1. Wave Propagation Over a Boltzmann Shock Profile
Shih-Hsien Yu
National University of Singapore
NTU MATH MONTH, JUNE 29-JULY 28, 2009

2. NTU MATH MONTH, JUNE 29-JULY 28, 2009

3. Background of the problems
Three non-fluid slips of the Grad’s asymptotic expansions for the Boltzmann equations
Shock slips, Boundary slips, and Initial slips.
Fluid regime
Fluid regime
Size of the slips: Magnitude of Knudsen number
NTU MATH MONTH, JUNE 29-JULY 28, 2009

4. NTU MATH MONTH, JUNE 29-JULY 28, 2009
The boundary slip is one of the important essence in the asymptotic theories by Sone and his collaborators to develop the theories on Condensation-Evaporation, Thermal Transpiration flows, Thermal Edge Flows, Ghost Effects, etc..
Kinetic Equations
Fluid Mechanics
NTU MATH MONTH, JUNE 29-JULY 28, 2009

5. Background of the problems
Global wave propagation structure for viscous conservation laws
Riemann Problem for compressible Navier-Stokes equations, etc.
Shock tube problem
NTU MATH MONTH, JUNE 29-JULY 28, 2009

6. NTU MATH MONTH, JUNE 29-JULY 28, 2009

7. Existence of Boltzmann Shock Profile
Conditions on end states of a shock wave:
Caflisch-Nicolaenko
Liu-Yu (By center-manifold theory to yield both existence and monotone properties on macroscopic variables.)
Liu-Yu (Under a zero total mass condition)
Stability of Boltzmann Shock Profile
(Planar wave perturbations)
NTU MATH MONTH, JUNE 29-JULY 28, 2009

8. Some Review on Stability of Viscous Shock Profiles
Goodman
NTU MATH MONTH, JUNE 29-JULY 28, 2009

9. Goodman’s Energy Estimates (for system of equations )
(Nishihara-Matsumura independently introduce this
notion)
NTU MATH MONTH, JUNE 29-JULY 28, 2009

10. NTU MATH MONTH, JUNE 29-JULY 28, 2009
Stability of Viscous Shock Profile by energy estimates. Goodman, Matsumura-Nishihara, Tai-Ping Liu, Matsumura-Nishihara-Kawashima, Goodman-Xin, Xin-Szepessy, etc.
NTU MATH MONTH, JUNE 29-JULY 28, 2009

11. NTU MATH MONTH, JUNE 29-JULY 28, 2009
A situation when Energy Estimates is stronger enough to give exponentially sharp a priori estimate
Example (Linearized Burgers equation around a shock wave)
NTU MATH MONTH, JUNE 29-JULY 28, 2009

12. Local structure vs Global structure
NTU MATH MONTH, JUNE 29-JULY 28, 2009