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December  2017, 10(4): 957-976. doi: 10.3934/krm.2017038

Grossly determined solutions for a Boltzmann-like equation

 Department of Mathematics, Bradley University, Bradley Hall 445, Peoria, IL 61625, USA

Received  October 2015 Revised  December 2016 Published  March 2017

Fund Project: The author is supported by NSF grant DMS 08-38434 "EMSW21-MCTP: Research Experience for Graduate Students" and from the Caterpillar Fellowship Grant at Bradley University.

In gas dynamics, the connection between the continuum physics model offered by the Navier-Stokes equations and the heat equation and the molecular model offered by the kinetic theory of gases has been understood for some time, especially through the work of Chapman and Enskog, but it has never been established rigorously. This paper established a precise bridge between these two models for a simple linear Boltzman-like equation. Specifically a special class of solutions, the grossly determined solutions, of this kinetic model are shown to exist and satisfy closed form balance equations representing a class of continuum model solutions.

Citation: Thomas Carty. Grossly determined solutions for a Boltzmann-like equation. Kinetic and Related Models, 2017, 10 (4) : 957-976. doi: 10.3934/krm.2017038
References:
 [1] R. Alonso, Boltzmann-type Equations and Their Applications Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2015. [2] R. Alonso and W. Sun, The radiative transfer equation in the forward-peaked regime, Comm. Math. Phys., 338 (2015), 1233-1286.  doi: 10.1007/s00220-015-2395-8. [3] R. J. Alonso and I. M. Gamba, Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section, J. Stat. Phys., 137 (2009), 1147-1165.  doi: 10.1007/s10955-009-9873-3. [4] L. Arkeryd, Stability in L1 for the spatially homogeneous Boltzmann equation, Archive for Rational Mechanics and Analysis, 103 (1988), 151-167.  doi: 10.1007/BF00251506. [5] T. E. Carty, Elementary Solutions for a model Boltzmann Equation in one-dimension and the connection to Grossly Determined Solutions, preprint, arXiv: 1608.03510. [6] K. M. Case, Elementary solutions of the transport equation and their applications, Annals of Physics, 9 (1960), 1-23.  doi: 10.1016/0003-4916(60)90060-9. [7] C. Cercignani, Methods of solution of the linearized Boltzmann equation for rarefied gas dynamics, Journal of Quantitative Spectroscopy and Radiative Transfer, 11 (1971), 973-985.  doi: 10.1016/0022-4073(71)90068-9. [8] C. Cercignani, H-theorem and trend to equilibrium in the kinetic theory of gases, Arch. Mech. (Arch. Mech. Stos.), 34 (1982), 231-241. [9] C. Cercignani, Elementary solutions of the linearized gas-dynamics Boltzmann equation and their application to the slip-flow problem, Annals of Physics, 20 (1962), 219-233.  doi: 10.1016/0003-4916(62)90199-9. [10] C. Cercignani, R. Illner and M. Pulvierenti, The Mathematical Theory of Dilute Gases Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8. [11] C. Cercignani and F. Sernagiotto, The method of elementary solutions for time-dependent problems in linearized kinetic theory, Annals of Physics, 30 (1964), 154-167.  doi: 10.1016/0003-4916(64)90308-2. [12] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math., 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9. [13] L. Desvillettes and C. Mouhot, Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials, Asymptot. Anal., 54 (2007), 235-245. [14] R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math., 130 (1989), 321-366.  doi: 10.2307/1971423. [15] E. Dolera, On the computation of the spectrum of the linearized Boltzmann collision operator for Maxwellian molecules, Boll. Unione Mat. Ital.(9), 4 (2011), 47-68. [16] S. Friedlander and D. Serre, Handbook of mathematical fluid dynamics Elsevier, 2002. [17] L. S. García-Colín, R. M. Velasco and F. J. Uribe, Beyond the Navier-Stokes equations: Burnett hydrodynamics, Phys. Rep., 465 (2008), 149-189. [18] L. Gosse, Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension, Kinetic and Related Models, 5 (2012), 283-323.  doi: 10.3934/krm.2012.5.283. [19] S. Harris, An Introduction to the Theory of the Boltzmann Equation Dover Books on Physics, 2012. [20] L. Hörmander, Linear Partial Differential Operators Springer Verlag, Berlin, 1976. [21] S. Jin, L. Pareschi and M. Slemrod, A relaxation scheme for solving the Boltzmann equation based on the Chapman-Enskog expansion, Acta Math. Appl. Sin. Engl. Ser., 18 (2002), 37-62.  doi: 10.1007/s102550200003. [22] S. Kaniel and M. Shinbrot, The Boltzmann equation. Ⅰ. Uniqueness and local existence, Comm. Math. Phys., 58 (1978), 65-84. [23] S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 467-501.  doi: 10.1016/S0294-1449(99)80025-0. [24] C. Mouhot, Quantitative linearized study of the Boltzmann collision operator and applications, Commun. Math. Sci., 1 (2007), 73-86.  doi: 10.4310/CMS.2007.v5.n5.a6. [25] M. Slemrod, Constitutive relations for monatomic gases based on a generalized rational approximation to the sum of the Chapman-Enskog expansion, Arch. Ration. Mech. Anal., 150 (1999), 1-22.  doi: 10.1007/s002050050178. [26] C. Truesdell and R. G. Muncaster, Fundamentals Of Maxwell's Kinetic Theory Of A Simple Monatomic Gas Treated as a branch of rational mechanics, Pure and Applied Mathematics, Vol. 83, Academic Press, 1980. [27] C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., 234 (2003), 455-490.  doi: 10.1007/s00220-002-0777-1.

