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A Variational Principle for KPP Front Speeds in Temporally Random Shear Flows

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Abstract

We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in temporally random shear flows with sufficiently decaying correlations. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. To prove the variational principle, we use the comparison principle of solutions, the path integral representation of solutions, and large deviation estimates of the associated stochastic flows. The variational principle then allows us to analytically bound the front speeds. The speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length, and the sublinear growth law if the temporal decorrelation is also large enough, the so-called bending phenomenon.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA

    James Nolen

  2. Department of Mathematics, University of California at Irvine, Irvine, CA, USA

    Jack Xin

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  1. James Nolen
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  2. Jack Xin
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Correspondence to James Nolen.

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Communicated by P. Constantin

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Nolen, J., Xin, J. A Variational Principle for KPP Front Speeds in Temporally Random Shear Flows. Commun. Math. Phys. 269, 493–532 (2007). https://doi.org/10.1007/s00220-006-0144-8

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  • DOI: https://doi.org/10.1007/s00220-006-0144-8

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