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November  2012, 11(6): 2445-2472. doi: 10.3934/cpaa.2012.11.2445

The rate of attraction of super-critical waves in a Fisher-KPP type model with shear flow

Patrick Martinez 1, and  Jean-Michel Roquejoffre 2,

1. 

Institut de Mathématiques (UMR CNRS 5219), Université Paul Sabatier, 31062 Toulouse Cedex 4, France
2. 

Laboratoire MIP, Université Paul Sabatier, 31062 Toulouse Cedex 9

Received  October 2010 Revised  June 2011 Published  April 2012

We consider in this paper the thermo-diffusive model for flame propagation, which is a reaction-diffusion equation of the KPP (Kolmogorov, Petrovskii, Piskunov) type, posed on an infinite cylinder. Such a model has a family of travelling waves of constant speed, larger than a critical speed $c_*$. The family of all supercritical waves attract a large class of initial data, and we try to understand how. We describe in this paper the fate of an initial datum trapped between two supercritical waves of the same velocity: the solution will converge to a whole set of translates of the same wave, and we identify the convergence dynamics as that of an effective drift, around which an effective diffusion process occurs.
Keywords: super critical waves, Fisher-KPP model with shear flow, Fourier analysis, fundamental solution, rate of attraction, stability.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35B35, 35K55, 42A38.
Citation: Patrick Martinez, Jean-Michel Roquejoffre. The rate of attraction of super-critical waves in a Fisher-KPP type model with shear flow. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2445-2472. doi: 10.3934/cpaa.2012.11.2445
References:
[1]

B. Audoly, H. Berestycki and Y. Pomeau, Réaction-diffusion en écoulement rapide, C. R. Acad. Sci.Paris, série II, 328 (2000), 255-262.

[2]

M. Bages, "Équations de réaction-diffusion de type KPP: ondes pulsatoires, dynamique non triviale et applications," Ph.D thesis, Université de Toulouse, 2007.

[3]

M. Bages, P. Martinez and J.-M. Roquejoffre, Dynamique en grand temps pour une classe d'équations de type KPP en milieu périodique, C.R. Acad. Sci. Paris, Ser. I, 346 (2008), 1051-1056.

[4]

M. Bages, P. Martinez and J.-M. Roquejoffre, How travelling waves attract solutions of KPP-type equations, Trans A.M.S., to appear.

[5]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, In "Perspectives in Nonlinear Partial Differential Equations," in honor of H. Brezis, Contemp. Math., 446 Amer. Math. Soc., 101-123.

[6]

F. Benkhaldoun and B. Larrouturou, Numerical analysis of the two-dimensional thermodiffusive model for flame propagation, RAIRO Modèl. Math. Anal. Numér., 22 (1988), 535-560.

[7]

H. Berestycki and B. Larrouturou, Quelques aspects mathématiques de la propagation des flammes prémélangées, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, vol. X, H. Brezis & J.-L. Lions eds., Pitman, London, (1991), 65-129.

[8]

H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré, Anal. non Linéaire, 9 (1992), 497-572.

[9]

M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Memoirs of the AMS, 44 (1983), Providence, R.I.

[10]

P. Collet and J. P. Eckmann, Space-time behaviour in problems of hydrodynamic type: a case study, Nonlinearity, 5 (1992), 1265-1302. doi: 10.1088/0951-7715/5/6/004.

[11]

U. Ebert and W. van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts, Physica D, 146 (2000), 1-99. doi: 10.1016/S0167-2789(00)00068-3.

[12]

A. Fannjiang and G. Papanicolaou, Convection Enhanced Diffusion for Periodic Flows, SIAM J. Appl. Math., 54 (1994), 333-408. doi: 10.1137/S0036139992236785.

[13]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\RR^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163. doi: 10.1007/PL00004238.

[14]

F. Hamel and L. Roques, Uniqueness and stability of monostable pulsating travelling fronts, J. European Math. Soc, to appear.

[15]

F. Hamel and L. Ryzhik, Non-adiabatic KPP fronts with an arbitrary Lewis number, Nonlinearity, 18 (2005), 2881-2902. doi: 10.1088/0951-7715/18/6/024.

[16]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840,1990, Springer.

[17]

A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection, Ann. de l'Inst. Henri Poincaré, C. Analyse non linéaire, 18 (2001), 309-358.

[18]

A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskowskogo Gos. Univ., 17 (1937), 1-26.

[19]

N. Maman and B. Larrouturou, Dynamical mesh adaption for two-dimensional reactive flow simulations, in "Numerical Grid Generation in Computational Fluid Dynamics and Related Fields" (Barcelona, 1991), 13-26, North-Holland, Amsterdam, 1991.

[20]

P. Martinez and J.-M. Roquejoffre, Convergence to critical waves in KPP equations, in preparation.

[21]

J.-F. Mallordy and J.-M. Roquejoffre, A parabolic equation of the KPP type in higher dimensions, SIAM J. Math. Anal., 26 (1995), 1-20. doi: 10.1137/S0036141093246105.

