Abstract.
Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing \(z_{0}=(x_{0}, t_{0})\), and let \(Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0})\) be a parabolic cylinder in the domain. We show that if either \(\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})\) with \(\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})\) with \(\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2\), where Lγ, αx,t denotes the Serrin type of class, then z0 is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.
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This research was supported partially by KRF Grant (MOEHRD, Basic Research Promotion Fund).
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Chae, D. On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations. J. Math. Fluid Mech. 12, 171–180 (2010). https://doi.org/10.1007/s00021-008-0280-3
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DOI: https://doi.org/10.1007/s00021-008-0280-3