The one-phase fractional Stefan problem. (English) Zbl 1473.80010

Summary: We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in \(\mathbb{R}^N\). In terms of the enthalpy \(h(x,t)\), the evolution equation reads \(\partial_th+ (- \Delta)^s\Phi(h)=0\), while the temperature is defined as \(u:=\Phi(h):=\max\{h-L,0\}\) for some constant \(L>0\) called the latent heat, and \((- \Delta)^s\) stands for the fractional Laplacian with exponent \(s\in(0,1)\).
We prove the existence of a continuous and bounded selfsimilar solution of the form \(h(x,t)=H(x t^{- 1 / ( 2 s )})\) which exhibits a free boundary at the change-of-phase level \(h(x,t)=L\). This level is located at the line (called the free boundary) \(x(t)= \xi_0 t^{1 / ( 2 s )}\) for some \(\xi_0>0\). The construction is done in 1D, and its extension to \(N\)-dimensional space is shown.
We also provide well-posedness and basic properties of very weak solutions for general bounded data \(h_0\) in several dimensions. The temperatures \(u\) of these solutions are continuous functions that have finite speed of propagation, with possible free boundaries. We obtain estimates on the growth in time of the support of \(u\) for solutions with compactly supported initial temperatures. Besides, we show the property of conservation of positivity for \(u\) so that the support never recedes. On the contrary, the enthalpy \(h\) has infinite speed of propagation and we obtain precise estimates on the tail.
The limits \(L\to 0^+\), \(L\to+\infty\), \(s\to 0^+\) and \(s\to 1^-\) are also explored, and we find interesting connections with well-studied diffusion problems. Finally, we propose convergent monotone finite-difference schemes and include numerical experiments aimed at illustrating some of the obtained theoretical results, as well as other interesting phenomena.


80A22 Stefan problems, phase changes, etc.
35D30 Weak solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35K65 Degenerate parabolic equations
35R09 Integro-partial differential equations
35R11 Fractional partial differential equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI arXiv


[1] Alibaud, N., Cifani, S. and Jakobsen, E. R., Optimal continuous dependence estimates for fractional degenerate parabolic equations, Arch. Ration. Mech. Anal.213 (2014) 705-762. · Zbl 1304.35742
[2] Ammar, K. and Wittbold, P., Existence of renormalized solutions of degenerate elliptic-parabolic problems, Proc. Roy. Soc. Edinburgh Sect. A133 (2003) 477-496. · Zbl 1077.35103
[3] Andreianov, B. and Brassart, M., Uniqueness of entropy solutions to fractional conservation laws with “fully infinite” speed of propagation, J. Differential Equations268 (2020) 3903-3935. · Zbl 1473.35619
[4] Athanasopoulos, I., Caffarelli, L. and Salsa, S., Regularity of the free boundary in parabolic phase-transition problems, Acta Math.176 (1996) 245-282. · Zbl 0891.35164
[5] Athanasopoulos, I. and Caffarelli, L. A., Continuity of the temperature in boundary heat control problems, Adv. Math.224 (2010) 293-315. · Zbl 1190.35125
[6] Barenblatt, G. I., Scaling, Self-similarity, and Intermediate Asymptotics, , Vol. 14. (Cambridge Univ. Press, 1996). · Zbl 0907.76002
[7] Bénilan, P., Brezis, H. and Crandall, M. G., A semilinear equation in \(L^1( R^N)\), Ann. Scuola Norm. Sup. Pisa Cl. Sci. \((4)2 (1975) 523-555\). · Zbl 0314.35077
[8] Biler, P., Imbert, C. and Karch, G., The nonlocal porous medium equation: Barenblatt profiles and other weak solutions, Arch. Ration. Mech. Anal.215 (2015) 497-529. · Zbl 1308.35197
