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 ( $$m=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ), is violated for shear-thinning fluids ( $$m&lt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ), and leads to an instantaneous regularization of acceleration waves for shear-thickening fluids ( $$m&gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ).