- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:04
Interfaces and Free Boundaries
Interfaces Free Bound.
IFB
1463-9963
1463-9971
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
10.4171/IFB
http://www.ems-ph.org/doi/10.4171/IFB
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
16
2014
2
Local solvability and turning for the inhomogeneous Muskat problem
Luigi
Berselli
Università di Pisa, PISA, ITALY
Diego
Córdoba
C/ Nicolás Cabrera, nº 13-15, MADRID, SPAIN
Rafael
Granero-Belinchón
University of California at Davis, DAVIS, UNITED STATES
Darcy’s law, inhomogeneous Muskat problem, well-posedness, blow-up, maximum principle
In this work we study the evolution of the free boundary between two different fluids in a porous medium where the permeability is a two dimensional step function. The medium can fill the whole plane $\mathbb R^2$ or a bounded strip $S=\mathbb R\times(-\pi/2,\pi/2)$. The system is in the stable regime if the denser fluid is below the lighter one. First, we show local existence in Sobolev spaces by means of energy method when the system is in the stable regime. Then we prove the existence of curves such that they start in the stable regime and in finite time they reach the unstable one. This change of regime (turning) was first proven in [5] for the homogeneus Muskat problem with infinite depth.
Partial differential equations
175
213
10.4171/IFB/317
http://www.ems-ph.org/doi/10.4171/IFB/317