Free Boundary Problems in Fluid Dynamics

✍ Scribed by Albert Ai , Thomas Alazard , Mihaela Ifrim , Daniel Tataru

• Publisher: Birkhäuser
• Year: 2024
• Tongue: English
• Leaves: 373
• Series: Oberwolfach Seminars 54
• Edition: 1
• Category: Library

✦ Synopsis

This book, originating from a seminar held at Oberwolfach in 2022, introduces to state-of-the-art methods and results in the study of free boundary problems which are arising from compressible as well as from incompressible Euler’s equations in general. A particular set of such problems is given by gaseous stars considered in a vacuum (modeled via the compressible Euler equations) as well as water waves in their full generality (seen as recasts of incompressible irrotational Euler equations). This is a broad research area which is highly relevant to many real life problems, and in which substantial progress has been made in the last decade.

• A comprehensive review of water waves
• In depth discussion of Dirichlet to Neuman map
  -Latest results on free boundary problems for compressible Euler

✦ Table of Contents

Preface
Contents
About the Authors
The Water-Wave Equations in Eulerian Coordinates
1 The Water-Wave Equations
1.1 Introduction
1.1.1 The Incompressible Euler Equations
1.1.2 Boundary Conditions on the Free Surface
1.1.3 Conserved Quantities
1.1.4 Arbitrary Space Dimension
1.2 Zakharov Equations
1.2.1 The Dirichlet-to-Neumann Operator
1.2.2 The Linearized Equation Around the Rest State
1.2.3 Dispersive Estimates
1.2.4 The Cauchy Problem
1.3 References
2 The Dirichlet-to-Neumann Operator
2.1 Definition
2.1.1 General Assumptions on the Domain
2.1.2 Harmonic Extension
2.1.3 About the Behavior at Infinite Depth
2.2 Definition of the Dirichlet-to-Neumann Operator
2.3 A Trace Estimate
3 Introduction to Paradifferential Calculus
3.1 Introduction
3.2 Symbolic Calculus
3.3 Paraproducts
3.4 References
4 Paralinearization of the Dirichlet-to-Neumann Operator
4.1 Classical Results
4.2 Main Statement
4.3 The Good Unknown of Alinhac
4.4 Paralinearization
4.4.1 Flattening of the Free Boundary
4.4.2 Elliptic Regularity
4.4.3 Paralinearization of the Interior Equation
4.4.4 Paralinearization of the Boundary Condition
4.5 Reduction to the Boundary
4.5.1 Formal Computations
4.5.2 Factorization
4.6 References
5 The Water-Wave Problem with Surface Tension
5.1 The Cauchy Problem
5.2 A Priori Estimates
5.2.1 Paralinearization
5.2.2 A Class of Symbols
5.2.3 Symmetrization of Equations
5.2.4 A Priori Estimates
5.3 The Smoothing Effect
5.4 References
6 Control and Stabilization of Water-Waves
6.1 Introduction
6.2 Extension to Periodic Functions
6.3 Controllability
References
Strichartz Estimates for Water Waves
1 Introduction
1.1 A Dispersive System
1.2 The Paradifferential Equation
1.3 Solutions at the Energy Threshold
1.4 Pure Gravity vs. Gravity-Capillary Waves
2 Parametrices and Estimates for General Dispersive Flows
2.1 The Parametrix Construction
2.2 Wave Packet Scale and Symbol Regularity
2.3 Strichartz Estimates for Rough Symbols
3 Equations with Rough Coefficients
3.1 The High Dispersion Case (Gravity-Capillary)
3.2 The Low Dispersion Case (Pure Gravity)
4 Application to Water Waves
4.1 Gravity-Capillary Water Waves
4.2 Gravity Water Waves
4.3 Integrating the Vector Field
5 The 2D Case for Gravity Water Waves
5.1 Integrating the Transport-Dispersive Symbol
5.2 Regularity of the Hamilton Flow
5.3 Geometry of the Characteristics
5.4 Local Smoothing Estimates
5.5 Characteristic Smoothing and the Eikonal Equation
5.6 Wave Packet Parametrix
5.7 Matching the Initial Data and Source
5.8 Strichartz Estimates
5.8.1 Packet Overlap
5.8.2 Counting Argument
