American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions
  • KRM Home
  • This Issue
  • Next Article
    A kinetic theory description of liquid menisci at the microscale
June  2015, 8(2): 255-280. doi: 10.3934/krm.2015.8.255

A Hamilton-Jacobi approach for front propagation in kinetic equations

Emeric Bouin 1,

1. 

UMR CNRS 5669 `UMPA' and INRIA project `NUMED', École Normale Supérieure de Lyon, 46, allée d'Italie, F-69364 Lyon Cedex 07, France

Received  July 2014 Revised  November 2014 Published  March 2015

In this paper we use the theory of viscosity solutions for Hamilton-Jacobi equations to study propagation phenomena in kinetic equations. We perform the hydrodynamic limit of some kinetic models thanks to an adapted WKB ansatz. Our models describe particles moving according to a velocity-jump process, and proliferating thanks to a reaction term of monostable type. The scattering operator is supposed to satisfy a maximum principle. When the velocity space is bounded, we show, under suitable hypotheses, that the phase converges towards the viscosity solution of some constrained Hamilton-Jacobi equation which effective Hamiltonian is obtained solving a suitable eigenvalue problem in the velocity space. In the case of unbounded velocities, the non-solvability of the spectral problem can lead to different behavior. In particular, a front acceleration phenomena can occur. Nevertheless, we expect that when the spectral problem is solvable one can extend the convergence result.
Keywords: Hopf-Cole transformation, Kinetic equations, geometric optics approximation., front propagation, hyperbolic limit, spectral problem.
Mathematics Subject Classification: Primary: 35Q92, 45K05, 35C0.
Citation: Emeric Bouin. A Hamilton-Jacobi approach for front propagation in kinetic equations. Kinetic & Related Models, 2015, 8 (2) : 255-280. doi: 10.3934/krm.2015.8.255
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Math., 446, Springer, Berlin, 1975, 5-49.  Google Scholar

[2]

C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284 (1984), 617-649. doi: 10.1090/S0002-9947-1984-0743736-0.  Google Scholar

[3]

G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, Mathématiques & Applications (Berlin) [Mathematics & Applications], 17, Springer-Verlag, Paris, 1994.  Google Scholar

[4]

G. Barles, L. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE, Duke Math. J., 61 (1990), 835-858. doi: 10.1215/S0012-7094-90-06132-0.  Google Scholar

[5]

G. Barles, S. Mirrahimi and B. Perthame, Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result, Methods Appl. Anal., 16 (2009), 321-340. doi: 10.4310/MAA.2009.v16.n3.a4.  Google Scholar

[6]

G. Barles and P. E. Souganidis, A remark on the asymptotic behavior of the solution of the KPP equation, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 679-684.  Google Scholar

[7]

E. Bouin and V. Calvez, A kinetic eikonal equation, C. R. Math. Acad. Sci. Paris, 350 (2012), 243-248. doi: 10.1016/j.crma.2012.03.009.  Google Scholar

[8]

, E. Bouin, V. Calvez, E. Grenier and G. Nadin,, work in progress., ().   Google Scholar

[9]

E. Bouin, V. Calvez, N. Meunier, S. Mirrahimi, B. Perthame, G. Raoul and R. Voituriez, Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration, C. R. Math. Acad. Sci. Paris, 350 (2012), 761-766. doi: 10.1016/j.crma.2012.09.010.  Google Scholar

[10]

E. Bouin, V. Calvez and G. Nadin, Front propagation in a kinetic reaction-transport equation, preprint, arXiv:1307.8325, 2013. Google Scholar

[11]

E. Bouin and S. Mirrahimi, A Hamilton-Jacobi approach for a model of population structured by space and trait, preprint, arXiv:1307.8332, 2013. Google Scholar

[12]

R. E. Caflisch and B. Nicolaenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys., 86 (1982), 161-194. doi: 10.1007/BF01206009.  Google Scholar

[13]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[14]

M. G. Crandall, L. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

[15]

C. M. Cuesta, S. Hittmeir and Ch. Schmeiser, travelling waves of a kinetic transport model for the KPP-Fisher equation, SIAM J. Math. Anal., 44 (2012), 4128-4146. doi: 10.1137/100795413.  Google Scholar

