AMSC 698L Description and Prerequisites

Course Description

AMSC 698L, Topics in Kinetic Theory II This is the second semester of a two-semester sequence designed to survey current understanding of the justification of fluid dynamics from kinetic theory. In the first semester there was a brief review of classical mechanics, various theories of fluid dynamics were intorduced from a traditional continuum perspective. The classical Boltzmann equation was then introduced followed by other kinetic theories. Fluid dynamic regimes were identified, and various fluid dynamical systems will be derived. Moment closure recipes were used to derive moment systems for transition regimes. The first semester had a formal and physical emphasis. The second will be more mathematical. A tentative outline for the entire year is given below.

Course Prerequisites

either a graduate level one semester course in partial differential equations, a theoretical graduate level course in an applied field such as fluid mechanics, or permission of instructor.

This course is intended for both students in mathematics and students in applied fields. Students should have some knowledge of partial differential equations (PDE), either directly or through courses such as fluid mechanics, quantum mechanics, or semiconductor design that use PDE extensively. Students should also have knowledge of the topics covered in the first semester. Please feel free to contact me concerning your background if you have any questions.

More Details

There will be no exams. Each student will be expected to produce two written reports on somes of the lectures.

Tentative Outline of First Term

Part 1: Preliminaries

Classical Mechanics
- Many-Body Problem
- Lagrangian Formulation
- Hamiltonian Formulation
- Rigid-Body Dynamics
- Vibrating-Body Dynamics
Fluid Dynamics
- Local Conservation and Balance Laws
- Compressible Euler Systems
- Hyperbolicity
- Equilibrium Thermodynamics and Entropy
- Compressible Navier-Stokes Systems
Incompressible Fluid Dynamics
- Incompressible Regimes
- Boussinesq-Balance Stokes, Navier-Stokes, and Euler Systems
- Dominant-Balance Stokes, Navier-Stokes, and Euler Systems

Part 2: Kinetic Theories

Kinetic Theories
- Microscopic Pictures
- Kinetic Regimes
- Collisionless Regimes
- Collisional Regimes
- The Boltzmann-Grad Limit
Maxwell-Boltzmann Theory
- The Classical Boltzmann Equation
  - Boltzmann Collision Operators
  - Elastic Binary Collisions
  - Gain and Loss Terms
  - Collision Kernels
  - Boundary Conditions
- Properties of the Boltzmann Equation
  - Galilean Symmetry
  - Boltzmann Identities
  - Collision Invariants
  - Local Conservation
  - Entropy, Local Dissipation, and Equilibria
  - The Maxwell Continuity Equation
- Recipes for Collision Kernels
  - Maxwell's Recipe for Classical Monatomic Molecules
  - Cut-off Kernels
  - Fermi's Recipe for Quantum Molecules
Classical Kinetic Theories
- Properties of Collision Operators
  - Galilean Symmetry
  - Local Conservation
  - Entropy, Local Dissipation, and Equilibria
- Example: Born-Green-Kirkwook Operators
- Example: Fokker-Planck-Landau Operators
- Example: Linear and Linearized Theories
More General Kinetic Theories
- Properties of Collision Operators
  - Local Conservation
  - Entropy, Local Dissipation, and Equilibria
- Example: Fermi-Dirac and Bose-Einstein Theories
- Example: Polyatomic Models
- Example: Multispecies Models
- Example: Discrete Velocity Models
- Example: Linear and Linearized Theories
- Properties of Boundary Operators