References
Local and Global Dynamics for Two Dimensional Gravity Water Waves
1 Introduction
1.1 Water Waves Equations
1.2 Water Waves in Holomorphic Coordinates
1.3 Function Spaces and Control Norms
1.4 The Main Results
2 Historical Notes
2.1 Local Well-Posedness
2.2 Almost Global, Global Well-Posedness and Modified Scattering
2.3 Other Aspects
3 Norms and Multilinear Estimates
3.1 Function Spaces
3.2 Coifman-Meyer and Moser Type Estimates
3.3 Paraproduct Estimates
4 Water Wave Related Bounds
4.1 and Pointwise Bounds
4.2 Material and Para-Material Derivative Bounds
5 The Linearized Equation
5.1 The Linearization of the Equation
5.2 The Linearization of the Equation
5.3 The Relation Between the Two Linearized Flows
6 Bounds for the Linearized Equation
6.1 Outline of the Reduction to the Paradifferential Equation
6.2 Bounds for the Paradifferential Equation
6.3 Bounds for the Paradifferential Equation
6.4 The Quadratic Bound for the Paradifferential Source Term
6.5 Normal Form Analysis for
6.6 Normal Form Analysis for
7 Energy Estimates for the Full Equation
7.1 An Outline of the Main Steps of the Argument
7.2 The Paradifferential Form of the Differentiated Equation
7.3 Well-Posedness for the Paradifferential Flow
7.4 The Normal Form Transformation
8 The Local Well-Posedness Theory
8.1 Bounds for Regular Solutions
8.2 Rough Solutions
8.3 Enhanced Cubic Lifespan Bounds
9 The Global Well-Posedness Theory
9.1 Energy Estimates
9.2 The Normal Form Reduction
9.3 Nonlinear Vector Field Sobolev Inequalities
9.4 Pointwise Bounds via Wave Packet Testing
References
Free Boundary Problems for Compressible Flows
1 Introduction
2 Compressible Fluid Flows
2.1 The Classical Model for an Isentropic, Ideal Gas
2.2 The Relativistic Model
2.3 Local Well-Posedness
3 Free Boundary Problems
3.1 Vacuum States
3.2 The Physical Vacuum Scenario
3.3 Good Variables
3.3.1 The Nonrelativistic Case
4 An Overview of Results
4.1 Energies and Function Spaces
4.2 The Scaling Law
4.3 Control Parameters
4.4 The Main Results
4.5 Historical Comments
4.5.1 Compressible Euler Flows
4.5.2 Vacuum States in Compressible Euler Flows
4.5.3 The Physical Vacuum Scenario
5 Our Approach: An Overview
5.1 Function Spaces and Interpolation
5.2 The Linearized Equation and Transition Operators
5.3 Difference Estimates and the Uniqueness Result
5.4 Higher Order Energy Estimates
5.5 Existence of Regular Solutions
5.6 Rough Solutions as Limits of Regular Solutions
6 Function Spaces
6.1 Weighted Sobolev Spaces
6.2 Weighted Sobolev Norms for Compressible Euler
6.3 The State Space
6.3.1 Sobolev Spaces and Control Norms
6.3.2 The Regularity of the Free Boundary
6.4 Regularization and Good Kernels
6.5 Interpolation Inequalities
7 The Linearized Equation
7.1 Energy Estimates and Well-Posedness
7.2 Second Order Transition Operators
8 Difference Estimates and Uniqueness
8.1 A Degenerate Difference Functional
8.2 The Energy Estimate
8.2.1 The Bound for J2
8.2.2 The Bound for J1: The Easier Case κ≤23
8.2.3 The Bound for J1: The General Case κ> 0
9 Good Variables and Energy Estimates
9.1 The Div-Curl Decomposition
9.2 Vector Fields
9.3 The Energy Functional
9.4 Energy Coercivity
9.5 Energy Estimates
10 Existence of Regular Solutions
10.1 The One Step Iteration
10.2 The One-Step Strategy
10.3 The Regularization Result
11 Nonlinear Littlewood-Paley Theory and Rough Solutions
11.1 Regularizing the Initial Data
11.2 Uniform Bounds and the Life-Span of Regular Solutions
11.3 The Limiting Solution
11.4 Continuous Dependence
11.5 The Lifespan of Rough Solutions
References

✦ Subjects

Water Waves, Free Boundary Problems, Compressible Euler, Strichartz Estimates, Paradifferential Calculus