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Lower bound for the Boltzmann equation whose regularity grows tempered with time
A spectral study of the linearized Boltzmann operator in $ L^2 $-spaces with polynomial and Gaussian weights
ENS de Paris - 45 rue d'Ulm, 75005, Paris, France |
The spectrum structure of the linearized Boltzmann operator has been a subject of interest for over fifty years and has been inspected in the space $ L^2\left( {\mathbb R}^d, \exp(|v|^2/4)\right) $ by B. Nicolaenko [
References:
[1] |
R. Alonso, I. M. Gamba and M. Tasković, Exponentially-tailed regularity and time asymptotic for the homogeneous Boltzmann equation, preprint, arXiv: 1711.06596. |
[2] |
R. Alonso, B. Lods and I. Tristani, Fluid dynamic limit of Boltzmann equation for granular hard-spheres in a nearly elastic regime, preprint, arXiv: 2008.05173. |
[3] |
R. Alonso, Y. Morimoto, W. Sun and T. Yang,
Non-cutoff Boltzmann equation with polynomial decay perturbations, Rev. Mat. Iberoam., 37 (2021), 189-292.
doi: 10.4171/rmi/1206. |
[4] |
R. J. Alonso, V. Bagland and B. Lods,
Convergence to self-similarity for ballistic annihilation dynamics, J. Math. Pures Appl. (9), 138 (2020), 88-163.
doi: 10.1016/j.matpur.2019.09.008. |
[5] |
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.
doi: 10.4171/RMI/436. |
[6] |
C. Bardos, F. Golse and D. Levermore,
Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys., 63 (1991), 323-344.
doi: 10.1007/BF01026608. |
[7] |
C. Bardos, F. Golse and D. Levermore,
Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46 (1993), 667-753.
doi: 10.1002/cpa.3160460503. |
[8] |
C. Bardos and S. Ukai,
The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci., 1 (1991), 235-257.
doi: 10.1142/S0218202591000137. |
[9] |
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. |
[10] |
M. Briant, S. Merino-Aceituno and C. Mouhot,
From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight, Anal. Appl. (Singap.), 17 (2019), 85-116.
doi: 10.1142/S021953051850015X. |
[11] |
T. Carleman, Problèmes Mathématiques dans la Théorie Cinétique des Gaz, Publ. Sci. Inst. Mittag-Leffler, 2, Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957. |
[12] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
[13] |
R. S. Ellis and M. A. Pinsky,
The first and second fluid approximations of the linearized Boltzmann equation, J. Math. Pures Appl. (9), 54 (1975), 125-156.
|
[14] |
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.
doi: 10.1007/b97696. |
[15] |
I. Gallagher and I. Tristani,
On the convergence of smooth solutions from Boltzmann to Navier-Stokes, Ann. H. Lebesgue, 3 (2020), 561-614.
doi: 10.5802/ahl.40. |
[16] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[17] |
F. Golse, The Boltzmann equation and its hydrodynamic limits, in Evolutionary Equations. Vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005,159–301.
doi: 10.1016/S1874-5717(06)80006-X. |
[18] |
H. Grad,
Asymptotic theory of the Boltzmann equation, Phys. Fluids, 6 (1963), 147-181.
doi: 10.1063/1.1706716. |
[19] |
M. P. Guadldani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential $H$-theorem, Mém. Soc. Math. Fr. (N.S.), 153 (2017), 137pp.
doi: 10.24033/msmf.461. |
[20] |
F. Hérau, D. Tonon and I. Tristani,
Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off, Comm. Math. Phys., 377 (2020), 697-771.
doi: 10.1007/s00220-020-03682-8. |
[21] |
D. Hilbert,
Begründung der kinetischen Gastheorie, Math. Ann., 72 (1912), 562-577.
doi: 10.1007/BF01456676. |
[22] |
T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. |
[23] |
B. Lods and M. Mokhtar-Kharroubi,
Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in $L^1$-spaces, Math. Methods Appl. Sci., 40 (2017), 6527-6555.
doi: 10.1002/mma.4473. |
[24] |
S. Mischler and C. Mouhot,
Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation, Arch. Ration. Mech. Anal., 221 (2016), 677-723.
doi: 10.1007/s00205-016-0972-4. |
[25] |
C. Mouhot,
Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Comm. Math. Phys., 261 (2005), 629-672.
doi: 10.1007/s00220-005-1455-x. |
[26] |
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. |
[27] |
B. Nicolaenko, Dispersion laws for plane wave propagation, in Boltzmann Equation, Courant Institute, 1971,125–172. |
[28] |
T. Nishida,
Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Comm. Math. Phys., 61 (1978), 119-148.
doi: 10.1007/BF01609490. |
[29] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[30] |
L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equation, Lecture Notes in Mathematics, 1971, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-92847-8. |
[31] |
I. Tristani,
Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting, J. Funct. Anal., 270 (2016), 1922-1970.
doi: 10.1016/j.jfa.2015.09.025. |
[32] |
I. Tristani,
Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off, J. Stat. Phys., 157 (2014), 474-496.
doi: 10.1007/s10955-014-1066-z. |
[33] |
S. Ukai,
On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184.
doi: 10.3792/pja/1195519027. |
[34] |
S. Ukai, Solutions of the Boltzmann equation, in Patterns and Waves, Stud. Math. Appl., 18, North-Holland, Amsterdam, 1986, 37–96.
doi: 10.1016/S0168-2024(08)70128-0. |
[35] |
S. Ukai and T. Yang, Mathematical Theory of Boltzmann Equation, Lecture Notes Series-No. 8, Hong Kong: Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong. Available from: http://www.cityu.edu.hk/rcms/publications/ln8.pdf. |
[36] |
T. Yang and H. Yu,
Spectrum analysis of some kinetic equations, Arch. Ration. Mech. Anal., 222 (2016), 731-768.