[22]

J. R. Norris, Long-time behaviour of heat flow : global estimates and exact asymptotics, Arch. Rational Mech. Anal., 140 (1997), 161-195. doi: 10.1007/s002050050063.

[23]

J. Ortega and E. Zuazua, Large time behaviour in RN for linear parabolic equations with periodic coefficients, Asymptotic Analysis, 22 (2000), 51-85.

[24]

J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling waves for semi-linear parabolic equations in cylinders, Ann. IHP, Analyse non linéaire, 14 (1997), 499-552.

[25]

K. Uchiyama, The behaviour of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.

[26]

J.-L. Vazqueza and E. Zuazua, Complexity of large time behaviour of evolution equations with bounded data, Chinese Annals of Mathematics, 23, ser. B (2002), 293-310. Special issue in honor of J. L. Lions.

show all references

References:
[1]

B. Audoly, H. Berestycki and Y. Pomeau, Réaction-diffusion en écoulement rapide, C. R. Acad. Sci.Paris, série II, 328 (2000), 255-262.

[2]

M. Bages, "Équations de réaction-diffusion de type KPP: ondes pulsatoires, dynamique non triviale et applications," Ph.D thesis, Université de Toulouse, 2007.

[3]

M. Bages, P. Martinez and J.-M. Roquejoffre, Dynamique en grand temps pour une classe d'équations de type KPP en milieu périodique, C.R. Acad. Sci. Paris, Ser. I, 346 (2008), 1051-1056.

[4]

M. Bages, P. Martinez and J.-M. Roquejoffre, How travelling waves attract solutions of KPP-type equations, Trans A.M.S., to appear.

[5]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, In "Perspectives in Nonlinear Partial Differential Equations," in honor of H. Brezis, Contemp. Math., 446 Amer. Math. Soc., 101-123.

[6]

F. Benkhaldoun and B. Larrouturou, Numerical analysis of the two-dimensional thermodiffusive model for flame propagation, RAIRO Modèl. Math. Anal. Numér., 22 (1988), 535-560.

[7]

H. Berestycki and B. Larrouturou, Quelques aspects mathématiques de la propagation des flammes prémélangées, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, vol. X, H. Brezis & J.-L. Lions eds., Pitman, London, (1991), 65-129.

[8]

H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré, Anal. non Linéaire, 9 (1992), 497-572.

[9]

M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Memoirs of the AMS, 44 (1983), Providence, R.I.

[10]

P. Collet and J. P. Eckmann, Space-time behaviour in problems of hydrodynamic type: a case study, Nonlinearity, 5 (1992), 1265-1302. doi: 10.1088/0951-7715/5/6/004.

[11]

U. Ebert and W. van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts, Physica D, 146 (2000), 1-99. doi: 10.1016/S0167-2789(00)00068-3.

[12]

A. Fannjiang and G. Papanicolaou, Convection Enhanced Diffusion for Periodic Flows, SIAM J. Appl. Math., 54 (1994), 333-408. doi: 10.1137/S0036139992236785.

[13]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\RR^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163. doi: 10.1007/PL00004238.

[14]

F. Hamel and L. Roques, Uniqueness and stability of monostable pulsating travelling fronts, J. European Math. Soc, to appear.

[15]

F. Hamel and L. Ryzhik, Non-adiabatic KPP fronts with an arbitrary Lewis number, Nonlinearity, 18 (2005), 2881-2902. doi: 10.1088/0951-7715/18/6/024.

[16]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840,1990, Springer.

[17]

A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection, Ann. de l'Inst. Henri Poincaré, C. Analyse non linéaire, 18 (2001), 309-358.

[18]

A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskowskogo Gos. Univ., 17 (1937), 1-26.

[19]

N. Maman and B. Larrouturou, Dynamical mesh adaption for two-dimensional reactive flow simulations, in "Numerical Grid Generation in Computational Fluid Dynamics and Related Fields" (Barcelona, 1991), 13-26, North-Holland, Amsterdam, 1991.

[20]

P. Martinez and J.-M. Roquejoffre, Convergence to critical waves in KPP equations, in preparation.

[21]

J.-F. Mallordy and J.-M. Roquejoffre, A parabolic equation of the KPP type in higher dimensions, SIAM J. Math. Anal., 26 (1995), 1-20. doi: 10.1137/S0036141093246105.

[22]

J. R. Norris, Long-time behaviour of heat flow : global estimates and exact asymptotics, Arch. Rational Mech. Anal., 140 (1997), 161-195. doi: 10.1007/s002050050063.

[23]

J. Ortega and E. Zuazua, Large time behaviour in RN for linear parabolic equations with periodic coefficients, Asymptotic Analysis, 22 (2000), 51-85.

[24]

J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling waves for semi-linear parabolic equations in cylinders, Ann. IHP, Analyse non linéaire, 14 (1997), 499-552.

[25]

K. Uchiyama, The behaviour of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.

[26]

J.-L. Vazqueza and E. Zuazua, Complexity of large time behaviour of evolution equations with bounded data, Chinese Annals of Mathematics, 23, ser. B (2002), 293-310. Special issue in honor of J. L. Lions.

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