The Oberbeck–Boussinesq approximation for the motion of two incompressible fluids

We consider the Oberbeck–Boussinesq approximation for unsteady motion of a drop in another fluid. On the unknown interface between the liquids, the surface tension is taken into account. We study this problem in Hölder classes of functions where local existence theorem for the problem is proved. The proof is based on the fact that the solvability of the problem with a temperature independent right-hand side was obtaind earlier. For a given velocity vector field of the fluids, we arrive at a diffraction problem for the heat equation which is solvable by well-known methods. Existence of a solution to the complete problem is proved by successive approximations. Bibl. – 10 titles.