# zbMATH — the first resource for mathematics

Conditions at infinity for Boltzmann’s equation. (English) Zbl 0799.35210
The paper studies boundary conditions at infinity for the Boltzmann equation: ${{\partial f} \over {\partial t}}+ v\cdot \nabla_ x f= Q(f,f), \qquad x\in \mathbb{R}^ N, \quad v\in \mathbb{R}^ N, \quad t\in (0,\infty), \tag{B}$ where $$N\geq 1$$ and $$Q(f,f)$$ is the so-called collision operator. An initial condition is given: (1) $$f|_{t=0}= f_ 0(x,\xi)$$ in $$\mathbb{R}^ N\times \mathbb{R}^ N$$. The following model of behaviour at infinity is studied: (2) $$f\to M$$ as $$| x|\to +\infty$$, where $$M$$ is a pure Maxwellian. This problem is called the case of a Maxwellian at infinity. Another problem consists of looking at (B), (1) in $$\Omega\times \mathbb{R}_ v^ N\times (0,\infty)$$, ($$\Omega={\mathcal O}^ c$$, where $${\mathcal O}$$ a bounded smooth open set of $$\mathbb{R}^ N$$), with (2) and a boundary condition on $$\partial\Omega\times \mathbb{R}_ v^ N\times (0,\infty)$$ like a specular reflection condition. This is called the exterior domain case.
Moreover, the “tube case” is studied, that is (B), (1) in $$\Omega= \mathbb{R}\times \omega$$ where $$\omega$$ is a bounded smooth open set in $$\mathbb{R}^{N-1}$$ with specular reflection condition in $$\partial\Omega= \mathbb{R}\times \partial\omega$$ and (2) replaced by (3) $$f\to M^ +$$ as $$x_ 1\to +\infty$$, $$f\to M^ -$$ as $$x_ 1\to +\infty$$, where $$M^ +$$, $$M^ -$$ are two pure Maxwellians of the following form $M^ \pm= \rho^ \pm \exp (| v_ 1- u_ 1^ \pm|^ 2+ | v_ 2|^ 2+ \dots+ | v_ N|^ 2/ 2T^ \pm) (2\pi T^ \pm)^{-N/2}\tag{4}$ with $$\rho^ +,\rho^ ->0$$, $$T^ +, T^ ->0$$, $$u_ 1^ +, u_ 1^ -\in \mathbb{R}$$.
Reviewer: S.Totaro (Firenze)

##### MSC:
 35Q72 Other PDE from mechanics (MSC2000) 35B40 Asymptotic behavior of solutions to PDEs 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 82C70 Transport processes in time-dependent statistical mechanics
##### Keywords:
Boltzmann equation; boundary conditions at infinity
Full Text:
##### References:
 [1] Arkeryd L., On the Boltzmann equation. Part I : Existence. 45 pp 1– (1972) · Zbl 0245.76059 [2] Arkeryd L., On the Boltzmann equation. Part II : The full initial-value problem. 45 pp 17– (1972) · Zbl 0245.76060 [3] Arkeryd L., L estimates for the space homogeneous Boltzmann equation. 31 pp 347– (1983) · Zbl 0584.35090 [4] DOI: 10.1007/BF01238905 · Zbl 0663.76080 [5] Asano, K. 1984.Local solutions to the initial and initial-boundary value problem for the Boltzmann equation., Vol. 24, 225–238. Journ. Math. Kyoto Univ. · Zbl 0571.76075 [6] Bardos, C., Golse, F. and Levermore, D. 1989.Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incom-pressibles., Vol. 309, 727–732. Paris: C.R. Acad. Sci. · Zbl 0697.35111 [7] Bardos C., Fluid dynamic limits of kinetic equations. I : Formal derivations. 63 pp 323– (1991) [8] Bardos C., Convergence proofs for the Boltzmann equation 3 [9] Bellomo N., On the Cauchy problem for the nonlinear Boltzmann equation. 