Holder Continuity of Solutions of 2D Navier-Stokes Equations with Singular Forcing

TL;DR

This paper investigates the regularity and continuity properties of solutions to the 2D Navier-Stokes equations under singular forcing conditions, relevant for complex fluid models involving coupled nonlinear equations.

Contribution

It establishes holder continuity of solutions when the Navier-Stokes equations are driven by singular forces, extending regularity results to more realistic complex fluid scenarios.

Findings

Solutions exhibit holder continuity despite singular forcing

The analysis applies to Navier-Stokes coupled with nonlinear Fokker-Planck equations

Provides a framework for understanding complex fluids with singular stress contributions

Abstract

We discuss the regularity of solutions of 2D incompressible Navier-Stokes equations forced by singular forces. The problem is motivated by the study of complex fluids modeled by the Navier-Stokes equations coupled to a nonlinear Fokker-Planck equation describing microscopic corpora embedded in the fluid. This leads naturally to bounded added stress and hence to $W^{-1,\infty}$ forcing of the Navier-Stokes equations.