On hyperbolic-dissipative systems of composite type

The Shizuta–Kawashima condition plays the fundamental role in guaranteeing global stability for systems of hyperbolic–parabolic/hyperbolic with relaxation. However, there are many important physical systems not satisfying this coupling condition, which are of composite type with regard to dissipation. The compressible Navier–Stokes equations with zero heat conductivity and Euler equations of adiabatic flow through porous media are two typical examples. In this paper, we construct the global unique solution near constant equilibria to these two systems in three dimensions for the small HℓHℓ(ℓ>3)(ℓ>3) initial data. Our proof is based on a reformation of the systems in terms of the pressure, velocity and entropy, a scaled energy estimates with minimal fractional   derivative counts in conjunction with the linear L2L2–L2L2 decay estimates to extract a fast enough decay of velocity gradient, which is used to close the energy estimates for the non-dissipative entropy. We also include an application to certain two-phase models.