Existence and uniqueness of solution of a thermoviscoelastic equation with time-dependent coefficients

Abstract

In this paper, we prove the existence and uniqueness of the deformation of a semi-transparent body Ω resulting from its radiative heating by a black source S surrounding it at absolute temperature TS(t) for 0≤t≤tf. This deformation is modeled during the thermal process by Maxwell's model of viscoelasticity theory with a viscosity in the constitutive equation depending on the absolute temperature T(x,t) solution of the heat transfer equations between S and Ω. The rate of thermal deformations is also taken into account in the constitutive equation. We consider a mixed formulation of the thermoviscoelasticity problem in the variables (σ,v,ρ) where σ is the Cauchy stress tensor field, v the velocity field, and ρ the vorticity tensor field. For the proof of the well posedness of this mixed formulation, we will need to prove that ∂T∂t exists and is continuous on Q¯, where Q:=Ω×]0,tf[. This regularity result plays a critical role in proving the boundedness of a family of linear operators that leads directly to establishing the time regularity of the stress fields of a family of auxiliary elasticity problems. It is precisely these auxiliary stress fields that will allow us to find the unique stress field for the dual mixed formulation of our non-isothermal Maxwell model.