Uniform global well-posedness of the Navier–Stokes–Coriolis system in a new critical space

We prove global well-posedness for the Navier–Stokes–Coriolis system (NSC) in a critical space whose definition is based on Fourier transform, namely the Fourier–Besov–Morrey space FN1,μ,∞μ-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {FN}_{1,\mu ,\infty }^{\mu -1}$$\end{document} with 0<μ<3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\mu <3$$\end{document}. The smallness condition on the initial data is uniform with respect to the angular velocity ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}. Our result provides a new class for the uniform global solvability of (NSC) and covers some previous ones. For μ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =0$$\end{document}, (NSC) is ill-posedness in FN1,μ,∞μ-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {FN}_{1,\mu ,\infty }^{\mu -1}$$\end{document} which shows the optimality of the results with respect to the space parameter μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu >0$$\end{document}. The lack of Hausdorff–Young inequality in Morrey spaces suggests that there are no inclusions between FN1,μ,∞μ-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {FN}_{1,\mu ,\infty }^{\mu -1}$$\end{document} and the largest previously known existence classes of Kozono–Yamazaki (Besov–Morrey space) and Koch–Tataru (BMO-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{BMO}^{-1}$$\end{document}) for Navier–Stokes equations (3DNS). So, taking in particular ω=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega =0$$\end{document}, we obtain a critical initial data class that seems to be new for global existence of solutions of (3DNS).