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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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].

[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.

[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.

[24]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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].

[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.

[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.

[24]

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

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