A New Prodi–Serrin Type Regularity Criterion in Velocity Directions

  • Benjamin Pineau
  • Xinwei Yu


In this article we generalize (Vasseur in Appl Math 54(1):47–52, 2009) to Lorentz spaces. More specifically, we prove the following. Let u be a Leray–Hopf solution to the Navier–Stokes equation with viscosity \(\nu \) and initial value \(u_0 \in L^2 ({\mathbb {R}}^3)\). Then there is \(c_0 > 0\) such that u is smooth beyond \(T > 0\) if
$$\begin{aligned} \left\| {{\mathrm{div}}} \left( \frac{u}{| u |} \right) \right\| _{L^{p, \infty } (0, T ; L^{q, \infty } ({\mathbb {R}}^3))} < c_0 \nu ^{1 - \frac{1}{p}} \Vert u_0 \Vert _{L^2}^{- 1} \end{aligned}$$
with \(\frac{2}{p} + \frac{3}{q} \leqslant \frac{1}{2}\), \(q > 6\). We also show that u remains smooth beyond \(T > 0\) if
$$\begin{aligned} \left\| {\mathrm{div}} \left( \frac{u}{| u |} \right) \right\| _{L^{p, r} (0, T ; L^{q, \infty } ({\mathbb {R}}^3))} < \infty \end{aligned}$$
with \(\frac{2}{p} + \frac{3}{q} \leqslant \frac{1}{2}, q > 6\) and \({1 \leqslant r < \infty }\).


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Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

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