The vanishing viscosity limit for incompressible fluids in two dimensions

Abstract

The Navier-Stokes equations describe the motion of an incompressible fluid of constant density and constant positive viscosity. With zero viscosity, the Navier-Stokes equations become the Euler equations. A question of longstanding interest to mathematicians and physicists is whether, as the viscosity goes to zero, a solution to the Navier-Stokes equations converges, in an appropriate sense, to a solution to the Euler equations: the so-called “vanishing viscosity” or “inviscid” limit. We investigate this question in three settings: in the whole plane, in a bounded domain in the plane, and for radially symmetric solutions in the whole plane. Working in the whole plane and in a bounded domain, we assume a particular bound on the growth of the L p -norms of the initial vorticity (curl of the velocity) with p, and obtain a bound on the convergence rate in the vanishing viscosity limit. This is the same class of initial vorticities considered by Yudovich and shown to imply uniqueness of the solution to the Euler equations in a bounded domain lying in Euclidean space of dimension 2 or greater. For radially symmetric initial vorticities we obtain a more precise bound on the convergence rate as a function of the smoothness of its initial vorticity as measured by its norm in a Sobolev space or in certain Besov spaces. We also consider the questions of existence, uniqueness, and regularity of solutions to the Navier-Stokes and Euler equations, as necessary, to make sense of the vanishing viscosity limit. In particular, we investigate properties of the flow for solutions to the Euler equations in the whole plane. We construct a specific example of an initial vorticity for which there exists a unique solution to the Euler equations whose associated flow lies in no H¨older space of positive exponent for any positive time. This example is an adaptation of a bounded-vorticity example of Bahouri and Chemin’s, which they show has a flow lying in no H¨older space of exponent greater than an exponentially decreasing function of time.