A Note on the Regularity Criterion of Weak Solutions of Navier-Stokes Equations in Lorentz Space

and Applied Analysis 3 satisfy the critical growth condition ũ ∈ LpLq, for 2 p 3 q 1, 3 < q ≤ ∞. 1.9 It should be mentioned that the weak solution remains regular if the single velocity component satisfies the higher subcritical growth conditions see Zhou 9 , Penel and Pokorný 10 , Kukavica and Ziane 11 , and Cao and Titi 12 . One may also refer to some interesting regularity criteria 13–15 for weak solutions of micropolar fluid flows. It seems difficult to show regularity of weak solutions by imposing Serrin’s growth condition on only one component of velocity field for both Navier-Stokes equations and micropolar fluid flows. However, whether or not the result 1.9 can be improved to the critical weak Lp spaces is an interesting and challenging problem, that is to say, when the weak critical growth condition is imposed to only two velocity components. The main difficulty lies in the lack of a priori estimates on two-velocity components ũ due to the special structure of the nonlinear convection term in monument equations. The aim of the present paper is to improve the two-component regularity criterion 1.9 from Lebesgue space to the critical Lorentz space see the definitions in Section 2 which satisfies the scaling invariance property. Before stating the main results, we firstly recall the definition of the Leray weak solutions. Definition 1.1 Temam, 16 . Let u0 ∈ L2 R3 and ∇ ·u0 0. A vector field u x, t is termed as a Leray weak solution of 1.1 on 0, T if u satisfies the following properties: i u ∈ L∞ 0, T ;L2 R3 ∩ L2 0, T ;H1 R3 ; ii ∂tu u · ∇ u ∇π Δu in the distribution space D′ 0, T × R3 ; iii ∇ · u 0 in the distribution space D′ 0, T × R3 ; iv u satisfies the energy inequality ‖u t ‖L2 2 ∫ t 0 ∫ R3 |∇u x, s | dxds ≤ ‖u0‖L2 , for 0 ≤ t ≤ T. 1.10 The main results now read as follows. Theorem 1.2. Suppose T > 0, u0 ∈ H1 R3 and ∇ · u0 0 in the sense of distributions. Assume that u is a Leray weak solution of the Navier-Stokes equations 1.1 in 0, T . If the horizontal velocity denoted by ũ u1, u2, 0 satisfies the following growth condition: ∫T 0 ‖ũ t ‖qLp,∞dt < ∞, for 2 q 3 p 1, 3 < p < ∞, 1.11 then u is a regular solution on 0, T . Remark 1.3. It is easy to verify that the spaces 1.11 satisfy the degree −1 growth conditions due to the scaling invariance property. Moreover, since the embedding relation Lp ↪→ Lp,∞, Theorem 1.2 is an important improvement of 1.9 . 4 Abstract and Applied Analysis Remark 1.4. Unlike the previous investigations via two components of vorticity see 17, 18 in weak space, of which the approaches are mainly based on the vorticity equations and seem not available in our case here due to the special structure of convection term, the present examination is directly based on the momentum equations. In order to make use of the structure of the nonlinear convection term u·∇ u, we study every component of u·∇ u,Δu and estimate them one by one with the aid of the identities ∇ · u 0. 2. Preliminaries and A Priori Estimates To start with, let us introduce the definitions of some functional spaces. Lp R3 , W R3 with k ∈ R, 1 ≤ p ≤ ∞ are usual Lebesgue space and Sobolev space. To define the Lorenz space Lp,q R3 with 1 ≤ p, q ≤ ∞, we consider a measurable function f and define for t ≥ 0 the Lebesgue measure m ( f, t ) : m { x ∈ R3 : ∣∣f x ∣∣ > t } , 2.1 of the set {x ∈ R3 : |f x | > t}. Then f ∈ Lp,q R3 if and only if