Systems of conservation laws with rotation and Galilean symmetry

We show that the structure of multidimensional systems of conservation laws, equipped with a single, convex entropy, and with both rotation and Galilean symmetries, is perhaps surprisingly limited. The following are established. The simplest examples of such systems, with only one vector field, are at most simple extensions of the familiar Euler systems. In such models of magnetohydrodynamic flow, including a solenoidal, Galilean invariant magnetic field, the property of finite signal propagation speeds is often lost. The only such system describing adiabatic multi-fluid flow, with the mass of each species conserved separately, corresponds to independent flow of each species. Whatever interaction between the different species occurs in an energy equation or through some other dependent variable, and not simply through expressions for the pressures in terms of the species' densities. A similar conclusion holds for incompressible flow.