Front propagation in heterogeneous media.

*(English)*Zbl 0951.35060In the article, the author presents a review of recent results on front propagation in reaction-diffusion-advection equations in homogeneous and heterogeneous media.

Beginning with the well-studied scalar homogeneous reaction-diffusion equations, the author exposes the basic properties of front solution, such as front existence and stability, front speed selection, and variational characterization. Many of these properties carry over to heterogeneous fronts.

A new theme associated with fronts in heterogeneous media is the understanding of multiple scales and their interaction. The author discusses how to apply homogenization ideas to front problems for the case of periodic media. Basic ideas of homogenization theory explained through concrete examples serves as useful guides.

Front propagation in random media are much less well understood and yet a largely open area. Randomness is a more physical and practical assumption in modeling heterogeneous porous structures and turbulent flows than periodicity. The author presents recent rigorous results on two noisy Burgers equations and three noisy Kolmogorov-Petrovsky-Piskunov equations. A new phenomenon is that in addition to the averaged front speed provided by a successful extension of homogenization into the random setting, front locations are random and undergo diffusion about the mean positions.

Another new phenomenon is that of front acceleration through a rough (on large scale) turbulent velocity field and the resulting anomalous scaling limits. The author also describes the related modeling activities in studying premixed turbulent flames as well as other noisy dynamics involving fronts.

Open problems are briefly discussed along the way.

Beginning with the well-studied scalar homogeneous reaction-diffusion equations, the author exposes the basic properties of front solution, such as front existence and stability, front speed selection, and variational characterization. Many of these properties carry over to heterogeneous fronts.

A new theme associated with fronts in heterogeneous media is the understanding of multiple scales and their interaction. The author discusses how to apply homogenization ideas to front problems for the case of periodic media. Basic ideas of homogenization theory explained through concrete examples serves as useful guides.

Front propagation in random media are much less well understood and yet a largely open area. Randomness is a more physical and practical assumption in modeling heterogeneous porous structures and turbulent flows than periodicity. The author presents recent rigorous results on two noisy Burgers equations and three noisy Kolmogorov-Petrovsky-Piskunov equations. A new phenomenon is that in addition to the averaged front speed provided by a successful extension of homogenization into the random setting, front locations are random and undergo diffusion about the mean positions.

Another new phenomenon is that of front acceleration through a rough (on large scale) turbulent velocity field and the resulting anomalous scaling limits. The author also describes the related modeling activities in studying premixed turbulent flames as well as other noisy dynamics involving fronts.

Open problems are briefly discussed along the way.

Reviewer: Vladimir N.Grebenev (Novosibirsk)

##### MSC:

35K55 | Nonlinear parabolic equations |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |

35K57 | Reaction-diffusion equations |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |