Fractional Laplacian with singular drift

Tomasz Jakubowski¹

¹ Institute of Mathematics and Computer Science Wrocław University of Technology Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland

Studia Mathematica, Tome 207 (2011) no. 3, pp. 257-273

Résumé

For $\alpha \in (1,2)$ we consider the equation $\partial _t u = \varDelta ^{\alpha /2} u + b \cdot \nabla u$, where $b$ is a time-independent, divergence-free singular vector field of the Morrey class $M_1^{1-\alpha }$. We show that if the Morrey norm $\| b\| _{M_1^{1-\alpha }}$ is sufficiently small, then the fundamental solution is globally in time comparable with the density of the isotropic stable process.

Détail

DOI : 10.4064/sm207-3-3

Keywords: alpha consider equation partial vardelta alpha cdot nabla where time independent divergence free singular vector field morrey class alpha morrey norm alpha sufficiently small fundamental solution globally time comparable density isotropic stable process