Locally space-time anisotropic regularity criteria for the Navier-Stokes equations in terms of two vorticity components
This interesting paper is concerning the Navier-Stokes equations in \(\mathbb{R}^3\) with initial conditions. The point is to get some regularity criteria for the global weak Leray solutions, in terms of the vorticity \(\omega\). To this end, the authors extend the previous result [\textit{D. Chae} and \textit{H.-J. Choe}, Electron. J. Differ. Equ. 1999, Paper No. 5, 7 p. (1999; Zbl 0923.35117)]. The new element is to suppose some ``good'' properties of the projection of \(\omega\) into the plane perpendicular to a unit vector \(v\), when the angle between two ``values'' of \(v\) verifies some regularity hypothesis -- see Theorem 3. The key idea is a particular decomposition of \(\omega\), used in the right-hand side of the energy equation verified by the vorticity -- see formula (7). The last part contains new regularity criteria in terms of vorticity direction. Some improvements of the result given in [\textit{Y. Giga} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 189, Article ID 111579, 11 p. (2019; Zbl 1447.35246)] are obtained.
scientific article; zbMATH DE number 7350150