A multipoint initial-final value problem for a linear model of plane-parallel thermal convection in viscoelastic incompressible fluid

S. A. Zagrebina

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 7 (2014) no. 3, pp. 5-22

Résumé

The linear model of plane-parallel thermal convection in a viscoelastic incompressible Kelvin–Voigt material amounts to a hybrid of the Oskolkov equations and the heat equations in the Oberbeck–Boussinesq approximation on a two-dimensional region with Bénard's conditions. We study the solvability of this model with the so-called multipoint initial-final conditions. We use these conditions to reconstruct the parameters of the processes in question from the results of multiple observations at various points and times. This enables us, for instance, to predict emergency situations, including the violation of continuity of thermal convection processes as a result of breaching technology, and so forth. For thermal convection models, the solvability of Cauchy problems and initial-final value problems has been studied previously. In addition, the stability of solutions to the Cauchy problem has been discussed. We study a multipoint initial-final value problem for this model for the first time. In addition, in this article we prove a generalized decomposition theorem in the case of a relatively sectorial operator. The main result is a theorem on the unique solvability of the multipoint initial-final value problem for the linear model of plane-parallel thermal convection in a viscoelastic incompressible fluid.

Détail

Keywords: multipoint initial-final value problem; Sobolev-type equation; generalized splitting theorem; linear model of plane-parallel thermal convection in viscoelastic incompressible fluid.