Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit

In this paper we prove the uniform-in-time L-p convergence in the inviscid limit of a family omega(nu) of solutions of the 2D Navier-Stokes equations towards a renormalized/Lagrangian solution co of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of omega(nu) to omega in L-p. Finally, we show that solutions of the Euler equations with L-p vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach.

Automated summary

In this paper, we prove the uniform-in-time L-p convergence in the inviscid limit of a family of solutions of the 2D Navier-Stokes equations towards a renormalized/Lagrangian solution of the Euler equations. We also show that a rate for the convergence of these solutions can be obtained in the class of solutions with bounded vorticity. Additionally, we demonstrate that solutions of the Euler equations with L-p vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. These proofs are derived using both a (stochastic) Lagrangian approach and an Eulerian approach.