Maximal Lorentz regularity for the Keller-Segel system of parabolic-elliptic type
The minimal parabolic-elliptic Keller-Segel system is studied in the whole space \(\mathbb R^n\), \(n\ge 2\), in the homogeneous Besov spaces \(\dot B^{-2+n/p}_{p,q}\) which are scale-invariant. The concept of the maximal parabolic Lorentz regularity in time is systematically applied to construct solutions with \(u_t,\, \Delta u\in L^{\alpha,q}((0,\infty);\dot B^s_{r,1}(\mathbb R^n))\) for \(2/\alpha+n/r=4+s\), \(1\le q\le \infty\). Local solutions are constructed as well as global-in-time ones for small initial data.
scientific article; zbMATH DE number 7452683
scientific article