October  2018, 11(5): 991-1010. doi: 10.3934/dcdss.2018058

Long-time behavior of the one-phase Stefan problem in periodic and random media

1. 

Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa, 920-1192, Japan

2. 

Graduate School of Natural Science and Technology, Kanazawa University, Kakuma, Kanazawa, 920-1192, Japan

* Corresponding author: Norbert Požár

Received  February 2017 Revised  May 2017 Published  June 2018

We study the long-time behavior of solutions of the one-phase Stefan problem in inhomogeneous media in dimensions n ≥ 2. Using the technique of rescaling which is consistent with the evolution of the free boundary, we are able to show the homogenization of the free boundary velocity as well as the locally uniform convergence of the rescaled solution to a self-similar solution of the homogeneous Hele-Shaw problem with a point source. Moreover, by viscosity solution methods, we also deduce that the rescaled free boundary uniformly approaches a sphere with respect to Hausdorff distance.

Citation: Norbert Požár, Giang Thi Thu Vu. Long-time behavior of the one-phase Stefan problem in periodic and random media. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 991-1010. doi: 10.3934/dcdss.2018058
References:
[1]

L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math., 139 (1977), 155-184.  doi: 10.1007/BF02392236.  Google Scholar

[2]

L. A. Caffarelli and A. Friedman, Continuity of the temperature in the Stefan problem, Indiana Univ. Math. J., 28 (1979), 53-70.  doi: 10.1512/iumj.1979.28.28004.  Google Scholar

[3]

L. A. Caffarelli and P. E. Souganidis, Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media, Invent. Math., 180 (2010), 301-360.  doi: 10.1007/s00222-009-0230-6.  Google Scholar

[4]

L. A. CaffarelliP. E. Souganidis and L. Wang, Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media, Comm. Pure Appl. Math., 58 (2005), 319-361.  doi: 10.1002/cpa.20069.  Google Scholar

[5]

B. Claudio, Sur un problème à frontière libre traduisant le filtrage de liquides à travers des milieux poreux, C. R. Acad. Sci. Paris Sér. A-B, 273 (1971), A1215-A1217.   Google Scholar

[6]

G. Duvaut, Résolution d'un problème de Stefan (fusion d'un bloc de glace à zéro degré), (French), C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A1461-A1463.   Google Scholar

[7]

C. M. Elliott and V. Janovský, A variational inequality approach to Hele-Shaw flow with a moving boundary, Proc. Roy. Soc. Edinburgh Sect. A, 88 (1981), 93-107.  doi: 10.1017/S0308210500017315.  Google Scholar

[8]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375.  doi: 10.1017/S0308210500018631.  Google Scholar

[9]

A. Friedman and D. Kinderlehrer, A one phase Stefan problem, Indiana Univ. Math. J., 24 (1974/75), 1005-1035.  doi: 10.1512/iumj.1975.24.24086.  Google Scholar

[10]

I. C. Kim, Uniqueness and existence results on the Hele-Shaw and the Stefan problems, Arch. Ration. Mech. Anal., 168 (2003), 299-328.  doi: 10.1007/s00205-003-0251-z.  Google Scholar

[11]

I. C. Kim, Homogenization of the free boundary velocity, Arch. Ration. Mech. Anal., 185 (2007), 69-103.  doi: 10.1007/s00205-006-0035-3.  Google Scholar

[12]

I. C. Kim, Homogenization of a model problem on contact angle dynamics, Comm. Partial Differential Equations, 33 (2008), 1235-1271.  doi: 10.1080/03605300701518273.  Google Scholar

[13]

I. C. Kim, Error estimates on homogenization of free boundary velocities in periodic media, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 999-1019.  doi: 10.1016/j.anihpc.2008.10.004.  Google Scholar

[14]

I. C. Kim and A. Mellet, Homogenization of a Hele-Shaw problem in periodic and random media, Arch. Ration. Mech. Anal., 194 (2009), 507-530.  doi: 10.1007/s00205-008-0161-1.  Google Scholar

