Weak solutions for a well-posed Hele-Shaw problem
Antontsev, S. N. ; Meirmanov, A. M. ; Yurinsky, V. V.
Bollettino dell'Unione Matematica Italiana, Tome 7-A (2004), p. 397-424 / Harvested from Biblioteca Digitale Italiana di Matematica

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.

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 BV, proviamo un teorema di comparazione, e mostriamo che la soluzione dipende in modo continuo dai dati iniziali e di contorno.

Publié le : 2004-06-01
@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/

[1] Damlamian, A., Some results on the multi-phase Stefan problem, Comm. Part. Diff. Eq., 2 (1977), 1017-1044. | MR 487015 | Zbl 0399.35054

[2] Elliott, C. M.-Janovsky, V., A variational inequality approach to the Hele-Shaw flow with a moving boundary, Proc. R. Soc. Edinb., 88A (1981), 97-107. | MR 611303 | Zbl 0455.76043

[3] Götz, I. G.-Zaltzman, B., Nonincrease of mushy region in nonhomogeneous Stefan problem, Quart. Appl. Math., Vol. XLIX, 4 (1991), 741-746. | MR 1134749 | Zbl 0756.35119

[4] Gustafsson, B., Applications of variational inequalities to a moving boundary problem for Hele-Shaw flows, SIAM J. Math. Anal., 16 (1985), 279-300. | MR 777468 | Zbl 0605.76043

[5] Howison, S. D., Complex variable methods in Hele-Shaw moving boundary problems, Eur. J. Appl. Math., 3, 3 (1992), 209-234. | MR 1182213 | Zbl 0759.76022

[6] Kamin, S. L. (KAMENOMOSTSKAYA), On the Stefan problem, Mat. Sb. (N.S.), 53 (1961), 489-514 (Russian). | Zbl 0102.09301

[7] Kinderlehrer, D.-Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. | MR 567696 | Zbl 0457.35001

[8] Kruzhkov, S. N., First order quasilinear equations in several independent variables, Math. USSR Sbornik, 10 (1970), 217-243. | Zbl 0215.16203

[9] Ladyzhenskaya, O. A.-Solonnikov, V. A.-Ural'Tseva, N. A., Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967 (English translation: series Transl. Math. Monographs, v. 23, AMS, Providence, 1968.) | Zbl 0174.15403

[10] Ladyzhenskaya, O. A.-Ural'Tseva, N. A., Linear and Quasilinear Elliptic Equations, Nauka, Moscow, 1973. (English ed: Mathematics in Science and Engineering (ed. by R. Bellman), vol. 46, Academic Press, New York, 1968. | Zbl 0164.13002

[11] Louro, B.-Rodrigues, J. F., Remarks on the quasisteady one-phase Stefan problem, Proc. R. Soc. Edinb., 102A (1986), 263-275. | MR 852360 | Zbl 0608.35081

[12] Meirmanov, A. M., The Stefan Problem, Walter de Gruyter, Berlin, New York, 1992. | MR 1154310 | Zbl 0751.35052

[13] Ockendon, J. R.-Howison, S. D.-Lacey, A. A., Mushy regions in negative squeeze films. (Submitted for publication) | Zbl 1034.76004

[14] Oleinik, O. A., A method of solution of the general Stefan problem, Dokl. Akad. Nauk. SSSR, 135 (1960), 1054-1057, Soviet Math. Dokl., 1 (1960), 1350-1354. | MR 125341 | Zbl 0131.09202

[15] Primicerio, M.-Rodrigues, J. F., The Hele-Shaw problem with nonlocal injection condition, Kawarada H. (ed.), Proc. Int. Conf. Nonlinear Math. Probl. in Industry, Tokyo, Gakkotosho, Gakuto Int. Ser. Math. Sci. Appl., 2 (1993), 375-390. | MR 1370478 | Zbl 0875.35157

[16] Rubinstein, L. I., The Stefan Problem, Zvaigne, Riga, 1967. (English ed.: Transl. Math. Monographs, Vol. 27, AMS, Providence, 1971.) | MR 222436 | Zbl 0219.35043

[17] Visintin, A., Models of Phase Transitions, Progress in Nonlinear Differential Equations and Their Applications, vol. 28. Birkhäuser, Boston-Basel-Berlin, 1996. | MR 1423808 | Zbl 0882.35004