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Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos
Chaotic distributions for relativistic particles
1. | Department of Mathematical sciences, Chalmers University of Technology and the University of Gothenburg, 412 96 GÖTEBORG, Sweden, Sweden |
References:
[1] |
E. A. Carlen, M. C. Carvalho, J. Le Roux, M. Loss and C. Villani, Entropy and chaos in the Kac model, Kinet. Relat. Models, 3 (2010), 85-122.
doi: 10.3934/krm.2010.3.85. |
[2] |
E. A. Carlen, P. Degond and B. Wennbrg, Kinetic limits for pair-interaction driven master equations and biological swarm models, Math. Models Methods Appl. Sci., 23 (2013), 1339-1376.
doi: 10.1142/S0218202513500115. |
[3] |
K. Carrapatoso, Quantitative and qualitative Kac's chaos on the Boltzmann's sphere, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 993-1039.
doi: 10.1214/14-AIHP612. |
[4] |
C. Cercignani and G. Medeiros Kremer, The Relativistic Boltzmann Equation: Theory and Applications, Birkhäuser Verlag, Basel, 2002.
doi: 10.1007/978-3-0348-8165-4. |
[5] |
J. T. Chang D. Pollard, Conditioning as disintegration, Statist. Neerlandica, 51 (1997), 287-317.
doi: 10.1111/1467-9574.00056. |
[6] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition, CRC Press, 2015. |
[7] |
I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013. |
[8] |
R. V. Gamkrelidze, Integral representations and asymptotic methods, in Encyclopaedia of Mathematical Sciences, Springer-Verlag, 1989.
doi: 10.1007/978-3-642-61310-4. |
[9] |
M. Kac, Foundations of kinetic theory, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3 (1956), 171-197. |
[10] |
O. E. Landford III, Time evolution of large classical systems, Dynamical Systems, Theory and Applications, 38 (1975), 1-111. |
[11] |
T. Lelièvre, M. Rousset and G. Stoltz, Free Energy Computations, Imperial College Press, London, 2010.
doi: 10.1142/9781848162488. |
[12] |
M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Rev. Math. Phys., 26 (2014), 1450001, 64pp.
doi: 10.1142/S0129055X14500019. |
[13] |
A.-S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989, 1464 (1991), 165-251.
doi: 10.1007/BFb0085169. |
[14] |
R. M. Strain, Coordinates in the relativistic Boltzmann theory, Kinet. Relat. Models, 4 (2011), 345-359.
doi: 10.3934/krm.2011.4.345. |
[15] |
R. M. Strain and S.-B. Yun, Spatially homogeneous Boltzmann equation for relativistic particles, SIAM J. Math. Anal., 46 (2014), 917-938.
doi: 10.1137/130923531. |
[16] |
M. Toda, R. Kubo and N. Saitô, Statistical Physics I, Equilibrium Statistical Mechanics, 2nd edition, Springer Verlag, 1992. |
show all references
References:
[1] |
E. A. Carlen, M. C. Carvalho, J. Le Roux, M. Loss and C. Villani, Entropy and chaos in the Kac model, Kinet. Relat. Models, 3 (2010), 85-122.
doi: 10.3934/krm.2010.3.85. |
[2] |
E. A. Carlen, P. Degond and B. Wennbrg, Kinetic limits for pair-interaction driven master equations and biological swarm models, Math. Models Methods Appl. Sci., 23 (2013), 1339-1376.
doi: 10.1142/S0218202513500115. |
[3] |
K. Carrapatoso, Quantitative and qualitative Kac's chaos on the Boltzmann's sphere, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 993-1039.
doi: 10.1214/14-AIHP612. |
[4] |
C. Cercignani and G. Medeiros Kremer, The Relativistic Boltzmann Equation: Theory and Applications, Birkhäuser Verlag, Basel, 2002.
doi: 10.1007/978-3-0348-8165-4. |
[5] |
J. T. Chang D. Pollard, Conditioning as disintegration, Statist. Neerlandica, 51 (1997), 287-317.
doi: 10.1111/1467-9574.00056. |
[6] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition, CRC Press, 2015. |
[7] |
I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013. |
[8] |
R. V. Gamkrelidze, Integral representations and asymptotic methods, in Encyclopaedia of Mathematical Sciences, Springer-Verlag, 1989.
doi: 10.1007/978-3-642-61310-4. |
[9] |
M. Kac, Foundations of kinetic theory, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3 (1956), 171-197. |
[10] |
O. E. Landford III, Time evolution of large classical systems, Dynamical Systems, Theory and Applications, 38 (1975), 1-111. |
[11] |
T. Lelièvre, M. Rousset and G. Stoltz, Free Energy Computations, Imperial College Press, London, 2010.
doi: 10.1142/9781848162488. |
[12] |
M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Rev. Math. Phys., 26 (2014), 1450001, 64pp.
doi: 10.1142/S0129055X14500019. |
[13] |
A.-S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989, 1464 (1991), 165-251.
doi: 10.1007/BFb0085169. |
[14] |
R. M. Strain, Coordinates in the relativistic Boltzmann theory, Kinet. Relat. Models, 4 (2011), 345-359.
doi: 10.3934/krm.2011.4.345. |
[15] |
R. M. Strain and S.-B. Yun, Spatially homogeneous Boltzmann equation for relativistic particles, SIAM J. Math. Anal., 46 (2014), 917-938.
doi: 10.1137/130923531. |
[16] |
M. Toda, R. Kubo and N. Saitô, Statistical Physics I, Equilibrium Statistical Mechanics, 2nd edition, Springer Verlag, 1992. |
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