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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 |
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.
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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