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 ( $$m=1$$ m = 1 ), is violated for shear-thinning fluids ( $$m<1$$ m < 1 ), and leads to an instantaneous regularization of acceleration waves for shear-thickening fluids ( $$m>1$$ m > 1 ).