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

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

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

Marc Briant

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 150^th 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.

Keywords: Boltzmann equation, perturbative theory, exponential trend to equilibrium, specular reflection boundary conditions, Maxwellian diffusion boundary conditions.

Mathematics Subject Classification: Primary: 35Q20, 35B20; Secondary: 76P05.

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.
[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.

show all references

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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