This topics course
covers broad issues on Analytical and Numerical issues associated to probability density
solutions associates to Boltzmann and Landau equations for conservative and
non-conservative systems and their connections to non-equilibrium statistical methods
applied to fluid dynamics, mechanics and social dynamics.
1)
Introduction and elementary properties. The Boltzmann transport equation for binary
elastic and inelastic collisional theory, time irreversibility, conservation
laws, H-theorem and energy inequalities for inelastic interactions. The grazing collision
limit for Coulomb interactions and the connection to the Landau Equation. Fourier representation of
the Boltzmann and Landau equations. Comon analytical structures for non-linear
collisional forms for classical and relativistic Boltzmann and Landau type
models.
2)
Space homogeneous problems. Fundamental estimates for
existence and uniqueness for ODE in the Banach space in L1k(Rd). Lower and upper moment estimates
inequalities for coerciveness and boundedness. Existence and Uniqueness Theorem
for solutions to the Cauchy problem. Summability of moments and exponential moments and tails.
Young’s type inequalities for the collisional form. The Carleman
integral representation and gain of integrability for propagation of Lp(Rd)
norms. Propagation
of Wm,p(Rd)
regularity theory, 1≤ p ≤ ∞.
3)
Space inhomogeneous problems. Kaniel-Shimbrot-Illner
iteration method for the classical Boltzmann equation and inelastic
interactions.
Initial and Boundary value problems. Scattering effects for solutions in all
space due to dispersion vs dissipation. Radiation Problems.
4)
Beyond Landau Damping: Weak turbulence problems in high
energy plasmas. Derivation for Quasilinear (particles) and Frequency waves
(plasmon) systems. Model reduction
for high energy states performed to Vlasov-Poisson and Vlasov-Maxwell systems
under ergodicity and Random Phase Approximation hypothesis, in 1- or 3-dimensional
in position and momenta space.
5)
Maxwell type of interactions and Graph theory. Maxwell interactions form in connection to
graph and graphons theory. Stationary and self-similar solutions for space
homogeneous problems. Connections to special solutions in Fourier space.
6) Numerical
approximations to kinetic particle systems. Deterministic solvers for
linear and non-linear collisional forms of classical and relativistic Boltzmann
and Landau type. Unconditional conservative spectral and FEM methods.
Galerkin-Petrov schemes and spherical harmonics for sparce solvers. Boundary
and initial conditions. Stability and error estimates. Applications to kinetic
models for plasmas and charge transport as well as to inverse problem in nano-scale. The Boltzmann-Poisson
system.
Prerequisites: Some knowledge of Methods of Applied Analysis and Mathematics and Partial Differential Equations.
Testing and examination plan and
policies: Attendance at lectures is expected. A 30-min prepared presentation on a topic to
be discussed with the instructor. There will be neither exams, nor tests for
this course.
This course is complementary to graduate Topics PDE II (M 393C)-Topics in Kinetic Theory by Dominic Wynter
Recommended
bibliography and references:
·
Class notes and several recent research papers to be
distributed in class.
·
Kip S. Thorne, Roger D. Blandford, Modern Classical Physics:
Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics,
Princeton University. (Focusing on Fluids, Plasmas Statistical Physics
material.)
·
Nicholson, D.R.; Introduction to Plasma Theory. John Wiley
and Sons, 1982.
·
Stix, T.H.; Waves in Plasmas, AIP, NY, 1992.
·
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, NY,1994.
·
Villani, C., A review of Mathematical topics in collisional
kinetic theory, Handbook of Fluid Mechanics, 2003.
·
Sone,Y, Kinetic Theory and Fluid Dynamics (Birkhäuser,2002):Click here to download supplementary notes&errata
·
Sone, Y.,
Molecular Gas Dynamics (Birkhäuser,
2007): Click here to download supplementary notes&errata
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.