# 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 & 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] T. Carleman, Problémes Mathématiques Dans La Théorie Cinétique Des Gaz Publ. Sci. Inst. Mittag-Leffler. 2, Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, arxiv: arXiv: 1006. 5523.Google Scholar [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. Google Scholar [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. Google Scholar [18] Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: 10.1002/cpa.10040. Google Scholar [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. Google Scholar [20] Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687. doi: 10.1002/cpa.20121. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar

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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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] T. Carleman, Problémes Mathématiques Dans La Théorie Cinétique Des Gaz Publ. Sci. Inst. Mittag-Leffler. 2, Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, arxiv: arXiv: 1006. 5523.Google Scholar [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. Google Scholar [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. Google Scholar [18] Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: 10.1002/cpa.10040. Google Scholar [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. Google Scholar [20] Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687. doi: 10.1002/cpa.20121. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar
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