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<p>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 <i>m</i>: it holds for Newtonian fluids (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(m=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>), is violated for shear-thinning fluids (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(m&lt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>), and leads to an instantaneous regularization of acceleration waves for shear-thickening fluids (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(m&gt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>&gt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>).</p>

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Acceleration waves and the K-condition in viscoelastic solids and non-Newtonian fluids

  • 期刊论文

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  • 发表日期：

    2026年3月11日

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  • Tommaso Ruggeri

摘要

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 ).