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Unique global solvability in two‐dimensional non‐linear thermoelasticity
Author(s) -
Pawłow Irena,
Zaja̧czkowski Wojciech M.
Publication year - 2004
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.582
Subject(s) - mathematics , uniqueness , dissipation , mathematical analysis , boundary value problem , elasticity (physics) , viscoelasticity , heat equation , fixed point theorem , a priori estimate , function (biology) , physics , materials science , evolutionary biology , biology , composite material , thermodynamics
The paper is concerned with initial‐boundary value problem in two‐dimensional (2‐D) non‐linear thermoelasticity which arises as a mathematical model of shape memory alloys. The problem has the form of viscoelasticity system with fourth order capillarity‐like term coupled with heat conduction equation with mechanical dissipation. The corresponding elastic energy is a nonconvex multiple‐well function of strain, with the shape changing qualitatively with temperature. Under assumption on the growth of this energy with respect to temperature we prove global in time existence and uniqueness of solutions for large data. The existence proof is based on parabolic decomposition of the elasticity system and application of the Leray–Schauder fixed point theorem. The main part of the proof consists in deriving a priori Hölder estimates by successive improvement of energy estimates. Copyright © 2004 John Wiley & Sons, Ltd.