Global well-posedness of the 3D incompressible porous media equation with critical dissipation in Triebel-Lizorkin spaces

Boundary value problems

Yanghai Yu¹  Xing Wu¹  Yanbin Tang¹ 

[1] School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, P.R. China

关键词: porous media equation;    global well-posedness;    fractional Laplacian;    Triebel-Lizorkin spaces;    35Q35;    76D03;    76S05;   

DOI  :  10.1186/s13661-016-0625-4

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In this paper, we study the global well-posedness of the 3D incompressible critical dissipative porous media equation with small initial data in the Triebel-Lizorkin spaceFp,qs(R3)$F^{s}_{p,q}(\mathbb{R}^{3})$. By a pointwise exponential decay estimate on the Poisson semigroupe−tν−Δ$e^{-t\nu\sqrt{-\Delta}}$and the Fourier localization technique, we generalize the global well-posedness in the Sobolev spacesHps(R3)=Fp,2s(R3)$H_{p}^{s}(\mathbb{R}^{3})=F^{s}_{p,2}(\mathbb{R}^{3})$andHs(R3)=F2,2s(R3)$H^{s}(\mathbb {R}^{3})=F^{s}_{2,2}(\mathbb{R}^{3})$into the general Triebel-Lizorkin spacesFp,qs(R3)$F^{s}_{p,q}(\mathbb{R}^{3})$withs>3p$s>\frac{3}{p}$,p,q∈(1,∞)$p, q\in (1,\infty)$.

【 授权许可】

CC BY