UDK 517.958 EXISTENCE AND UNIQUENESS RESULTS FOR A COUPLED PROBLEM IN CONTINUUM THERMOMECHANICS V. G. Zvyagin, V. P. Orlov (Voronezh State University) Поступила в редакцию 22.03.2014 г. <...> Abstract: this paper presents results on solvability of multidimensional systems of equations of thermoviscoelasticity. <...> Both compressible and incompressible continua are considered. <...> The existence and uniqueness of regular, weak and weak-renormalized solutions are given, both local and global. <...> The systems under consideration are coupled systems of a motion equation and an energy equation. <...> The results are based on the reduction of the system of equations of thermoviscoelasticity to an operator equation in the suitable Banach space. <...> The operator equation is constructed by means of successive solving of the motion equation and the energy equation. <...> The corresponding apriori estimates admit to obtain the solvability of operator equations by means of application of various fix-point theorems. <...> The theory of anisotropic Sobolev spaces with a mixed norm is used. <...> Key words and phrases: Thermoviscoelastic continuum, successive approximations, a priori estimates, fix-point theorem, weak solution, weak-renormalized solution, Oberbeck-Boussinesq type system. <...> МАТЕМАТИКА. 2014. № 2 Existence and uniqueness results for a coupled problem in continuum thermomechanics 1. <...> INTRODUCTION In the present paper we review existence results for multidimensional thermoviscoelastic systems. <...> Thermoviscoelastic system represents balance laws for the linear momentum and energy. <...> The linear momentum balance equation is specified by the stress tensor given by a rheology law of the appropriate type. <...> The energy equation is governed by thermodynamic potentials ( the free energy, the internal energy, the dissipation potential et.c.) which characterize the material. <...> Different assumptions about the form of the rheology law and the thermodynamic potentials generate a variety of system of thermoviscoelasticity. <...> The existence of solutions to such systems is established by a suitable successive approximation method to regularized (if needed) system in a suitable functional space, proof of solvability of which is based on appropriately chosen successive approximations, global a priori estimates, application of a fixed point theorem and pass to the limit. <...> In this paper we present the series of results on multidimensional mathematical models of thermoviscoelasticity. <...> The review does not claim to be exhaustive in its domain and is only intended to demonstrate wildly used applications of the based on fix-point arguments methods in the theory <...>