Front Quenching in G-equation Model Induced by Straining of Cellular Flow

Jack Xin University of California at Irvine Yifeng Yu University of California at Irvine

Analysis of PDEs mathscidoc:1608.03003

Distinguished Paper Award in 2017

Archive for Rational Mechanics and Analysis, 214, (1), 2014
We study homogenization of G-equation with a ow straining term (or the strain G-equation) in two dimensional periodic cellular ow. The strain G-equation is a highly non-coercive and non-convex level set Hamilton-Jacobi equation. The main objective is to investigate how the ow induced straining (the nonconvex term) in uences front propagation as the ow intensity A increases. Three distinct regimes are identified. When A is below the critical level, homogenization holds and the turbulent ame speed sT (effective Hamiltonian) is well-defined for any periodic ow with small divergence and is enhanced by the cellular ow as $s_T \ge O(A/logA)$. In the second regime where A is slightly above the critical value, homogenization breaks down, and $s_T$ is not well defined along any direction. Solutions become a mixture of fast moving part and a stagnant part. When $A$ is sufficiently large, the whole ame front ceases to propagate forward due to the flow induced straining. In particular, along directions $p = (1; 0)$ and $(0;1)$, $s_T$ is well-defined again with a value of zero (trapping). A partial homogenization result is also proved. If we consider a similar but relatively simpler Hamiltonian, the trapping occurs along all directions. The analysis is based on the two-player di erential game representation of solutions, selection of game strategies and trapping regions, and construction of connecting trajectories.
strain G-equation, cellular fl
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@inproceedings{jack2014front,
  title={Front Quenching in G-equation Model Induced by Straining of Cellular Flow},
  author={Jack Xin, and Yifeng Yu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160818135517594293230},
  booktitle={Archive for Rational Mechanics and Analysis},
  volume={214},
  number={1},
  year={2014},
}
Jack Xin, and Yifeng Yu. Front Quenching in G-equation Model Induced by Straining of Cellular Flow. 2014. Vol. 214. In Archive for Rational Mechanics and Analysis. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160818135517594293230.
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