Variational Principles for Propagation Speeds in Inhomogeneous Media

An important problem in reactive flows is how to estimate the speed of front propagation, especially when inhomogeneities are present. Here we prove a variational characterization of the front speed for reaction-diffusion-advectionequations in periodically varying heterogeneous media. This formulation makes it possible to calculate sharp estimates for the speed explicitly. The method can be applied to any problem obeying a maximum principle. Three examples will be analyzed in detail: a shear flow problem, a problem with rapidly oscillating coefficients, and a discretized diffusion problem. In all cases the effects of the inhomogeneous medium on the speed are discussed in comparison to the homogeneous problem. For the shear flow problem, enhancement of the speed results.

[1]  Peter Kuchment,et al.  Waves in Periodic and Random Media , 2003 .

[2]  Henri Berestycki,et al.  Travelling fronts in cylinders , 1992 .

[3]  Erik S. Van Vleck,et al.  Analysis and computation of travelling wave solutions of bistable differential-difference equations , 1999 .

[4]  J. McLeod,et al.  The approach of solutions of nonlinear diffusion equations to travelling front solutions , 1977 .

[5]  Yi Li,et al.  Travelling Fronts in Cylinders and Their Stability , 1997 .

[6]  J. Bell,et al.  Study of propagation along nonuniform excitable fibers. , 1994, Mathematical biosciences.

[7]  James P. Keener,et al.  Homogenization and propagation in the bistable equation , 2000 .

[8]  Adam M. Oberman,et al.  Bulk Burning Rate in¶Passive–Reactive Diffusion , 1999, math/9907132.

[9]  B. Zinner,et al.  Stability of traveling wavefronts for the discrete Nagumo equation , 1991 .

[10]  荻原俊子.  Stability analysis in order-preserving systems in the presence of symmetry(対称性の入った順序保存力学系における安定性解析) , 1996 .

[11]  X. Xin,et al.  Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity , 1991 .

[12]  Brian D. Sleeman,et al.  WAVE FRONT PROPAGATION AND ITS FAILURE IN COUPLED SYSTEMS OF DISCRETE BISTABLE CELLS MODELLED BY FITZHUGH-NAGUMO DYNAMICS , 1995 .

[13]  A. Volpert,et al.  Traveling Wave Solutions of Parabolic Systems: Translations of Mathematical Monographs , 1994 .

[14]  Gábor Fáth,et al.  Propagation failure of traveling waves in a discrete bistable medium , 1997, patt-sol/9711003.

[15]  Jack Xin,et al.  Front Propagation in Heterogeneous Media , 2000, SIAM Rev..

[16]  B. Zinner,et al.  Existence of traveling wavefront solutions for the discrete Nagumo equation , 1992 .

[17]  K. P. Hadeler,et al.  Travelling fronts in nonlinear diffusion equations , 1975 .

[18]  P. Lions,et al.  Multi-dimensional travelling-wave solutions of a flame propagation model , 1990 .

[19]  Jack Xin,et al.  Existence of planar flame fronts in convective-diffusive periodic media , 1992 .

[20]  George Papanicolaou,et al.  Reaction-diffusion fronts in periodically layered media , 1991 .

[21]  F. Hamel.  Formules min-max pour les vitesses d'ondes progressives multidimensionnelles , 1999 .

[22]  J. Roquejoffre.  Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders , 1997 .

[23]  R. Benguria,et al.  Speed of Fronts of the Reaction-Diffusion Equation. , 1995, Physical review letters.

[24]  James P. Keener,et al.  Propagation and its failure in coupled systems of discrete excitable cells , 1987 .

[25]  John Mallet-Paret,et al.  Traveling Wave Solutions for Systems of ODEs on a Two-Dimensional Spatial Lattice , 1998, SIAM J. Appl. Math..

[26]  G. Barles,et al.  A New Approach to Front Propagation Problems: Theory and Applications , 1998 .

[27]  Jack Xin,et al.  Quenching and propagation of bistable reaction-diffusion fronts in multidimensional periodic media , 1995 .