[16]

C. Cuesta and C. Schmeiser, Kinetic profiles for shock waves of scalar conservation laws, Bull. of the Inst. of Math., Acad. Sinica, 2 (2007), 391-408.  Google Scholar

[17]

P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49 (2000), 1175-1198.  Google Scholar

[18]

L. C. Evans, Partial Differential Equations, Second edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.  Google Scholar

[19]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. R. Soc. Edinb. Sec. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631.  Google Scholar

[20]

L. C. Evans and P. E. Souganidis, A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana Univ. Math. J., 38 (1989), 141-172. doi: 10.1512/iumj.1989.38.38007.  Google Scholar

[21]

S. Fedotov, travelling waves in a reaction-diffusion system: diffusion with finite velocity and Kolmogorov-Petrovskii-Piskunov kinetics, Phys. Rev. E, 58 (1998), 5143-5145. doi: 10.1103/PhysRevE.58.5143.  Google Scholar

[22]

S. Fedotov, Wave front for a reaction-diffusion system and relativistic Hamilton-Jacobi dynamics, Phys. Rev. E, 59 (1999), 5040-5044. doi: 10.1103/PhysRevE.59.5040.  Google Scholar

[23]

W. H. Fleming and P. E. Souganidis, PDE-viscosity solution approach to some problems of large deviations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1986), 171-192.  Google Scholar

[24]

M. Freidlin, Functional Integration and Partial Differential Equations, Annals of Mathematics Studies, 109, Princeton University Press, Princeton, NJ, 1985.  Google Scholar

[25]

F. Golse, Shock profiles for the Perthame-Tadmor kinetic model, Comm. Partial Differential Equations, 23 (1998), 1857-1874,  Google Scholar

[26]

A. Henkel, J. Müller and C. Pötzsche, Modeling the spread of Phytophthora, J. Math. Biol., 65 (2012), 1359-1385. doi: 10.1007/s00285-011-0492-7.  Google Scholar

[27]

A. Kolmogorov, Über die Analytischen Methoden in der Wahrscheinlichkeitsrechnung, (On Analytical Methods in the Theory of Probability), 1931. Google Scholar

[28]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Moskow Univ. Math. Bull., 1 (1937), 1-25. Google Scholar

[29]

M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Translation, 1950 (1950), 128pp.  Google Scholar

[30]

T.-P. Liu and Shih-Hsien Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2.  Google Scholar

[31]

A. Lorz, S. Mirrahimi and B. Perthame, Dirac mass dynamics in multidimensional nonlocal parabolic equations, Comm. Partial Differential Equations, 36 (2011), 1071-1098. doi: 10.1080/03605302.2010.538784.  Google Scholar

[32]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Basel, 2007.  Google Scholar

[33]

J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguin and P. Silberzan, Directional persistence of chemotactic bacteria in a travelling concentration wave, PNAS, 108 (2011), 16235-16240. doi: 10.1073/pnas.1101996108.  Google Scholar

[34]

P. E. Souganidis, Front propagation: Theory and applications, in Viscosity Solutions and Applications (Montecatini Terme, 1995), Lecture Notes in Math., 1660, Springer, Berlin, 1997, 186-242. doi: 10.1007/BFb0094298.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Math., 446, Springer, Berlin, 1975, 5-49.  Google Scholar

[2]

C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284 (1984), 617-649. doi: 10.1090/S0002-9947-1984-0743736-0.  Google Scholar

[3]

G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, Mathématiques & Applications (Berlin) [Mathematics & Applications], 17, Springer-Verlag, Paris, 1994.  Google Scholar

[4]

G. Barles, L. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE, Duke Math. J., 61 (1990), 835-858. doi: 10.1215/S0012-7094-90-06132-0.  Google Scholar

[5]

G. Barles, S. Mirrahimi and B. Perthame, Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result, Methods Appl. Anal., 16 (2009), 321-340. doi: 10.4310/MAA.2009.v16.n3.a4.  Google Scholar

[6]

