Front Propagation in Heterogeneous Media

A review is presented of recent results on front propagation in reaction-diffusion-advection equations in homogeneous and heterogeneous media. Formal asymptotic expansions and heuristic ideas are used to motivate the results wherever possible. The fronts include constant-speed monotone traveling fronts in homogeneous media, periodically varying traveling fronts in periodic media, and fluctuating and fractal fronts in random media. These fronts arise in a wide range of applications such as chemical kinetics, combustion, biology, transport in porous media, and industrial deposition processes. Open problems are briefly discussed along the way.

[1]  M. Freidlin.  Functional Integration And Partial Differential Equations , 1985 .

[2]  N. Shigesada,et al.  Traveling periodic waves in heterogeneous environments , 1986 .

[3]  M. Bramson.  Convergence of solutions of the Kolmogorov equation to travelling waves , 1983 .

[4]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[5]  P. Souganidis,et al.  Differential Games and Representation Formulas for Solutions of Hamilton-Jacobi-Isaacs Equations. , 1983 .

[6]  L. Nirenberg,et al.  Some Qualitative Properties of Solutions of Semilinear Elliptic Equations in Cylindrical Domains , 1990 .

[7]  Andrew J. Majda,et al.  Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales , 1994 .

[8]  Todd Kapitula,et al.  STABILITY OF TRAVELING WAVES FOR NONCONVEX SCALAR VISCOUS CONSERVATION-LAWS , 1993 .

[9]  Zhang,et al.  Dynamic scaling of growing interfaces. , 1986, Physical review letters.

[10]  P. Ronney.  Some open issues in premixed turbulent combustion , 1995 .

[11]  F. Torcaso,et al.  Wave propagation in a lattice KPP equation in random media , 1997 .

[12]  Kapral,et al.  Biscale chaos in propagating fronts. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  Jürgen Gärtner,et al.  Bistable Reaction‐Diffusion Equations and Excitable Media , 1983 .

[14]  John Billingham,et al.  The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form travelling waves , 1991, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[15]  Henri Berestycki,et al.  Traveling Wave Solutions to Combustion Models and Their Singular Limits , 1985 .

[16]  S. Scott,et al.  Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis , 1994, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[17]  Jack Xin,et al.  Front Speed in the Burgers Equation with a Random Flux , 1997 .

[18]  P. Clavin,et al.  Linear Stability Analysis of Nonadiabatic Flames: Diffusional-Thermal Model , 1979 .

[19]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[20]  Kerstein,et al.  Propagation rate of growing interfaces in stirred fluids. , 1992, Physical review letters.

[21]  P. Fife.  Dynamics of Internal Layers and Diffusive Interfaces , 1988 .

[22]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[23]  D. Aronson,et al.  Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation , 1975 .

[24]  P. Souganidis,et al.  A PDE approach to geometric optics for certain semilinear parabolic equations , 1989 .

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

[26]  Structural stability and selection of propagating fronts in semilinear parabolic partial differential equations. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[27]  Bounds on enhanced turbulent flame speeds for combustion with fractal velocity fields , 1996 .

[28]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[29]  U. Seifert,et al.  Front propagation in the pearling instability of tubular vesicles , 1995, cond-mat/9510093.

[30]  R. Benguria,et al.  Validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation. , 1994, Physical review letters.

[31]  J. Xin.  Stability of Traveling Waves of a Solute Transport Equation , 1997 .

[32]  G. Barles,et al.  Wavefront propagation for reaction-diffusion systems of PDE , 1990 .

[33]  Klaus Kirchgässner,et al.  On the nonlinear dynamics of travelling fronts , 1992 .

[34]  Andrew J. Majda,et al.  Effective geometric front dynamics for premixed turbulent combustion with separated velocity scales , 1994 .

[35]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[36]  J. Keller.  Genetic variability due to geographical inhomogeneity , 1984 .

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

[38]  G. Marsily.  Quantitative Hydrogeology: Groundwater Hydrology for Engineers , 1986 .

[39]  L. Evans.  Periodic homogenisation of certain fully nonlinear partial differential equations , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[40]  Norbert Peters,et al.  A spectral closure for premixed turbulent combustion in the flamelet regime , 1992, Journal of Fluid Mechanics.

[41]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[42]  J. Hale,et al.  Stable Equilibria in a Scalar Parabolic Equation with Variable Diffusion , 1985 .

[43]  R. Martin,et al.  Nonlinear Reaction-Diffusion Systems , 1992 .

[44]  Vitaly Volpert,et al.  Traveling Wave Solutions of Parabolic Systems , 1994 .

[45]  F. Williams,et al.  Theory of premixed-flame propagation in large-scale turbulence , 1979, Journal of Fluid Mechanics.

[46]  L. Richardson,et al.  Atmospheric Diffusion Shown on a Distance-Neighbour Graph , 1926 .

[47]  Nigel Goldenfeld,et al.  Selection, stability and renormalization , 1994 .

