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(Arch. Rat. Mech. Ana. 76, 97-133 (1981))
Georg Friedrich Bernhard Riemann (1826 – 1866)
Peter David Lax (1926--)
Riemann (1860)
Lax (1957)
Mathematical Theory of Conservation Laws
Commun. Pure Appl. Math., 10 (1957)
Simple Waves In the case the characteristic field is genuinely nonlinear
Shock wave :
Rarefaction Wave :
Rankine-Hugoniot Condition + Entropy Condition
Simple Waves : In the case the characteristic field is linearly degenerate
Contact discontinuity :
General Riemann Solution
A linear superposition of simple waves
An example
Asymptotic state under viscous effect
・Rarefaction wave
・Shock wave Viscous shock wave
・Contact discontinuity Viscous contact wave
Viscous shock wave :
Viscous contact wave:
Traveling wave relaxed by viscosity
Diffusion wave relaxed by viscosity
Rarefaction wave
An example
Riemann Solution:
Asymptotic Solution:
Known results on Viscous and Heat-conductive case :
Rarefaction + (Rarefaction) Kawashima-Nishihara-M (1986)
Single Shock Kawashima-M (1985) (zero mass initial perturbations)
Shock +Shock
Rarefaction+Contact discontinuity+Shock Open
Liu (1997) Zumbrum (2004) Liu-Zen (2009)
Huang-M (2009)
Contact discontinuity +Shocks Open
Contact discontinuity+Rarefactions Huang-Li-M (2010)
Single Contact discontinuity
Huang-Xin-Yang (2008)
Huanag-Xin-M (2006) (zero mass initial perturbation)
Rarefaction+Shock Open
Thank You!