Abstract
We develop a framework for a unified treatment of well-posedness for the Stefan problem with or without surface tension. In the absence of surface tension, we establish well-posedness in Sobolev spaces for the classical Stefan problem. We introduce a new velocity variable which extends the velocity of the moving free-boundary into the interior domain. The equation satisfied by this velocity is used for the analysis in place of the heat equation satisfied by the temperature. Solutions to the classical Stefan problem are then constructed as the limit of solutions to a carefully chosen sequence of approximations to the velocity equation, in which the moving free-boundary is regularized and the boundary condition is modified in a such a way as to preserve the basic nonlinear structure of the original problem. With our methodology, we simultaneously find the required stability condition for well-posedness and obtain new estimates for the regularity of the moving free-boundary. Finally, we prove that solutions of the Stefan problem with positive surface tension \({\sigma}\) converge to solutions of the classical Stefan problem as \({\sigma \to 0}\).
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Almgren F., Wang L.: Mathematical existence of crystal growth with Gibbs–Thomson curvature effects. J. Geom. Anal. 10(1), 1–100 (2000)
Ambrose D.M., Masmoudi N.: The zero surface tension limit of three-dimensional water waves. Indiana U. Math. J. 58, 479–522 (2009)
Ambrose D.M., Masmoudi N.: The zero surface tension limit of two-dimensional water waves. Comm. Pure Appl. Math 58, 1287–1315 (2005)
Athanasopoulos I., Caffarelli L. A., Salsa S.: Regularity of the free-boundary in parabolic phase-transition problems. Acta Math. 176, 245–282 (1996)
Athanasopoulos I., Caffarelli L. A., Salsa S.: Phase transition problems of parabolic type: flat free-boundaries are smooth. Comm. Pure Appl. Math. 51, 77–112 (1998)
Caffarelli L.A.: Some aspects of the one-phase Stefan problem. Indiana Univ. Math. J. 27, 73–77 (1978)
Caffarelli L.A., Evans L.C.: Continuity of the temperature in the two-phase Stefan problem. Arch. Rational Mech. Anal. 81, 199–220 (1983)
Caffarelli, L.A., Salsa, S.: A geometric approach to free-boundary problems. American Mathematical Society, Providence, RI, 2005
Cheng C.H.A., Coutand D., Shkoller S.: Global existence and decay for solutions of the Hele-Shaw flow with injection. Interfaces Free Bound. 16, 297–338 (2014)
Cheng C.H.A., Granero-Belinchón R., Shkoller S.: Well-posedness of the Muskat problem with \({H^2}\) initial data, Adv. Math. 286, 32–104 (2016)
Cheng, C.H.A., Shkoller, S.: Solvability and regularity for an elliptic system prescribing the curl, divergence, and partial trace of a vector field on Sobolev-class domains. http://arxiv.org/abs/1408.2469 (2014)
Choi S., Kim I.: The two-phase Stefan problem: regularization near Lipschitz initial data by phase dynamics. Anal. PDE 5(5), 1063–1103 (2012)
Choi S., Kim I.: Regularity of one-phase Stefan problem near Lipschitz initial data. Am. J. Math. 132(6), 1693–1727 (2010)
Constantin P., Córdoba D., Gancedo F., Strain R. M.: On the global existence for the Muskat problem. J. Eur. Math. Soc 15(1), 201–227 (2013)
Córdoba A., Córdoba D., Gancedo F.: Interface evolution: the Hele-Shaw and Muskat problems. Annals of Math. 173(1), 477–542 (2011)
Córdoba A., Córdoba D., Gancedo F.: Porous media: the Muskat problem in 3D. Analysis & PDE, 6(2), 447–497 (2013)
Coutand D., Hole J., Shkoller S.: Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit. SIAM J. Math. Anal. 45, 3690–3767 (2013)
Coutand D., Shkoller S.: On the interaction between quasilinear elastodynamics and the Navier-Stokes equations. Arch. Rational Mech. Anal. 179(3), 303–352 (2006)
Coutand D., Shkoller S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20(3), 829–930 (2007)
Coutand D., Shkoller S.: A simple proof of well-posedness for the free surface incompressible Euler equations. Discr. Cont. Dyn. Systems, Series S 3(3), 429–449 (2010)
