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Construction of  Green's functions for the Boltzmann equations

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

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Construction of  Green's functions for the Boltzmann equations

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

  2. 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

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

  4. Initial value problem • Particle-like wave-like decomposition

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

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

  7. 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):

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

  9. An approximation to the full boundary data.

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

  11. Separation of wave structures Transversal wave Compressive wave

  12. 1. Shift data

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

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

  15. 6. Approximation to Respond, Compressive Operator

  16. 6. T-C scheme for An estimates

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

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

  19. Applications of the Green’s functions • Milne’s problme

  20. Sone’s Diagram for Condensation-Evaporation

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