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

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.