Brief description:

The transport Liouville equation. The initial value problem. The Boltzmann equation for diluted gases (elastic-interactions) and granular media (inelastic interactions). Elementary properties of the solutions. Irreversivility, conservation laws, H-theorem and energy inequalities. Driven flows. Pseudo Maxwell molecules vs. hard sphere models. Stationary solutions. Space homogenoeus solutions. Boundary conditions and space dependant solutions. The Fokker Plank equation for grazing collisions. Issues on existence, uniqueness and stability properties. Trends to equilibrium.

Derivation of kinetic models for charge transport. The Boltzmann-Poisson system. Modeling inhomogeneous small devices. Boltzmann-Poisson vs. Fokker-Plank-Poisson systems.

From kinetic to fluid dynamical models. Small mean free path, Hilbert and Chapmann expansions. Moment Methods. Derivation of fluid level equations. Low field approximations: Drift-Diffusion models. High field approximations. Hydrodynamic models. The initial--boundary value problem.

Some topics on numerical simulations of particle kinetic systems: Direct Simulations of Montecarlo (DSMC) vs. deterministic solvers.

Prerequisite: Some knowledge of methods of applied mathematics and differential equations

Bibliography:

Cercignani C., The Boltzmann Equation and its Applications, Springer, New York, 1988.

Cercignani C., Illner, R. and Pulverenti, M., The Mathematical Theory of Diluted Gases, Springer, New York, 1994.

Villani, C., A review of Mathematical topics in collisional kinetic theory, Handbook of fluid mechanics, (2003).

Markovich, P., Ringhofer, C.A., and Schmeiser, C., Semiconductor Equations, Springer, Wien-New York, 1989.

Several recent papers to be distributed in class.

Permission of instructor not required.:

Irene M. Gamba

RLM 10.166

471-7150

Email: gamba@math.utexas.edu

Homepage: http://www.math.utexas.edu/users/gamba