Pointwise Green function bounds for shock profiles of systems with real viscosity

Following the pointwise semigroup approach of [ZH, MZ.1], we establish sharp pointwise Green function bounds and consequent linearized stability for viscous shock profiles of general hyperbolic-parabolic systems of conservation laws of dissipative type, under the necessary assumptions ([Z.1, Z.3, Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave; transversality of the connecting profile; and hyperbolic stability of the corresponding ideal shock of the associated inviscid system, with no additional assumptions on the structure or strength of the shock. These bounds are used in a companion paper [MZ.2] to establish nonlinear stability of small-amplitude Lax shocks of symmetrizable hyperbolic-parabolic systems.