Existence of a variational solution for the stationary Boussinesq equations with thermocapillary effect and nonhomogenous boundary conditions

Abstract

In this work, we prove the existence of at least one variational solution $(\mathbf {u},{\theta },{p})$ of the stationary Boussinesq equations with thermocapillary effect on the surface and nonhomogenous boundary conditions for the velocity and the temperature. We assume that the domain is plane, bounded and polygonal. The velocity field of the fluid is denoted by $\mathbf {u}$, the temperature $\theta$ and the pressure p. To get the existence result we use the Leray–Schauder principle. Firstly, the variational problem is reduced to the existence of a fixed point for a completely continuous map in a Banach space. Then, we establish a priori estimates needed to apply the Leray–Schauder principle. We use also lifting trace results, e.g. an adaptation of Hopf’s lemma to our setting, to achieve our goal.