. A P ] 7 J an 2 01 4 ON THE INVISCID LIMIT OF THE 2 D EULER EQUATIONS WITH VORTICITY ALONG THE ( L α mo ) α SCALE
In [C2], Chemin shows that solutions of the Navier-Stokes equations in R for an incompressible fluid whose initial vorticity lies in L ∩ L∞ converge in the zero-viscosity limit in the L–norm to a solution of the Euler equations, convergence being uniform over any finite time interval. In [Y2], Yudovich assumes an initial vorticity lying in L for all p ≥ p0, and establishes the uniqueness of solutions to the Euler equations for an incompressible fluid in a bounded domain of R , assuming a particular bound on the growth of the L–norm of the initial vorticity as p grows large. We combine these two approaches to establish, in R, 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 p0 = 2. The resulting bounded rate of convergence can be arbitrarily slow as a function of the viscosity ν.