[15]

I. C. Kim and A. Mellet, Homogenization of one-phase Stefan-type problems in periodic and random media, Trans. Amer. Math. Soc., 362 (2010), 4161-4190.  doi: 10.1090/S0002-9947-10-04945-7.  Google Scholar

[16]

D. Kinderlehrer and L. Nirenberg, The smoothness of the free boundary in the one phase Stefan problem, Comm. Pure Appl. Math., 31 (1978), 257-282.  doi: 10.1002/cpa.3160310302.  Google Scholar

[17]

N. Požár, Long-time behavior of a Hele-Shaw type problem in random media, Interfaces Free Bound., 13 (2011), 373-395.  doi: 10.4171/IFB/263.  Google Scholar

[18]

N. Požár, Homogenization of the Hele-Shaw problem in periodic spatiotemporal media, Arch. Ration. Mech. Anal., 217 (2015), 155-230.  doi: 10.1007/s00205-014-0831-0.  Google Scholar

[19]

F. Quirós and J. L. Vázquez, Asymptotic convergence of the Stefan problem to Hele-Shaw, Trans. Amer. Math. Soc., 353 (2001), 609-634.  doi: 10.1090/S0002-9947-00-02739-2.  Google Scholar

[20]

J. F. Rodrigues, Free boundary convergence in the homogenization of the one-phase Stefan problem, Trans. Amer. Math. Soc., 274 (1982), 297-305.  doi: 10.2307/1999510.  Google Scholar

[21]

J. F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, vol. 134, North-Holland Publishing Co., Amsterdam, 1987, Notas de Matemática [Mathematical Notes].  Google Scholar

[22]

J. F. Rodrigues, The Stefan problem revisited, in Mathematical Models for Phase Change Problems, (Óbidos, 1988), Internat. Ser. Numer. Math., vol. 88, Basel Birkhäuser, 1989, 129–190.  Google Scholar

[23]

J. F. Rodrigues, Variational methods in the Stefan problem, in Phase Transitions and Hysteresis, (Montecatini Terme, 1993), Lecture Notes in Math., vol. 1584, Springer, Berlin, 1994, 147–212. doi: 10.1007/BFb0073397.  Google Scholar

[24]

P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptot. Anal., 20 (1999), 1-11.   Google Scholar

show all references

References:
[1]

L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math., 139 (1977), 155-184.  doi: 10.1007/BF02392236.  Google Scholar

[2]

L. A. Caffarelli and A. Friedman, Continuity of the temperature in the Stefan problem, Indiana Univ. Math. J., 28 (1979), 53-70.  doi: 10.1512/iumj.1979.28.28004.  Google Scholar

[3]

L. A. Caffarelli and P. E. Souganidis, Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media, Invent. Math., 180 (2010), 301-360.  doi: 10.1007/s00222-009-0230-6.  Google Scholar

[4]

L. A. CaffarelliP. E. Souganidis and L. Wang, Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media, Comm. Pure Appl. Math., 58 (2005), 319-361.  doi: 10.1002/cpa.20069.  Google Scholar

[5]

B. Claudio, Sur un problème à frontière libre traduisant le filtrage de liquides à travers des milieux poreux, C. R. Acad. Sci. Paris Sér. A-B, 273 (1971), A1215-A1217.   Google Scholar

[6]

G. Duvaut, Résolution d'un problème de Stefan (fusion d'un bloc de glace à zéro degré), (French), C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A1461-A1463.   Google Scholar

[7]

C. M. Elliott and V. Janovský, A variational inequality approach to Hele-Shaw flow with a moving boundary, Proc. Roy. Soc. Edinburgh Sect. A, 88 (1981), 93-107.  doi: 10.1017/S0308210500017315.  Google Scholar

[8]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375.  doi: 10.1017/S0308210500018631.  Google Scholar

[9]

