Survey of admissible shock waves for $2 \times 2$ systems of conservation laws with an umbilic point

Abstract

In this survey note, we consider $2 \times 2$ systems of conservation laws with an umbilic point. At the umbilic point, the characteristic speeds coincide and the Jacobian matrix of the flux functions is diagonalizable. It is shown that if there are no undercompressive shocks with viscous shock profile then any compressive shock has a viscous shock profile and any overcompressive shock has a infinitely many viscous shock profiles.