# Liouville type theorems for 3D stationary Navier-Stokes equations in weighted mixed-norm Lebesgue spaces

## TL;DR

This paper establishes Liouville type theorems for 3D stationary Navier-Stokes solutions in mixed-norm Lebesgue spaces, showing under certain conditions that solutions must be zero, even with slow decay rates.

## Contribution

It introduces new Liouville theorems for solutions in weighted mixed-norm Lebesgue spaces, extending previous results and providing novel estimates for Navier-Stokes equations.

## Key findings

- Solutions are identically zero under certain mixed-norm conditions
- Results cover solutions with slow decay rates in different directions
- New mixed-norm and weighted estimates for Navier-Stokes established

## Abstract

This work studies the system of $3D$ stationary Navier-Stokes equations. Several Liouville type theorems are established for solutions in mixed-norm Lebesgue spaces and weighted mixed-norm Lebesgue spaces. In particular, we show that, under some sufficient conditions in mixed-norm Lebesgue spaces, solutions of the stationary Navier-Stokes equations are identically zero. This result covers the important case that solutions may decay to zero with different rates in different spatial directions, and some these rates could be significantly slow. In the un-mixed norm case, the result recovers available results. With some additional geometric assumptions on the supports of solutions, this work also provides several other important Liouville type theorems for solutions in weighted mixed-norm Lebesgue spaces. To prove the results, we establish some new results on mixed-norm and weighted mixed-norm estimates for Navier-Stokes equations. All of these results are new and could be useful in other studies.