Construction of Green's functions for the Boltzmann equations Shih-Hsien Yu Department of Mathematics National University of Singapore

Motivation to investigate Green’s function for Boltzmann equation before 2003 Nonlinear time-asymptotic stability of a Boltzmann shock profile Zero total macroscopic perturbations Nonlinear time-asymptotic stability of a Knudsen layer for the Boltzmann Equation Mach number <-1

Green’s function of linearized equation around a global Maxwellian, Fourier transformation The inverse transformation

Initial value problem Particle-like wave-like decomposition

Pointwise of structure of the Green’s function Space dimension=3 Space dimension=1

Macroscopic wave structure of 1-D Green’s function Application: Pointwise time-asymptotic stability of a global Maxwellian state in 1-D.

Green’s function of linearized equation around a global Maxwellian M,, in a half-space problem x>0. Green identity: Boundary value estimates ( a priori estimate):

Approximate boundary data for case |Mach(M)|<1 Upwind damping approximation to the boundary data

An approximation to the full boundary data.

Green’s function of linearized equation around a stationary shock profile.

Separation of wave structures Transversal wave Compressive wave

1. Shift data

2. Hyperbolic Decomposition Transversal wave Compressive wave 3. Transverse Operator and Local Wave Front tracing

4. Coupling of T and D operators 5. Respond to Coupling

6. Approximation to Respond, Compressive Operator

6. T-C scheme for An estimates

A Diagram for A Diagram for general pattern + extra time decaying rate in microscopic component nonlinear stability of Boltzmann shock profile

Applications of the Green’s functions Nonlinear invariant manifolds for steady Boltzmann flow

Applications of the Green’s functions Milne’s problme

Sone’s Diagram for Condensation-Evaporation