A new convergence proof for approximations of the Stefan problem

Robert Eymard; Thierry Gallouët

doi:10.3934/dcds.2022094

Article Contents

2023, Volume 43, Issue 3&4: 1383-1399. Doi: 10.3934/dcds.2022094

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A new convergence proof for approximations of the Stefan problem

Robert Eymard^1, and
Thierry Gallouët^2, ,

1.
LAMA, Université Gustave Eiffel, France
2.
I2M, Aix-Marseille Université, France

^*Corresponding author: Thierry Gallouët
^*Corresponding author: Thierry Gallouët

Received: February 2022

Revised: June 2022

Early access: July 15, 2022

Published online: July 15, 2022

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Abstract

We consider the Stefan problem, firstly with regular data and secondly with irregular data. In both cases is given a proof for the convergence of an approximation obtained by regularising the problem. These proofs are based on weak formulations and on compactness results in some Sobolev spaces with negative exponents.

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Mathematics Subject Classification: Primary: 35A01.

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References

[1]	H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. doi: 10.1007/BF01176474.
[2]	B. Andreianov, C. Cancès and A. Moussa, A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs, J. Funct. Anal., 273 (2017), 3633-3670. doi: 10.1016/j.jfa.2017.08.010.
[3]	P. Bénilan and F. Bouhsiss, Une remarque sur l'unicité des solutions pour l'opérateur de Serrin, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 611-616. doi: 10.1016/S0764-4442(97)84770-2.
[4]	L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0.
[5]	J. Droniou, Intégration et Espaces de Sobolev à Valeurs Vectorielles., working paper or preprint, Apr. 2001.
[6]	R. Eymard and D. Maltese., Convergence of nonlinear numerical approximations for an elliptic linear problem with irregular data, ESAIM Math. Model. Numer. Anal., 55 (2021), 3043-3089. doi: 10.1051/m2an/2021079.
[7]	T. Gallouët and R. Herbin, Equations aux dérivées partielles, Lecture notes. https://hal.archives-ouvertes.fr/cel-01196782, Sept. 2015.
[8]	T. Gallouët, A. Larcher and J. C. Latché, Convergence of a finite volume scheme for the convection-diffusion equation with $L^1$ data, Math. Comp., 81 (2012), 1429-1454. doi: 10.1090/S0025-5718-2011-02571-8.
[9]	T. Gallouët and J.-C. Latché, Compactness of discrete approximate solutions to parabolic PDEs–-application to a turbulence model, Commun. Pure Appl. Anal., 11 (2012), 2371-2391. doi: 10.3934/cpaa.2012.11.2371.
[10]	N. Igbida, K. Sbihi and P. Wittbold, Renormalized solution for Stefan type problems: existence and uniqueness, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 69-93. doi: 10.1007/s00030-009-0040-y.
[11]	A. Moussa, Some variants of the classical Aubin-Lions lemma, J. Evol. Equ., 16 (2016), 65-93. doi: 10.1007/s00028-015-0293-3.