Unique global solvability of 1D Fried–Gurtin model

Zenon Kosowski¹

¹ Institute of Mathematics and Cryptology Faculty of Cybernetics Military University of Technology Kaliskiego 2 00-908 Warszawa, Poland

Applicationes Mathematicae, Tome 34 (2007) no. 3, pp. 269-288.

Résumé

We investigate a 1-dimensional simple version of the Fried–Gurtin 3-dimensional model of isothermal phase transitions in solids. The model uses an order parameter to study solid-solid phase transitions. The free energy density has the Landau–Ginzburg form and depends on a strain, an order parameter and its gradient. The problem considered here has the form of a coupled system of one-dimensional elasticity and a relaxation law for a scalar order parameter. Under some physically justified assumptions on the strain energy and data we prove the existence and uniqueness of a regular solution to the problem. The proof is based on the Leray–Schauder fixed point theorem.

Détail

DOI : 10.4064/am34-3-2

Keywords: investigate dimensional simple version fried gurtin dimensional model isothermal phase transitions solids model uses order parameter study solid solid phase transitions energy density has landau ginzburg form depends strain order parameter its gradient problem considered here has form coupled system one dimensional elasticity relaxation law scalar order parameter under physically justified assumptions strain energy prove existence uniqueness regular solution problem proof based leray schauder fixed point theorem