Propagation of logarithmic regularity and inviscid limit for the 2D Euler equations

Abstract

The aim of this note is to study the Cauchy problem for the 2D Euler equations under very low regularity assumptions on the initial datum. We prove propagation of regularity of logarithmic order in the class of weak solutions with $L^p$ initial vorticity, provided that $p\ge 4$. We also study the inviscid limit from the 2D Navier-Stokes equations for vorticity with logarithmic regularity in the Yudovich class, showing a rate of convergence of order $|\log \nu {|}^{-\alpha /2}$ with $\alpha >0$.

Submitted to "Mathematics in Engineering" for the special issue "Math aspects of classical and quantum fluid dynamics" dedicated to Pierangelo Marcati for his 70th birthday

References in corpus (7)

• Strong convergence of the vorticity for the 2D Euler Equations in the inviscid limit
• Advection diffusion equations with Sobolev velocity field
• On the advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity
• Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I
• A note on the vanishing viscosity limit in the Yudovich class
• An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces
• Propagation of regularity for transport equations. A Littlewood-Paley approach