On the pointwise jump condition at the free boundary in the 1-phase Stefan problem

Previous Article
A result on singularly perturbed elliptic problems
CPAA Home
This Issue
Next Article
Existence and stability of periodic travelling-wavesolutions of the Benjamin equation

June 2005, 4(2): 357-366. doi: 10.3934/cpaa.2005.4.357

On the pointwise jump condition at the free boundary in the 1-phase Stefan problem

Donatella Danielli ^1, and Marianne Korten ^2,

1.	Department of Mathematics, Purdue University, United States
2.	Department of Mathematics, Kansas State University, Manhattan, KS 66506, United States

Received March 2004 Revised November 2004 Published March 2005

Abstract
Related Papers

In this paper we obtain the jump (or Rankine-Hugoniot) condition at the interphase for solutions in the sense of distributions to the one phase Stefan problem $u_t= \Delta (u-1)_+.$ We do this by approximating the free boundary with level sets, and using methods from the theory of bounded variation functions. We show that the spatial component of the normal derivative of the solution has a trace at the free boundary that is picked up in a natural sense. The jump condition is then obtained from the equality of the $n$-density of two different disintegrations of the free boundary measure. This is done under an additional condition on the $n$-density of this measure. In the last section we show that this condition is optimal, in the sense that its satisfaction depends on the geometry of the initial data.

Keywords: finite perimeter., mushy regions, Stefan problem, level sets, jump condition, freebound-aries, discontinuous solutions.

Mathematics Subject Classification: 35K65, 35R35.

Citation: Donatella Danielli, Marianne Korten. On the pointwise jump condition at the free boundary in the 1-phase Stefan problem. Communications on Pure & Applied Analysis, 2005, 4 (2) : 357-366. doi: 10.3934/cpaa.2005.4.357

[1]	Luigi Ambrosio, Michele Miranda jr., Diego Pallara. Sets with finite perimeter in Wiener spaces, perimeter measure and boundary rectifiability. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 591-606. doi: 10.3934/dcds.2010.28.591
[2]	Alessandro Ferriero, Nicola Fusco. A note on the convex hull of sets of finite perimeter in the plane. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 103-108. doi: 10.3934/dcdsb.2009.11.103
[3]	Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216
[4]	Fanghua Lin, Dan Liu. On the Betti numbers of level sets of solutions to elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4517-4529. doi: 10.3934/dcds.2016.36.4517
[5]	Marianne Korten, Charles N. Moore. Regularity for solutions of the two-phase Stefan problem. Communications on Pure & Applied Analysis, 2008, 7 (3) : 591-600. doi: 10.3934/cpaa.2008.7.591
[6]	Michael L. Frankel, Victor Roytburd. A Finite-dimensional attractor for a nonequilibrium Stefan problem with heat losses. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 35-62. doi: 10.3934/dcds.2005.13.35
[7]	Adriana C. Briozzo, María F. Natale, Domingo A. Tarzia. The Stefan problem with temperature-dependent thermal conductivity and a convective term with a convective condition at the fixed face. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1209-1220. doi: 10.3934/cpaa.2010.9.1209
[8]	Mario Roldan. Hyperbolic sets and entropy at the homological level. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3417-3433. doi: 10.3934/dcds.2016.36.3417
[9]	Michael L. Frankel, Victor Roytburd. Fractal dimension of attractors for a Stefan problem. Conference Publications, 2003, 2003 (Special) : 281-287. doi: 10.3934/proc.2003.2003.281
[10]	Lincoln Chayes, Inwon C. Kim. The supercooled Stefan problem in one dimension. Communications on Pure & Applied Analysis, 2012, 11 (2) : 845-859. doi: 10.3934/cpaa.2012.11.845
[11]	Piotr B. Mucha. Limit of kinetic term for a Stefan problem. Conference Publications, 2007, 2007 (Special) : 741-750. doi: 10.3934/proc.2007.2007.741
[12]	Marvin S. Müller. Approximation of the interface condition for stochastic Stefan-type problems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-23. doi: 10.3934/dcdsb.2019121
[13]	Jan Prüss, Jürgen Saal, Gieri Simonett. Singular limits for the two-phase Stefan problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5379-5405. doi: 10.3934/dcds.2013.33.5379
[14]	Karl P. Hadeler. Stefan problem, traveling fronts, and epidemic spread. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 417-436. doi: 10.3934/dcdsb.2016.21.417
[15]	Renata Bunoiu, Claudia Timofte. Homogenization of a thermal problem with flux jump. Networks & Heterogeneous Media, 2016, 11 (4) : 545-562. doi: 10.3934/nhm.2016009
[16]	Elena Kosygina. Brownian flow on a finite interval with jump boundary conditions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 867-880. doi: 10.3934/dcdsb.2006.6.867
[17]	Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129
[18]	Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431
[19]	Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595
[20]	Vincenzo Ambrosio. Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2265-2284. doi: 10.3934/dcds.2017099

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences