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Asymptotic spreading of KPP reactive fronts in incompressible space-time random flows

James Nolen Jack Xin

Analysis of PDEs mathscidoc:1912.43847

Annales de l'IHP Analyse non linaire, 26, (3), 815-839, 2009
We study the asymptotic spreading of KolmogorovPetrovskyPiskunov (KPP) fronts in spacetime random incompressible flows in dimension d> 1. We prove that if the flow field is stationary, ergodic, and obeys a suitable moment condition, the large time front speeds (spreading rates) are deterministic in all directions for compactly supported initial data. The flow field can become unbounded at large times. The front speeds are characterized by the convex rate function governing large deviations of the associated diffusion in the random flow. Our proofs are based on the Harnack inequality, an application of the sub-additive ergodic theorem, and the construction of comparison functions. Using the variational principles for the front speed, we obtain general lower and upper bounds of front speeds in terms of flow statistics. The bounds show that front speed enhancement in incompressible flows can grow at most linearly in the root mean square amplitude of the flows, and may have much slower growth due to rapid temporal decorrelation of the flows.
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@inproceedings{james2009asymptotic,
  title={Asymptotic spreading of KPP reactive fronts in incompressible space-time random flows},
  author={James Nolen, and Jack Xin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210153075860411},
  booktitle={Annales de l'IHP Analyse non linaire},
  volume={26},
  number={3},
  pages={815-839},
  year={2009},
}
James Nolen, and Jack Xin. Asymptotic spreading of KPP reactive fronts in incompressible space-time random flows. 2009. Vol. 26. In Annales de l'IHP Analyse non linaire. pp.815-839. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210153075860411.
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