\(L_2\)-theory for two incompressible fluids separated by a free interface
It is a very interesting and technical paper. The 3D non-steady flow of two viscous, incompressible, immiscible fluids in a bounded container is considered, related with the motion of a drop in a liquid. A surface tension exists on the interface between the fluids and the flow is governed by the Navier-Stokes system. Some methods introduced in [the second author, Lect. Notes Math. 1812, 123--175 (2003; Zbl 1038.35063)], [the authors, J. Math. Sci., New York 185, No. 5, 668--686 (2012; Zbl 1278.35175); translation from Zap. Nauchn. Semin. POMI 397, 20--52 (2011)], [the second author, J. Math. Sci., New York 198, No. 6, 761--827 (2014; Zbl 1309.35093); translation from Probl. Mat. Anal. 75, 93--152 (2014)] are used, giving new estimates of the solution. The problem is treated in a particular Sobolev-Slobodetskii space. A modified Hanzawa transformation is used to obtain a flow in two domains separated by a specific sphere $S_{R_0}(0)$ centered in origin. The stability of a rest state is studied, for small velocities and mass forces. The transformed problem is linearized. An exponential inequality for the corresponding energy is obtained. The results for the linearized problem and some estimates for the nonlinear terms give an exponential decay of a global solution. The authors specify that the (unsteady) interface tends to a sphere which in general is different from $S_{R_0}(0)$.
scientific article; zbMATH DE number 7029868
scientific article