The stationary Boltzmann equation in $\mathbb {R}^n$ with given indata

Arkeryd, Leif; Nouri, Anne

The stationary Boltzmann equation in

ℝ^{n}

with given indata

Arkeryd, Leif; Nouri, Anne

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 1 (2002) no. 2, p. 359-385

Abstract

An $L^{1}$ -existence theorem is proved for the nonlinear stationary Boltzmann equation for soft and hard forces in $ℝ^{n}$ with given indata on the boundary, when the collision operator is truncated for small velocities.

MR 1991144 | Zbl 1170.76350 | 1 citation in Numdam

Classification: 76P05

@article{ASNSP_2002_5_1_2_359_0,
     author = {Arkeryd, Leif and Nouri, Anne},
     title = {The stationary Boltzmann equation in $\mathbb {R}^n$ with given indata},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 1},
     number = {2},
     year = {2002},
     pages = {359-385},
     zbl = {1170.76350},
     mrnumber = {1991144},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2002_5_1_2_359_0}
}

Arkeryd, Leif; Nouri, Anne. The stationary Boltzmann equation in $\mathbb {R}^n$ with given indata. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 1 (2002) no. 2, pp. 359-385. http://www.numdam.org/item/ASNSP_2002_5_1_2_359_0/

References

[1] L. Arkeryd, On the stationary Boltzmann equation in $ℝ^{n}$ , IMRN 12 (2000), 626-641. | 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 - R. Illner, Measure solutions of the steady Boltzmann equation in a slab, Comm. Math. Phys. 142 (1991), 285-296. | MR 1137065 | Zbl 0733.76063

[4] 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

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

[6] L. Arkeryd - A. Nouri, On the stationary Povzner equation in three space variables, J. Math. Kyoto Univ. 39 (1999), 115-153. | MR 1684160 | Zbl 1010.35022

[7] L. Arkeryd - A. Nouri, On the Milne problem and the hydrodynamic limit for a steady Boltzmann equation model, J. Statist. Phys. 99 (2000), 993-1019. | MR 1766902 | Zbl 0959.82022

[8] A. V. Bobylev - G. Spiga, On a class of exact two-dimensional stationary solutions for the Broadwell model of the Boltzmann equation, J. Phys. A 27 (1994), 7451-7459. | MR 1310280 | Zbl 0844.76084

[9] A. V. Bobylev - G. Toscani, Two-dimensional half space problems for the Broadwell discrete velocity model, Contin. Mech. Thermodyn. 8 (1996), 257-274. | MR 1416192 | Zbl 0880.76068

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

[11] C. Cercignani, Measure solutions for the steady nonlinear Boltzmann equation in a slab, Transport Theory Statist. Phys. 27 (1998), 257-271. | MR 1636275 | MR 1646503 | Zbl 0914.76071

[12] C. Cercignani - R. Illner - M. Shinbrot, “A boundary value problem for the 2-dimensional Broadwell model”, Comm. Math. Phys. 114 (1985), 687-698. | MR 929135 | Zbl 0668.76091

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

[14] C. Cercignani - M. Giurin, Measure solutions for the steady linear Boltzmann equation in a slab, Transport Theory Statist. Phys. 28 (1999), 521-529. | MR 1705622 | Zbl 0940.35165

[15] H. Cornille, Exact $(2 + 1)$ -dimensional solutions for two discrete velocity models with two independent densities, J. Phys. A 20 (1987), 1063-1067. | MR 924710

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

[17] R. J. Diperna - P. L. Lions - Y. Meyer, $L^{p}$ regularity of velocity averages, Anal. Non Lin. 8 (1991), 271-287. | Numdam | MR 1127927 | Zbl 0763.35014

[18] L. Falk, Existence of solutions to the stationary linear Boltzmann equation, Thesis, Gothenburg, 2000. | Zbl 1082.82011

[19] H. Grad, High frequency sound recording according to Boltzmann equation, SIAM J. Appl. Math. 14 (1966), 935-955. | MR 208969 | Zbl 0163.23203

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

[21] A. Heintz, Solvability of a boundary problem for the non linear Boltzmann equation in a bounded domain, In: “Molecular Gas Dynamics” (in Russian), Aerodynamics of rarefied gases 10, 16-24, Leningrad, 1980.

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

[23] N. Maslova, “Non linear evolution equations, Kinetic approach”, Series on Advances in Mathematics for Applied Sciences Vol. 10, World Scientific, 1993. | Zbl 0846.76002

[24] N. Maslova, The solvability of internal stationary problems for Boltzmann's equation at large Knudsen numbers, USSR Comp. Math. Math. Phys. 17 (1977), 194-204. | MR 459451 | Zbl 0383.35063

[25] V. Panferov, “On the existence of stationary solutions to the Povzner equation in a bounded domain”, 2000, submitted.

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

[27] R. Pettersson, On convergence to equilibrium for the linear Boltzmann equation without detailed balance assumptions, Rarefied gas dynamics, Oxford UP, 19 (1995), 107-113.

[28] L. Triolo, A formal generalization of the H-theorem in kinetic theory, Report, Roma Tor Vergata, 1993.

[29] S. Ukai, Stationary solutions of the BGK model equation on a finite interval with large boundary data, Transport Theory Statist. Phys. 21 (1992), 487-500. | MR 1194459 | Zbl 0791.76076

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