Ill-posedness for subcritical hyperdissipative Navier-Stokes equations in the largest critical spaces

TL;DR

This paper demonstrates the ill-posedness of subcritical hyperdissipative Navier-Stokes equations in the largest critical spaces by constructing discontinuous solutions from small initial data.

Contribution

It establishes the existence of discontinuous Leray-Hopf solutions in the largest critical space for these equations, highlighting ill-posedness issues.

Findings

Discontinuous solutions exist for small initial data.

Ill-posedness occurs in the largest critical space.

Constructs solutions with fractional Laplacian dissipation.

Abstract

We study the incompressible Navier-Stokes equations with a fractional Laplacian and prove the existence of discontinuous Leray-Hopf solutions in the largest critical space with arbitrarily small initial data.