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

TL;DR

This work analyzes suitable weak solutions to the stationary Navier–Stokes system in $\mathbb{R}^5$ and proves that a finite lower limit of the local Reynolds energy $M(r)$ inhibits blow-up into a self-similar profile of degree $-1$. The authors develop a monotonicity formula tied to an energy defect and project solutions onto the self-similar subspace $\mathcal{H}(R)$, coupled with a classification of degree $-1$ Euler self-similar solutions in five dimensions, which yields a zero Bernoulli pressure scenario. They establish pressure bounds, control the cubic energy flux via a projection-based smallness condition, and use a dyadic-iteration argument to obtain bounded energy across scales, culminating in regularity at the origin. The results extend partial regularity insights to five dimensions under weaker scale-invariant smallness hypotheses by leveraging projection theory and Euler-flow classification in conjunction with a monotonicity framework. Overall, the paper provides a robust mechanism to preclude certain self-similar blow-ups and to guarantee regularity for stationary flows in higher dimensions.

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.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Lemma 3.4
• proof
• Theorem 4.1
• proof