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A kinetic theory description of liquid menisci at the microscale
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 |
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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