On the Liouville theorem for the Navier-Stokes and the Euler equations in $\Bbb R^N$

TL;DR

This paper discusses the limitations of a method for proving Liouville theorems for Navier-Stokes and Euler equations in br^N, highlighting an issue with a key inequality that prevents the proof from succeeding.

Contribution

The paper critically examines a proposed approach to Liouville theorems, identifying a fundamental flaw in the method due to an incompatible inequality sign.

Findings

The method cannot establish the vanishing result due to the sign issue.

The paper clarifies the constraints on proving Liouville theorems for these equations.

It emphasizes the need for alternative approaches to prove such theorems.

Abstract

This paper has been withdrawn by the author due to the fact that the negative sign of the exponent of $R$ in (2.20) is not allowed by the second inequality of (2.2), and thus the desired vanishing in (2.20) could not be obtained by this method.