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

TL;DR

This paper investigates the regularity of solutions to the 2D Navier-Stokes equations with singular forcing, demonstrating local gain of two derivatives using Fourier analysis techniques, despite the breakdown of classical methods.

Contribution

It introduces a Fourier-based approach to show local two-derivative gain for solutions with singular forcing, extending classical results to less regular forces.

Findings

Solutions gain two derivatives locally in time with singular forcing

Classical techniques fail for singular forces, new Fourier methods succeed

Interval of regularity depends only on initial data size

Abstract

It has long been known, for the autonomous 2D Navier-Stokes equations with singular forcing, that there exist unique solutions which gain one derivative, globally. On the other hand, if the forcing term smooth enough, it is known that the solution gains two derivatives globally. In this paper, we explore classical techniques to show that if the force is sufficiently smooth, then the solution gains two derivatives globally. These methods break down when the force becomes singular. In this scenario, we use a Littlewood-Paley decomposition in Fourier space to show that a solutions gain two derivatives locally in time and the interval of time depends only on the size of the initial data.