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Navier-Stokes equations: Some questions related to the direction of the vorticity

To Professor Vicentiu Rǎdulescu on the occasion of his 60th birthday

Partially supported by FCT (Portugal) under grant UID/MAT/04561/3013

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  • We consider solutions $u$ to the Navier-Stokes equations in the whole space. We set $\omega = \nabla × u, $ the vorticity of $u$. Our study concerns relations between $\beta -$Hölder continuity assumptions on the direction of the vorticity and induced integrability regularity results, a significant research field starting from a pioneering 1993 paper by P. Constantin and Ch. Fefferman. Nowadays it is know that if $\beta = \frac{1}{2}$ then $\omega ∈ L^{∞}(L^2), $ a 2002 result by L.C. Berselli and the author. This conclusion implies smoothness of solutions. Assume now that one is able to prove that a strictly decreasing perturbation of $\beta $ near $\frac{1}{2}$ induces a strictly decreasing perturbation for $r$ near $2$. Since regularity holds if merely $\omega ∈ L^{∞}(L^r), $ for some $r≥ \frac32, $ the above assumption would imply regularity for values $\beta <\frac{1}{2}.$ The aim of the present note is to go deeper into this study and related open problems. The approach developed below reinforces the conjecture on the particular significance of the value $\beta = \frac{1}{2}.$

    Mathematics Subject Classification: Primary: 35Q30, 76D05; Secondary: 76D03.


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