Global regularity for generalized anisotropic Navier–Stokes equations

Abstract

We consider the global Cauchy problem for the three-dimensional generalized anisotropic Navier–Stokes system with the viscosity term \documentclass[12pt]{minimal}\begin{document}$\Delta _h u-M^2u$\end{document}Δhu−M2u, where M is a Fourier multiplier whose symbol \documentclass[12pt]{minimal}\begin{document}$m:\mathbb {R}\rightarrow \mathbb {R}^+$\end{document}m:R→R+ is non-negative. The case \documentclass[12pt]{minimal}\begin{document}$m(\xi _3)=|\xi _3|$\end{document}m(ξ3)=|ξ3| is essentially the Navier–Stokes system, while the case m = 0 is the anisotropic Navier–Stokes system. We obtain the global regularity in the case that \documentclass[12pt]{minimal}\begin{document}$m(\xi _3)\ge {|\xi _3|^{\frac{3}{2}}}/{g(|\xi _3|)}$\end{document}m(ξ3)≥|ξ3|32/g(|ξ3|) for all sufficiently large \documentclass[12pt]{minimal}\begin{document}$|\xi _3|$\end{document}|ξ3| where \documentclass[12pt]{minimal}\begin{document}$g:=\mathbb {R}^+\rightarrow \mathbb {R}^+$\end{document}g:=R+→R+ is a nondecreasing function such that \documentclass[12pt]{minimal}\begin{document}$\int ^\infty _1\frac{ds}{ g^2(s) s\ln s}=+\infty$\end{document}∫1∞dsg2(s)slns=+∞. Then, using the method of descent, we can obtain the classical global regularity result of the two-dimensional Navier–Stokes system.