(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!