3 years ago

# Localized Anisotropic Regularity Conditions for the Navier–Stokes Equations

Walter Rusin, Igor Kukavica, Mohammed Ziane

### Abstract

We establish a sufficient regularity condition for local solutions of the Navier–Stokes equations. For a suitable weak solution (up) on a domain D we prove that if $$\partial _3 u$$ belongs to the space $$L_t^{s_0}L_x^{r_0}(D)$$ where $$2/s_0 + 3/r_0 \le 2$$ and $$9/4 \le r_0\le 5/2$$ , then the solution is Hölder continuous in D.

DOI: 10.1007/s00332-017-9382-5

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