Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity

Abstract

We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in L1(R2) for positive times is entirely determined by the trace of the vorticity at t=0, which is a finite measure. When combined with previous existence results by Cottet, by Giga, Miyakawa & Osada, and by Kato, this uniqueness property implies that the Cauchy problem for the vorticity equation in R2 is globally well-posed in the space of finite measures. In particular, this provides an example of a situation where the Navier-Stokes equation is well-posed for arbitrary data in a function space that is large enough to contain the initial data of some self-similar solutions.