[9] Bogdan, K., Grzywny, T. and Ryznar, M., Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Probab.38 (2010) 1901-1923. · Zbl 1204.60074
[10] Bonforte, M., Sire, Y. and Vázquez, J. L., Optimal existence and uniqueness theory for the fractional heat equation, Nonlinear Anal.153 (2017) 142-168. · Zbl 1364.35416
[11] Brändle, C., Chasseigne, E. and Quirós, F.. Phase transitions with midrange interactions: A nonlocal Stefan model, SIAM J. Math. Anal.44 (2012) 3071-3100. · Zbl 1387.35594
[12] Caffarelli, L. and Vazquez, J. L., Nonlinear porous medium flow with fractional potential pressure, Arch. Ration. Mech. Anal.202 (2011) 537-565. · Zbl 1264.76105
[13] Caffarelli, L. A. and Evans, L. C., Continuity of the temperature in the two-phase Stefan problem, Arch. Ration. Mech. Anal.81 (1983) 199-220. · Zbl 0516.35080
[14] Caffarelli, L. A. and Friedman, A., Continuity of the temperature in the Stefan problem, Indiana Univ. Math. J.28 (1979) 53-70. · Zbl 0406.35032
[15] Cao, J.-F., Du, Y., Li, F. and Li, W.-T., The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, Commun. Pure Appl. Anal.19 (2020) 3651-3672. · Zbl 1445.35211
[16] Chang-Lara, H. and Dávila, G., Regularity for solutions of nonlocal, nonsymmetric equations, Ann. Inst. H. Poincaré Anal. Non Linéaire29 (2012) 833-859. · Zbl 1317.35278
[17] Chang-Lara, H. and Dávila, G., Regularity for solutions of nonlocal parabolic equations, Calc. Var. Partial Differential Equations49 (2014) 139-172. · Zbl 1292.35068
[18] Chang-Lara, H. and Dávila, G., Regularity for solutions of nonlocal parabolic equations II, J. Differential Equations256 (2014) 130-156. · Zbl 1320.35124
[19] Chasseigne, E. and Sastre-Gómez, S., A nonlocal two-phase Stefan problem, Differential Integral Equations26 (2013) 1335-1360. · Zbl 1313.35352
[20] Chen, H. and Véron, L., Semilinear fractional elliptic equations involving measures, J. Differential Equations257 (2014) 1457-1486. · Zbl 1290.35305
[21] Chen, Z.-Q., Kim, P. and Song, R., Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc.12 (2010) 1307-1329. · Zbl 1203.60114
[22] Cortázar, C., Quirós, F. and Wolanski, N., A nonlocal diffusion problem with a sharp free boundary, Interf. Free Bound.21 (2019) 441-462. · Zbl 1430.35240
[23] Crandall, M. and Pierre, M., Regularizing effects for \(u_t+A\varphi(u)=0\) in \(L^1\), J. Funct. Anal.45 (1982) 194-212. · Zbl 0483.35076
[24] Crandall, M. G. and Liggett, T. M., Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math.93 (1971) 265-298. · Zbl 0226.47038
[25] de Pablo, A., Quirós, F., Rodríguez, A. and Vázquez, J. L., A fractional porous medium equation, Adv. Math.226 (2011) 1378-1409. · Zbl 1208.26016
[26] de Pablo, A., Quirós, F., Rodríguez, A. and Vázquez, J. L., A general fractional porous medium equation, Comm. Pure Appl. Math.65 (2012) 1242-1284. · Zbl 1248.35220
[27] del Teso, F., Endal, J. and Jakobsen, E. R., On distributional solutions of local and nonlocal problems of porous medium type, C. R. Math. Acad. Sci. Paris355 (2017) 1154-1160. · Zbl 1382.35243
[28] del Teso, F., Endal, J. and Jakobsen, E. R., Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type, Adv. Math.305 (2017) 78-143. · Zbl 1349.35311
[29] del Teso, F., Endal, J. and Jakobsen, E. R., Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments, SIAM J. Numer. Anal.56 (2018) 3611-3647. · Zbl 06995703
[30] del Teso, F., Endal, J. and Jakobsen, E. R., Robust numerical methods for nonlocal (and local) equations of porous medium type, Part I: Theory, SIAM J. Numer. Anal.57 (2019) 2266-2299. · Zbl 1428.65005
