Optimal bounds for the inviscid limit of Navier–Stokes equations

Abstract

We consider the inviscid limit of incompressible two‐dimensional fluids with initial vorticity in L^∞ and in some Besov space B^η_2,∞ with low regularity index. We obtain a general result of strong convergence in L² which applies to the case of vortex patches with smooth boundaries. The rate of convergence we find is (νt)^3/4 (where ν stands for the viscosity and t, for the time). It improves the (νt)^1/2 rate given by P. Constantin and J. Wu in (Nonlinearity 8 (1995), 735–742). Besides, it is shown to be optimal in the case of circular vortex patches.