On regularity criteria for the Navier-Stokes equations based on one directional derivative of the velocity or one diagonal entry of the velocity gradient
The regularity of the global weak solution \(\mathbf{u}\) for Navier-Stokes equations in whole \(\mathbb{R}^3\) it is studied. Previous regularity criteria exist, by using some ``good'' properties of the derivatives of \(\mathbf{u}\). In this paper, the authors improve some results given in [\textit{Z. Zhang} et al., Appl. Math., Praha 64, No. 3, 301--308 (2019; Zbl 07088742)]. In formula (16) are given the known results concerning the regularity, when \(\partial_3 \mathbf{u} \in L^p (0, T ; L^q (\mathbb{R}^3 ))\), as function of \((2/p+3/q)\) for \(q\) in some ranges. The main point is an improvement of (16), for \(q \in (3,4)\). Moreover, a more useful regularity criterion in terms of \(\partial_3 u_3\) is also obtained. A very interesting comparison with previous regularity criteria in this field is given in Figures 1,2. The basic tools are the strong energy inequality, the Riesz transform, interpolation inequality, multiplicative Sobolev inequalities, the Gagliardo-Nirenberg inequality. An particular Hölder inequality is used -- see the formula (34). A large volume of calculations is performed in a very elegant way.
scientific article; zbMATH DE number 7349791
scientific article