A variational principle for KPP front speeds in temporally random shear flows

James Nolen Jack Xin

Analysis of PDEs mathscidoc:1912.43857

Communications in Mathematical Physics, 269, (2), 493-532, 2007.1
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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@inproceedings{james2007a,
  title={A variational principle for KPP front speeds in temporally random shear flows},
  author={James Nolen, and Jack Xin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210235401043421},
  booktitle={Communications in Mathematical Physics},
  volume={269},
  number={2},
  pages={493-532},
  year={2007},
}
James Nolen, and Jack Xin. A variational principle for KPP front speeds in temporally random shear flows. 2007. Vol. 269. In Communications in Mathematical Physics. pp.493-532. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210235401043421.
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