Characterize joint convexity for general pressure laws and material variables

Characterize conditions under which standard pressure constitutive laws, including the gamma-law pressure, are jointly convex with respect to the flow variables and advected material variables in a multi-material interface-capturing formulation.

Background

The symmetric-hyperbolic strategy requires a strictly convex extension in the conservative variables. When material parameters such as the bulk modulus and the adiabatic exponent are themselves transported variables, establishing this convexity becomes substantially more difficult.

The paper explicitly notes that the difficulty persists even for the standard gamma-law pressure, so a general characterization of admissible constitutive laws and exponent choices remains unresolved.