AMSC 698L Description and Prerequisites

Course Description

AMSC 698L, Mathematics of Kinetic Theory This course is designed to survey current understanding of the justification of fluid dynamics from kinetic theory. After a brief review of classical mechanics, various theories of fluid dynamics will be intorduced from a traditional continuum perspective. The classical Boltzmann equation then will be introduced followed by other kinetic theories. Fluid dynamic regimes will be identified, and various fluid dynamical systems will be derived. Moment closure recipes will be used to derive moment systems for transition regimes. Applications to rarefied gases, semiconductor modeling, radiative transport, and plasmas might be given. A more detailed tentative outline 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. Some knowledge of classical mechanics or thermodynamics would also be helpful, but is not necessary. 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

I. Maxwell-Boltzmann Theory

Rarefied Gases and the Boltzmann Equation
- Gaseous Regimes
- Ideal Gas Limits
- Kinetic Equations
Maxwell's Recipe for the Collision Operator
- Elastic Binary Collisions
- Collision Operators
- Gain and Loss Terms
- Collision Kernels
Properties of the Boltzmann Equation
- Dilation and Galilean Symmetries
- Boltzmann Identities
- Collision Invariants
- Local Conservation
- Entropy, Local Dissipation, and Equilibria
Relation to Euler and Navier-Stokes Systems
- Knudsen Number
- Compressible Euler Limit
- Deviation from Local Maxwellian
- Compressible Navier-Stokes Approximation
Linearized Collision Operator
- Null Space and Coercivity
- Compactness for the Loss Term
- Compactness for the Gain Term
- Fredholm Alternative
- Pseudoinverse
Boundary Conditions
- Perfectly Reflecting Stationary Boundaries
- Absorbing-Emitting Stationary Boundaries
- Simple Models
- Conservation and Dissipation Laws
- Moving Boundaries: Prescribed and Free

II. Other Kinetic Theories

Classical Kinetic Equations
- Properties of Collision Operators
- Example: Born-Green-Kirkwook Model
- Example: Ellipsoidal Approximation Model
- Example: Fokker-Planck-Landau Operators
- Example: Fokker-Planck Model
Linear and Linearized Kinetic Equations
- Interaction with a Thermal Background
- Photon and Neutron Transport
- Diffusion Approximation for Transport
- Linearized Kinetic Equations
- Linearized Fluid Approximations
Analytic Methods for Linear Kinetic Equations
- Integral Operators
- Wiener-Hopf Method
- Generalized Eigenfunction Method
General Kinetic Theories
- Properties of Collision Operators
- Example: Fermi-Dirac and Bose-Einstein Theories
- Example: Polyatomic Models
- Example: Discrete Velocity Models
- Example: Multispecies Models
General Euler and Navier-Stokes Systems
- Fluid Dynamical Regimes
- General Compressible Euler Systems
  - Potential Formulation
  - Density Formulation
  - Characteristic Velocities
- General Compressible Navier-Stokes Systems
  - Entropy Dissipation
  - Hypocoercivity

III. Fluid Dynamical Approximations

Hilbert and Chapman-Enskog Expansions
- Assumptions on the Linearized Collision Operator
- Hilbert Expansions
- Chapman-Enskog Expansions
- Application to Justifying Fluid Approximations
- Failure Beyond Navier-Stokes
Initial and Boundary Layers
- Initial Layer Expansions
- Initial Conditions for Fluid Approximations
- Boundary Layaer Expansions
- Half-Space Problems
- Boundary Conditions for Fluid Approximations
Beyond Navier-Stokes
- Temporal Approximations in a Linearized Setting
- Spatial Approximations in a Linearized Setting
- Balance
- First Correction to the Navier-Stokes System
Linear and Weakly Nonlinear Fluid Systems
- General Setting
- Moment-Based Derivations
- Derivation of Acoustic Systems
- Derivation of Boussinesq-Balance Systems
- Derivation of Dominant-Balance Systems
Nonstandard Fluid Systems
- Thermal-Stress Systems

IV. Global Solutions for Fluid Systems

Linear Fluid Systems
- Well-Posedness of Acoustic Systems
- Well-Posedness of Stokes Systems
- Regularity for Stokes Systems
- Well-Posedness of Compressiblr Stokes Systems
- Regularity for Compressiblr Stokes Systems
Leray Theory for Incompressible Navier-Stokes
- Weak Solutions
- Constuction of Approximate Solutions
- Compactness Results
- Passing to the Limit
- Weak-Strong Uniqueness

V. Global Solutions for Boltzmann Equations

Linear Cases
- Characteristics for the Transport Equation
- Convexity Estimates for the Transport Equation
- L^2 Theory for Linearized Boltzmann
Kaniel-Shinbrot Theory
- Construction of Super and Subsolutions
- Beginning Condition
- Near Vacuum Case
- Near Maxwellian Cases
Entropy and Dissipation Bounds
- Compactness in Weak $L^1$ Topologies
- Compactness from Relative Entropy
- Compactness from Entropy Dissipation
Velicity Averaging
- L^2 Theory
- L^p Theory
- L^1 Theory
DiPerna-Lions Theory
- Renormalized Solutions
- Construction of Approximate Solutions
- Compactness Results
- Passing to the Limit
- Weak-Strong Uniqueness

VI. Global Fluid Limits for Boltzmann Equations

Linear Cases
- Diffusion Limit for Linear Transport
- Acoustic Limit for Linearized Boltzmann
- Stokes Limit for Linerarized Boltzmann
Control of Fluctuations
- Compactness from Relative Entropy
- Compactness from Entropy Dissipation
- Infinitesimal Maxwellians
- Limiting Dissipation Inequality
- Relative Entropy Cutoff Control
- Compactness from Velocity Averaging
Limits to Linear Systems
- Linearized Boltzmann Limit
- Control of Conservation Defects
- Acoustic Limit
- Incompressible Stokes Limit
- Passing to the Limit in Diffusive Terms
Incompressible Navier-Stokes Limit
- Statement of the Theorems
- Passing to the Limit in Convection Terms
- Strong Convergence to Classical Solutions
Incompressible Euler Limit
- Statement of the Theorems
- Relative Entropy Method
- Passing to the Limit in Convection Terms