Leibenson's equation on Riemannian manifolds


Speaker:    Alexander Grigoryan (Universität Bielefeld)

Time:    2022.09.29 Thu    17:00-18:30

Venue:  BIMSA 1120

Zoom:    518 868 7656 (PW: BIMSA)




We consider on arbitrary Riemannian manifolds the Leibenson equation:
$$\partial _{t}u=\Delta _{p}u^{q}.$$
This equation comes from hydrodynamics where it describes filtration of a turbulent compressible liquid in porous medium.  Here $u(x,t)$ is the fraction of the volume that the liquid takes in porous medium at time $t$ at point $x$. The parameter $p$ characterizes the turbulence of a flow, while $q$ describes the compressibility of the liquid. 
We prove that if $p>2$ and $1/(p-1)<q\leq 1$ then solutions of this equation have finite propagation speed. On complete manifolds with non-negative Ricci curvature, we obtain a sharp propagation rate that matches that of Barenblatt solution.


Speaker Intro:

Professor Alexander Grigor'yan received his PhD from Lomonosov Moscow University in 1982. Since then, he worked at the State University of Volgograd and the Institute of Control Sciences in Moscow with receiving his habilitation in 1989. He has been a Humboldt Fellow at Bielefeld University in 1992-93 and a Guest Scholar at Harvard University in 1993/94. After working from lecturer to professor at Imperial College London in 1994-2005, he became a professor at Bielefeld University from 2005. 
Professor Grigor'yan won a gold medal at International Mathematical Olympiad when he was 17, awarded by the Prize of the Moscow Mathematical Society in 1988, and received the Whitehead Prize of London Mathematical Society in 1997. He was an invited speaker of European Congress of Mathematicians in Barcelona in 2000. 
Professor Grigor'yan is one of the top experts in geometric analysis on Riemannian manifolds, metric spaces and graphs, with making various important contributions on the topics.


AMSS-YMSC-BIMSA Joint Seminar on Progress of Topology and Its Applications