \mathsf{s)}$ is the pair formed by the normal and tangent vectors to the boundary. We establish the solvability of this problem in the class of solutions with $L_{p}-$bounded \ vorticity, $p\in (2,\infty ].$ To prove the solvability we realize the passage to the limit in Navier-Stokes equations with vanishing viscosity.