A. Friedman and D. Kinderlehrer, A one phase Stefan problem, Indiana Univ. Math. J., 24 (1974/75), 1005-1035.  doi: 10.1512/iumj.1975.24.24086.  Google Scholar

[10]

I. C. Kim, Uniqueness and existence results on the Hele-Shaw and the Stefan problems, Arch. Ration. Mech. Anal., 168 (2003), 299-328.  doi: 10.1007/s00205-003-0251-z.  Google Scholar

[11]

I. C. Kim, Homogenization of the free boundary velocity, Arch. Ration. Mech. Anal., 185 (2007), 69-103.  doi: 10.1007/s00205-006-0035-3.  Google Scholar

[12]

I. C. Kim, Homogenization of a model problem on contact angle dynamics, Comm. Partial Differential Equations, 33 (2008), 1235-1271.  doi: 10.1080/03605300701518273.  Google Scholar

[13]

I. C. Kim, Error estimates on homogenization of free boundary velocities in periodic media, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 999-1019.  doi: 10.1016/j.anihpc.2008.10.004.  Google Scholar

[14]

I. C. Kim and A. Mellet, Homogenization of a Hele-Shaw problem in periodic and random media, Arch. Ration. Mech. Anal., 194 (2009), 507-530.  doi: 10.1007/s00205-008-0161-1.  Google Scholar

[15]

I. C. Kim and A. Mellet, Homogenization of one-phase Stefan-type problems in periodic and random media, Trans. Amer. Math. Soc., 362 (2010), 4161-4190.  doi: 10.1090/S0002-9947-10-04945-7.  Google Scholar

[16]

D. Kinderlehrer and L. Nirenberg, The smoothness of the free boundary in the one phase Stefan problem, Comm. Pure Appl. Math., 31 (1978), 257-282.  doi: 10.1002/cpa.3160310302.  Google Scholar

[17]

N. Požár, Long-time behavior of a Hele-Shaw type problem in random media, Interfaces Free Bound., 13 (2011), 373-395.  doi: 10.4171/IFB/263.  Google Scholar

[18]

N. Požár, Homogenization of the Hele-Shaw problem in periodic spatiotemporal media, Arch. Ration. Mech. Anal., 217 (2015), 155-230.  doi: 10.1007/s00205-014-0831-0.  Google Scholar

[19]

F. Quirós and J. L. Vázquez, Asymptotic convergence of the Stefan problem to Hele-Shaw, Trans. Amer. Math. Soc., 353 (2001), 609-634.  doi: 10.1090/S0002-9947-00-02739-2.  Google Scholar

[20]

J. F. Rodrigues, Free boundary convergence in the homogenization of the one-phase Stefan problem, Trans. Amer. Math. Soc., 274 (1982), 297-305.  doi: 10.2307/1999510.  Google Scholar

[21]

J. F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, vol. 134, North-Holland Publishing Co., Amsterdam, 1987, Notas de Matemática [Mathematical Notes].  Google Scholar

[22]

J. F. Rodrigues, The Stefan problem revisited, in Mathematical Models for Phase Change Problems, (Óbidos, 1988), Internat. Ser. Numer. Math., vol. 88, Basel Birkhäuser, 1989, 129–190.  Google Scholar

[23]

J. F. Rodrigues, Variational methods in the Stefan problem, in Phase Transitions and Hysteresis, (Montecatini Terme, 1993), Lecture Notes in Math., vol. 1584, Springer, Berlin, 1994, 147–212. doi: 10.1007/BFb0073397.  Google Scholar

[24]

P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptot. Anal., 20 (1999), 1-11.   Google Scholar

[1]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[2]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[3]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[4]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[5]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336

[6]

Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299

[7]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[8]

Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112

[9]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[10]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108

[11]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[12]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[13]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075

[14]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[15]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[16]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[17]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099

[18]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102

[19]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[20]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (73)
  • HTML views (122)
  • Cited by (0)

Other articles
by authors

[Back to Top]