The Duhem Model

Abstract

The Duhem operator M: (u, w°) ↦ w is defined by a Cauchy problem of the form 1 {\%} MathType!MTEF!2!1!+-{\%} feaagCart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm{\%} Wu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9{\%} q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir{\%} -Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGa{\%} aeqabaWaaeaaeaaakabaaaaaaaaapeqaamaaceaapaqaauaabaqace{\%} aaaeaapeGaem4DaCNaeyypa0Jaem4zaC2damaaBaaaleaapeGaeGym{\%} aedapaqabaGcpeGaeiikaGIaemyDauNaeiilaWIaem4DaCNaeiykaK{\%} IaeiikaGIafmyDauNbaiaacqGGPaqkpaWaaWbaaSqabeaapeGaey4k{\%} aScaaOGaeyOeI0Iaem4zaC2damaaBaaaleaapeGaeGOmaidapaqaba{\%} GcpeGaeiikaGIaemyDauNaeiilaWIaem4DaCNaeiykaKIaeiikaGIa{\%} fmyDauNbaiaacqGGPaqkpaWaaWbaaSqabeaapeGaeyOeI0caaaGcpa{\%} qaa8qacqWG3bWDcqGGOaakcqaIWaamcqGGPaqkcqGH9aqpcqWG3bWD{\%} paWaaWbaaSqabeaapeGaeGimaadaaOGaeiilaWcaaaGaay5EaaGaem{\%} yAaKMaemOBa42aaKWiaeaacqaIWaamcqGGSaalcqWGubavaiaaw2fa{\%} caGLBbaacqGGSaalaaa!5F0F! {\$}{\$}$\backslash$left$\backslash${\{} {\{}$\backslash$begin{\{}array{\}}{\{}*{\{}20{\}}{\{}l{\}}{\}} {\{}w = {\{}g{\_}1{\}}(u,w){\{}{\{}($\backslash$dot u){\}}{\^{}} + {\}} - {\{}g{\_}2{\}}(u,w){\{}{\{}($\backslash$dot u){\}}{\^{}} - {\}}{\}} $\backslash$$\backslash$ {\{}w(0) = {\{}w{\^{}}0{\}},{\}} $\backslash$end{\{}array{\}}{\}} $\backslash$right.in$\backslash$left] {\{}0,T{\}} $\backslash$right[,{\$}{\$} with gl and g2 given continuous functions. Continuity properties of M in the Sobolev spaces W1,P(0,T) (1 ≤ p {\textless} +∞), endowed with the strong topology, and existence of a continuous extension of M to C0([0, T]) n BV(0,T) are proved.}