On the interior boundary-value problem for the stationary Povzner equation with hard and soft interactions
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 4, p. 771-825

The Povzner equation is a version of the nonlinear Boltzmann equation, in which the collision operator is mollified in the space variable. The existence of stationary solutions in ${L}^{1}$ is established for a class of stationary boundary-value problems in bounded domains with smooth boundaries, without convexity assumptions. The results are obtained for a general type of collision kernels with angular cutoff. Boundary conditions of the diffuse reflection type, as well as the given incoming profile, are treated. The method is based on establishing the weak compactness of approximate solutions by using estimates of the entropy production.

Classification:  82C40,  35Q72,  45K05
@article{ASNSP_2004_5_3_4_771_0,
title = {On the interior boundary-value problem for the stationary Povzner equation with hard and soft interactions},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 3},
number = {4},
year = {2004},
pages = {771-825},
zbl = {1121.82035},
mrnumber = {2124588},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2004_5_3_4_771_0}
}

Panferov, Vladislav A. On the interior boundary-value problem for the stationary Povzner equation with hard and soft interactions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 4, pp. 771-825. http://www.numdam.org/item/ASNSP_2004_5_3_4_771_0/

[1] L. Arkeryd, On the stationary Boltzmann equation in ${ℝ}^{n}$, Internat. Math. Res. Notices (2000), 625-641. | MR 1772079 | Zbl 0965.35126

[2] L. Arkeryd - C. Cercignani, On the convergence of solutions of the Enskog equation to solutions of the Boltzmann equation, Comm. Partial Differential Equations 14 (1989), 1071-1089. | MR 1017064 | Zbl 0688.76053

[3] L. Arkeryd - C. Cercignani, Global existence in ${L}^{1}$ for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation, J. Statist. Phys. 59 (1990), 845-867. | MR 1063185 | Zbl 0780.76066

[4] L. Arkeryd - C. Cercignani - R. Illner, Measure solutions of the steady Boltzmann equation in a slab, Comm. Math. Phys. 142 (1991), 285-296. | MR 1137065 | Zbl 0733.76063

[5] L. Arkeryd - A. Nouri, A compactness result related to the stationary Boltzmann equation in a slab, with applications to the existence theory, Indiana Univ. Math. J. 44 (1995), 815-839. | MR 1375351 | Zbl 0853.45015

[6] L. Arkeryd - A. Nouri, The stationary Boltzmann equation in the slab with given weighted mass for hard and soft forces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), 533-556. | Numdam | MR 1677990 | Zbl 0936.76076

[7] L. Arkeryd - A. Nouri, On the stationary Povzner equation in ${ℝ}^{n}$, J. Math. Kyoto Univ. 39 (1999), 115-153. | MR 1684160 | Zbl 1010.35022

[8] L. Arkeryd - A. Nouri, ${L}^{1}$ solutions to the stationary Boltzmann equation in a slab, Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), 375-413. | Numdam | MR 1842024 | Zbl 0991.45005

[9] L. Arkeryd - A. Nouri, The stationary Boltzmann equation in ${ℝ}^{n}$ with given indata, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), 359-385. | Numdam | MR 1991144 | Zbl 1170.76350

[10] P. Broman, On a class of boundary value problems for the Povzner equation, preprint no. 1993:27, Chalmers Univ. Tech., Göteborg (1993).

[11] S. Caprino - M. Pulvirenti - W. Wagner, Stationary particle systems approximating stationary solutions to the Boltzmann equation, SIAM J. Math. Anal. 29 (1998), 913-934. | MR 1617710 | Zbl 0911.60081

[12] C. Cercignani, The Grad limit for a system of soft spheres, Comm. Pure Appl. Math. 36 (1983), 479-494. | MR 709645 | Zbl 0505.76088

[13] C. Cercignani, Measure solutions for the steady nonlinear Boltzmann equation in a slab, Comm. Math. Phys. 197 (1998), 199-210. | MR 1646503 | Zbl 0920.35156

[14] C. Cercignani - R. Illner - M. Pulvirenti, “The mathematical theory of dilute gases”, Springer-Verlag, New York, 1994. | MR 1307620 | Zbl 0813.76001

[15] C. Cercignani - R. Illner - M. Pulvirenti - M. Shinbrot, On nonlinear stationary half-space problems in discrete kinetic theory, J. Statist. Phys. 52 (1988), 885-896. | MR 968962 | Zbl 1084.82561

[16] C. Cercignani - R. Illner - M. Shinbrot, A boundary value problem for the two-dimensional Broadwell model, Comm. Math. Phys. 114 (1988), 687-698. | MR 929135 | Zbl 0668.76091

[17] J. Darrozes - J.-P. Guiraud, Généralisation formelle du théorème $H$ en présence de parois. Applications, C. R. Acad. Sc. Paris 262 (1966), 1368-1371.