[48]  J. Xin.  Multidimensional Stability of Traveling Waves in a Bistable Reaction–Diffusion Equation, I , 1992 .

[49]  John Billingham,et al.  A note on the properties of a family of travelling-wave solutions arising in cubic autocatalysis , 1991 .

[50]  Gregory I. Sivashinsky,et al.  Cascade-renormalization theory of turbulent flame speed , 1988 .

[51]  J. Xin,et al.  A numerical study of fronts in random media using a reactive solute transport model , 1997 .

[52]  R. Durrett.  Probability: Theory and Examples , 1993 .

[53]  T. Gallay,et al.  Existence and stability of propagating fronts for an autocatalytic reaction-diffusion system , 1997, patt-sol/9705008.

[54]  ECOLE POLYTECHNIQUE , 1994 .

[55]  A. Kerstein,et al.  Field equation for interface propagation in an unsteady homogeneous flow field. , 1988, Physical review. A, General physics.

[56]  Wave propagation in a lattice KPP equation in random media , 1998 .

[57]  George Fennemore,et al.  Wetting Fronts in One-Dimensional Periodically Layered Soils , 1998, SIAM J. Appl. Math..

[58]  Henri Berestycki,et al.  On the method of moving planes and the sliding method , 1991 .

[59]  P. Souganidis,et al.  A PDE approach to certain large deviation problems for systems of parabolic equations , 1989 .

[60]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[61]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

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

[63]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[64]  Sjoerd E. A. T. M. van der Zee,et al.  Transport of reacting solute in a one-dimensional, chemically heterogeneous porous medium. , 1993 .

[65]  J. Bricmont,et al.  Stability of moving fronts in the Ginzburg-Landau equation , 1993, chao-dyn/9306009.

[66]  H. Osada.  Diffusion processes with generators of generalized divergence form , 1987 .

[67]  David H. Sattinger,et al.  On the stability of waves of nonlinear parabolic systems , 1976 .

[68]  Srinivasa Varadhan,et al.  Asymptotics for the wiener sausage , 1975 .

[69]  Renormalization group theory and variational calculations for propagating fronts. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

[71]  Robert H. Kraichnan,et al.  Small‐Scale Structure of a Scalar Field Convected by Turbulence , 1968 .

[72]  M. Marion,et al.  Qualitative properties of a nonlinear system for laminar flames without ignition temperature , 1985 .

[73]  K. Showalter,et al.  Instabilities in propagating reaction-diffusion fronts , 1993 .

[74]  Björn Sandstede,et al.  Stability of N -fronts bifurcating from a twisted heteroclinic loop and an application to the FitzHugh-Nagumo equation , 1998 .

[75]  S. Tuljapurkar,et al.  Mathematical Aspects of Reacting and Diffusing Systems. Lecture Notes in Biomathematics, Volume 28.Paul C. Fife , S. Levin , 1980 .

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

[77]  Shunsaku Nii.  Stability of travelling multiple-front (multiple-back) wave solutions of the Fitzhugh-Nagumo equations , 1997 .

[78]  Jack Xin,et al.  Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media , 1993 .

[79]  M. Freidlin.  Limit Theorems for Large Deviations and Reaction-Diffusion Equations , 1985 .

[80]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[81]  A. Fannjiang,et al.  Convection-enhanced diffusion for random flows , 1997 .

[82]  P. Knabner,et al.  Solute transport in porous media with equilibrium and non-equilibrium multiple-site adsorpition: Travelling weaves. , 1991 .

[83]  L. Sainsaulieu,et al.  On the stability of multiple steady planar flames when the Lewis number is lees than 1 , 1993 .

[84]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[85]  C. J. van Duijn,et al.  Large Time Asymptotics in Contaminant Transport in Porous Media , 1996, SIAM J. Appl. Math..

[86]  Victor Yakhot,et al.  Propagation Velocity of Premixed Turbulent Flames , 1988 .

[87]  G. Sivashinsky,et al.  Instabilities, Pattern Formation, and Turbulence in Flames , 1983 .

[88]  E. Perkins,et al.  The compact support property for solutions to the heat equation with noise , 1992 .

[89]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

[90]  John Billingham,et al.  The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. III. Large time development in quadratic autocatalysis , 1992 .

[91]  A. Volpert,et al.  Existence and stability of stationary solutions for a class of semilinear parabolic systems , 1993 .

[92]  J. Xin,et al.  Global Solutions to a Reactive Boussinesq System with Front Data on an Infinite Domain , 1998 .

[93]  Bertram Zinner,et al.  EXISTENCE OF TRAVELING WAVES FOR REACTION DIFFUSION EQUATIONS OF FISHER TYPE IN PERIODIC MEDIA , 1995 .

[94]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[95]  J. Pauwelussen.  Nerve impulse propagation in a branching nerve system: A simple model , 1981 .

[96]  Gregory I. Sivashinsky,et al.  SHORT COMMUNICATION On Flame Propagation Through Periodic Flow Fields , 1991 .

[97]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[98]  C. V. Duijn,et al.  Plume development of a nonlinearly adsorbing solute in heterogeneous porous formations , 1996 .

