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March  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

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.
Citation: Tomasz Komorowski. Long time asymptotics of a degenerate linear kinetic transport equation. Kinetic and 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, National Bureau of Standards, Applied Mathematics Series 55, 1972. [2] K. Aoki and F. Golse, On the speed of approach to equilibrium for a collisionless gas, Kinet. Relat. Models, 4 (2011), 87-107. 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), 151-167. 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), 819-841. 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), 53-157. 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), 969-989. 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), 1175-1198. [8] L. Desvillettes and S. Salvarani, Asymptotic behavior of degenerate linear transport equations, Bull. Sci. Math., 133 (2009), 848-858. 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), 1-42. 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'' 12th Conference on Waves and Stability in Continuous Media, 153-165, World Sci. Publ., River Edge, NJ, 2004. 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, New York-Heidelberg, 1974. vii+326 pp. [12] J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, available at http://fr.arxiv.org/abs/1005.1495 (2010) to appear in TAMS. [13] R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189. 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, New York, 1988. [15] K. Engel and R. Nagel, One-Parameter Semigroups for linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 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, New York., 1986. doi: 10.1002/9780470316658. [17] S. R. Foguel, The Ergodic Theory of Markov processes, Van Nostrand Mathematical Studies, No. 21. Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. [18] Y. Katznelson, An Introduction to Harmonic Analysis, Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2004. [19] P. D. Lax, Functional Analysis, Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. [20] A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Rat. Mech. Anal., 199 (2011), 493-525. 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., New York-Auckland-Düsseldorf, 1976. [22] W. R. Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill Book Co., New York, 1987. [23] E. M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, 30 Princeton University Press, Princeton, N.J, 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), 518-543. 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), 179-184. 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., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1967. [27] F. Zhang, Matrix Theory Basic Results and Techniques, Second Ed. Universitext, Springer, 2011. doi: 10.1007/978-1-4614-1099-7.

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##### References:
 [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Tenth priniting, National Bureau of Standards, Applied Mathematics Series 55, 1972. [2] K. Aoki and F. Golse, On the speed of approach to equilibrium for a collisionless gas, Kinet. Relat. Models, 4 (2011), 87-107. 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), 151-167. 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), 819-841. 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), 53-157. 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), 969-989. 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), 1175-1198. [8] L. Desvillettes and S. Salvarani, Asymptotic behavior of degenerate linear transport equations, Bull. Sci. Math., 133 (2009), 848-858. 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), 1-42. 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'' 12th Conference on Waves and Stability in Continuous Media, 153-165, World Sci. Publ., River Edge, NJ, 2004. 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, New York-Heidelberg, 1974. vii+326 pp. [12] J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, available at http://fr.arxiv.org/abs/1005.1495 (2010) to appear in TAMS. [13] R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189. 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, New York, 1988. [15] K. Engel and R. Nagel, One-Parameter Semigroups for linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 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, New York., 1986. doi: 10.1002/9780470316658. [17] S. R. Foguel, The Ergodic Theory of Markov processes, Van Nostrand Mathematical Studies, No. 21. Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. [18] Y. Katznelson, An Introduction to Harmonic Analysis, Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2004. [19] P. D. Lax, Functional Analysis, Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. [20] A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Rat. Mech. Anal., 199 (2011), 493-525. 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., New York-Auckland-Düsseldorf, 1976. [22] W. R. Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill Book Co., New York, 1987. [23] E. M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, 30 Princeton University Press, Princeton, N.J, 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), 518-543. 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), 179-184. 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., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1967. [27] F. Zhang, Matrix Theory Basic Results and Techniques, Second Ed. Universitext, Springer, 2011. doi: 10.1007/978-1-4614-1099-7.
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