[18] R. Diperna - P. L. Lions, On the Cauchy problem for the Boltzmann equation: Global existence and weak stability, Ann. of Math. 130 (1989), 321-366. | MR 1014927 | Zbl 0698.45010

[19] M. J. Esteban - B. Perthame, On the modified Enskog equation for elastic and inelastic collisions. Models with spin, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), 289-308. | Numdam | MR 1127928 | Zbl 0850.70141

[20] F. Golse - L. Saint-Raymond, Velocity averaging in ${L}^{1}$ for the transport equation, C. R. Math. Acad. Sci. Paris 334 (2002), 557-562. | MR 1903763 | Zbl 1154.35326

[21] J.-P. Guiraud, Problème aux limites intérieur pour l'équation de Boltzmann en régime stationnaire, faiblement non linéaire, J. Mécanique 11 (1972), 183-231. | MR 406275 | Zbl 0245.76061

[22] N. M. Günter, “Potential theory and its applications to basic problems of mathematical physics”, Frederick Ungar Publ. Co., New York, 1967. | MR 222316 | Zbl 0164.41901

[23] A. Heintz, Initial-boundary value problems in irregular domains for nonlinear kinetic equations of Boltzmann type, Transport Theory Statist. Phys. 28 (1999), 105-134. | MR 1669053 | Zbl 0942.35003

[24] R. Illner - J. Struckmeier, Boundary value problems for the steady Boltzmann equation, J. Statist. Phys. 85 (1996), 427-454. | MR 1413668 | Zbl 0930.76075

[25] A. I. Khisamutdinov, A simulation method for statistical modeling of rarefied gases, Dokl. Akad. Nauk SSSR 291 (1986), 1300-1304. | MR 870923 | Zbl 0624.76107

[26] M. Lachowicz - M. Pulvirenti, A stochastic system of particles modelling the Euler equations, Arch. Ration. Mech. Anal. 109 (1990), 81-93. | MR 1019171 | Zbl 0682.76002

[27] N. B. Maslova, Stationary problems for the Boltzmann equation in the case of large Knudsen numbers, Dokl. Akad. Nauk SSSR 229 (1976), 593-596. | MR 459450 | Zbl 0355.45012

[28] N. B. Maslova, “Nonlinear evolution equations. Kinetic approach”, vol. 10 of Series on Advances in Mathematics for Applied Sciences, World Scientific Publishing Co. Inc., River Edge, NJ, 1993. | MR 1218176 | Zbl 0846.76002

[29] S. Mischler, On weak-weak convergences and applications to the initial-boundary value problems for kinetic equations, preprint no. 35 of the University of Versailles, (1999).

[30] A. Nouri, Private communication, Dec., 1999.

[31] V. A. Panferov, Two problems on existence and approximation related to the Boltzmann equation, Ph. D. thesis, Chalmers University of Technology, Göteborg, Sweden, 1999.

[32] Y.-P. Pao, Boundary-value problems for the linearized and weakly nonlinear Boltzmann equation, J. Mathematical Phys. 8 (1967), 1893-1898. | MR 230532 | Zbl 0155.32603

[33] A. Y. Povzner, On the Boltzmann equation in the kinetic theory of gases, Mat. Sbornik (N. S.) 58/100 (1962), 65-86. Translated in Amer. Math. Soc. Transl., Ser. 2, 47, pp. 193-216, AMS, Providence RI, (1965). | MR 142362 | Zbl 0188.21204

[34] P. Résibois, $H$ theorem for the (modified) nonlinear Enskog equation, J. Statist. Phys. 19 (1978), 593-609. | MR 471846

[35] P. Résibois - M. De Leener, “Classical Kinetic Theory of Fluids”, John Wiley & Sons, New York, 1977.

[36] S. Ukai, Solutions of the Boltzmann equation, in Patterns and waves, North-Holland, Amsterdam (1986), 37-96. | MR 882376 | Zbl 0633.76078

[37] S. Ukai - K. Asano, Steady solutions of the Boltzmann equation for a gas flow past an obstacle. I. Existence, Arch. Ration. Mech. Anal. 84 (1983), 249-291. | MR 714977 | Zbl 0538.76070

[38] W. Wagner, A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation, J. Statist. Phys. 66 (1992), 1011-1044. | MR 1151989 | Zbl 0899.76312