August  2021, 14(4): 725-747. doi: 10.3934/krm.2021022

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

Received  October 2020 Revised  April 2021 Published  August 2021 Early access  June 2021

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 [27] in the case of hard spheres, then generalized to hard and Maxwellian potentials by R. Ellis and M. Pinsky [13], and S. Ukai proved the existence of a spectral gap for large frequencies [33]. The aim of this paper is to extend to the spaces $ L^2\left( {\mathbb R}^d, (1+|v|)^{k}\right) $ the spectral studies from [13,33]. More precisely, we look at the Fourier transform in the space variable of the inhomogeneous operator and consider the dual Fourier variable as a fixed parameter. We then perform a precise study of this operator for small frequencies (by seeing it as a perturbation of the homogeneous one) and also for large frequencies from spectral and semigroup point of views. Our approach is based on Kato's perturbation theory for linear operators [22] as well as enlargement arguments from [25,19].

Citation: Pierre Gervais. A spectral study of the linearized Boltzmann operator in $ L^2 $-spaces with polynomial and Gaussian weights. Kinetic & Related Models, 2021, 14 (4) : 725-747. doi: 10.3934/krm.2021022
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. Google Scholar

[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. Google Scholar

[3]

R. AlonsoY. MorimotoW. Sun and T. Yang, Non-cutoff Boltzmann equation with polynomial decay perturbations, Rev. Mat. Iberoam., 37 (2021), 189-292.  doi: 10.4171/rmi/1206.  Google Scholar

[4]

R. J. AlonsoV. 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.  Google Scholar

[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.  Google Scholar

[6]

C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608.  Google Scholar

[7]

C. BardosF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. BriantS. 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.  Google Scholar

[11]

T. Carleman, Problèmes Mathématiques dans la Théorie Cinétique des Gaz, Publ. Sci. Inst. Mittag-Leffler, 2, Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

H. Grad, Asymptotic theory of the Boltzmann equation, Phys. Fluids, 6 (1963), 147-181.  doi: 10.1063/1.1706716.  Google Scholar

[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.  Google Scholar

[20]

F. HérauD. 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.  Google Scholar

[21]

D. Hilbert, Begründung der kinetischen Gastheorie, Math. Ann., 72 (1912), 562-577.  doi: 10.1007/BF01456676.  Google Scholar

[22]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

B. Nicolaenko, Dispersion laws for plane wave propagation, in Boltzmann Equation, Courant Institute, 1971,125–172. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

R. AlonsoY. MorimotoW. Sun and T. Yang, Non-cutoff Boltzmann equation with polynomial decay perturbations, Rev. Mat. Iberoam., 37 (2021), 189-292.  doi: 10.4171/rmi/1206.  Google Scholar

[4]

R. J. AlonsoV. 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.  Google Scholar

[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.  Google Scholar

[6]

C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608.  Google Scholar

[7]

C. BardosF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. BriantS. 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.  Google Scholar

[11]

T. Carleman, Problèmes Mathématiques dans la Théorie Cinétique des Gaz, Publ. Sci. Inst. Mittag-Leffler, 2, Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

H. Grad, Asymptotic theory of the Boltzmann equation, Phys. Fluids, 6 (1963), 147-181.  doi: 10.1063/1.1706716.  Google Scholar

[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.  Google Scholar

[20]

F. HérauD. 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.  Google Scholar

[21]

D. Hilbert, Begründung der kinetischen Gastheorie, Math. Ann., 72 (1912), 562-577.  doi: 10.1007/BF01456676.  Google Scholar

[22]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

B. Nicolaenko, Dispersion laws for plane wave propagation, in Boltzmann Equation, Courant Institute, 1971,125–172. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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