Integro-differential equations and functional analysis

Solution of the inverse problem describing slow thermal convection in a rotating layer

Viktor K. Andreev^a, Liliya I. Latonova<sup>b

a</sup> Institute of Computational Modeling SB RAS, Krasnoyarsk, Russian Federation
^b Siberian Federal University, Krasnoyarsk, Russian Federation

Abstract: The linear inverse initial-boundary value problem arising when modeling the rotational motion of a viscous heat-conducting liquid in a flat layer is solved. It is shown that the problem has two different integral identities. Based on these identities, a priori estimates of the solution in a uniform metric are obtained and its uniqueness is proved. The conditions for the input data are also determined, under which this solution goes to the stationary mode with increasing time according to the exponential law. In the final part, the existence of a unique classical solution of the inverse problem is proved. To do this, differentiating the problem by a spatial variable, we come to a direct non-classical problem with two integral conditions instead of the usual boundary conditions. The new problem is solved by the method of separation of variables, which makes it possible to find a solution in the form of rapidly converging series on a special basis.

Keywords: thermal convection, liquid motion, inverse problem, stationary solution.

UDC: 517.956: 532.5.032

MSC: 31B20, 76D05

Received: 30.11.2022
Revised: 06.04.2023
Accepted: 14.04.2023

DOI: 10.26516/1997-7670.2023.44.3