Gaining two derivatives on a singular force in the 2D Navier-Stokes equations

Abstract

It has long been known, for the autonomous 2D Navier-Stokes equations with singular forcing in H−1, that there exists a unique solution, globally bounded in L2, i.e., with a gain of one derivative on the force in terms of regularity. These classical techniques also show us that the solution is almost everywhere in H1. On the other hand, if the forcing term is in L2, it is known that the solution is bounded in H2 globally – a gain of two derivatives. In this paper we prove new maximal regularity results for the 2D Navier-Stokes equations. First, we explore classical techniques to show that if the force is in Hα for α∈(−1,0), then the solution gains two derivatives globally. These methods break down for forces in H−1. In this scenario, we use the Littlewood-Paley decomposition in Fourier space, and an appropriate frequency splitting, to show that for any initial data in H1 there exists a bounded in H1 solutions for a short time interval.