Part 3: Fluid Dynamical Approximations and Beyond

Compressible Euler Limits
- Fluid Dynamical Regimes
  - The Knudsen Number
  - Classical Compressible Euler Limits
  - General Compressible Euler Limits
- General Compressible Euler Systems
  - Potential Formulation
  - Density Formulation
  - Characteristic Velocities
Beyond Compressible Euler Limits
- The Hilbert Approach
  - Hilbert Expansions
  - Assumptions on the Linearized Collision Operator
  - Order-by-Order Solution
  - The Navier-Stokes Approximation
  - The Hilbert-Grad Theory for Cut-Off Hard Kernels
  - The Golse-Poupaud Theory for Cut-Off Soft Kernels
- The Chapman-Enskog Approach
  - Chapman-Enskog Equations
  - Chapman-Enskog Expansions
  - Assumptions on the Linearized Collision Operator
  - Order-by-Order Solution
  - The Navier-Stokes Approximation
  - Problems with Formal Well-Posedness at Higher Order
- Self-Consistency of Classical Fluid Systems
  - Classical Fluid Systems and Moments
  - Moment Realizability
  - Moment-Based Self-Consistency Criteria
- Moment Systems
  - The Moment Closure Problem
  - Grad Closures
  - Entropy-Based Closures
  - Recovering Fluid Dynamics
Linear and Weakly Nonlinear Regimes
- General Setting
  - Nondimensional Form
  - Conservation and Dissipation Laws
  - Fluctuations
  - Moment-Based Derivations
- Derivation of Acoustic Systems
  - Infinitesimal Maxwellians
  - The Acoustic System
- Derivation of Boussinesq-Balance Systems
  - Long-Time Scaling
  - The Incompressibility and Boussinesq Relations
  - The von Karman Relation
  - The Key Ideas of the Derivation
  - Assumptions on the Linearized Collision Operator
  - The Stokes, Navier-Stokes, and Euler Dynamics
- Derivation of Dominant-Balance Systems
  - The Odd-Even Splitting
  - The Incompressibility and Pressure Relations
  - The Key Ideas of the Derivation
  - Assumptions on the Collision Operator
  - Identities for the Collision Operator
  - The Stokes, Navier-Stokes, and Euler Dynamics
- Derivation of Thermal-Stress Systems
  - Thermal-Stress Regimes
  - The Incompressibility and Pressure Relations
  - The Key Ideas of the Derivation
  - Assumptions on the Collision Operator
  - The Dynamics

Rough Tentative Outline of Second Term

Part 4: Mathematical Theories for Fluids

Global Existence and Uniqueness Theories for Fluids
- Well-Posedness of the Acoustic and Stokes Equations
- Leray Existence Theory for the Incompressible Navier-Stokes Equations
- Leray Uniqueness Theory for Classical Solutions of the Incompressible Navier-Stokes Equations
- Dissipative Solutions of the Incompressible Euler Equations
Regularity Theories for Fluids
- Regularity of Solutions of the Incompressible Euler Equations
- Regularity of Solutions of the Incompressible Navier-Stokes Equations

Part 5: Mathematical Theories for Kinetic Models

Analytical Tools for Kinetic Equations
- Characteristics, $L^p$ and $L^\Phi$ Estimates for the Transport Equation
- Compactness in the Weak $L^1$ Topology
- Velocity Averaging
- Example: Parabolic Scalings and the Rosseland Approximation
- More Velocity Averaging
- Example: Hyperbolic Scalings and the Perthame-Tadmor BGK model
Global Existence and Uniqueness Theories for Kinetic Models
- Space Homogeneous Solutions of the Boltzmann Equation
- Kaniel-Shinbrot Near Vacuum Theory
- Near Equilibrium Theory: $L^2$ Theory of the Linearized Boltzmann Equation (Grad)
- Global Existence for the BGK Model
DiPerna-Lions Theory
- Renormalized Solution of the Boltzmann Equation (DiPerna-Lions)
- Regularity of the Gain Term
- Propagation of Singularities for the Boltzmann Equation
- Proof of the DiPerna-Lions Theorem
- Extensions the DiPerna-Lions Result
- Uniqueness of Classical Solutions

Part 6: Mathematical Theories for Fluid Dynamical Limits

Local Hydrodynamic Limits for Smooth Solutions
- Results Based on the Hilbert Expansion
- Results Based on the Cauchy-Kovalevska Theorem
- The Bardos-Ukai Theorem on the Incompressible Navier-Stokes Limit for Small Initial Data
Global Hydrodynamic Limits: the BGL Program
- Fluctuation Theory for Renormalized Solutions
- The Linearized Boltzmann Limit
- The Acoustic Limit
- An Incompressible Stokes-Fourier Limit
- More Fluctuation Theory
- An Incompressible Navier-Stokes-Fourier Limit
The Relative Entropy Method
- Evolution of the Relative Entropy: Formal Computations
- An Incompressible Euler Limit
- The Compressible Stokes Limit