On the Liouville type theorems for self-similar solutions to the Navier-Stokes equations

TL;DR

This paper establishes Liouville type theorems for self-similar solutions to the Navier-Stokes equations, extending previous results and eliminating certain blow-up scenarios with specific profile conditions.

Contribution

It generalizes existing Liouville theorems for self-similar Navier-Stokes solutions and removes a blow-up scenario under new profile assumptions.

Findings

Proved generalized Liouville theorems for self-similar solutions.

Eliminated asymptotically self-similar blow-up scenarios for certain profiles.

Extended previous results by Nečas-Růžička-Sverák and Tsai.

Abstract

We prove Liouville type theorems for the self-similar solutions to the Navier-Stokes equations. One of our results generalizes the previous ones by Ne\v{c}as-R\.{u}\v{z}i\v{c}ka-\v{S}verak and Tsai. Using the Liouville type theorem we also remove a scenario of asymtotically self-similar blow-up for the Navier-Stokes equations with the profile belonging to $L^{p, \infty} (\Bbb R^3)$ with $p> \frac{3}{2}$.