Spring 2014   -    Kinetic Theory: analysis, applications and numerical issues
M 393C and CAM 393C   
Unique 57210

Instructor: Prof. Irene M. Gamba
Office: RLM 10.166, Phone: 471-7150
E-Mail: gamba@math.utexas.edu
Office hours: by appointment

Meeting Hours:   RLM 10.176,    T-Th 11:00-12:30pm

Extra We will have an extra hour discussion time when needed on Wednesdays at RLM 9.176 sometime between 4:00 to 6:30pm.
This room and time will also be available for our class as a discussion forum place.

Kinetic theory: analysis, applications and numerical issues

This topics course covers issues on the Boltzmann and Smoluchowski type equations for conservative and non-conservative systems and connection to non-equilibrium statistical mechanics.
More specifically, introductory elementary properties of the solutions, time irreversibility, conservation laws, H-theorem and energy inequalities. Conservative and non-conservative kinetic problems for interacting kernels from variable potentials and angular cross sections. Connection to the Landau Equation.
Space Homogeneous problems. Linearized Boltzmann Equation. Povzner type lemmas. Existence and uniqueness properties. Carleman integral representation. Moment inequalities. Comparisons for point-wise bounds to solutions. Convolution inequalities for collision Operators.
Fourier representation of the Boltzmann equation. Kinetic equations of Maxwell type, stationary and self-similar solutions for space homogeneous problems. Connections to dynamical scaling and connections to stable laws from continuous probability theory to non-Gaussian states. Applications to information propagation problems.
Non-conservative kinetic problems for interacting kernels from variable potentials. Existence and uniqueness properties. Comparisons for point-wise bounds to solutions of Boltzmann equations. The space inhomogeneous problem. The space inhomogeneous problem. The Kaniel-Shimbrot iteration method. Averaging lemmas and renormalized DiPerna-Lions Solutions
From kinetic to fluid dynamical models. Small mean free path, Hilbert and Chapman expansions. Moment Methods. Derivation of fluid level equations. Low field approximations: Drift-Diffusion models. High field approximations. Hydrodynamic models.
Numerical methods of kinetic particle systems: deterministic solvers for linear and non-linear collisional forms. Spectral and FEM methods
Topics on kinetic models for plasmas and charge transport. The Boltzmann-Poisson system. Boltzmann-Poisson vs. Fokker-Plank-Poisson systems.


Prerequisites: Some knowledge of methods of applied mathematics and differential equations.

The following is a suggested bibliography:
Cercignani C., "The Boltzmann Equation and its Applications", Springer, New York, 1988.
Cercignani C., Illner, R. and Pulvirenti, 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, Handbook of Fluid Mechanics, (2003).

Class notes and several recent papers to be distributed in class.

The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.