This paper is concerned with the regularity criteria in terms of the middle eigenvalue of the deformation (strain) tensor \(\mathcal {D}(u)\) to the 3D Navier–Stokes equations in Lorentz spaces. It is shown that a Leray–Hopf weak solution is regular on (0, T] provided that the norm \(\Vert \lambda _{2}^{+}\Vert _{L^{p,\infty }(0,T; L ^{q,\infty }(\mathbb {R}^{3}))} \) with \( {2}/{p}+{3}/{q}=2\) \(( {3}/{2}<q\le \infty )\) is small. This generalizes the corresponding works of Neustupa–Penel and Miller.

中文翻译：

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Lorentz空间中Navier-Stokes方程的变形张量正则性

本文关注的变形准则是关于Lorentz空间中3D Navier–Stokes方程的变形（应变）张量\（\ mathcal {D}（u）\）的中间特征值。结果表明，一个的Leray-的Hopf弱溶液是常规上（0，  Ť提供]，该规范\（\ Vert的\拉姆达_ {2} ^ {+} \ Vert的_ {L ^ {P，\ infty}（0 ，T; L ^ {q，\ infty}（\ mathbb {R} ^ {3}）}} \）与\（{2} / {p} + {3} / {q} = 2 \） \（ （{3} / {2} <q \ le \ infty} \）很小，这概括了Neustupa–Penel和Miller的相应著作。