This mathematical monograph details the authors' results on solutions to problems governing the simultaneous motion of two incompressible fluids. Featuring a thorough investigation of the unsteady motion of one fluid in another, researchers will find this to be a valuable resource when studying non-coercive problems to which standard techniques cannot be applied. As authorities in the area, the authors offer valuable insight into this area of research, which they have helped pioneer. This volume will offer pathways to further research for those interested in the active field of free boundary problems in fluid mechanics, and specifically the two-phase problem for the Navier-Stokes equations. The authors' main focus is on the evolution of an isolated mass with and without surface tension on the free interface. Using the Lagrange and Hanzawa transformations, local well-posedness in the Hölder and Sobolev-Slobodeckij on L2 spaces is proven as well. Global well-posedness for small data is also proven, as is the well-posedness and stability of the motion of two phase fluid in a bounded domain.

Motion of a Drop in an Incompressible Fluid will appeal to researchers and graduate students working in the fields of mathematical hydrodynamics, the analysis of partial differential equations, and related topics.

Contenu
Introduction.- A Model Problem with Plane Interface and with Positive Surface Tension Coefficient.- The Model Problem Without Surface Tension Forces.- A Linear Problem with Closed Interface Under Nonnegative Surface Tension.- Local Solvability of the Problem in Weighted Hölder Spaces.- Global Solvability in the Hölder Spaces for the Nonlinear Problem without Surface Tension.- Global Solvability of the Problem Including Capillary Forces. Case of the Hölder Spaces.- Thermocapillary Convection Problem.- Motion of Two Fluids in the Oberbeck - Boussinesq Approximation.- Local L**2-solvability of the Problem with Nonnegative Coefficient of Surface Tension.- Global L**2-solvability of the Problem without Surface Tension.- L**2-Theory for Two-Phase Capillary Fluid.