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References:
 [1] R. Alonso, Boltzmann-type Equations and Their Applications Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2015. [2] R. Alonso and W. Sun, The radiative transfer equation in the forward-peaked regime, Comm. Math. Phys., 338 (2015), 1233-1286.  doi: 10.1007/s00220-015-2395-8. [3] R. J. Alonso and I. M. Gamba, Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section, J. Stat. Phys., 137 (2009), 1147-1165.  doi: 10.1007/s10955-009-9873-3. [4] L. Arkeryd, Stability in L1 for the spatially homogeneous Boltzmann equation, Archive for Rational Mechanics and Analysis, 103 (1988), 151-167.  doi: 10.1007/BF00251506. [5] T. E. Carty, Elementary Solutions for a model Boltzmann Equation in one-dimension and the connection to Grossly Determined Solutions, preprint, arXiv: 1608.03510. [6] K. M. Case, Elementary solutions of the transport equation and their applications, Annals of Physics, 9 (1960), 1-23.  doi: 10.1016/0003-4916(60)90060-9. [7] C. Cercignani, Methods of solution of the linearized Boltzmann equation for rarefied gas dynamics, Journal of Quantitative Spectroscopy and Radiative Transfer, 11 (1971), 973-985.  doi: 10.1016/0022-4073(71)90068-9. [8] C. Cercignani, H-theorem and trend to equilibrium in the kinetic theory of gases, Arch. Mech. (Arch. Mech. Stos.), 34 (1982), 231-241. [9] C. Cercignani, Elementary solutions of the linearized gas-dynamics Boltzmann equation and their application to the slip-flow problem, Annals of Physics, 20 (1962), 219-233.  doi: 10.1016/0003-4916(62)90199-9. [10] C. Cercignani, R. Illner and M. Pulvierenti, The Mathematical Theory of Dilute Gases Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8. [11] C. Cercignani and F. Sernagiotto, The method of elementary solutions for time-dependent problems in linearized kinetic theory, Annals of Physics, 30 (1964), 154-167.  doi: 10.1016/0003-4916(64)90308-2. [12] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math., 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9. [13] L. Desvillettes and C. Mouhot, Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials, Asymptot. Anal., 54 (2007), 235-245. [14] R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math., 130 (1989), 321-366.  doi: 10.2307/1971423. [15] E. Dolera, On the computation of the spectrum of the linearized Boltzmann collision operator for Maxwellian molecules, Boll. Unione Mat. Ital.(9), 4 (2011), 47-68. [16] S. Friedlander and D. Serre, Handbook of mathematical fluid dynamics Elsevier, 2002. [17] L. S. García-Colín, R. M. Velasco and F. J. Uribe, Beyond the Navier-Stokes equations: Burnett hydrodynamics, Phys. Rep., 465 (2008), 149-189. [18] L. Gosse, Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension, Kinetic and Related Models, 5 (2012), 283-323.  doi: 10.3934/krm.2012.5.283. [19] S. Harris, An Introduction to the Theory of the Boltzmann Equation Dover Books on Physics, 2012. [20] L. Hörmander, Linear Partial Differential Operators Springer Verlag, Berlin, 1976. [21] S. Jin, L. Pareschi and M. Slemrod, A relaxation scheme for solving the Boltzmann equation based on the Chapman-Enskog expansion, Acta Math. Appl. Sin. Engl. Ser., 18 (2002), 37-62.  doi: 10.1007/s102550200003. [22] S. Kaniel and M. Shinbrot, The Boltzmann equation. Ⅰ. Uniqueness and local existence, Comm. Math. Phys., 58 (1978), 65-84. [23] S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 467-501.  doi: 10.1016/S0294-1449(99)80025-0. [24] C. Mouhot, Quantitative linearized study of the Boltzmann collision operator and applications, Commun. Math. Sci., 1 (2007), 73-86.  doi: 10.4310/CMS.2007.v5.n5.a6. [25] M. Slemrod, Constitutive relations for monatomic gases based on a generalized rational approximation to the sum of the Chapman-Enskog expansion, Arch. Ration. Mech. Anal., 150 (1999), 1-22.  doi: 10.1007/s002050050178. [26] C. Truesdell and R. G. Muncaster, Fundamentals Of Maxwell's Kinetic Theory Of A Simple Monatomic Gas Treated as a branch of rational mechanics, Pure and Applied Mathematics, Vol. 83, Academic Press, 1980. [27] C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., 234 (2003), 455-490.  doi: 10.1007/s00220-002-0777-1.
The graph of $|\boldsymbol{\xi}|=\Xi(c)$
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