Coutand D., Shkoller S.: On the finite-time splash and splat singularities for the 3-D free-surface Euler equations. Commun. Math. Phys. 325, 143–183 (2014)
De Giorgi E.: \({\Gamma}\)-convergenza e G-convergenza. Boll. Un. Mat. Ital. 5-B, 213–220 (1977)
Escher J., Prüss J., Simonett G.: Analytic solutions for a Stefan problem with Gibbs–Thomson correction. J. Reine Angew. Math. 563, 1–52 (2003)
Friedman A.: Variational Principles and free-boundary problems. Wiley, New York (1982)
Friedman A.: The Stefan problem for a hyperbolic heat equation. J. Math. Anal. Appl. 138, 249–279 (1989)
Friedman A., Kinderlehrer D.: A one phase Stefan problem. Indiana Univ. Math. J. 25, 1005–1035 (1975)
Friedman A., Reitich F.: The Stefan problem with small surface tension. Trans. Amer. Math. Soc. 328, 465–515 (1991)
Frolova E. V., Solonnikov V.A.: \({L_p}\)-theory for the Stefan problem. J. Math. Sci. 99(1), 989–1006 (2000)
Hadžić M.: Orthogonality conditions and asymptotic stability in the Stefan problem with surface tension. Arch. Rational Mech. Anal. 203(3), 719–745 (2012)
Hadžić M., Guo Y.: Stability in the Stefan problem with surface tension (I). Commun. Partial Diff. Eqns. 35(2), 201–244 (2010)
Hadžić M., Shkoller S.: Global stability and decay for the classical Stefan problem. Comm. Pure Appl. Math. 68, 689–757 (2015)
Hadžić, M., Shkoller, S.: Global stability and decay for the classical Stefan problem for general boundary shapes. Philosophical Transactions of Royal Society A, 373, pp. 2050, 2015
Hadžić, M., Navarro, G., Shkoller, S.: Local well-posedness and global stability of the two-phase stefan problem (Preprint)
Hanzawa E.I.: Classical solution of the Stefan problem. Tohoku Math, J. 33, 297–335 (1981)
Kamenomostskaya S. L.: On the Stefan problem. Mat. Sb. 53, 489–514 (1961)
Kim I.: Uniqueness and existence of Hele-Shaw and Stefan problem. Arch. Rat. Mech. Anal. 168, 299–328 (2003)
Kim I., Požar N.: Viscosity solutions for the two-phase Stefan problem. Comm. PDE 36(1), 42–66 (2011)
Ladyženskaja, O.A., Solonnikov, V.A., Uralõceva, N.N.: Linear and quasilinear equations of parabolic type. Trans. Math. Monographs 23, Am. Math. Soc., Providence, RI (1968), Russian edition: Nauka, Moscow 1967.
Luckhaus S.: Solutions for the two-phase Stefan problem with the Gibbs–Thomson law for the melting temperature. Eur. J. Appl. Math. 1, 101–111 (1990)
Meirmanov, A. M.: The Stefan Problem. De Gruyter Expositions in Mathematics. 3, 1992
Prüss J., Saal J., Simonett G.: Existence of analytic solutions for the classical Stefan problem. Math. Ann. 338, 703–755 (2007)
Prüss J., Simonett G., Zacher R.: Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension. Arch. Ration. Mech. Anal. 207, 611–667 (2013)
Radkevich E.V.: Gibbs-Thomson law and existence of the classical solution of the modified Stefan problem. Soviet Dokl. Acad. Sci. 316, 1311–1315 (1991)
Lord Rayleigh.: On the instability of jets. Proc. London Math. Soc. 1 s1–10, 4–13, 1878
Röger M.: Solutions for the Stefan problem with Gibbs–Thomson law by a local minimisation. Interfaces Free Bound. 6, 105–133 (2004)
Taylor, G.: The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I Proc. R. Soc. Lond. A 201 no. 1065, 192–196, 1950
Taylor, M.E.: Partial differential equations. III. Nonlinear equations. Corrected reprint of the 1996 original. Applied Mathematical Sciences, 117, Springer-Verlag, New York, 1997
Visintin, A.: Models of phase transitions. Progr. Nonlin. Diff. Equ. Appl. 28. Birkhauser, Boston, 1996
Visintin, A.: Introduction to Stefan-type problems. Handbook of differential equations, evolutionary equations, 4, 377-484, Elsevier B.V., North-Holland, 2008
Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130(1), 39–72 (1997)
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Hadžić, M., Shkoller, S. Well-posedness for the Classical Stefan Problem and the Zero Surface Tension Limit. Arch Rational Mech Anal 223, 213–264 (2017). https://doi.org/10.1007/s00205-016-1041-8
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DOI: https://doi.org/10.1007/s00205-016-1041-8