Exponential Convergence to Equilibrium for Solutions of the Homogeneous Boltzmann Equation for Maxwellian Molecules
Abstract
1. Introduction
1.1. List of Symbols
1.2. The Equation and Its Linearization
2. Main Results
3. Proofs
3.1. Proof of Theorem 1
3.2. Proof of Theorem 2
3.3. Proof of Proposition 1
4. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
References
- Cercignani, C. The Boltzmann Equation and Its Applications; Springer: New York, NY, USA, 1988. [Google Scholar]
- Truesdell, C.; Muncaster, R. Fundamentals of Maxwell’s Kinetic Theory of a Simple Monoatomic Gas; Academic Press: New York, NY, USA, 1980. [Google Scholar]
- Villani, C. A review of mathematical topics in collisional kinetic theory. In Handbook of Mathematical Fluid Dynamics; Friedlander, S., Serre, D., Eds.; Elsevier: Amsterdam, The Netherlands, 2002; Volume I, pp. 71–305. [Google Scholar]
- Kox, A.J. HA Lorentz’s contributions to kinetic gas theory. Ann. Sci. 1990, 47, 591–606. [Google Scholar] [CrossRef]
- Lorentz, H.A. Over de Entropie eener Gasmassa. German translation: Über die Entropie eines Gases. In Abhandlungen über Theoretische Physik; B. G. Teubner Verlag: Leipzig, Germany, 1907. [Google Scholar]
- Barbaroux, J.M.; Hundertmark, D.; Ried, T.; Vugalter, S. Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction. Kinet. Relat. Models 2017, 10, 901–924. [Google Scholar] [CrossRef][Green Version]
- Bassetti, F.; Matthes, D.; Ladelli, L. Infinite energy solutions to inelastic homogeneous Boltzmann equations. Electron. J. Probab. 2015, 20, 34. [Google Scholar] [CrossRef]
- Bobylev, A.V.; Gamba, I.M. Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules. Kinet. Relat. Models 2017, 10, 573–585. [Google Scholar] [CrossRef]
- Desvillettes, L.; Furioli, G.; Terraneo, E. Propagation of Gevrey Regularity for Solutions of the Boltzmann Equation for Maxwellian Molecules. Trans. Amer. Math. Soc. 2009, 361, 1731–1747. [Google Scholar] [CrossRef]
- Pulvirenti, A.; Wennberg, B. A Maxwellian lower bound for solutions to the Boltzmann equation. Commun. Math. Phys. 1997, 183, 145–160. [Google Scholar] [CrossRef]
- Carlen, E.A.; Gabetta, E.; Toscani, G. Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Commun. Math. Phys. 1999, 199, 521–546. [Google Scholar] [CrossRef]
- Carlen, E.A.; Lu, X. Fast and slow convergence to equilibrium for Maxwellian molecules via Wild Sums. J. Stat. Phys. 2003, 112, 59–134. [Google Scholar] [CrossRef]
- Dolera, E. Mathematical treatment of the homogeneous Boltzmann equation for Maxwellian molecules in the presence of singular kernels. Ann. Mat. Pura Appl. 2015, 194, 1707–1732. [Google Scholar] [CrossRef]
- Dolera, E.; Regazzini, E. Proof of a McKean conjecture on the rate of convergence of Boltzmann–equation solutions. Probab. Theory Related Fields 2014, 160, 315–389. [Google Scholar] [CrossRef]
- Villani, C. Cercignani’s conjecture is sometimes true and always almost true. Comm. Math. Phys. 2003, 234, 455–490. [Google Scholar] [CrossRef]
- Grad, H. Asymptotic theory of the Boltzmann equation, II. In Rarefied Gas Dynamics, Proceedings of the Third International Symposium on Rarified Gas Dynamics, Held at the Palais de L’Unesco, Paris, 1962; Academic Press: New York, NY, USA, 1962; Volume I, pp. 26–59. [Google Scholar]
- Hilbert, D. Begründung der kinetischen Gastheorie. Math. Ann. 1912, 72, 562–577. [Google Scholar] [CrossRef]
- Cercignani, C. H-theorem and trend to equilibrium in the kinetic theory of gases. Arch. Mech. 1982, 34, 231–241. [Google Scholar]
