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June  2015, 8(2): 255-280. doi: 10.3934/krm.2015.8.255

A Hamilton-Jacobi approach for front propagation in kinetic equations

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.
Citation: Emeric Bouin. A Hamilton-Jacobi approach for front propagation in kinetic equations. Kinetic and 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[8]

, E. Bouin, V. Calvez, E. Grenier and G. Nadin, work in progress.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[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.

[27]

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

[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.

[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.

[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.

[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.

[32]

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

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[8]

, E. Bouin, V. Calvez, E. Grenier and G. Nadin, work in progress.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[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.

[27]

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

[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.

[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.

[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.

[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.

[32]

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

[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.

[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.

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