Geodesic
Interface evolution: the Hele-Shaw and Muskat problems
Antonio Córdoba1 ; Diego Córdoba2 ; Francisco Gancedo3
1 Departamento de Matemáticas<br/>Universidad Autónoma de Madrid<br/> 28049 Madrid<br/>Spain
2 Instituto de Ciencias Matemáticas<br/> Consejo Superior de Investigaciones Científicas<br/>28006 Madrid<br/> Spain
3 Department of Mathematics<br/>University of Chicago<br/> Chicago, IL 60637
Annals of mathematics, Tome 173 (2011) no. 1, pp. 477-542.

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We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy’s law. The free boundary is given by the discontinuity among the densities and viscosities of the fluids. This physical scenario is known as the two-dimensional Muskat problem or the two-phase Hele-Shaw flow. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition.
DOI : 10.4007/annals.2011.173.1.10

Antonio Córdoba 1 ; Diego Córdoba 2 ; Francisco Gancedo 3

1 Departamento de Matemáticas<br/>Universidad Autónoma de Madrid<br/> 28049 Madrid<br/>Spain
2 Instituto de Ciencias Matemáticas<br/> Consejo Superior de Investigaciones Científicas<br/>28006 Madrid<br/> Spain
3 Department of Mathematics<br/>University of Chicago<br/> Chicago, IL 60637 
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Antonio Córdoba; Diego Córdoba; Francisco Gancedo. Interface evolution: the Hele-Shaw and Muskat problems. Annals of mathematics, Tome 173 (2011) no. 1, pp. 477-542. doi : 10.4007/annals.2011.173.1.10. http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.173.1.10/

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