On the Deformation Tensor Regularity for the Navier–Stokes Equations in Lorentz Spaces

Abstract

This paper is concerned with the regularity criteria in terms of the middle eigenvalue of the deformation (strain) tensor $$\mathcal {D}(u)$$ 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 $$\Vert \lambda _{2}^{+}\Vert _{L^{p,\infty }(0,T; L ^{q,\infty }(\mathbb {R}^{3}))} $$ ‖ λ 2 + ‖ L p , ∞ ( 0 , T ; L q , ∞ ( R 3 ) ) with $$ {2}/{p}+{3}/{q}=2$$ 2 / p + 3 / q = 2 $$( {3}/{2}<q\le \infty )$$ ( 3 / 2 < q ≤ ∞ ) is small. This generalizes the corresponding works of Neustupa–Penel and Miller.