Weak solutions for the Navier-Stokes equations with $B^{-1(\ln )}_{\infty \infty }+B_{X_r}^{-1+r,\frac{2}{1-r}}+L^2$B∞∞−1(ln)+BXr−1+r,21−r+L2 initial data

Abstract

One of the major topics in the study of nonlinear partial differential equations of the evolutionary type is to look for as large as possible initial value spaces so that as many as possible solutions of such equations can be obtained. In the book “Recent Developments in the Navier-Stokes Problems,” Lemarié-Rieusset proved that the Navier-Stokes equations have global weak solutions for initial data in the space \documentclass[12pt]{minimal}\begin{document}$B_{\widetilde{X}_r}^{-1+r,\frac{2}{1-r}} (\mathbb {R}^N)+L^2(\mathbb {R}^N)$\end{document}BX̃r−1+r,21−r(RN)+L2(RN) (0 < r < 1), where Xr is the space of functions whose pointwise products with Hr functions belong to L2, \documentclass[12pt]{minimal}\begin{document}$\widetilde{X}_r$\end{document}X̃r denotes the closure of \documentclass[12pt]{minimal}\begin{document}$C_0^\infty (\mathbb {R}^N)$\end{document}C0∞(RN) in Xr, and \documentclass[12pt]{minimal}\begin{document}$B_{\widetilde{X}_r}^{-1+r,\frac{2}{1-r}}(\mathbb {R}^N)$\end{document}BX̃r−1+r,21−r(RN) is the Besov space over \documentclass[12pt]{minimal}\begin{document}$\widetilde{X}_r$\end{document}X̃r. In this paper we partially extend this result of Lemarié-Rieusset to the larger initial value space \documentclass[12pt]{minimal}\begin{document}$B^{-1(\ln )}_{\infty \infty } (\mathbb {R}^N)+B_{\widetilde{X}_r}^{-1+r,\frac{2}{1-r}}(\mathbb {R}^N)+ L^2(\mathbb {R}^N)$\end{document}B∞∞−1(ln)(RN)+BX̃r−1+r,21−r(RN)+L2(RN) (0 < r < 1), where \documentclass[12pt]{minimal}\begin{document}$B^{-1(\ln )}_{\infty \infty }(\mathbb {R}^N)$\end{document}B∞∞−1(ln)(RN) is a logarithmically modified version of the usual Besov space \documentclass[12pt]{minimal}\begin{document}$B^{-1}_{\infty \infty } (\mathbb {R}^N)$\end{document}B∞∞−1(RN).