Volume 7, no. 3Pages 69 - 76 On the Strong Solutions in an Oldroyd-Type Model of Thermoviscoelasticity
V.P. Orlov, M.I. ParshinFor the initial-boundary value problem in a dynamic Oldroyd-type model of thermoviscoelasticity, we establish the local existence theorem for strong solutions in the planar case. The continuum under consideration is a plane bounded domain with sufficiently smooth boundary. The corresponding system of equations generalizes the Navier-Stokes-Fourier system by having an additional integral term in the stress tensor responsible for the memory of the continuum. In our proof, we study firstly the initial-boundary value problem for an Oldroyd-type viscoelasticity system with variable viscosity. Then we consider the initial-boundary value problem for the equation of energy conservation with a variable heat conductivity coefficient and an integral term. We establish the solvability of these problems by reducing them to operator equations and applying the fixed-point theorem. For the original thermoviscoelasticity system, we construct an iterative process consisting in a consecutive solution of auxiliary problems. Suitable a priori estimates ensure that the iterative process converges on a sufficiently small interval of time. The proof relies substantially on Consiglieri's results on the solvability of the corresponding Navier - Stokes - Fourier system.
Full text- Keywords
- Navier - Stokes equation; Oldroyd-type model; thermoviscoelastic; strong solutions; fixed point.
- References
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