G. Barles and P. E. Souganidis, A remark on the asymptotic behavior of the solution of the KPP equation, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 679-684.  Google Scholar

[7]

E. Bouin and V. Calvez, A kinetic eikonal equation, C. R. Math. Acad. Sci. Paris, 350 (2012), 243-248. doi: 10.1016/j.crma.2012.03.009.  Google Scholar

[8]

, E. Bouin, V. Calvez, E. Grenier and G. Nadin,, work in progress., ().   Google Scholar

[9]

E. Bouin, V. Calvez, N. Meunier, S. Mirrahimi, B. Perthame, G. Raoul and R. Voituriez, Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration, C. R. Math. Acad. Sci. Paris, 350 (2012), 761-766. doi: 10.1016/j.crma.2012.09.010.  Google Scholar

[10]

E. Bouin, V. Calvez and G. Nadin, Front propagation in a kinetic reaction-transport equation, preprint, arXiv:1307.8325, 2013. Google Scholar

[11]

E. Bouin and S. Mirrahimi, A Hamilton-Jacobi approach for a model of population structured by space and trait, preprint, arXiv:1307.8332, 2013. Google Scholar

[12]

R. E. Caflisch and B. Nicolaenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys., 86 (1982), 161-194. doi: 10.1007/BF01206009.  Google Scholar

[13]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[14]

M. G. Crandall, L. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

[15]

C. M. Cuesta, S. Hittmeir and Ch. Schmeiser, travelling waves of a kinetic transport model for the KPP-Fisher equation, SIAM J. Math. Anal., 44 (2012), 4128-4146. doi: 10.1137/100795413.  Google Scholar

[16]

C. Cuesta and C. Schmeiser, Kinetic profiles for shock waves of scalar conservation laws, Bull. of the Inst. of Math., Acad. Sinica, 2 (2007), 391-408.  Google Scholar

[17]

P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49 (2000), 1175-1198.  Google Scholar

[18]

L. C. Evans, Partial Differential Equations, Second edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.  Google Scholar

[19]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. R. Soc. Edinb. Sec. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631.  Google Scholar

[20]

L. C. Evans and P. E. Souganidis, A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana Univ. Math. J., 38 (1989), 141-172. doi: 10.1512/iumj.1989.38.38007.  Google Scholar

[21]

S. Fedotov, travelling waves in a reaction-diffusion system: diffusion with finite velocity and Kolmogorov-Petrovskii-Piskunov kinetics, Phys. Rev. E, 58 (1998), 5143-5145. doi: 10.1103/PhysRevE.58.5143.  Google Scholar

[22]

S. Fedotov, Wave front for a reaction-diffusion system and relativistic Hamilton-Jacobi dynamics, Phys. Rev. E, 59 (1999), 5040-5044. doi: 10.1103/PhysRevE.59.5040.  Google Scholar

[23]

W. H. Fleming and P. E. Souganidis, PDE-viscosity solution approach to some problems of large deviations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1986), 171-192.  Google Scholar

[24]

M. Freidlin, Functional Integration and Partial Differential Equations, Annals of Mathematics Studies, 109, Princeton University Press, Princeton, NJ, 1985.  Google Scholar

[25]

F. Golse, Shock profiles for the Perthame-Tadmor kinetic model, Comm. Partial Differential Equations, 23 (1998), 1857-1874,  Google Scholar

[26]

A. Henkel, J. Müller and C. Pötzsche, Modeling the spread of Phytophthora, J. Math. Biol., 65 (2012), 1359-1385. doi: 10.1007/s00285-011-0492-7.  Google Scholar

[27]

A. Kolmogorov, Über die Analytischen Methoden in der Wahrscheinlichkeitsrechnung, (On Analytical Methods in the Theory of Probability), 1931. Google Scholar

[28]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Moskow Univ. Math. Bull., 1 (1937), 1-25. Google Scholar

[29]

M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Translation, 1950 (1950), 128pp.  Google Scholar

[30]

T.-P. Liu and Shih-Hsien Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2.  Google Scholar

[31]