[99]  D. Aronson,et al.  Multidimensional nonlinear di u-sion arising in population genetics , 1978 .

[100]  A. Barabasi,et al.  Fractal concepts in surface growth , 1995 .

[101]  A. Fannjiang,et al.  Diffusion in turbulence , 1996 .

[102]  Stanley Osher,et al.  L1 stability of travelling waves with applications to convective porous media flow , 1982 .

[103]  J. Xin.  Existence of multidimensional traveling waves in the transport of reactive solutes through periodic porous media , 1994 .

[104]  G. Matheron,et al.  Is transport in porous media always diffusive? A counterexample , 1980 .

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

[106]  J. Eckmann,et al.  The non-linear stability of front solutions for parabolic partial differential equations , 1994 .

[107]  Jack Xin,et al.  White noise perturbation of the viscous shock fronts of the Burgers equation , 1996 .

[108]  J. Roquejoffre,et al.  Global stability of traveling fronts and convergence towards stacked families of waves in monotone parabolic systems , 1996 .

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

[110]  J. Keller,et al.  Fast reaction, slow diffusion, and curve shortening , 1989 .

[111]  J. Roquejoffre.  Stability of travelling fronts in a model for flame propagation Part II: Nonlinear stability , 1992 .

[112]  S.E.A.T.M. van der Zee,et al.  Transport of reactive solute in spatially variable soil systems , 1987 .

[113]  H. McKean.  Application of brownian motion to the equation of kolmogorov-petrovskii-piskunov , 1975 .

[114]  G. Sivashinsky,et al.  Pockets in Premixed Flames and Combustion Rate , 1991 .

[115]  Christopher Jones,et al.  Stability of the travelling wave solution of the FitzHugh-Nagumo system , 1984 .

[116]  Nikolas Provatas,et al.  Scaling, propagation, and kinetic roughening of flame fronts in random media , 1995, cond-mat/9506017.

[117]  M. Freidlin,et al.  Wave front propagation and large deviations for diffusion –transmutation process , 1996 .

[118]  Jean-Michel Roquejoffre,et al.  A parabolic equation of the KPP type in higher dimensions , 1995 .

[119]  X. Xin,et al.  Existence and uniqueness of travelling waves in a reaction−diffusion equation with combustion nonlinearity , 1991 .

[120]  Andrew J. Majda,et al.  Mathematical models with exact renormalization for turbulent transport , 1990 .

[121]  Rhys,et al.  Front propagation rates in randomly stirred media. , 1995, Physical review letters.

[122]  J. Bricmont,et al.  Renormalizing partial differential equations , 1994, chao-dyn/9411015.

[123]  J. Roquejoffre,et al.  Stability of travelling fronts in a model for flame propagation part I: Linear analysis , 1992 .

[124]  S. Heinze.  Wave Solutions to Reaction-Diffusion Systems in Perforated Domains , 2001 .

[125]  T. Gallay,et al.  Local stability of critical fronts in nonlinear parabolic partial differential equations , 1994 .

[126]  G. Batchelor,et al.  The theory of homogeneous turbulence , 1954 .

[127]  R. Gardner.  Existence of multidimensional travelling wave solutions of an initial-boundary value problem , 1986 .

[128]  Paul D. Ronney,et al.  Fractal properties of propagating fronts in a strongly stirred fluid , 1995 .

[129]  Clinton N Dawson,et al.  ASYMPTOTIC PROFILES WITH FINITE MASS IN ONE-DIMENSIONAL CONTAMINANT TRANSPORT THROUGH POROUS MEDIA: THE FAST REACTION CASE , 1994 .

[130]  Richard B. Sowers,et al.  Random Travelling Waves for the KPP Equation with Noise , 1995 .

[131]  J. Roquejoffre.  Convergence to Travelling Waves for Solutions of a Class of Semilinear Parabolic Equations , 1994 .

[132]  J. Xin,et al.  Global large time self-similarity of a thermal-diffusive combustion system with critical nonlinearity , 1995 .

[133]  L. Rogers,et al.  Diffusions, Markov processes, and martingales , 1979 .

[134]  Andrew J. Majda,et al.  Comparison of turbulent flame speeds from complete averaging and the G‐equation , 1995 .

[135]  D. Stroock,et al.  A new proof of Moser's parabolic harnack inequality using the old ideas of Nash , 1986 .

[136]  J. Nash.  Continuity of Solutions of Parabolic and Elliptic Equations , 1958 .

[137]  Andrew J. Majda,et al.  Mathematical models with exact renormalization for turbulent transport, II: Fractal interfaces, non-Gaussian statistics and the sweeping effect , 1992 .

[138]  S. Varadhan.  Large Deviations and Applications , 1984 .

[139]  M. Freidlin,et al.  Large deviation principle for the diffusion-transmutation processes and dirichlet problem for PDE systems with small parameter , 1996 .

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

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

[142]  R. Benguria,et al.  Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation , 1994, patt-sol/9408001.