Abstract
Half-space problems in the kinetic theory of gases are of great importance in the study of the asymptotic behavior of solutions of boundary value problems for the Boltzmann equation for small Knudsen numbers. They provide the boundary conditions for the fluid-dynamic-type equations and Knudsen-layer corrections to the solution of the fluid-dynamic-type equations in a neighborhood of the boundary. These problems are well studied for single species, including some important contributions by Carlo Cercignani, and it is well-known that the number of additional conditions needed to be imposed depends on different regimes for the Mach number (corresponding to subsonic/supersonic evaporation/condensation). However, the case of mixtures is not as well studied in the literature. We will address some extensions of the results for half-space problems for single species to the case of multicomponent mixtures.
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References
Aoki, K., Bardos, C., Takata, S.: Knudsen layer for gas mixtures. J. Stat. Phys. 112, 629–655 (2003)
Babovsky, H.: Shocks in the light of discrete velocity models. AIP Conf. Proc. 2132, 060002 (2019)
Bardos, C., Yang, X.: The classification of well-posed kinetic boundary layer for hard sphere gas mixtures. Commun. Partial Differ. Equ. 37, 1286–1314 (2012)
Bardos, C., Golse, F., Sone, Y.: Half-space problems for the Boltzmann equation: a survey. J. Stat. Phys. 124, 275–300 (2006)
Bernhoff, N.: On half-space problems for the linearized discrete Boltzmann equation. Riv. Mat. Univ. Parma 9, 73–124 (2008)
Bernhoff, N.: On half-space problems for the weakly non-linear discrete Boltzmann equation. Kinet. Relat. Models 3, 195–222 (2010)
Bernhoff, N.: Boundary layers and shock profiles for the discrete Boltzmann equation for mixtures. Kinet. Relat. Models 5, 1–19 (2012)
Bernhoff, N.: Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles. AIP Conf. Proc. 1786, 040005 (2016)
Bernhoff, N.: Boundary layers for discrete kinetic models: multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations. Kinet. Relat. Models 10, 925–955 (2017)
Bernhoff, N.: Discrete velocity models for polyatomic molecules without nonphysical collision invariants. J. Stat. Phys. 172, 742–761 (2018)
Bernhoff, N.: Linear half-space problems in kinetic theory: abstract formulation and regime transitions (2022). arXiv: 2201.03459
Bernhoff, N.: Linearized Boltzmann collision operator: I. Polyatomic molecules modeled by a discrete internal energy variable and multicomponent mixtures (2022). arXiv: 2201.01365
Bernhoff, N., Golse, F.: On the boundary layer equations with phase transition in the kinetic theory of gases. Arch. Ration. Mech. Anal. 240, 51–98 (2021)
Bernhoff, N., Vinerean, M.C.: Discrete velocity models for multicomponent mixtures without nonphysical collision invariants. J. Stat. Phys. 165, 434–453 (2016)
Bobylev, A.V., Bernhoff, N.: Discrete velocity models and dynamical systems. In: Lecture Notes on the Discretization of the Boltzmann Equation, pp. 203–222. World Scientific, Singapore (2003)
Bobylev, A.V., Cercignani, C.: Discrete velocity models for mixtures. J. Stat. Phys. 91, 327–341 (1998)
Bobylev, A.V., Cercignani, C.: Discrete velocity models without non-physical invariants. J. Stat. Phys. 97, 677–686 (1999)
Boudin, L., Grec, B., Pavić, M., Salvarani, F.: Diffusion asymptotics of a kinetic model for gaseous mixtures. Kinet. Relat. Models 6, 137–157 (2013)
Briant, M., Daus, E.S.: The Boltzmann equation for a multi-species mixture close to global equilibrium. Arch. Ration. Mech. Anal. 222, 1367–1443 (2016)
Cercignani, C.: Half-space problems in the kinetic theory of gases. In: Trends in Applications of Pure Mathematics to Mechanics, pp. 35–50. Springer, Berlin (1986)
Cercignani, C.: The Boltzmann Equation and its Applications. Springer, Berlin (1988)
Cercignani, C.: Rarefied Gas Dynamics. Cambridge University Press, Cambridge (2000)
Daus, E.S., Jungel, A., Mouhot, C., Zamponi, S.: Hypocoercivity for a linearized multispecies Boltzmann system. SIAM J. Math. Anal. 48, 538–568 (2016)
Golse, F.: Analysis of the boundary layer equation in the kinetic theory of gases. Bull. Inst. Math. Acad. Sin. 3, 211–242 (2008)
Liu, T.P., Yu, S.H.: Invariant manifolds for steady Boltzmann flows and applications. Arch. Ration. Mech. Anal. 209, 869–997 (2013)
Sone, Y.: Kinetic Theory and Fluid Dynamics. Birkhauser, Basel (2002)
Sone, Y.: Molecular Gas Dynamics. Birkhauser, Basel (2007)
Ukai, S., Yang, T., Yu, S.H.: Nonlinear boundary layers of the Boltzmann equation: I. Existence. Commun. Math. Phys. 236, 373–393 (2003)
Acknowledgements
The author is indebted to Prof. F. Golse for valuable discussions and kind hospitality during visits to Paris and acknowledges the support by French Institute in Sweden (through the FRÖ program in 2016) and SveFUM (in 2017 and 2019) for his visits.
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Bernhoff, N. (2023). Half-Space Problems for the Boltzmann Equation of Multicomponent Mixtures. In: Barbante, P., Belgiorno, F.D., Lorenzani, S., Valdettaro, L. (eds) From Kinetic Theory to Turbulence Modeling. INdAM 2021. Springer INdAM Series, vol 51. Springer, Singapore. https://doi.org/10.1007/978-981-19-6462-6_4
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DOI: https://doi.org/10.1007/978-981-19-6462-6_4
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