The linearized collision operator of the Boltzmann equation can in a natural way be written as a sum of a positive multiplication operator, the collision frequency, and an integral operator. Compactness of the integral operator for monatomic single species is a classical result, while corresponding results for mixtures and polyatomic single species where the polyatomicity is modeled by a discrete internal energy variable, are more recently obtained. In this work the compactness of the integral operator for polyatomic single species, for which the number of internal degrees of freedom is greater or equal to two and the polyatomicity is modeled by a continuous internal energy variable, is studied. Compactness of the integral operator is obtained by proving that its terms are, or, at least, can be approximated by, Hilbert-Schmidt integral operators, under some assumptions on the collision kernel. Self-adjointness of the linearized collision operator follows. Moreover, bounds on - including coercivity of - the collision frequency, are obtained for some particular collision kernels - corresponding to hard sphere like models, but also hard potential with cut-off like models. Then it follows that the linearized collision operator is a Fredholm operator.
Citation: |
[1] | C. Baranger, M. Bisi, S. Brull and L. Desvillettes, On the Chapman-Enskog asymptotics for a mixture of monatomic and polyatomic rarefied gases, Kinet. Relat. Models, 11 (2018), 821-858. doi: 10.3934/krm.2018033. |
[2] | N. Bernhoff, Linear half-space problems in kinetic theory: Abstract formulation and regime transitions, preprint, 2022, arXiv: 2201.03459. |
[3] | N. Bernhoff, Linearized Boltzmann collision operator: I. Polyatomic molecules modeled by a discrete internal energy variable and multicomponent mixtures, Acta Appl. Math., 183 (2023), 1-45. doi: 10.1007/s10440-022-00550-6. |
[4] | V. C. Boffi, V. Protopopescu and G. Spiga, On the equivalence between the probabilistic kinetic, and scattering kernel formulations of the Boltzmann equation, Phys. A, 164 (1990), 400-410. doi: 10.1016/0378-4371(90)90203-5. |
[5] | T. Borsoni, L. Boudin and F. Salvarani, Compactness property of the linearized Boltzmann operator for a polyatomic gas undergoing resonant collisions, J. Math. Anal. Appl., 517 (2023), 126579. doi: 10.1016/j.jmaa.2022.126579. |
[6] | L. Boudin, B. Grec, M. Pavić and F. Salvarani, Diffusion asymptotics of a kinetic model for gaseous mixtures, Kinet. Relat. Models, 6 (2013), 137-157. doi: 10.3934/krm.2013.6.137. |
[7] | L. Boudin, A. Rossi and F. Salvarani, A kinetic model of polyatomic gas with resonant collisions, Ricerche Mat., 2022. doi: 10.1007/s11587-022-00733-1. |
[8] | J.-F. Bourgat, L. Desvillettes, P. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, Eur. J. Mech. B, 13 (1994), 237-254. |
[9] | C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1039-9. |
[10] | H. B. Drange, The linearized Boltzmann collision operator for cut-off potentials, SIAM J. Appl. Math., 29 (1975), 665-676. doi: 10.1137/0129054. |
[11] | I. Gamba and M. Pavic-Colic, On the Cauchy problem for Boltzmann equation modelling polyatomic gas, J. Math. Phys., 64 (2023), 013303. doi: 10.1063/5.0103621. |
[12] | R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, 1996. doi: 10.1137/1.9781611971477. |
[13] | H. Grad, Asymptotic theory of the Boltzmann equation II, in Rarefied Gas Dynamics, volume 1, Academic Press, 1963, 26-59. |
[14] | T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1980. |
[15] | M. N. Kogan, Rarefied Gas Dynamics, Plenum Press, 1969. doi: 10.1007/978-1-4899-6381-9. |
[16] | D. Levermore and W. Sun, Compactness of the gain parts of the linearized Boltzmann operator with weakly cutoff kernels, Kinet. Relat. Models, 3 (2010), 335-351. doi: 10.3934/krm.2010.3.335. |
[17] | M. Renardy and R. C. Rogers, An Introduktion to Partial Differential Equations, 2$^{nd}$ edition, Springer-Verlag, 1993. |
[18] | G. Spiga, T. Nonnenmacher and V. C. Boffi, Moment equations for the diffusion of the particles of a mixture via the scattering kernel formulation of the nonlinear Boltzmann equation, Phys. A, 131 (1985), 431-448. doi: 10.1016/0378-4371(85)90007-X. |
[19] | K. Yoshida, Functional Analysis, Springer-Verlag, 1965. |
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Typical collision of