Long time asymptotics of a degenerate linear kinetic transport equation

Previous Article
Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation
KRM Home
This Issue
Next Article
Stability of solutions of kinetic equations corresponding to the replicator dynamics

2014, 7(1): 79-108. doi: 10.3934/krm.2014.7.79

Long time asymptotics of a degenerate linear kinetic transport equation

1.	IMPAN, ul. Śniadeckich 8, 00-956 Warsaw, Poland

Received January 2013 Revised June 2013 Published December 2013

Abstract
Related Papers

In the present article we prove an algebraic rate of decay towards the equilibrium for the solution of a non-homogeneous, linear kinetic transport equation. The estimate is of the form $C(1+t)^{-a}$ for some $a>0$. The total scattering cross-section $R(k)$ is allowed to degenerate but we assume that $R^{-a}(k)$ is integrable with respect to the invariant measure.

Keywords: Kinetic transport equation, scattering kernel., Fredholm determinant.

Mathematics Subject Classification: Primary: 82C40, 82C70; Secondary: 35B40, 82D7.

Citation: Tomasz Komorowski. Long time asymptotics of a degenerate linear kinetic transport equation. Kinetic & Related Models, 2014, 7 (1) : 79-108. doi: 10.3934/krm.2014.7.79

References:

[1]	M. Abramowitz and I. Stegun, Handbook of Mathematical Functions,, Tenth priniting, (1972).
[2]	K. Aoki and F. Golse, On the speed of approach to equilibrium for a collisionless gas,, Kinet. Relat. Models, 4 (2011), 87. doi: 10.3934/krm.2011.4.87.
[3]	L. Arkeryd, Stability in $L^1$ for the spatially homogeneous Boltzmann equation,, Arch. Rat. Mech. Anal., 103* (1988), 151. doi: 10.1007/BF00251506.*
[4]	C. Baranger and C. Mouhot, Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials,, Rev. Mat. Iberoamericana, 21 (2005), 819. doi: 10.4171/RMI/436.
[5]	A. Bensoussan, J-L. Lions and G. C. Papanicolaou, Boundary layers and homogenization of transport processes,, Publ. Res. Inst. Math. Sci., 15 (1979), 53. doi: 10.2977/prims/1195188427.
[6]	M. Caceres, J. Carrillo and T. Goudon, Equilibration Rate for the Linear Inhomogeneous Relaxation-Time Boltzmann Equation for Charged Particles,, Comm in PDE, 28 (2003), 969. doi: 10.1081/PDE-120021182.
[7]	P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro- reversible processes,, Indiana Univ. Math. J., 49 (2000), 1175.
[8]	L. Desvillettes and S. Salvarani, Asymptotic behavior of degenerate linear transport equations,, Bull. Sci. Math., 133* (2009), 848. doi: 10.1016/j.bulsci.2008.09.001.*
[9]	L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation,, Comm. Pure Appl. Math., 54 (2001), 1. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.
[10]	L. Desvillettes and C. Villani, Rate of convergence toward the equilibrium in degenerate settings,, "WASCOM 2003'', (2003), 153. doi: 10.1142/9789812702937_0020.
[11]	G. Doetsch, Introduction to the theory and application of the Laplace transform,, Translated from the second German edition by Walter Nader. Springer-Verlag, (1974).
[12]	J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass,, available at , (2010).
[13]	R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations,, Nonlinearity, 24 (2011), 2165. doi: 10.1088/0951-7715/24/8/003.
[14]	N. Dunford and J. T. Schwartz, Linear Operators. Part I,, General theory. With the assistance of William G. Bade and Robert G. Bartle. Reprint of the 1958 original. Wiley Classics Library. A Wiley-Interscience Publication. Wiley & Sons, (1958).
[15]	K. Engel and R. Nagel, One-Parameter Semigroups for linear Evolution Equations,, Graduate Texts in Mathematics, 194 (2000).
[16]	S. Ethier and T. Kurtz, Markov Processes,, Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley & Sons, (1986). doi: 10.1002/9780470316658.
[17]	S. R. Foguel, The Ergodic Theory of Markov processes,, Van Nostrand Mathematical Studies, (1969).
[18]	Y. Katznelson, An Introduction to Harmonic Analysis,, Third edition. Cambridge Mathematical Library. Cambridge University Press, (2004).
[19]	P. D. Lax, Functional Analysis,, Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], (2002).
[20]	A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations,, Arch. Rat. Mech. Anal., 199* (2011), 493. doi: 10.1007/s00205-010-0354-2.*
[21]	W. R. Rudin, Principles of Mathematical Analysis,, Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., (1976).
[22]	W. R. Rudin, Real and Complex Analysis,, Third Edition, (1987).
[23]	E. M. Stein, Singular Integrals and Differentiability Properties of Functions., Princeton Mathematical Series, 30 (1970).
[24]	T. Tsuji, K. Aoki and F. Golse, Relaxation of a free-molecular gas to equilibrium caused by interaction with vessel wall,, J. Stat Phys, 140* (2010), 518. doi: 10.1007/s10955-010-9997-5.*
[25]	S. Ukai, On the existence of global solutions of mixed problems for non-linear Boltzmann equation,, Proc. Japan Acad., 50 (1974), 179. doi: 10.3792/pja/1195519027.
[26]	A. C. Zaanen, Integration. Completely Revised Edition of An Introduction to the Theory of Integration,, North-Holland Publishing Co., (1967).
[27]	F. Zhang, Matrix Theory Basic Results and Techniques,, Second Ed. Universitext, (2011). doi: 10.1007/978-1-4614-1099-7.

