Geodesic
Weak solutions for a well-posed Hele-Shaw problem
Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 2, pp. 397-424

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We analyze existence and uniqueness of weak solutions to the well-posed Hele-Shaw problem under general conditions on the fixed boundaries and non-homogeneous governing equation in the unknown domain and non-homogeneous dynamic condition on the free boundary. Our approach allows us also to minimize the restrictions on the boundary and initial data. We derive several estimates on the solutions in $BV$ spaces, prove a comparison theorem, and show that the solution depends continuously on the initial and boundary data. 
@article{BUMI_2004_8_7B_2_a8,
     author = {Antontsev, S. N. and Meirmanov, A. M. and Yurinsky, V. V.},
     title = {Weak solutions for a well-posed {Hele-Shaw} problem},
     journal = {Bollettino della Unione matematica italiana},
     pages = {397--424},
     publisher = {mathdoc},
     volume = {Ser. 8, 7B},
     number = {2},
     year = {2004},
     zbl = {1177.76398},
     mrnumber = {MR2072944},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2004_8_7B_2_a8/}
}
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Antontsev, S. N.; Meirmanov, A. M.; Yurinsky, V. V. Weak solutions for a well-posed Hele-Shaw problem. Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 2, pp. 397-424. http://geodesic.mathdoc.fr/item/BUMI_2004_8_7B_2_a8/