Fractional porous medium and mean field equations in Besov spaces

In this article, we consider the evolution model $$ \partial_t{u} -\nabla\cdot(u\nabla Pu)=0,\quad Pu=(-\Delta)^{-s}u, \quad 0< s\leq 1,\; x\in\mathbb{R}^d,\; t>0. $$ We show that when $s\in[1/2,1)$, $\alpha>d+1$, $d\geq 2$, the equation has a unique local in time solution for any initial data in $B^\alpha_{1,\infty}$. Moreover, in the critical case $s=1$, the solution exists in $B^\alpha_{p,\infty}$, $2\leq p\leq\infty$, $\alpha> d/p$, $d\geq3$.