On an almost sharp Liouville type theorem for fractional Navier-Stokes equations

TL;DR

This paper explores existence, uniqueness, and regularity of solutions for fractional Navier-Stokes equations with fractional Laplacian, establishing near-sharp conditions for trivial solutions and highlighting open problems in certain regimes.

Contribution

It provides new existence and Liouville theorems for fractional Navier-Stokes equations, including near-sharp conditions for trivial solutions and regularity results for specific fractional powers.

Findings

Weak solutions exist in Sobolev spaces for fractional Navier-Stokes.

Under additional integrability, zero is the unique smooth solution for certain fractional powers.

Regularity gains are shown for 1<α<2, but open problems remain for 0<α≤1.

Abstract

We investigate existence, Liouville type theorems and regularity results for the 3D stationary and incompressible fractional Navier-Stokes equations: in this setting the usual Laplacian is replaced by its fractional power $(-\Delta)^{\frac{\alpha}{2}}$ with $0<\alpha<2$. By applying a fixed point argument, weak solutions can be obtained in the Sobolev space $\dot{H}^{\frac{\alpha}{2}}(\mathbb{R})$ and if we add an extra integrability condition, stated in terms of Lebesgue spaces, then we can prove for some values of $\alpha$ that the zero function is the unique smooth solution. The additional integrability condition is almost sharp for $3/5<\alpha<5/3$. Moreover, in the case $1<\alpha<2$ a gain of regularity is established under some conditions, however the study of regularity in the regime $0<\alpha\leq 1$ seems for the moment to be an open problem.