[31] Teso, F. del, Endal, J. and Vázquez, J. L., On the two-phase fractional Stefan problem, Adv. Nonlinear Stud.20 (2020) 437-458. · Zbl 1434.80006
[32] Duvaut, G., Résolution d’un problème de Stefan (fusion d’un bloc de glace à zéro degré), C. R. Acad. Sci. Paris Sér. A-B276 (1973) A1461-A1463. · Zbl 0258.35037
[33] Fernández-Real, X. and Ros-Oton, X., Boundary regularity for the fractional heat equation, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM110 (2016) 49-64. · Zbl 1334.35386
[34] Friedman, A., The Stefan problem in several space variables, Trans. Amer. Math. Soc.133 (1968) 51-87. · Zbl 0162.41903
[35] Friedman, A., Variational Principles and Free-Boundary Problems, (John Wiley & Sons, Inc., 1982). · Zbl 0564.49002
[36] Gómez-Castro, D. and Vázquez, J. L., The fractional Schrödinger equation with singular potential and measure data, Discrete Contin. Dyn. Syst.39 (2019) 7113-7139. · Zbl 1425.35212
[37] Grillo, G., Muratori, M. and Punzo, F., Uniqueness of very weak solutions for a fractional filtration equation, Adv. Math.365 (2020) 107041. · Zbl 1439.35531
[38] Gupta, S. C., The Classical Stefan Problem. Basic Concepts, Modelling and Analysis with Quasi-analytical Solutions and Methods (Elsevier, 2018). · Zbl 1390.80001
[39] Kamenomostskaja (Kamin), S. L., On Stefan’s problem, Mat. Sb. (N.S.)53 (1961) 489-514. · Zbl 0102.09301
[40] Landkof, N. S., Foundations of Modern Potential Theory (Springer-Verlag, 1972). · Zbl 0253.31001
[41] Meirmanov, A. M., The Stefan Problem, , Vol. 3 (Walter de Gruyter & Co., 1992). · Zbl 0751.35052
[42] Quirós, F. and Vázquez, J. L., Asymptotic convergence of the Stefan problem to Hele-Shaw, Trans. Amer. Math. Soc.353 (2001) 609-634. · Zbl 0956.35136
[43] Roscani, S. D., Caruso, N. D., and Tarzia, D. A., Explicit solutions to fractional Stefan-like problems for Caputo and Riemann-Liouville derivatives, Commun. Nonlinear Sci. Numer. Simul.90 (2020) 105361. · Zbl 1450.35302
[44] Ros-Oton, X., Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat.60 (2016) 3-26. · Zbl 1337.47112
[45] Ros-Oton, X. and Serra, J., The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. \((9)101 (2014) 275-302\). · Zbl 1285.35020
[46] Rubenšteĭn, L. I., The Stefan Problem (Amer. Math. Soc., 1971). · Zbl 0219.35043
[47] Ryszewska, K., A space-fractional Stefan problem, Nonlinear Anal.199 (2020) 112027. · Zbl 1456.35237
[48] Silvestre, L., Hölder estimates for advection fractional-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. \((5)11 (2012) 843-855\). · Zbl 1263.35056
[49] Stan, D., del Teso, F. and Vázquez, J. L., Finite and infinite speed of propagation for porous medium equations with nonlocal pressure, J. Differential Equations260 (2016) 1154-1199. · Zbl 1379.35253
[50] Stan, D., del Teso, F. and Vázquez, J. L., Existence of weak solutions for a general porous medium equation with nonlocal pressure, Arch. Ration. Mech. Anal.233 (2019) 451-496. · Zbl 1437.35430
[51] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, , Vol. 30 (Princeton Univ. Press, 1970). · Zbl 0207.13501
[52] Vázquez, J. L., The Porous Medium Equation. Mathematical Theory. (Oxford Univ. Press, 2007). · Zbl 1107.35003
[53] Vázquez, J. L., de Pablo, A., Quirós, F. and Rodríguez, A., Classical solutions and higher regularity for nonlinear fractional diffusion equations, J. Eur. Math. Soc.19 (2017) 1949-1975. · Zbl 1371.35331
[54] Voller, V. R., Fractional Stefan problems, Int. J. Heat Mass Transf.74 (2014) 269-277.
[55] Vuik, C., Some historical notes on the Stefan problem, Nieuw Arch. Wisk. \((4)11 (1993) 157-167\). · Zbl 0801.35002
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.