Acceleration Waves and the K-Condition in Viscoelastic Solids and Non-Newtonian Fluids

Abstract

The K-condition introduced by Shizuta and Kawashima provides a sufficient criterion for the global existence of smooth solutions to dissipative hyperbolic systems. For genuinely nonlinear characteristic fields, a weaker K-condition becomes necessary, although not sufficient. In this paper, we analyze this weaker K-condition through the study of acceleration waves propagating in an equilibrium state. We investigate two classes of hyperbolic models: one describing viscoelasticity with linear dissipation, and the other non-Newtonian fluids asymptotically converging to a power-law behavior. For viscoelastic models, the weaker K-condition is always satisfied and acceleration waves remain bounded. For non-Newtonian fluids, the validity of the condition depends on the power-law index $m$: it holds for Newtonian fluids ($m=1$), is violated for shear-thinning fluids ($m<1$), and leads to an instantaneous regularization of acceleration waves for shear-thickening fluids ($m>1$).