Analysis of a nonlinear degenerating PDE system for phase transitions in thermoviscoelastic materials |
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Authors: | Elisabetta Rocca |
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Institution: | a Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy b Dipartimento di Matematica, Università di Brescia, Via Valotti 9, 25133 Brescia, Italy |
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Abstract: | We address the analysis of a nonlinear and degenerating PDE system, proposed by M. Frémond for modelling phase transitions in viscoelastic materials subject to thermal effects. The system features an internal energy balance equation, governing the evolution of the absolute temperature ?, an evolution equation for the phase change parameter χ, and a stress-strain relation for the displacement variable u. The main novelty of the model is that the equations for χ and u are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for u which needs to be carefully handled.In this paper, we first prove a local (in time) well-posedness result for (a suitable initial-boundary value problem for) the above mentioned PDE system, in the (spatially) three-dimensional setting. Secondly, we restrict to the one-dimensional case, in which, for the same initial-boundary value problem, we indeed obtain a global well-posedness theorem. |
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Keywords: | 76A10 35L80 35A05 35B45 35B50 80A22 |
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