Exponent bounds for a convolution inequality in Euclidean space with applications to the Navier-Stokes equations
The convolution inequality $h*h(ξ) \leq B |ξ|^θh(ξ)$ 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 $θ\geq n/2$, 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.
math.CA↗