On the problem of thermocapillary convection for two incompressible fluids separated by a closed interface

Abstract

We consider the unsteady motion of a drop in another incompressible fluid. On the unknown interface between the liquids, the surface tension is taken into account. Moreover, the coe cient of surface tension depends on the temperature. We study this problem of the thermocapillary convection by M.V. Lagunova and V.A. Solonnikov’s technique developed for a single liquid. The local existence theorem for the problem is proved in Hölder classes of functions. The proof is based on the fact that the solvability of the problem with a constant coe cient of surface tension was obtain earlier. For a given velocity vector field of the fluids, we arrive at a di raction problem for the heat equation which solvability is established by well-known methods. Existence of a solution to the complete problem is proved by successive approximations.