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 spaces, prove a comparison theorem, and show that the solution depends continuously on the initial and boundary data.
Analizziamo l'esistenza e l'unicità di soluzioni deboli del problema ben posto di Hele-Shaw con condizioni generali sul contorno assegnato, equazione governante non-omogenea nel dominio incognito e condizione dinamica non-omogenea a contorno libero. Il nostro approccio permette anche di indebolire le restrizioni sui dati iniziali e di contorno. Otteniamo infine alcune stime per la soluzione negli spazi , proviamo un teorema di comparazione, e mostriamo che la soluzione dipende in modo continuo dai dati iniziali e di contorno.
@article{BUMI_2004_8_7B_2_397_0, author = {S. N. Antontsev and A. M. Meirmanov and V. V. Yurinsky}, title = {Weak solutions for a well-posed Hele-Shaw problem}, journal = {Bollettino dell'Unione Matematica Italiana}, volume = {7-A}, year = {2004}, pages = {397-424}, zbl = {1177.76398}, mrnumber = {2072944}, language = {en}, url = {http://dml.mathdoc.fr/item/BUMI_2004_8_7B_2_397_0} }
Antontsev, S. N.; Meirmanov, A. M.; Yurinsky, V. V. Weak solutions for a well-posed Hele-Shaw problem. Bollettino dell'Unione Matematica Italiana, Tome 7-A (2004) pp. 397-424. http://gdmltest.u-ga.fr/item/BUMI_2004_8_7B_2_397_0/
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