Optimality of logarithmic interpolation inequalities and extension criteria to the Navier–Stokes and Euler equations in Vishik spaces

We show the logarithmic interpolation inequality by means of the Vishik space V˙q,σ,θs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{V}}^{s}_{q,\sigma ,\theta }$$\end{document} which is larger than the homogeneous Besov space B˙q,σs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{B}}^{s}_{q,\sigma }$$\end{document}. We emphasize that V˙q,σ,θs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{V}}^{s}_{q,\sigma ,\theta }$$\end{document} may be the largest normed space that satisfies the logarithmic interpolation inequality. As an application of this inequality, we prove that the strong solution to the Navier–Stokes and Euler equations can be extended if the scaling invariant quantity of vorticity in the Vishik space is finite. Namely, the Beirão da Veiga- and Beale–Kato–Majda-type regularity criteria are improved in the terms of the Vishik space.