AMSC 698K Description and Prerequisites

Course Description

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.

More Details

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