# American Institute of Mathematical Sciences

June  2017, 10(2): 329-371. doi: 10.3934/krm.2017014

## Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions

 Sorbonne Universités, UPMC Univ. Paris 06/ CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Received  July 2015 Revised  July 2016 Published  November 2016

Fund Project: The author was supported by the 150th Anniversary Postdoctoral Mobility Grant of the London Mathematical Society and by the Division of Applied Mathematics of Brown University.

We study the Boltzmann equation near a global Maxwellian in the case of bounded domains. We consider the boundary conditions to be either specular reflections or Maxwellian diffusion. Starting from the reference work of Guo [21] in $L_{x,v}^\infty \left( {{{\left( {1 + |v|} \right)}^\beta }{e^{|v{|^2}/4}}} \right)$, we prove existence, uniqueness, continuity and positivity of solutions for less restrictive weights in the velocity variable; namely, polynomials and stretch exponentials. The methods developed here are constructive.

Citation: Marc Briant. Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions. Kinetic and Related Models, 2017, 10 (2) : 329-371. doi: 10.3934/krm.2017014
##### References:
 [1] 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, URLhttp://projecteuclid.org/getRecord?id=euclid.rmi/1136999132. doi: 10.4171/RMI/436. [2] R. Beals and V. Protopopescu, Abstract time-dependent transport equations, J. Math. Anal. Appl., 121 (1987), 370-405.  doi: 10.1016/0022-247X(87)90252-6. [3] M. Briant, Instantaneous Filling of the Vacuum for the Full Boltzmann Equation in Convex Domains, Arch. Ration. Mech. Anal., 218 (2015), 985-1041.  doi: 10.1007/s00205-015-0874-x. [4] M. Briant, Stability of global equilibrium for the multi-species Boltzmann equation in ${L}^∞$ settings, Discrete and Continuous Dynamical Systems -Series A, 36 (2016), 6669-6688.  doi: 10.3934/dcds.2016090. [5] M. Briant, From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: A quantitative error estimate, J. Differential Equations, 259 (2015), 6072-6141.  doi: 10.1016/j.jde.2015.07.022. [6] M. Briant, Instantaneous exponential lower bound for solutions to the boltzmann equation with maxwellian diffusion boundary conditions, Kin. Rel. Mod., 8 (2015), 281-308.  doi: 10.3934/krm.2015.8.281. [7] M. Briant and E. Daus, The Boltzmann equation for multi-species mixture close to global equilibrium, Arch. Ration. Mech. Anal., 222 (2016), 1367-1443.  doi: 10.1007/s00205-016-1023-x. [8] T. Carleman, Problémes Mathématiques Dans La Théorie Cinétique Des Gaz Publ. Sci. Inst. Mittag-Leffler. 2, Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957. [9] C. Cercignani, The Boltzmann Equation and Its Applications vol. 67 of Applied Mathematical Sciences, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [10] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases vol. 106 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8. [11] R. Esposito, Y. Guo, C. Kim and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Comm. Math. Phys., 323 (2013), 177-239.  doi: 10.1007/s00220-013-1766-2. [12] 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. [13] H. Grad, Principles of the kinetic theory of gases, in Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin, 1958,205–294. [14] P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.  doi: 10.1090/S0894-0347-2011-00697-8. [15] M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, arxiv: arXiv: 1006. 5523. [16] Y. Guo, C. Kim, D. Tonon and A. Trescases, Regularity of the boltzmann equation in convex domains, Inventiones Mathematicae, (2016), 1-76.  doi: 10.1007/s00222-016-0670-8. [17] Y. Guo, C. Kim, D. Tonon and A. Trescases, BV-regularity of the Boltzmann equation in non-convex domains, Arch. Ration. Mech. Anal., 220 (2016), 1045-1093.  doi: 10.1007/s00205-015-0948-9. [18] Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.  doi: 10.1002/cpa.10040. [19] Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353.  doi: 10.1007/s00205-003-0262-9. [20] Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.  doi: 10.1002/cpa.20121. [21] Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal., 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y. [22] C. Kim, Formation and propagation of discontinuity for Boltzmann equation in non-convex domains, Comm. Math. Phys., 308 (2011), 641-701.  doi: 10.1007/s00220-011-1355-1. [23] C. Kim, Boltzmann equation with a large potential in a periodic box, Comm. Partial Differential Equations, 39 (2014), 1393-1423.  doi: 10.1080/03605302.2014.903278. [24] C. Kim and S.-B. Yun, The Boltzmann equation near a rotational local Maxwellian, SIAM J. Math. Anal., 44 (2012), 2560-2598.  doi: 10.1137/11084981X. [25] O. E. Lanford Ⅲ, Time evolution of large classical systems, in Dynamical systems, theory and applications (Recontres, Battelle Res. Inst., Seattle, Wash., 1974), Springer, Berlin, Lecture Notes in Phys., 38 (1975), 1–111. [26] C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348.  doi: 10.1080/03605300600635004. [27] C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011. [28] 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. [29] S. Ukai, On the existence of global solutions of mixed problem for non-linear {B}oltzmann equation, Proc. Japan Acad., 50 (1974), 179-184.  doi: 10.3792/pja/1195519027. [30] S. Ukai, Solutions of the Boltzmann equation, in Patterns and Waves, vol. 18 of Stud. Math. Appl., North-Holland, Amsterdam, 1986, 37–96. doi: 10.1016/S0168-2024(08)70128-0. [31] S. Ukai and T. Yang, Mathematical Theory of the Boltzmann Equation 2006, Lecture Notes Series, no. 8, Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong. [32] C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, 71–305. doi: 10.1016/S1874-5792(02)80004-0.

