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

Abstract

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

arXiv comment18 pages, 1 Figure, to appear in Proceedings of the AMS