Singular vorticity solutions of the incompressible Euler equation via inviscid limits

TL;DR

This paper constructs singular vorticity solutions for the 3D incompressible Euler equation by analyzing inviscid limits of modified Navier-Stokes equations with damping, revealing solutions that become singular at boundary points.

Contribution

It introduces a novel method to obtain singular solutions of the Euler equation through inviscid limits of damped Navier-Stokes type equations, satisfying the BKM criterion.

Findings

Constructed solutions satisfy the BKM criterion.

Solutions become singular at boundary points.

Method links viscous and inviscid flow behaviors.

Abstract

Singular vorticty solutions of the incompressible 3D-Euler equation are constructed which satisfy the BKM criterion (cf. [2]). The construction is done by inviscid limits of vorticity solutions of transformed incompressible Navier Stokes type equations with a damping potential term, where the latter equations admit a global regular solution for positive viscosity. The inviscid limit vorticity solution of the incompressible Euler vorticity equation becomes singular at a point of the boundary of a finite domain.