The interaction of waves for the ultra-relativistic Euler equations

The interactions between nonlinear waves for the ultra-relativistic Euler equations for an ideal gas are studied. These equations are described in terms of the pressure p and the spatial part u∈R3 of the dimensionless four-velocity. A new function, which measures the strengths of the waves of the ultra-relativistic Euler equations is presented, and sharp estimates for these strengths are derived. The interpretation of the strength for the Riemann solution is given. This function has the important implication that the strength is non increasing for the interactions of waves for the system. This study of interaction estimates also allows to determine the type of the outgoing Riemann solutions.