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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 & 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). Google Scholar [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. Google Scholar [3] L. Arkeryd, Stability in $L^1$ for the spatially homogeneous Boltzmann equation,, Arch. Rat. Mech. Anal., 103 (1988), 151. doi: 10.1007/BF00251506. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro- reversible processes,, Indiana Univ. Math. J., 49 (2000), 1175. Google Scholar [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. Google Scholar [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. Google Scholar [10] L. Desvillettes and C. Villani, Rate of convergence toward the equilibrium in degenerate settings,, "WASCOM 2003'', (2003), 153. doi: 10.1142/9789812702937_0020. Google Scholar [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). Google Scholar [12] J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass,, available at , (2010). Google Scholar [13] R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations,, Nonlinearity, 24 (2011), 2165. doi: 10.1088/0951-7715/24/8/003. Google Scholar [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). Google Scholar [15] K. Engel and R. Nagel, One-Parameter Semigroups for linear Evolution Equations,, Graduate Texts in Mathematics, 194 (2000). Google Scholar [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. Google Scholar [17] S. R. Foguel, The Ergodic Theory of Markov processes,, Van Nostrand Mathematical Studies, (1969). Google Scholar [18] Y. Katznelson, An Introduction to Harmonic Analysis,, Third edition. Cambridge Mathematical Library. Cambridge University Press, (2004). Google Scholar [19] P. D. Lax, Functional Analysis,, Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], (2002). Google Scholar [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. Google Scholar [21] W. R. Rudin, Principles of Mathematical Analysis,, Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., (1976). Google Scholar [22] W. R. Rudin, Real and Complex Analysis,, Third Edition, (1987). Google Scholar [23] E. M. Stein, Singular Integrals and Differentiability Properties of Functions., Princeton Mathematical Series, 30 (1970). Google Scholar [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. Google Scholar [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. Google Scholar [26] A. C. Zaanen, Integration. Completely Revised Edition of An Introduction to the Theory of Integration,, North-Holland Publishing Co., (1967). Google Scholar [27] F. Zhang, Matrix Theory Basic Results and Techniques,, Second Ed. Universitext, (2011). doi: 10.1007/978-1-4614-1099-7. Google Scholar

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##### References:
 [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions,, Tenth priniting, (1972). Google Scholar [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. Google Scholar [3] L. Arkeryd, Stability in $L^1$ for the spatially homogeneous Boltzmann equation,, Arch. Rat. Mech. Anal., 103 (1988), 151. doi: 10.1007/BF00251506. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro- reversible processes,, Indiana Univ. Math. J., 49 (2000), 1175. Google Scholar [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. Google Scholar [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. Google Scholar [10] L. Desvillettes and C. Villani, Rate of convergence toward the equilibrium in degenerate settings,, "WASCOM 2003'', (2003), 153. doi: 10.1142/9789812702937_0020. Google Scholar [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). Google Scholar [12] J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass,, available at , (2010). Google Scholar [13] R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations,, Nonlinearity, 24 (2011), 2165. doi: 10.1088/0951-7715/24/8/003. Google Scholar [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). Google Scholar [15] K. Engel and R. Nagel, One-Parameter Semigroups for linear Evolution Equations,, Graduate Texts in Mathematics, 194 (2000). Google Scholar [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. Google Scholar [17] S. R. Foguel, The Ergodic Theory of Markov processes,, Van Nostrand Mathematical Studies, (1969). Google Scholar [18] Y. Katznelson, An Introduction to Harmonic Analysis,, Third edition. Cambridge Mathematical Library. Cambridge University Press, (2004). Google Scholar [19] P. D. Lax, Functional Analysis,, Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], (2002). Google Scholar [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. Google Scholar [21] W. R. Rudin, Principles of Mathematical Analysis,, Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., (1976). Google Scholar [22] W. R. Rudin, Real and Complex Analysis,, Third Edition, (1987). Google Scholar [23] E. M. Stein, Singular Integrals and Differentiability Properties of Functions., Princeton Mathematical Series, 30 (1970). Google Scholar [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. Google Scholar [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. Google Scholar [26] A. C. Zaanen, Integration. Completely Revised Edition of An Introduction to the Theory of Integration,, North-Holland Publishing Co., (1967). Google Scholar [27] F. Zhang, Matrix Theory Basic Results and Techniques,, Second Ed. Universitext, (2011). doi: 10.1007/978-1-4614-1099-7. Google Scholar
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