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

This paper is concerned with the regularity criteria in terms of the middle eigenvalue of the deformation (strain) tensor D(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}(u)$$\end{document} 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 ‖λ2+‖Lp,∞(0,T;Lq,∞(R3))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \lambda _{2}^{+}\Vert _{L^{p,\infty }(0,T; L ^{q,\infty }(\mathbb {R}^{3}))} $$\end{document} with 2/p+3/q=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {2}/{p}+{3}/{q}=2$$\end{document}(3/2<q≤∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$( {3}/{2}<q\le \infty )$$\end{document} is small. This generalizes the corresponding works of Neustupa–Penel and Miller.

(  1 Here, u and denote the unknown velocity vector and the pressure, respectively. The initial datum u 0 is given and satisfies the divergence-free condition. The pioneering work involving the global finite energy weak solutions is due to Leray [12] and Hopf [8]. Precisely, they showed that for any given divergence-free datum u 0 ∈ L 2 ( ), there exists a global weak solution u of the 3D Navier-Stokes equations such that u ∈ L ∞ (0, ∞; L 2 ( ))∩L 2 (0, ∞; W 1,2 ( )) where ⊆ R 3 . However, full regularity of Leary-Hopf weak solutions to the 3D Navier-Stokes system is still a fundamental open question. Starting from Serrin's famous work, regularity criteria of Leray-Hopf weak solutions are extensively studied (see [1][2][3]5,7,[9][10][11]13,[16][17][18][19][20][21][23][24][25][26]28,29] and references therein). The so-called Serrin type regularity criteria is that a weak solution u is regular on Beirao da Veiga found that condition (1.2) can be replaced by the following It is clear that the gradient of the velocity field ∇u can be decomposed into a symmetric part D and an antisymmetric part , that is, D(u) and (u) are usually called the deformation tensor (or rate-of-strain tensor) and the rotation tensor, respectively (See, e.g., [14]). Denote λ 1 , λ 2 , λ 3 by the eigenvalue of the deformation tensor D(u). In a series of works [17][18][19][20][21], Neustupa-Penel obtained regularity criteria via only the middle eigenvalue λ 2 of the deformation tensor below Very recently, a new alternative proof (1.4) was presented by Miller in [16]. Subsequently, Wu [28] generalized (1.4) to the anisotropic Lebesgue spaces in the 3D double-diffusive convection equations. Since the Lorentz spaces L r ,s ( ) (s ≥ r ) are larger than the Lebesgues spaces L r ( ), much effort is devoted to the regularity criteria in terms of velocity, gradient, the pressure, velocity directions in Lorentz spaces (see [3,5,7,[9][10][11]25,26,29] and references therein). Motivated by the references mentioned above, we extend (1.4) to Lorentz spaces. The main result of this paper reads as follows.
Theorem 1.1 Suppose that u is a weak solution to (1.1) with the divergence-free initial datum u 0 (x) ∈ W 1,2 (R 3 ). Then, there exists a positive constant ε 1 such that u(x, t) is a regular solution on (0, T ] provided holds λ + 2 ∈ L p,∞ (0, T ; L q,∞ (R 3 )) and λ + 2 L p,∞ (0,T ;L q,∞ (R 3 )) ≤ ε 1 , with 2/ p + 3/q = 2, 3/2 < q ≤ ∞; Remark 1.1 Theorem 1.1 is an improvement of regularity criteria based on only the middle eigenvalue of the deformation tensor (1.4). It seems that a slightly modified the technique in Theorem 1.1 can be applied to other incompressible fluid equations such as Navier-Stokes equations with fractional dissipation and the double-diffusive convection equations in [28].
As a by-production of Theorem 1.1, by the absolute continuity of norm in Lorentz space L p,r (0, T ) with r < ∞, we have

Notations and auxiliary lemmas
In this section, we give some preliminaries on functional settings and some useful lemmas, which will be used in proof of our main results.
. | | represents the n-dimensional Lebesgue measure of a set ⊂ R n . We will use the summation convention on repeated indices.
We now present some basic facts on Lorentz spaces. For p, q ∈ [1, ∞], we define Moreover, Similarly, one can define Lorentz spaces L p, Now, we list some properties of Lorentz spaces.
To apply the above lemma, we need the following fact obtained in [9].

Lemma 2.2 ([9]
). Assume that the pair ( p, q) satisfies 2 p + 3 q = a with a, q ≥ 1 and p > 0. Then, for every κ ∈ [0, 1] and given b, c 0 ≥ 1, there exist p κ > 0 and min{q, b} ≤ q κ ≤ max{q, b} such that (2.5) To proceed further, we recall the Kronecker symbol δ i j and the three-dimensional Levi-Civita symbol ε i jk , respectively, Then, a useful identity (Levi-Civita and Kronecker Delta identity) holds We denote a i j = 1 2 (∂ j u i − ∂ i u j ) and d i j = 1 2 (∂ i u j + ∂ j u i ). Next, we recall the following result stated in [17]. Since this result plays an important role in the derivation of regularity criteria via only the middle eigenvalue of the deformation tensor, we present its proof here.
Multiplying both side of (2.7) by ε imn , we obtain This together with the aforementioned Levi-Civita and Kronecker Delta identity yields that where the definition of a i j was used.
As a consequence, we infer that This completes the proof of Lemma 2.3.

Proof of Theorem 1.1
This section is devoted to proving Theorem 1.1.

Proof of Theorem 1.1
Multiplying both side of the Navier-Stokes equations (1.1) by u, integrating by parts and using the divergence-free condition div u = 0, we obtain Using the divergence-free condition div u = 0, we write the right-hand side of (3.1) as Interchanging i with k, we have Hence, we see that Arguing in the same manner as the above, we observe that

Substituting (3.3) and (3.4) into (3.2),
We derive from Lemma 2.3 that Collecting the above estimates, we see that As in [17], we invoke the vorticity equation to control the first term in the right hand side of (3.6). Recall that the vorticity equation of the Navier-Stokes system is given by Multiplying Eq. (3.7) by ω, we infer that Notice that, Thus, we obtain Multiplying Eq. (3.9) by − 1 4 and summing with (3.6), it immediately follows that Without loss of generality, we assume λ 1 ≤ λ 2 ≤ λ 3 . The tensor (d i j ) is symmetric so that the corresponding eigenvectors are orthogonal. Due to the fact divu = 0, we deduce that and The quantity λ 1 λ 2 λ 3 = |d i j | is independent of choice of the system of coordinates. Hence, we obtain Using the Hölder inequality (2.2) or interpolation characteristic (2.1), Sobolev embedding (2.4) in Lorentz spaces and the Young inequality, we have (3.14) Inserting (3.14) into (3.13), we get With the help of interpolation characteristic of Lorentz spaces (2.1) or the Hölder inequality (2.2), applying Lemma 2.2 with a = b = 2, c 0 = 4 and (2.3), we see that Since the pair ( p κ , q κ ) also meets 2/ p κ + 3/q κ = 2, we insert (3.16) into (3.15) to obtain d dt ∇u 2 L 2 (R 3 ) dx ≤ C · λ + Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.