A Monotonicity formula for almost self-similar suitable weak solutions to the stationary Navier-Stokes equations in $\mathbb R^5$

TL;DR

This paper establishes a monotonicity formula for suitable weak solutions to the stationary Navier-Stokes equations in five dimensions, showing they cannot mimic self-similar solutions of degree -1 under finite Reynolds number conditions.

Contribution

It introduces a novel monotonicity formula approach and classifies homogeneous solutions to the Euler equations in $\

Findings

Suitable weak solutions cannot behave like degree -1 self-similar solutions with finite Reynolds number.

Develops a new method combining monotonicity formulas, classification of Euler solutions, and projection theorems.

Provides insights into the structure and behavior of solutions in higher-dimensional Navier-Stokes problems.

Abstract

In this paper we show that a suitable weak solution to the stationary Navier-Stokes system in $\mathbb R^5$, cannot behave like a self-similar function of degree negative one if the lower limit of the local Reynolds number is finite. To prove the result we develop a method that uses a monotonicity formula approach, classification of homogenous solutions to the incompressible Euler equations in $\mathbb R^5$, and a projection theorem.