Transport Phenomena and Kinetic Theory Applications to Gases, Semico
The study of kinetic equations related to gases, semiconductors, photons, traffic flow, and other systems has developed rapidly in recent years because of its role as a mathematical tool in many applications in areas such as engineering, meteorology, biol
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H.G. Of/)/ne/'(Mattiematical Biology) Department of Matliematics University of Minnesota 270A Vincent Hall Minneapolis, MN 55455, USA othmerSmath.umn.edu
K.J. Bathe {SoM IVIectianlcs) Department of IVIechanlcal Engineering iVIassachusetts institute of Teciinology Cambridge, MA 02139, USA kjbSmit.edu
L P/ez/os/(industrial Matliematics) Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino, Italy luigi.preziosifipolito.it
P. Degond (Semiconductor & Transport Modeling) Math§matiques pour I'industrie et la Pliysique Universite P. Sabatier Toulouse 3 118 Route deNarbonne 31062 Toulouse Cedex, France degondSmlp.ups-tlse.fr Andreas Deutsch (Complex Systems in tlie Life Sciences) Center for Higti Performance Computing Teclinical University of Dresden D-01062 Dresden, Germany deutschQzhr.tu-dresden.de M.A. Herrero Garcia (Mathematical Methiods) Departamento de Matematica Aplicada Universidad Complutense de Madrid Avenida Complutense s/n 28040 Madrid, Spain herrero@suima4. mat. ucm. es W. Kliemann (Stochastic Modeling) Department of Mathematics Iowa State University 400 Carver Hall Ames, lA 50011, USA kliemannSiastate.edu
V. Protopopescu (Competitive Systems, Epidemiology) CSMD Oal< Ridge National Laboratory Oal< Ridge, TN 37831-6363, USA vvpSepmnas. epm. o m l . gov K.R. /?ayapopa/(Multipliase Flows) Department of Mechanical Engineering Texas A&M University College Station, TX 77843, USA KRaj agopalSmengr.tamu.edu Y. Sone (Fluid Dynamics in Engineering Sciences) Professor Emeritus Kyoto University 230-133 lwal 0 and we pass to the limit → 0 due to (1.4.6)). In the noncutoff case we approximate the solution by cutting off the angles close to π/2 and the small relative speeds. In this way we can obtain a sequence fn formally approximating the solution f whose existence we want to prove. Lemma 5.1 Let {fn } be a sequence of solutions to an approximating problem. There is a subsequence such that for each T > 0 (i) fn dv → f dv a.e. and in L1 ((0, T ) × R3 ), (ii) R3
|V|2 fn∗ dv∗ →
R3
|V|2 f∗ dv∗
in L1 ((0, T ) × R3 × BR ) for all R > 0, and a.e.,
12
Carlo Cercignani
(iii)
gn (x, t) =
|V|2 fn fn∗ dvdv∗ → 1 + fn dv
R3 ×R3
|V|2 f f∗ dvdv∗ = g(x, t) 1 + f dv (1.5.1)
R3 ×R3
weakly in L1 ((0, T ) × (0, 1)). Proof. (i) is immediate. (ii) uses an argument well known in the DiPerna–Lions proof with the estimate supn fn (1 + |v|2 ) dv < ∞ to reduce the problem to bounded domains with respect to v∗ . For (iii) we use (i) and the fact that fn converges weakly, but the factor multiplying it in the integral converges a.e. because of (ii). Now we remark that gn (x, t) converges weakly to g(x, t) and ρn (x, t) converges a.e. to ρ(x, t) and the integral ρn gn dxdt is uniformly bounded to conclude with the following lemma. Lemma 5.2 Let {fn } be a sequence of solutions to an approximating problem. There is a subsequence such that for each T > 0 2 |V| fn fn∗ dμdt → |V|2 f f∗ dμdt. (1.5.2) (0,T )×(0,1)×R3 ×R3
(0,T )×(0,1)×R3 ×R3
We can now prove the basic result. Le
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