May  2008, 7(3): 591-600. doi: 10.3934/cpaa.2008.7.591

Regularity for solutions of the two-phase Stefan problem

1. 

Department of Mathematics, Kansas State University, Manhattan, KS 66506, United States

Received  March 2007 Revised  October 2007 Published  February 2008

We consider the two-phase Stefan problem $u_t=\Delta\alpha(u)$ where $\alpha(u) =u+1$ for $u<-1$, $\alpha(u) =0$ for $-1 \leq u \leq 1$, and $\alpha(u)=u-1$ for $u \geq 1$. We show that if $u$ is an $L_{l o c}^2$ distributional solution then $\alpha(u)$ has $L_{l o c}^2$ derivatives in time and space. We also show $|\alpha(u)|$ is subcaloric and conclude that $\alpha(u)$ is continuous.
Citation: 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
[1]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[2]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

[3]

Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure & Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617

[4]

Chi-Cheung Poon. Blowup rate of solutions of a degenerate nonlinear parabolic equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5317-5336. doi: 10.3934/dcdsb.2019060

[5]

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

[6]

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

[7]

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

[8]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

[9]

Alexandre Montaru. Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 231-256. doi: 10.3934/dcdsb.2014.19.231

[10]

M. Sango. Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 885-905. doi: 10.3934/dcdsb.2007.7.885

[11]

Michael Winkler. Nontrivial ordered ω-limit sets in a linear degenerate parabolic equation. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 739-750. doi: 10.3934/dcds.2007.17.739

[12]

Goro Akagi, Kazumasa Suzuki. On a certain degenerate parabolic equation associated with the infinity-laplacian. Conference Publications, 2007, 2007 (Special) : 18-27. doi: 10.3934/proc.2007.2007.18

[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]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-26. doi: 10.3934/dcds.2019226

[16]

Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022

[17]

Xin Li, Chunyou Sun, Na Zhang. Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7063-7079. doi: 10.3934/dcds.2016108

[18]

Lingwei Ma, Zhong Bo Fang. A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1697-1706. doi: 10.3934/cpaa.2017081

[19]

Jingxue Yin, Chunhua Jin. Critical exponents and traveling wavefronts of a degenerate-singular parabolic equation in non-divergence form. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 213-227. doi: 10.3934/dcdsb.2010.13.213

[20]

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

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]