Strong convergence of the vorticity for the 2D Euler Equations in the inviscid limit

Abstract

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 $\omega$ 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.

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Cited by in corpus (8)

• On the advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity
• Statistical solutions of the incompressible Euler equations
• Vanishing viscosity in mean-field optimal control
• Probabilistic Descriptions of Fluid Flow: A Survey
• Energy balance for forced two-dimensional incompressible ideal fluid flow
• Propagation of logarithmic regularity and inviscid limit for the 2D Euler equations
• Energy conservation in the limit of filtered solutions for the 2D Euler equations
• On Maximum Enstrophy Dissipation in 2D Navier-Stokes Flows in the Limit of Vanishing Viscosity