On Diffusive Equilibria in Generalized Kinetic Theory
- 54 Downloads
We investigate the solvability of equations
Q(f,f)+∈ 2 Δf=0
in term of nonnegative integrable densities f∈L1+(R3). Here, Q(f, f) is a generalized collision operator. If Q is the Boltzmann operator, the only solution is 0. In contrast, we show that if Q is the pseudo-Maxwellian collision operator for granular flow, then there are non -trivial weak solutions of (★).
Unable to display preview. Download preview PDF.
- 1.R. J. DiPerna and P. L. Lions, On Fokker–Planck–Boltzmann equations, Commun. Math. Phys. (1988).Google Scholar
- 2.A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic, I. Modeling, and II. Numerical Investigations, SIAM J. Appl. Math. 59:983–1001 (1999).Google Scholar
- 3.M. Günther, A. Klar, Th. Materne, and R. Wegener, An Explicit Solvable Kinetic Model for Vehicular Traffic and Associated Macroscopic Equations, Math. Models and Methods in the Appl. Sci., to appear.Google Scholar
- 4.R. Illner and C. Stoica, Kinetic Equilibria in Traffic flow Models, preprint.Google Scholar
- 5.N. Sela and I. Goldhirsch, Hydrodynamic equations for rapid flows of smooth inelastic spheres to Burnett order, J. Fluid Mech. 361:41–74 (1998).Google Scholar
- 6.A. V. Bobylev, J. Carrillo, and I. A. Gamba, On some kinetic properties and hydrodynamics equations for inelastic interactions, J. of Stat. Phys. 98(3/4):743–773 (2000).Google Scholar
- 7.C. Cercignani, Shear flow of a granular material, J. Stat. Phys., to appear.Google Scholar
- 8.N. Dunford and L. Schwartz, Linear Operators I, Interscience Publ. (1967), Section IV.8.Google Scholar