26 pp 334– (1985) · Zbl 0561.35074 [10] Boltzmann, L. 1872.Weitere studien über das wärme gleichgenicht unfer gasmoleküler., Edited by: Brush, S.G. Vol. 66, 275–670. Wien: Sitzungsberichte der Akademie der Wissenschaften. Translation :Further Studies on the thermal equilibrium of gaz Molecules, 88-174; In Kinetic Theory 2, Pergamen, Oxford, 1966 [11] Caflish R., The Boltzmann equation with a soft potential. Part I. 74 pp 71– (1980) [12] Carleman T., Problèmes mathématiques dans la théorie cinétique des gaz. Notes rédigées par L. Carleson et O. Frostman. (1957) [13] Cercignani, C. 1975. ”Theory and application of the Boltzmann equation.”. Edinburgh: Scottish Academic Press. · Zbl 0403.76065 [14] Cercignani, C. 1988. ”The Boltzman equation and its applications”. Berlin: Springer. · Zbl 0646.76001 [15] Cercignani C., A remarkable estimate for the solutions of the Boltzmann equations 5 pp 59– (1992) · Zbl 0762.35090 [16] Chapman, S. and Cowling, T.G. 1939. ”The mathematical theory of nonuniform gases”. 1952Cambridge Univ. Press. · Zbl 0063.00782 [17] DiPerna R.J., On the Cauchy problem for Boltzmann equations : Global existence and weak stability. 130 pp 321– (1989) · Zbl 0698.45010 [18] DiPerna, R.J. and Lions, P.L. 1988. Vol. 306, 343–346. Paris: C.R. Acad. Sci. announced in [19] DiPerna R.J., Global solutions of Boltzmann ’s equation and the entropy inequality. 114 pp 47– (1991) · Zbl 0724.45011 [20] Elmroth T., Global boundedness of moments of solutions of the Boltzmann equation for forces of inifnite rante. 82 pp 1– (1983) · Zbl 0503.76091 [21] Grad, H. 1958.Principles of the kinetic theory of gases, 205–294. Berlin: in Flügge’s Handbuch der Physik XII Springer. [22] Gustaffson T., Lp-estimates for nonlinear space homogeneous Boltzmann equation. (1985) [23] Hamdache K., Global existence for weak solutions for the initial boundary value problems of Boltzmann equation. 119 pp 309– (1992) · Zbl 0777.76084 [24] Maxwell, J.C. 1866.On the dynamical theory of gases.Vol. 157, 49–8. London Phil. Trans. Roy. Soc. [25] Maxwell, J.C. 1890. ”Scientific papers.”. Vol. 2, Cambridge: Cambridge Univ. Press. Dover Publications,New-York reprinted by · JFM 22.0023.01 [26] Morgenstern, D. 1954.General existence and uniqueness proof for spatially homogeneous solutions of Maxwell-Boltzmann equation in the case of Maxwellian molecules., Vol. 40, 719–721. USA: Proc. Nat. Acad. Sci. · Zbl 0056.20506 [27] Nishida, T. and Umai, K. 1976.Global solutions to the initial-value problem for the nonlinear Boltzmann equation., Vol. 12, 229–239. Publ. R.I.M.S. Kyoto Univ. · Zbl 0344.35003 [28] Povsner A. Ya., The Boltzmann equation in the kinetic theory of gases. 47 pp 193– (1962) [29] Snitman, A.S. 1984.Equations de type Boltzmann spatialement homogènes., Vol. 66, 559–592. Z. Wahrs. verw. Gebeite. · Zbl 0553.60069 [30] Truesdell, C. and Muncaster, R.G. 1963. ”Fundamentals of Maxwell’s kinetic theory of a simple monoatomic gas”. New-York: Academic Press. [31] Ukai, S. and Asano, K. 1976.Steady solutions of the Boltzmann equation for a gas flow past an obstacle. II. Stability., Vol. 12, 229–239. Publ. R.I.M.S. Kyoto Univ.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.