doi: 10.1007/s00205-016-1010-2. |
show all references
References:
[1] |
R. Alonso, I. M. Gamba and M. Tasković, Exponentially-tailed regularity and time asymptotic for the homogeneous Boltzmann equation, preprint, arXiv: 1711.06596. |
[2] |
R. Alonso, B. Lods and I. Tristani, Fluid dynamic limit of Boltzmann equation for granular hard-spheres in a nearly elastic regime, preprint, arXiv: 2008.05173. |
[3] |
R. Alonso, Y. Morimoto, W. Sun and T. Yang,
Non-cutoff Boltzmann equation with polynomial decay perturbations, Rev. Mat. Iberoam., 37 (2021), 189-292.
doi: 10.4171/rmi/1206. |
[4] |
R. J. Alonso, V. Bagland and B. Lods,
Convergence to self-similarity for ballistic annihilation dynamics, J. Math. Pures Appl. (9), 138 (2020), 88-163.
doi: 10.1016/j.matpur.2019.09.008. |
[5] |
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.
doi: 10.4171/RMI/436. |
[6] |
C. Bardos, F. Golse and D. Levermore,
Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys., 63 (1991), 323-344.
doi: 10.1007/BF01026608. |
[7] |
C. Bardos, F. Golse and D. Levermore,
Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46 (1993), 667-753.
doi: 10.1002/cpa.3160460503. |
[8] |
C. Bardos and S. Ukai,
The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci., 1 (1991), 235-257.
doi: 10.1142/S0218202591000137. |
[9] |
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. |
[10] |
M. Briant, S. Merino-Aceituno and C. Mouhot,
From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight, Anal. Appl. (Singap.), 17 (2019), 85-116.
doi: 10.1142/S021953051850015X. |
[11] |
T. Carleman, Problèmes Mathématiques dans la Théorie Cinétique des Gaz, Publ. Sci. Inst. Mittag-Leffler, 2, Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957. |
[12] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
[13] |
R. S. Ellis and M. A. Pinsky,
The first and second fluid approximations of the linearized Boltzmann equation, J. Math. Pures Appl. (9), 54 (1975), 125-156.
|
[14] |
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.
doi: 10.1007/b97696. |
[15] |
I. Gallagher and I. Tristani,
On the convergence of smooth solutions from Boltzmann to Navier-Stokes, Ann. H. Lebesgue, 3 (2020), 561-614.
doi: 10.5802/ahl.40. |
[16] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[17] |
F. Golse, The Boltzmann equation and its hydrodynamic limits, in Evolutionary Equations. Vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005,159–301.
doi: 10.1016/S1874-5717(06)80006-X. |
[18] |
H. Grad,
Asymptotic theory of the Boltzmann equation, Phys. Fluids, 6 (1963), 147-181.
doi: 10.1063/1.1706716. |
[19] |
M. P. Guadldani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential $H$-theorem, Mém. Soc. Math. Fr. (N.S.), 153 (2017), 137pp.
doi: 10.24033/msmf.461. |
[20] |
F. Hérau, D. Tonon and I. Tristani,
Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off, Comm. Math. Phys., 377 (2020), 697-771.
doi: 10.1007/s00220-020-03682-8. |
[21] |
D. Hilbert,
Begründung der kinetischen Gastheorie, Math. Ann., 72 (1912), 562-577.
doi: 10.1007/BF01456676. |
[22] |
T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. |
[23] |
B. Lods and M. Mokhtar-Kharroubi,
Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in $L^1$-spaces, Math. Methods Appl. Sci., 40 (2017), 6527-6555.
doi: 10.1002/mma.4473. |
[24] |
S. Mischler and C. Mouhot,
Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation, Arch. Ration. Mech. Anal., 221 (2016), 677-723.
doi: 10.1007/s00205-016-0972-4. |
[25] |
C. Mouhot,
Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Comm. Math. Phys., 261 (2005), 629-672.
doi: 10.1007/s00220-005-1455-x. |
[26] |
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. |
[27] |
B. Nicolaenko, Dispersion laws for plane wave propagation, in Boltzmann Equation, Courant Institute, 1971,125–172. |
[28] |
T. Nishida,
Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Comm. Math. Phys., 61 (1978), 119-148.
doi: 10.1007/BF01609490. |
[29] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[30] |
L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equation, Lecture Notes in Mathematics, 1971, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-92847-8. |
[31] |
I. Tristani,
Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting, J. Funct. Anal., 270 (2016), 1922-1970.
doi: 10.1016/j.jfa.2015.09.025. |
[32] |
I. Tristani,
Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off, J. Stat. Phys., 157 (2014), 474-496.
doi: 10.1007/s10955-014-1066-z. |
[33] |
S. Ukai,
On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184.
doi: 10.3792/pja/1195519027. |
[34] |
S. Ukai, Solutions of the Boltzmann equation, in Patterns and Waves, Stud. Math. Appl., 18, North-Holland, Amsterdam, 1986, 37–96.
doi: 10.1016/S0168-2024(08)70128-0. |
[35] |
S. Ukai and T. Yang, Mathematical Theory of Boltzmann Equation, Lecture Notes Series-No. 8, Hong Kong: Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong. Available from: http://www.cityu.edu.hk/rcms/publications/ln8.pdf. |
[36] |
T. Yang and H. Yu,
Spectrum analysis of some kinetic equations, Arch. Ration. Mech. Anal., 222 (2016), 731-768.
doi: 10.1007/s00205-016-1010-2. |
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