Conduction-convection problems with change of phase. (English) Zbl 0593.35085

In the usual simplest formulation of free boundary problems one considers only the conduction, here a formulation is studied where also convective motions are taking into account. Two problems are studied: a ground freezing one, in which the water is subjected to Darcy’s law, and a two phases stationary Stefan problem, where the motion of water is governed by the Navier-Stokes equations. Results on regularity of the solutions and of the free boundary are given in both cases.
Reviewer: M.Biroli


35R35 Free boundary problems for PDEs
35B65 Smoothness and regularity of solutions to PDEs
35Q30 Navier-Stokes equations
76R99 Diffusion and convection
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI


[1] Bear, J, Dynamics of fluids in porous media, (1972), American Elsevier New York · Zbl 1191.76001
[2] Caffarelli, L.A; Friedman, A, A free boundary problem in the Thomas-Fermi atomic model, J. differential equations, 32, 335-356, (1974) · Zbl 0408.35083
[3] Cannon, J.R; DiBenedetto, E; Knightly, G.H, The steady state Stefan problem with convection, Arch. rational mech. anal., 73, 79-97, (1980) · Zbl 0436.76056
[4] Carey, V.P; Gebhart, B; Mollendorf, J, Buoyancy force reversal in vertical natural convection flows in cold water, J. fluid mech., 97, 279-297, (1980) · Zbl 0444.76061
[5] Collins, R.E, Theory of flow through porous media, (1978), Interscience New York
[6] DiBenedetto, E; Elliott, C.H, An existence theorem for a problem in ground freezing, Nonlinear analysis, 9, 953-967, (1985) · Zbl 0592.76131
[7] Fife, P, The Bénard problem for general fluid dynamical equations and remarks on Boussinesq approximation, Indiana univ. math. J., 20, 303-326, (1970) · Zbl 0211.29302
[8] Gehring, F.W, The Lp integrability of partial derivatives of a quasi conformal mapping, Acta math., 130, 265-277, (1973) · Zbl 0258.30021
[9] Giaquinta, M, Multiple integrals in the calculus of variations and nonlinear elliptic systems, () · Zbl 1006.49030
[10] Gilbarg, D; Trudinger, N.S, Elliptic partial differential equations of second order, (1977), Springer-Verlag Heidelberg /New York · Zbl 0691.35001
[11] Hartman, P; Wintner, A, On the local behavior of non-parabolic partial differential equations, Amer. J. math., 85, 449-476, (1953) · Zbl 0052.32201
[12] Isakov, V.M, Inverse theorems concerning the smoothness of potentials, Differential equations, 11, 50-56, (1974) · Zbl 0328.31010
[13] Isakov, V.M, Analyticity of solutions of nonlinear transmission problems, Differential equations, 12, 41-47, (1976) · Zbl 0343.35035
[14] Kinderlehrer, D, A free boundary problem determined by the solution to a differential equation, Indiana univ. math. J., 25, 195-208, (1976) · Zbl 0336.35031
[15] Kinderlehrer, D; Nirenberg, L, Regularity in free boundary value problems, Ann. scuola norm. sup. Pisa cl. sci. (4), 4, 373-391, (1977) · Zbl 0352.35023
[16] Kinderlehrer, D; Spruck, J, Regularity in free boundary problems, Ann. scuola norm. sup. Pisa cl. sci. (4), 5, 131-148, (1978) · Zbl 0402.35045
[17] Ladyzhenskaya, O.A; Ural’tseva, N.N, Linear and quasilinear elliptic equations, (1968), Academic Press New York · Zbl 0164.13002
[18] Morrey, C.B, Multiple “integrals in the calculus of variations”, (1966), Springer-Verlag New York · Zbl 0142.38701
[19] Rabinowitz, P.H, Existence and nonuniqueness of rectangular solutions of the Bénard problem, Arch. rational mech. anal., 29, 32-57, (1968) · Zbl 0164.28704
[20] Ramilison, J.M; Gebhart, B, Buoyancy induced transport in porous media saturated with pure or saline water at low temperature, Heat mass transfer, 23, 1521-1530, (1980) · Zbl 0457.76080
[21] Schauder, J, Potentialtheoretische untersuchungen, erste abhandlunger, Math. Z., 33, 602-640, (1931) · JFM 57.0572.01
[22] Schröder, K, Über die ableitungen biharmonischer funktionen am rande, Math. Z., 49, 110-147, (1943/1944) · Zbl 0028.06801
[23] Stredulinsky, E.W, Higher integrability from reverse Hölder inequalities, Indiana univ. math. J., 29, 408-417, (1980) · Zbl 0442.35064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.