show all references

References:

[1]	M. Abramowitz and I. Stegun, Handbook of Mathematical Functions,, Tenth priniting, (1972).
[2]	K. Aoki and F. Golse, On the speed of approach to equilibrium for a collisionless gas,, Kinet. Relat. Models, 4 (2011), 87. doi: 10.3934/krm.2011.4.87.
[3]	L. Arkeryd, Stability in $L^1$ for the spatially homogeneous Boltzmann equation,, Arch. Rat. Mech. Anal., 103* (1988), 151. doi: 10.1007/BF00251506.*
[4]	C. Baranger and C. Mouhot, Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials,, Rev. Mat. Iberoamericana, 21 (2005), 819. doi: 10.4171/RMI/436.
[5]	A. Bensoussan, J-L. Lions and G. C. Papanicolaou, Boundary layers and homogenization of transport processes,, Publ. Res. Inst. Math. Sci., 15 (1979), 53. doi: 10.2977/prims/1195188427.
[6]	M. Caceres, J. Carrillo and T. Goudon, Equilibration Rate for the Linear Inhomogeneous Relaxation-Time Boltzmann Equation for Charged Particles,, Comm in PDE, 28 (2003), 969. doi: 10.1081/PDE-120021182.
[7]	P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro- reversible processes,, Indiana Univ. Math. J., 49 (2000), 1175.
[8]	L. Desvillettes and S. Salvarani, Asymptotic behavior of degenerate linear transport equations,, Bull. Sci. Math., 133* (2009), 848. doi: 10.1016/j.bulsci.2008.09.001.*
[9]	L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation,, Comm. Pure Appl. Math., 54 (2001), 1. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.
[10]	L. Desvillettes and C. Villani, Rate of convergence toward the equilibrium in degenerate settings,, "WASCOM 2003'', (2003), 153. doi: 10.1142/9789812702937_0020.
[11]	G. Doetsch, Introduction to the theory and application of the Laplace transform,, Translated from the second German edition by Walter Nader. Springer-Verlag, (1974).
[12]	J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass,, available at , (2010).
[13]	R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations,, Nonlinearity, 24 (2011), 2165. doi: 10.1088/0951-7715/24/8/003.
[14]	N. Dunford and J. T. Schwartz, Linear Operators. Part I,, General theory. With the assistance of William G. Bade and Robert G. Bartle. Reprint of the 1958 original. Wiley Classics Library. A Wiley-Interscience Publication. Wiley & Sons, (1958).
[15]	K. Engel and R. Nagel, One-Parameter Semigroups for linear Evolution Equations,, Graduate Texts in Mathematics, 194 (2000).
[16]	S. Ethier and T. Kurtz, Markov Processes,, Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley & Sons, (1986). doi: 10.1002/9780470316658.
[17]	S. R. Foguel, The Ergodic Theory of Markov processes,, Van Nostrand Mathematical Studies, (1969).
[18]	Y. Katznelson, An Introduction to Harmonic Analysis,, Third edition. Cambridge Mathematical Library. Cambridge University Press, (2004).
[19]	P. D. Lax, Functional Analysis,, Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], (2002).
[20]	A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations,, Arch. Rat. Mech. Anal., 199* (2011), 493. doi: 10.1007/s00205-010-0354-2.*
[21]	W. R. Rudin, Principles of Mathematical Analysis,, Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., (1976).
[22]	W. R. Rudin, Real and Complex Analysis,, Third Edition, (1987).
[23]	E. M. Stein, Singular Integrals and Differentiability Properties of Functions., Princeton Mathematical Series, 30 (1970).
[24]	T. Tsuji, K. Aoki and F. Golse, Relaxation of a free-molecular gas to equilibrium caused by interaction with vessel wall,, J. Stat Phys, 140* (2010), 518. doi: 10.1007/s10955-010-9997-5.*
[25]	S. Ukai, On the existence of global solutions of mixed problems for non-linear Boltzmann equation,, Proc. Japan Acad., 50 (1974), 179. doi: 10.3792/pja/1195519027.
[26]	A. C. Zaanen, Integration. Completely Revised Edition of An Introduction to the Theory of Integration,, North-Holland Publishing Co., (1967).
[27]	F. Zhang, Matrix Theory Basic Results and Techniques,, Second Ed. Universitext, (2011). doi: 10.1007/978-1-4614-1099-7.