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##### References:
 [1] 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, URLhttp://projecteuclid.org/getRecord?id=euclid.rmi/1136999132. doi: 10.4171/RMI/436. [2] R. Beals and V. Protopopescu, Abstract time-dependent transport equations, J. Math. Anal. Appl., 121 (1987), 370-405.  doi: 10.1016/0022-247X(87)90252-6. [3] M. Briant, Instantaneous Filling of the Vacuum for the Full Boltzmann Equation in Convex Domains, Arch. Ration. Mech. Anal., 218 (2015), 985-1041.  doi: 10.1007/s00205-015-0874-x. [4] M. Briant, Stability of global equilibrium for the multi-species Boltzmann equation in ${L}^∞$ settings, Discrete and Continuous Dynamical Systems -Series A, 36 (2016), 6669-6688.  doi: 10.3934/dcds.2016090. [5] M. Briant, From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: A quantitative error estimate, J. Differential Equations, 259 (2015), 6072-6141.  doi: 10.1016/j.jde.2015.07.022. [6] M. Briant, Instantaneous exponential lower bound for solutions to the boltzmann equation with maxwellian diffusion boundary conditions, Kin. Rel. Mod., 8 (2015), 281-308.  doi: 10.3934/krm.2015.8.281. [7] M. Briant and E. Daus, The Boltzmann equation for multi-species mixture close to global equilibrium, Arch. Ration. Mech. Anal., 222 (2016), 1367-1443.  doi: 10.1007/s00205-016-1023-x. [8] T. Carleman, Problémes Mathématiques Dans La Théorie Cinétique Des Gaz Publ. Sci. Inst. Mittag-Leffler. 2, Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957. [9] C. Cercignani, The Boltzmann Equation and Its Applications vol. 67 of Applied Mathematical Sciences, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [10] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases vol. 106 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8. [11] R. Esposito, Y. Guo, C. Kim and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Comm. Math. Phys., 323 (2013), 177-239.  doi: 10.1007/s00220-013-1766-2. [12] 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. [13] H. Grad, Principles of the kinetic theory of gases, in Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin, 1958,205–294. [14] P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.  doi: 10.1090/S0894-0347-2011-00697-8. [15] M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, arxiv: arXiv: 1006. 5523. [16] Y. Guo, C. Kim, D. Tonon and A. Trescases, Regularity of the boltzmann equation in convex domains, Inventiones Mathematicae, (2016), 1-76.  doi: 10.1007/s00222-016-0670-8. [17] Y. Guo, C. Kim, D. Tonon and A. Trescases, BV-regularity of the Boltzmann equation in non-convex domains, Arch. Ration. Mech. Anal., 220 (2016), 1045-1093.  doi: 10.1007/s00205-015-0948-9. [18] Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.  doi: 10.1002/cpa.10040. [19] Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353.  doi: 10.1007/s00205-003-0262-9. [20] Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.  doi: 10.1002/cpa.20121. [21] Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal., 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y. [22] C. Kim, Formation and propagation of discontinuity for Boltzmann equation in non-convex domains, Comm. Math. Phys., 308 (2011), 641-701.  doi: 10.1007/s00220-011-1355-1. [23] C. Kim, Boltzmann equation with a large potential in a periodic box, Comm. Partial Differential Equations, 39 (2014), 1393-1423.  doi: 10.1080/03605302.2014.903278. [24] C. Kim and S.-B. Yun, The Boltzmann equation near a rotational local Maxwellian, SIAM J. Math. Anal., 44 (2012), 2560-2598.  doi: 10.1137/11084981X. [25] O. E. Lanford Ⅲ, Time evolution of large classical systems, in Dynamical systems, theory and applications (Recontres, Battelle Res. Inst., Seattle, Wash., 1974), Springer, Berlin, Lecture Notes in Phys., 38 (1975), 1–111. [26] C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348.  doi: 10.1080/03605300600635004. [27] C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011. [28] 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. [29] S. Ukai, On the existence of global solutions of mixed problem for non-linear {B}oltzmann equation, Proc. Japan Acad., 50 (1974), 179-184.  doi: 10.3792/pja/1195519027. [30] S. Ukai, Solutions of the Boltzmann equation, in Patterns and Waves, vol. 18 of Stud. Math. Appl., North-Holland, Amsterdam, 1986, 37–96. doi: 10.1016/S0168-2024(08)70128-0. [31] S. Ukai and T. Yang, Mathematical Theory of the Boltzmann Equation 2006, Lecture Notes Series, no. 8, Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong. [32] C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, 71–305. doi: 10.1016/S1874-5792(02)80004-0.
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