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

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

2012-01-31

arxiv.org/abs/1201.6433v1

Mathematics

Classical Analysis and ODEs

18 pages, 1 Figure, to appear in Proceedings of the AMS

Scientific paper

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.

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