Exponent bounds for a convolution inequality in Euclidean space with applications to the Navier-Stokes equations

TL;DR

This paper establishes bounds on convolution inequalities relevant to Navier-Stokes solutions, showing nonexistence of certain positive solutions for specific parameters and linking probabilistic solution spaces to classical function spaces.

Contribution

It proves nonexistence of positive solutions for a key convolution inequality when a parameter exceeds a threshold, and connects probabilistic solution spaces to classical function spaces for Navier-Stokes.

Findings

Nonexistence of positive solutions for $ heta \,\geq\, n/2$

Embedding of probabilistic solution spaces into $BMO^{-1}$ and $BMO_T^{-1}$

Application to Navier-Stokes regularity theory

Abstract

The convolution inequality $h*h(\xi) \leq B |\xi|^\theta h(\xi)$ defined on $\Rn$ arises from a probabilistic representation of solutions of the $n$-dimensional Navier-Stokes equations, $n \geq 2$. Using a chaining argument, we establish the nonexistence of strictly positive fully supported solutions of this inequality if $\theta \geq n/2$, in all dimensions $n \geq 1$. We use this result to describe a chain of continuous embeddings from spaces associated with probabilistic solutions to the spaces $BMO^{-1}$ and $BMO_T^{-1}$ associated with the Koch-Tataru solutions of the Navier-Stokes equations.