An anisotropic regularity condition for the 3D incompressible Navier-Stokes equations for the entire exponent range

Abstract

We show that a suitable weak solution to the incompressible Navier-Stokes equations on ${\mathbb{R}}^3\times (-1,1)$ is regular on ${\mathbb{R}}^3\times (0,1]$ if ${\partial }_{3}u$ belongs to $M^{2p/(2p-3),\alpha }((-1,0);L^p({\mathbb{R}}^3))$ for any $\alpha >1$ and $p\in (3/2,\infty )$, which is a logarithmic-type variation of a Morrey space in time. For each $\alpha >1$ this space is, up to a logarithm, critical with respect to the scaling of the equations, and contains all spaces $L^q((-1,0);L^p({\mathbb{R}}^3))$ that are subcritical, that is for which $2/q+3/p<2$.

arXiv comment10 pages

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Cited by in corpus (1)

• Quantitative transfer of regularity of the incompressible Navier-Stokes equations from ${\mathbb{R}}^3$ to the case of a bounded domain