A. Lorz, S. Mirrahimi and B. Perthame, Dirac mass dynamics in multidimensional nonlocal parabolic equations, Comm. Partial Differential Equations, 36 (2011), 1071-1098. doi: 10.1080/03605302.2010.538784.  Google Scholar

[32]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Basel, 2007.  Google Scholar

[33]

J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguin and P. Silberzan, Directional persistence of chemotactic bacteria in a travelling concentration wave, PNAS, 108 (2011), 16235-16240. doi: 10.1073/pnas.1101996108.  Google Scholar

[34]

P. E. Souganidis, Front propagation: Theory and applications, in Viscosity Solutions and Applications (Montecatini Terme, 1995), Lecture Notes in Math., 1660, Springer, Berlin, 1997, 186-242. doi: 10.1007/BFb0094298.  Google Scholar

[1]

Bo Su and Martin Burger. Global weak solutions of non-isothermal front propagation problem. Electronic Research Announcements, 2007, 13: 46-52.

[2]

Margarita Arias, Juan Campos, Cristina Marcelli. Fastness and continuous dependence in front propagation in Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 11-30. doi: 10.3934/dcdsb.2009.11.11
[3]

Luisa Malaguti, Cristina Marcelli, Serena Matucci. Continuous dependence in front propagation of convective reaction-diffusion equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1083-1098. doi: 10.3934/cpaa.2010.9.1083
[4]

Hongyu Liu, Ting Zhou. Two dimensional invisibility cloaking via transformation optics. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 525-543. doi: 10.3934/dcds.2011.31.525
[5]

Piotr B. Mucha. Limit of kinetic term for a Stefan problem. Conference Publications, 2007, 2007 (Special) : 741-750. doi: 10.3934/proc.2007.2007.741
[6]

Arnaud Debussche, Julien Vovelle. Diffusion limit for a stochastic kinetic problem. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2305-2326. doi: 10.3934/cpaa.2012.11.2305
[7]

Julian Koellermeier, Roman Pascal Schaerer, Manuel Torrilhon. A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods. Kinetic & Related Models, 2014, 7 (3) : 531-549. doi: 10.3934/krm.2014.7.531
[8]

Mohar Guha, Keith Promislow. Front propagation in a noisy, nonsmooth, excitable medium. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 617-638. doi: 10.3934/dcds.2009.23.617
[9]

Yana Nec, Vladimir A Volpert, Alexander A Nepomnyashchy. Front propagation problems with sub-diffusion. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 827-846. doi: 10.3934/dcds.2010.27.827
[10]

Qian Liu, Shuang Liu, King-Yeung Lam. Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3683-3714. doi: 10.3934/dcds.2020050
[11]

Daomin Cao, Ezzat S. Noussair, Shusen Yan. On the profile of solutions for an elliptic problem arising in nonlinear optics. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 649-666. doi: 10.3934/dcds.2004.11.649
[12]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[13]

Monica Conti, Vittorino Pata, M. Squassina. Singular limit of dissipative hyperbolic equations with memory. Conference Publications, 2005, 2005 (Special) : 200-208. doi: 10.3934/proc.2005.2005.200
[14]

Naoufel Ben Abdallah, Antoine Mellet, Marjolaine Puel. Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach. Kinetic & Related Models, 2011, 4 (4) : 873-900. doi: 10.3934/krm.2011.4.873
[15]

Benoît Perthame, P. E. Souganidis. Front propagation for a jump process model arising in spacial ecology. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1235-1246. doi: 10.3934/dcds.2005.13.1235
[16]

Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1769-1781. doi: 10.3934/dcdsb.2014.19.1769
[17]

Mikhail Kuzmin, Stefano Ruggerini. Front propagation in diffusion-aggregation models with bi-stable reaction. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 819-833. doi: 10.3934/dcdsb.2011.16.819
[18]

Jong-Shenq Guo, Chang-Hong Wu. Front propagation for a two-dimensional periodic monostable lattice dynamical system. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 197-223. doi: 10.3934/dcds.2010.26.197
[19]

Peter Howard, Bongsuk Kwon. Spectral analysis for transition front solutions in Cahn-Hilliard systems. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 125-166. doi: 10.3934/dcds.2012.32.125
[20]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences