Bénard problem for slightly compressible fluids: existence and nonlinear stability in 3D (Q2026247)

Summary: This paper shows the existence, uniqueness, and asymptotic behavior in time of regular solutions (a la Ladyzhenskaya) to the Bénard problem for a heat-conducting fluid model generalizing the classical Oberbeck-Boussinesq one. The novelty of this model, introduced by \textit{A. Corli} and the author [Mediterr. J. Math. 16, No. 1, Paper No. 18, 24 p. (2019; Zbl 1409.76134)], and the author and \textit{T. Ruggeri} [``The Bénard problem for quasi-thermal-incompressible materials: a linear analysis'', Int. J. Nonlinear Mech. 67, 178--185 (2014; \url{doi:10.1016/j.ijnonlinmec.2014.09.003})], consists in allowing the density of the fluid to also depend on the pressure field, which, as shown by the author and Ruggeri [loc. cit.] is a necessary request from a thermodynamic viewpoint when dealing with convective problems. This property adds to the problem a rather interesting mathematical challenge that is not encountered in the classical model, thus requiring a new approach for its resolution.