Electron. J. Diff. Equ., Vol. 2014 (2014), No. 199, pp. 1-14.

Fractional porous medium and mean field equations in Besov spaces

Xuhuan Zhou, Weiliang Xiao, Jiecheng Chen

Abstract:
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 $B^\alpha_{p,\infty}$, $2\leq p\leq\infty$, $\alpha> d/p$, $d\geq3$.

Submitted April 10, 2014. Published September 23, 2014.
Math Subject Classifications: 35K55, 35K65, 76S05.
Key Words: Fractional porous medium equation; mean field equation; local solution; Besov space.

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Xuhuan Zhou
Department of Mathematics, Zhejiang University
310027 Hangzhou, China
email: zhouxuhuan@163.com
Weiliang Xiao
Department of Mathematics, Zhejiang University
310027 Hangzhou, China
email: xwltc123@163.com
Jiecheng Chen
Department of Mathematics, Zhejiang Normal University
321004 Jinhua, China
email: jcchen@zjnu.edu.cn

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