Sunto
Questa è la terza parte di una serie di lavori dedicati all'esistenza, unicità, monotonia e proprietà asintotiche delle soluzioni d'onda di propagazione per leggi di conservazione diffusive-dispersive. In questa parte, l'attenzione è focalizzata su un modello iperbolico non convesso di due leggi di conservazione che sorgono in elastodinamica non lineare, che tengono conto della viscosità non lineare e dei termini di capillarità. Da una parte, utilizzando le tecniche precedentemente sviluppate, studiamo le proprietà delle corrispondenti onde d'urto classiche e non classiche e le loro corrispondenti relazioni cinetiche. Diverse nuove proprietà sono state trovate per questo modello (iperbolico). Innanzitutto, qui distinguiamo tra una funzione cinetica ed una funzione cinetica inversa, quest'ultima essendo sempre definita globalmente ma possibilimente non sempre globalmente iinvertibile. In secondo luogo, mostriamo che onde d'urto con ampiezza sufficientemente piccola sono sempre classiche, per un valore fissato del rapporto tra diffusione e dispersione. In ultimo, determiniamo il comportamento asintotico della funzione cinetica per onde d'urto aventi sia ampiezza grande sia piccola.
Abstract
This is the third part of a series devoted to the existence, uniqueness, monotonicity, and asymptotic properties of the traveling wave solutions of diffusive-dispersive conservation law. In this part, we focus attention on a nonconvex hyperbolic model of two conservation laws arising in nonlinear elastodynamics and including nonlinear viscosity and capillarity terms. On one hand, using the techniques developed earlier, we study the properties of the corresponding classical and nonclassical shock waves and their corresponding kinetic relation. Several new features are found for this (hyperbolic) model: First of all, we distinguish here between a kinetic function and an inverse kinetic function; the latter is always globally defined but may fail to be globally invertible. Second, we show that shock waves with sufficiently small amplitude are always classical, for a fixed ratio of diffusion and dispersion. Third, we determine here the asymptotic behavior of the kinetic function for both shocks with large and small amplitudes.
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Bedjaoui, N., Lefloch, P.G. Diffusive-dispersive traveling waves and kinetic relations III. An hyperbolic model of elastodynamics. Ann. Univ. Ferrara 47, 117–144 (2001). https://doi.org/10.1007/BF02838179
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DOI: https://doi.org/10.1007/BF02838179