[1]	Wolfgang Wagner. Some properties of the kinetic equation for electron transport in semiconductors. Kinetic & Related Models, 2013, 6 (4) : 955-967. doi: 10.3934/krm.2013.6.955
[2]	John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181
[3]	Yin Yang, Yunqing Huang. Spectral Jacobi-Galerkin methods and iterated methods for Fredholm integral equations of the second kind with weakly singular kernel. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 685-702. doi: 10.3934/dcdss.2019043
[4]	Pedro Teixeira. Dacorogna-Moser theorem on the Jacobian determinant equation with control of support. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4071-4089. doi: 10.3934/dcds.2017173
[5]	Alfredo Lorenzi, Eugenio Sinestrari. Identifying a BV-kernel in a hyperbolic integrodifferential equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1199-1219. doi: 10.3934/dcds.2008.21.1199
[6]	Charles Nguyen, Stephen Pankavich. A one-dimensional kinetic model of plasma dynamics with a transport field. Evolution Equations & Control Theory, 2014, 3 (4) : 681-698. doi: 10.3934/eect.2014.3.681
[7]	Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283
[8]	Indranil Biswas, Georg Schumacher, Lin Weng. Deligne pairing and determinant bundle. Electronic Research Announcements, 2011, 18: 91-96. doi: 10.3934/era.2011.18.91
[9]	Philippe Laurençot, Barbara Niethammer, Juan J.L. Velázquez. Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel. Kinetic & Related Models, 2018, 11 (4) : 933-952. doi: 10.3934/krm.2018037
[10]	Fabrizio Colombo, Davide Guidetti. Identification of the memory kernel in the strongly damped wave equation by a flux condition. Communications on Pure & Applied Analysis, 2009, 8 (2) : 601-620. doi: 10.3934/cpaa.2009.8.601
[11]	Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285
[12]	Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022
[13]	Kazuhiro Ishige, Tatsuki Kawakami, Kanako Kobayashi. Global solutions for a nonlinear integral equation with a generalized heat kernel. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 767-783. doi: 10.3934/dcdss.2014.7.767
[14]	Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[15]	Jorge Clarke, Christian Olivera, Ciprian Tudor. The transport equation and zero quadratic variation processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2991-3002. doi: 10.3934/dcdsb.2016083
[16]	Alexander Bobylev, Raffaele Esposito. Transport coefficients in the $2$-dimensional Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 789-800. doi: 10.3934/krm.2013.6.789
[17]	Bertram Düring, Ansgar Jüngel, Lara Trussardi. A kinetic equation for economic value estimation with irrationality and herding. Kinetic & Related Models, 2017, 10 (1) : 239-261. doi: 10.3934/krm.2017010
[18]	Leif Arkeryd. A kinetic equation for spin polarized Fermi systems. Kinetic & Related Models, 2014, 7 (1) : 1-8. doi: 10.3934/krm.2014.7.1
[19]	José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[20]	Shengfan Zhou, Jinwu Huang, Xiaoying Han. Compact kernel sections for dissipative non-autonomous Zakharov equation on infinite lattices. Communications on Pure & Applied Analysis, 2010, 9 (1) : 193-210. doi: 10.3934/cpaa.2010.9.193

2017 Impact Factor: 1.219

American Institute of Mathematical Sciences