Knudsen type group for time in \(\mathbb{R}\) and related Boltzmann type equations. (English) Zbl 1512.35431
Summary: We consider certain Boltzmann type equations on a bounded physical and a bounded velocity space under the presence of both reflective as well as diffusive boundary conditions. We introduce conditions on the shape of the physical space and on the relation between the reflective and the diffusive part in the boundary conditions such that the associated Knudsen type semigroup can be extended to time \(t\in\mathbb{R}\). Furthermore, we provide conditions under which there exists a unique global solution to a Boltzmann type equation for time \(t\geq 0\) or for time \(t\in[\tau_0,\infty)\) for some \(\tau_0<0\) which is independent of the initial value at time 0. Depending on the collision kernel, \(\tau_0\) can be arbitrarily small.
MSC:
| 35Q20 | Boltzmann equations |
| 35Q49 | Transport equations |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 76N15 | Gas dynamics (general theory) |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
References:
| [1] | Aoki, K. and Golse, F., On the speed of approach to equilibrium for a collisionless gas, Kinet. Relat. Models4 (2011) 87-107. · Zbl 1218.82019 |
| [2] | Barbu, V., Nonlinear Differential Equations of Monotone Types in Banach Spaces, (Springer, 2010). · Zbl 1197.35002 |
| [3] | Bernou, A., A semigroup approach to the convergence rate of a collisionless gas, Kinet. Relat. Models13 (2020) 1071-1106. · Zbl 1464.35277 |
| [4] | Bobylev, A. V. and Cercignani, C., Exact eternal solutions of the Boltzmann equation, J. Stat. Phys.106 (2002) 1019-1038. · Zbl 1001.82090 |
| [5] | Bodineau, T., Gallagher, I., Saint-Raymond, L. and Simonella, S., One-sided convergence in the Boltzmann-Grad limit, Ann. Fac. Sci. Toulouse Math.27 (2018) 985-1022. · Zbl 1416.35174 |
| [6] | Briant, M., Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions, Kinet. Relat. Models8 (2015) 281-308. · Zbl 1330.35282 |
| [7] | Briant, M., Instantaneous filling of the vacuum for the full Boltzmann equation in convex domains, Arch. Ration. Mech. Anal.218 (2015) 985-1041. · Zbl 1457.35023 |
| [8] | Caprino, S., Pulvirenti, M. and Wagner, W., Stationary particle systems approximating stationary solutions to the Boltzmann equation, SIAM J. Math. Anal.29 (1998) 913-934. · Zbl 0911.60081 |
| [9] | Cercignani, C., Illner, R. and Pulvirenti, M., The Mathematical Theory of Dilute Gases, , Vol. 106 (Springer, 1994). · Zbl 0813.76001 |
| [10] | Chen, H., Cercignani-Lampis boundary in the Boltzmann theory, Kinet. Relat. Models13 (2020) 549-597. · Zbl 1447.76029 |
| [11] | Chernov, N. and Markarian, R., Introduction to the Ergodic Theory of Chaotic Billiards, 2nd edn. (Facultad de Ingeniería Universidad de la República, Uruguay, 2003), https://www.fing.edu.uy/imerl/grupos/ssd /publicaciones/pdfs/articulos/ 2003/2003CheMarIntrod.pdf. · Zbl 1037.37017 |
| [12] | Engel, K.-J. and Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, with contributions by Brendle, S., Campiti, M., Hahn, T., Metafune, G., Nickel, G., Pallara, D., Perazzoli, C., Rhandi, A., Romanelli, S. and Schnaubelt, R., , Vol. 194 (Springer, 2000). · Zbl 0952.47036 |
| [13] | Engel, K.-J. and Nagel, R., A Short Course on Operator Semigroups, (Springer, 2006). · Zbl 1106.47001 |
| [14] | Esposito, R. and Marra, R., Stationary non equilibrium states in kinetic theory, J. Stat. Phys.180 (2020) 773-809. · Zbl 1447.82026 |
| [15] | Fournier, N., Strict positivity of the solution to a 2-dimensional spatially homogeneous Boltzmann equation without cutoff, Ann. Inst. H. Poincaré Probab. Stat.37 (2001) 481-502. · Zbl 0981.60056 |
| [16] | Gamba, I. M., Panferov, V. and Villani, C., Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal.194 (2009) 253-282. · Zbl 1273.76373 |
| [17] | Gikhman, I. I. and Skorokhod, A. V., Introduction to the Theory of Random Processes (W. B. Saunders Company, 1969). · Zbl 0573.60003 |
| [18] | Imbert, C., Mouhot, C. and Silvestre, L., Gaussian lower bounds for the Boltzmann equation without cutoff, SIAM J. Math. Anal.52 (2020) 2930-2944. · Zbl 1442.35050 |
| [19] | Janson, S., Gaussian Hilbert Spaces, , Vol. 129 (Cambridge University Press, 1997). · Zbl 0887.60009 |
| [20] | Kuo, H.-W., Liu, T.-P. and Tsai, L.-C., Free molecular flow with boundary effect, Commun. Math. Phys.318 (2013) 375-409. · Zbl 1263.82043 |
| [21] | Löbus, J.-U., Boundedness of the stationary solution to the Boltzmann equation with spatial smearing, diffusive boundary conditions, and Lions’ collision kernel, SIAM J. Math. Anal.50 (2018) 5761-5782. · Zbl 1408.35116 |
| [22] | Lods, B., Mokhtar-Kharroubi, M. and Rudnicki, R., Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators, Ann. Inst. H. Poincaré Anal. Non Linéaire37 (2020) 877-923. · Zbl 1439.82037 |
| [23] | Maxwell, C., Stresses in rarefied gases arising from inequalities of temperature, Philos. Trans. Roy. Soc. London170 (1879) 231-256. · JFM 11.0777.01 |
| [24] | Mischler, S., Kinetic equations with Maxwell boundary conditions, Ann. Sci. Éc. Norm. Supér. (4)43 (2010) 719-760. · Zbl 1228.35249 |
| [25] | Mouhot, C., Quantitative lower bounds for the full Boltzmann equation I. Periodic boundary conditions, Comm. Partial Differential Equations30 (2005) 881-917. · Zbl 1112.76061 |
| [26] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, 1983). · Zbl 0516.47023 |
| [27] | Pulvirenti, A. and Wennberg, B., A Maxwellian lower bound for solutions to the Boltzmann equation, Commun. Math. Phys.183 (1997) 145-160. · Zbl 0866.76077 |
| [28] | Rezakhanlou, F. and Villani, C., Entropy Methods for the Boltzmann Equation, , Vol. 1916 (Springer, 2008). · Zbl 1125.76001 |
| [29] | Rjasanow, S. and Wagner, W., Stochastic Numerics for the Boltzmann Equation, , Vol. 37 (Springer, 2005). · Zbl 1155.82021 |
| [30] | Saint-Raymond, L., Hydrodynamic Limits of the Boltzmann Equation, , Vol. 1971 (Springer, 2009). · Zbl 1171.82002 |
| [31] | Webb, G. F., Continuous nonlinear perturbations of linear accretive operators in Banach spaces, J. Funct. Anal.10 (1972) 191-203. · Zbl 0245.47052 |
| [32] | Wei, J. and Zhang, X., Eternal solutions of the Boltzmann equation near travelling Maxwellians, J. Math. Anal. Appl.314 (2006) 219-232. · Zbl 1090.35151 |
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