Motion of a Drop in an Incompressible Fluid

Description

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. Contents 1 Introduction 1.1 Statement of the Problem, Definition of theHölder Spaces 2 A Model Problem with Plane Interface and with Positive Surface Tension Coefficient 2.1 Auxiliary Propositions 2.2 An Explicit Solution of a Homogeneous Model Problem 2.3 Theorems on the Fourier Multipliers in the Hölder Spaces 2.4 An Estimate of the Solution of Problem (2.2.1) 2.5 The Problem for the Inhomogeneous Stokes System 3 The Model Problem Without Surface Tension Forces 3.1 Statement of the Problem and Formulation of Existence Theorem 3.2 Preliminary Considerations 3.3 The Homogeneous Problem: An Explicit Solution 3.4 The Proof of Theorem 3.3.1 3.4.1 The Analysis of d 3.4.2 The Estimate of π 3.4.3 The Estimate of the Vector f=-1ρp in D3T 3.5 The Proof of Theorem 3.1.1 4 A Linear Problem with Closed Interface Under Nonnegative Surface Tension 4.1 Auxiliary Propositions: The Statement of Results 4.2 A Priori Estimates of the Solution of Problem (1.1.7) 4.3 The Solvability of Problem (1.1.7): Constructing a Regularizer 5 Local Solvability of the Problem in Weighted Hölder Spaces 5.1 Weighted Hölder Spaces: Formulation of the Local Existence Theorem for the Nonlinear Problem 5.2 Weighted Estimates for Linear Problem (1.1.7) 5.3 Solvability of a Linearized Problem on a Finite Time Interval 5.4 The Proof of the Solvability of Nonlinear Problem (5.1.1) 6 Global Solvability in the Hölder Spaces for the Nonlinear Problem Without Surface Tension 6.1 Statement of the Main Result 6.2 A Linear Problem with Closed InterfaceBetween the Fluids 6.3 A Linearized Problem 6.4 Global Solvability of Problem (1.1.1) with σ=0 7 Global Solvability of the Problem Including Capillary Forces: Case of the Hölder Spaces 7.1 Setting of the Problem, Statement of the Main Result 7.2 Energy Estimate of the Solution 7.3 A Linearized Problem 7.4 Global Classical Solvability of Problem (7.1.3), (1.0.3) 8 Thermocapillary Convection Problem 8.1 Setting of the Problem and Statement of Results 8.2 Linearized Problems 8.3 The Solvability of Problem (8.1.2) 8.4 The Problem in R3 with a Constant Temperature Value at Infinity 9 Motion of Two Fluids in the Oberbeck-Boussinesq Approximation 9.1 Setting of the Problem and the Statement of Its Local Solvability 9.2 A Linearized Problem and Estimates ofthe Initial Pressure 9.3 Local Solvability of Problem (9.1.3), (9.1.4) 9.4 Global Solvability of the Oberbek-Boussinesq Problem 9.4.1 Energy Estimate of the Solution 9.4.2 Global Solvability ofProblem (9.1.1), (9.4.1), (1.0.3) 10 Local L2 -Solvability of the Problem with Nonnegative Coefficient of Surface Tension 10.1 The Definition of the Sobolev-Slobodetskiǐ Spaces and the Introduction of Equivalent Norms 10.2 L2-estimates of a Solution of the Model Problem with Plane Interface Between the Fluids 10.2.1 The Homogeneous Problem 10.2.2 The Nonhomogeneous Problem withPlane Interface 10.3 A Priori Estimates of a Solution of the Problem with a Closed Interface 10.4 L2-Solvability of the Linearized Problem with Closed Interface in R3 10.4.1 The Solvability of Problem (1.1.7) in a Simplified Case 10.4.2 The Case of Nonzero r, b' and v0=0 10.4.3 Problem (1.1.7) in the General Case 10.5 Local Solvability of the Nonlinear Problemin the L2 -Setting 11 Global L2 -Solvability of the Problem Without Surface Tension 11.1 The Statement of Global Existence Theorem 11.2 Auxiliary Propositions 11.3 The Proof of the Existence of a Global Solution 12 L2-Theory for Two-Phase Capillary Fluid 12.1 The Statement of the Theorem on Global Solvability 12.2 Global Solvability of a Linear Problem 12.3 The Nonlinear Problem Conclusions References