The inviscid limit for two-dimensional incompressible fluids with unbounded vorticity

TL;DR

This paper proves the convergence of solutions of the 2D Navier-Stokes equations to Euler solutions in the inviscid limit for unbounded vorticity, extending previous results to broader initial conditions.

Contribution

It combines Chemin's and Yudovich's methods to establish zero-viscosity convergence and uniqueness for solutions with unbounded vorticity in 2D fluids.

Findings

Established convergence of Navier-Stokes solutions to Euler solutions for unbounded vorticity.

Proved uniqueness of Euler solutions under Yudovich's vorticity growth conditions.

Demonstrated that convergence rate can be arbitrarily slow as viscosity approaches zero.

Abstract

Chemin has shown that solutions of the Navier-Stokes equations in the plane for an incompressible fluid whose initial vorticity is bounded and lies in L^2 converge in the zero-viscosity limit in the L^2-norm to a solution of the Euler equations, convergence being uniform over any finite time interval. Yudovich, assuming an initial vorticity lying in L^p for all p >= q for some q, established the uniqueness of solutions to the Euler equations for an incompressible fluid in a bounded domain of n-space assuming a particular bound on the growth of the L^p-norm of the initial vorticity as p grows large. We combine these two approaches to establish, in the plane, the uniqueness of solutions to the Euler equations and the same zero-viscosity convergence as Chemin, but under Yudovich's assumptions on the vorticity with q = 2. The resulting bounded rate of convergence can be arbitrarily slow as a function of the viscosity.