The shape of a free boundary driven by a line of fast diffusion

Luis A. Caffarelli; Jean-Michel Roquejoffre; Luis A. Caffarelli; Jean-Michel Roquejoffre

doi:10.3934/mine.2021010

Mathematics in Engineering

2021, Volume 3, Issue 1: 1-25. doi: 10.3934/mine.2021010

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The shape of a free boundary driven by a line of fast diffusion

Luis A. Caffarelli ¹,
Jean-Michel Roquejoffre ^{2
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1.
The University of Texas at Austin, Mathematics Department RLM 8.100, 2515 Speedway Stop C1200, Austin, Texas 78712-1202, U.S.A
2.
Institut de Mathématiques de Toulouse (UMR CNRS 5219), Université Toulouse III, 118 route de Narbonne, 31062 Toulouse cedex, France

Received: 27 March 2020 Accepted: 23 October 2020 Published: 09 November 2020

We complete the description, initiated in ^[6], of a free boundary traveling at constant speed in a half plane, where the propagation is controlled by a line having a large diffusion on its own. The main result of this work is that the free boundary is asymptotic to a line at infinity, whose angle to the horizontal is dictated by the velocity of the wave on the line. This helps understanding some counter-intuitive simulations of ^[7].
- free boundary,
- travelling wave,
- line of fast diffusion
Citation: Luis A. Caffarelli, Jean-Michel Roquejoffre. The shape of a free boundary driven by a line of fast diffusion[J]. Mathematics in Engineering, 2021, 3(1): 1-25. doi: 10.3934/mine.2021010

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Abstract

We complete the description, initiated in ^[6], of a free boundary traveling at constant speed in a half plane, where the propagation is controlled by a line having a large diffusion on its own. The main result of this work is that the free boundary is asymptotic to a line at infinity, whose angle to the horizontal is dictated by the velocity of the wave on the line. This helps understanding some counter-intuitive simulations of ^[7].

References

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[8]	Hamel F, Monneau R (2002) Existence and uniqueness of solutions of a conical-shaped free boundary problem in $\mathbb{R}.2$. Interface Free Bound 4: 167-210.
[9]	Hamel F, Monneau R, Roquejoffre JM (2004) Stability of travelling waves in a model for conical flames in two space dimensions. Ann Sci Ecole Norm Sup 37: 469-506.
[10]	Hamel F, Monneau R, Roquejoffre JM (2006) Uniqueness and classification of travelling waves with Lipschitz level lines in bistable reaction-diffusion equations. Discrete Contin Dyn Syst A 14: 75-92.

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