The leading edge of a free boundary interacting with a line of fast diffusion
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- by L. A. Caffarelli and J.-M. Roquejoffre
- St. Petersburg Math. J. 32, 499-522
- DOI: https://doi.org/10.1090/spmj/1658
- Published electronically: May 11, 2021
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Abstract:
Our goal in this work is to explain an unexpected feature of the expanding level sets of the solutions of a system where a half-plane in which reaction-diffusion phenomena occur exchanges mass with a line having a large diffusion of its own. The system was proposed by H. Berestycki, L. Rossi, and the second author as a model of enhancement of biological invasions by a line of fast diffusion. It was observed numerically by A.-C. Coulon that the leading edge of the front, rather than being located on the line, was in the lower half-plane.
We explain this behavior for a closely related free boundary problem. We construct travelling waves for this problem, and the analysis of their free boundary near the line confirms the predictions of the numerical simulations.
References
- H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144. MR 618549
- H. Berestycki, L. A. Caffarelli, and L. Nirenberg, Uniform estimates for regularization of free boundary problems, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 567–619. MR 1044809
- Henri Berestycki, Basil Nicolaenko, and Bruno Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal. 16 (1985), no. 6, 1207–1242. MR 807905, DOI 10.1137/0516088
- Henri Berestycki, Jean-Michel Roquejoffre, and Luca Rossi, The influence of a line with fast diffusion on Fisher-KPP propagation, J. Math. Biol. 66 (2013), no. 4-5, 743–766. MR 3020920, DOI 10.1007/s00285-012-0604-z
- Henri Berestycki, Jean-Michel Roquejoffre, and Luca Rossi, The shape of expansion induced by a line with fast diffusion in Fisher-KPP equations, Comm. Math. Phys. 343 (2016), no. 1, 207–232. MR 3475665, DOI 10.1007/s00220-015-2517-3
- Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, Graduate Studies in Mathematics, vol. 68, American Mathematical Society, Providence, RI, 2005. MR 2145284, DOI 10.1090/gsm/068
- Luis A. Caffarelli and Juan L. Vázquez, A free-boundary problem for the heat equation arising in flame propagation, Trans. Amer. Math. Soc. 347 (1995), no. 2, 411–441. MR 1260199, DOI 10.1090/S0002-9947-1995-1260199-7
- A.-C. Coulon Chalmin, Fast propagation in reaction-diffusion equations with fractional diffusion, PhD thesis, 2014. Online manuscript: thesesups.ups-tlse.fr/2427/
- Laurent Dietrich, Existence of travelling waves for a reaction–diffusion system with a line of fast diffusion, Appl. Math. Res. Express. AMRX 2 (2015), 204–252. MR 3394265, DOI 10.1093/amrx/abv002
- Laurent Dietrich and Jean-Michel Roquejoffre, Front propagation directed by a line of fast diffusion: large diffusion and large time asymptotics, J. Éc. polytech. Math. 4 (2017), 141–176 (English, with English and French summaries). MR 3611101, DOI 10.5802/jep.40
- David Jerison and Nikola Kamburov, Free boundaries subject to topological constraints, Discrete Contin. Dyn. Syst. 39 (2019), no. 12, 7213–7248. MR 4026187, DOI 10.3934/dcds.2019301
- Yousong Luo and Neil S. Trudinger, Linear second order elliptic equations with Venttsel′boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 118 (1991), no. 3-4, 193–207. MR 1121663, DOI 10.1017/S0308210500029048
- G. I. Sivashinsky, Instabilities, pattern formation and turbulence in flames, Ann. Rev. Fluid Mech. 15 (1983), 179–199.
- Ya. B. Zel′dovich, G. I. Barenblatt, V. B. Librovich, and G. M. Makhviladze, The mathematical theory of combustion and explosions, Consultants Bureau [Plenum], New York, 1985. Translated from the Russian by Donald H. McNeill. MR 781350, DOI 10.1007/978-1-4613-2349-5
Bibliographic Information
- L. A. Caffarelli
- Affiliation: The University of Texas at Austin, Mathematics Department RLM 8.100, 2515 Speedway Stop C1200, Austin, Texas 78712-1202
- MR Author ID: 44175
- Email: caffarel@math.utexas.edu
- J.-M. Roquejoffre
- Affiliation: Institut de Mathématiques, de Toulouse (UMR CNRS 5219), Université Toulouse III, 118 route de Narbonne, 31062 Toulouse cedex, France
- Email: jean-michel.roquejoffre@math.univ-toulouse.fr
- Received by editor(s): July 29, 2019
- Published electronically: May 11, 2021
- Additional Notes: The first author was supported by NSF grant DMS-1160802. The research of the second author has received funding from the ERC under the European Union’s Seventh Frame work Programme (FP/2007-2013) / ERC Grant Agreement 321186 – ReaDi. He also acknowledges J.T. Oden fellowships, for visits at the University of Texas
- © Copyright 2021 American Mathematical Society
- Journal: St. Petersburg Math. J. 32, 499-522
- MSC (2020): Primary 35R35
- DOI: https://doi.org/10.1090/spmj/1658
- MathSciNet review: 4099097
Dedicated: We dedicate this article to Nina Ural’tseva, a great mathematician and wonderful person.