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

Orum, Chris; Ossiander, Mina

doi:10.1090/S0002-9939-2013-11662-X

Exponent bounds for a convolution inequality in Euclidean space with applications to the Navier-Stokes equations
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by Chris Orum and Mina Ossiander PDF

Proc. Amer. Math. Soc. 141 (2013), 3883-3897 Request permission

Abstract:

The convolution inequality $h*h(\xi ) \leq B |\xi |^\theta h(\xi )$ defined on $\mathbf {R}^n$ arises from a probabilistic representation of solutions of the $n$-dimensional Navier-Stokes equations, $n \geq 2$. Using a chaining argument, we establish in all dimensions $n \geq 1$ the nonexistence of strictly positive fully supported solutions of this inequality for $\theta \geq n/2.$ 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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