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Speed-up of combustion fronts in shear flows

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Abstract

This paper is concerned with the analysis of speed-up of reaction-diffusion-advection traveling fronts in infinite cylinders with periodic boundary conditions. The advection is a shear flow with a large amplitude and the reaction is nonnegative, with either positive or zero ignition temperature. The unique or minimal speeds of the traveling fronts are proved to be asymptotically linear in the flow amplitude as the latter goes to infinity, solving an open problem from Berestycki (Nonlinear PDEs in condensed matter and reactive flows, Kluwer, Doordrecht, 2003). The asymptotic growth rate is characterized explicitly as the unique or minimal speed of traveling fronts for a limiting degenerate problem, and the convergence of the regular traveling fronts to the degenerate ones is proved for positive ignition temperatures under an additional Hörmander-type condition on the flow.

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Notes

  1. For \(\zeta =(\zeta _1,\ldots ,\zeta _{N-1})\in \mathbb{N }^{N-1}\), we let \(|\zeta |=\zeta _1+\cdots +\zeta _{N-1}\) and \(D^{\zeta }\alpha (y)=\frac{\partial ^{|\zeta |}\alpha }{\partial y_1^{\zeta _1}\cdots \partial y_{N-1}^{\zeta _{N-1 }}}(y)\).

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Acknowledgments

We thank Tom Kurtz and Daniel Stroock for useful discussions and pointers to references. FH is indebted to the Alexander von Humboldt Foundation for its support. His work was also supported by the French Agence Nationale de la Recherche through the project PREFERED. AZ was supported in part by NSF grants DMS-1113017 and DMS-1056327, and by an Alfred P. Sloan Research Fellowship. Part of this work was carried out during visits by FH to the Departments of Mathematics of the Universities of Chicago and Wisconsin and by AZ to the Faculté des Sciences et Techniques, Aix-Marseille Université, the hospitality of which is gratefuly acknowledged.

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Authors and Affiliations

  1. Faculté des Sciences, Aix-Marseille Université & Institut Universitaire de France LATP, 39 rue F. Joliot-Curie, 13453,  Marseille Cedex 13, France

    François Hamel

  2. Department of Mathematics, University of Wisconsin, Madison, WI,  53706, USA

    Andrej Zlatoš

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  1. François Hamel
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  2. Andrej Zlatoš
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Correspondence to François Hamel.

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Hamel, F., Zlatoš, A. Speed-up of combustion fronts in shear flows. Math. Ann. 356, 845–867 (2013). https://doi.org/10.1007/s00208-012-0877-y

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