- 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.

- 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

- 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

- 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

- 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

- 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

- 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