Volume 7, no. 3Pages 69 - 76

On the Strong Solutions in an Oldroyd-Type Model of Thermoviscoelasticity

V.P. Orlov, M.I. Parshin
For 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
Navier - Stokes equation; Oldroyd-type model; thermoviscoelastic; strong solutions; fixed point.
1. Consiglieri L. Weak Solution for a Class of Non-Newtonian Fluids with Energy Transfer. J. Math. Fluid, Mech., 2000, vol. 2, pp. 267-293. DOI: 10.1007/PL00000952
2. Orlov V.P., Parshin M.I. On One Problem of Dynamics of Thermoviscoelastic Medium of Oldroyd Type. Russian Mathematics, 2014, vol. 58, no. 5, pp. 57-62. DOI: 10.3103/S1066369X14050089
3. Temam R. Navier - Stokes Equations. North-Holland Publishing Company, Amsterdam, New York, Oxford, 1977.
4. Agranovich Yu.Ya., Sobolevskii P.E. [Research of Mathematical Models of Viscoelastic Liquids]. Dokl. Akad. Nauk UkrSSR. Series A, 1989, no. 10, pp. 71-74.
5. Agranovich Yu.Ya., Sobolevskii P.E. [Research of Weak Solutions of Model of Oldroyd of Viscoelastic Liquid]. Kachestvennye metody issledovaniya operatornykh uravneniy [Qualitative Methods of Research of the Operator Equations], Yaroslavl, 1991, pp. 39-43. (in Russian)