- Carlen, E.A.; Carvalho, M.C. Probabilistic methods in kinetic theory. Riv. Mat. Univ. Parma 2003, 7, 101–149. [Google Scholar]
- Carlen, E.A.; Carvalho, M.C.; Gabetta, E. On the relation between rates of relaxation and convergence of Wild sums for solutions of the Kac equation. J. Funct. Anal. 2005, 220, 362–387. [Google Scholar] [CrossRef][Green Version]
- Carlen, E.A.; Carvalho, M.C.; Gabetta, E. Central limit theorem for Maxwellian molecules and truncation of the Wild expansion. Commun. Pure Appl. Math. 2000, 53, 370–397. [Google Scholar] [CrossRef]
- McKean, H.P., Jr. Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas. Arch. Ration. Mech. Anal. 1966, 21, 343–367. [Google Scholar] [CrossRef]
- Dolera, E.; Regazzini, E. The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation. Ann. Appl. Probab. 2010, 20, 430–461. [Google Scholar] [CrossRef][Green Version]
- Dolera, E.; Gabetta, E.; Regazzini, E. Reaching the best possible rate of convergence to equilibrium for solutions of Kac’s equation via central limit theorem. Ann. Appl. Probab. 2009, 19, 186–209. [Google Scholar] [CrossRef]
- Dolera, E. Estimates of the approximation of weighted sums of conditionally independent random variables by the normal law. J. Inequal. Appl. 2013, 2013, 320. [Google Scholar] [CrossRef][Green Version]
- Cercignani, C.; Lampis, M.; Sgarra, C. L2-Stability near equilibrium of the solution of the homogeneous Boltzmann equation in the case of Maxwellian molecules. Meccanica 1988, 23, 15–18. [Google Scholar] [CrossRef]
- Mouhot, C. Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials. Commun. Math. Phys. 2006, 261, 629–672. [Google Scholar] [CrossRef]
- Maxwell, J.C. On the dynamical theory of gases. Philos. Trans. R. Soc. Lond. Ser. 1866, 157, 49–88. [Google Scholar]
- Morgenstern, D. General existence and uniqueness proof for the spatially homogeneous solution of the Maxwell-Boltzmann equation in the case of Maxwellian molecules. Proc. Natl. Acad. Sci. USA 1954, 40, 719–721. [Google Scholar] [CrossRef]
- Grünbaum, F.A. Linearization of the Boltzmann equation. Trans. Am. Math. Soc. 1972, 165, 425–449. [Google Scholar] [CrossRef]
- Dolera, E. On the computation of the spectrum of the linearized Boltzmann collision operator for Maxwellian molecules. Boll. Unione Mat. Ital. 2011, 4, 47–68. [Google Scholar]
- Ladas, G.E.; Lakshmikantham, V. Differential Equations in Abstract Spaces; Academic Press: New York, NY, USA, 1972. [Google Scholar]
- Martin, R.H., Jr. Non-Linear Operators and Differential Equations in Banach Spaces; Wiley: New York, NY, USA, 1976. [Google Scholar]
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Dolera, E. Exponential Convergence to Equilibrium for Solutions of the Homogeneous Boltzmann Equation for Maxwellian Molecules. Mathematics 2022, 10, 2347. https://doi.org/10.3390/math10132347
Dolera E. Exponential Convergence to Equilibrium for Solutions of the Homogeneous Boltzmann Equation for Maxwellian Molecules. Mathematics. 2022; 10(13):2347. https://doi.org/10.3390/math10132347
Chicago/Turabian StyleDolera, Emanuele. 2022. "Exponential Convergence to Equilibrium for Solutions of the Homogeneous Boltzmann Equation for Maxwellian Molecules" Mathematics 10, no. 13: 2347. https://doi.org/10.3390/math10132347
APA StyleDolera, E. (2022). Exponential Convergence to Equilibrium for Solutions of the Homogeneous Boltzmann Equation for Maxwellian Molecules. Mathematics, 10(13), 2347. https://doi.org/10.3390/math10132347