Trend to equilibrium and spectral localization properties for the linear Boltzmann equation

Han-Kwan, Daniel; Léautaud, Matthieu

Séminaire Laurent Schwartz — EDP et applications (2013-2014), Exposé no. 7, 15 p.

L’objectif de cette note est de présenter les résultats de [11, 12], qui concernent l’équation de Boltzmann linéaire, posée dans un domaine borné et en présence d’une force extérieure. Une spécificité de ces travaux réside dans la prise en compte d’opérateurs de collision dégénérés aux deux sens suivants : (1) le noyau de collision associé peut s’annuler sur un grand sous-ensemble de l’espace des phases ; (2) le noyau de collision n’est pas supposé être minoré par une maxwellienne à l’infini en vitesse.

Nous étudions :

• le comportement en temps grand des solutions l’équation de Boltzmann linéaire, en donnant des critères (inspirés par la théorie du contrôle) pour assurer la convergence vers un équilibre et quand cela est possible, convergence à un taux exponentiel [11] ;
• les propriétés de localisation du spectre de l’opérateur associé [12].

The aim of this note is to present the results from [11, 12], which deal with the linear Boltzmann equation, set in a bounded domain and in the presence of an external force. A specificity of these works is that the collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity.

We study:

• the large time behavior of solutions of the linear Boltzmann equation, by giving criteria (inspired from control theory) which ensure converge towards an equilibrium and when possible, convergence at an exponential rate [11] ;
• some properties of localization for the spectrum of the associated operator [12].

[1] C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim., 30:1024–1065, 1992. | MR | Zbl

[2] É. Bernard and F. Salvarani. On the convergence to equilibrium for degenerate transport problems. Arch. Ration. Mech. Anal., 208(3):977–984, 2013. | MR

[3] É. Bernard and F. Salvarani. On the exponential decay to equilibrium of the degenerate linear Boltzmann equation. J. Funct. Anal., 265(9):1934–1954, 2013. | MR

[4] L. Desvillettes and F. Salvarani. Asymptotic behavior of degenerate linear transport equations. Bull. Sci. Math., 133(8):848–858, 2009. | MR | Zbl

[5] L. Desvillettes and C. Villani. On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math., 54(1):1–42, 2001. | MR | Zbl

[6] L. Desvillettes and C. Villani. On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math., 159(2):245–316, 2005. | MR | Zbl

[7] R. J. DiPerna, P.-L. Lions, and Y. Meyer. $L^p$ regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire, 8(3-4):271–287, 1991. | Numdam | MR | Zbl

[8] J. Dolbeault, C. Mouhot, and C. Schmeiser. Hypocoercivity for linear kinetic equations conserving mass. to appear in Trans. AMS, 2010. | arXiv

[9] F. Golse, P.-L. Lions, B. Perthame, and R. Sentis. Regularity of the moments of the solution of a transport equation. J. Funct. Anal., 76(1):110–125, 1988. | MR | Zbl

[10] Y. Guo. Decay and continuity of the Boltzmann equation in bounded domains. Arch. Ration. Mech. Anal., 197(3):713–809, 2010. | MR

[11] D. Han-Kwan and M. Léautaud. Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium. 2014. | arXiv

[12] D. Han-Kwan and M. Léautaud. Geometric analysis of the linear Boltzmann equation II. Localization properties of the spectrum. 2014.

[13] F. Hérau. Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot. Anal., 46(3-4):349–359, 2006. | MR | Zbl

[14] F. Hérau and F. Nier. Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Ration. Mech. Anal., 171(2):151–218, 2004. | MR | Zbl

[15] H. Koch and D. Tataru. On the spectrum of hyperbolic semigroups. Comm. Partial Differential Equations, 20(5-6):901–937, 1995. | MR | Zbl

[16] G. Lebeau. Équation des ondes amorties. In Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), volume 19 of Math. Phys. Stud., pages 73–109. Kluwer Acad. Publ., Dordrecht, 1996. | MR | Zbl

[17] C. Mouhot and L. Neumann. Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity, 19(4):969–998, 2006. | MR | Zbl

[18] J. Rauch and M. Taylor. Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J., 24:79–86, 1974. | MR | Zbl

[19] S. Ukai, N. Point, and H. Ghidouche. Sur la solution globale du problème mixte de l’équation de Boltzmann nonlinéaire. J. Math. Pures Appl. (9), 57(3):203–229, 1978. | MR | Zbl

[20] I. Vidav. Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator. J. Math. Anal. Appl., 22:144–155, 1968. | MR | Zbl

[21] I. Vidav. Spectra of perturbed semigroups with applications to transport theory. J. Math. Anal. Appl., 30:264–279, 1970. | MR | Zbl

[22] C. Villani. Hypocoercivity. Mem. Amer. Math. Soc., 202(950):iv+141, 2009. | MR | Zbl