Laplacian Growth on Branched Riemann Surfaces (Lecture Notes in Mathematics)

✍ Scribed by Björn Gustafsson

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 163
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

This book studies solutions of the Polubarinova–Galin and Löwner–Kufarev equations, which describe the evolution of a viscous fluid (Hele-Shaw) blob, after the time when these solutions have lost their physical meaning due to loss of univalence of the mapping function involved. When the mapping function is no longer locally univalent interesting phase transitions take place, leading to structural changes in the data of the solution, for example new zeros and poles in the case of rational maps.

This topic intersects with several areas, including mathematical physics, potential theory and complex analysis. The text will be valuable to researchers and doctoral students interested in fluid dynamics, integrable systems, and conformal field theory.

✦ Table of Contents

Preface
Contents
1 Introduction
1.1 General Background
1.2 Loss of Univalence, Several Scenarios
1.3 On the Construction of a Branched Riemann Surface
1.4 Moment Coordinates and the String Equation
1.5 Outlooks to Physics
1.6 Acknowledgements
2 The Polubarinova-Galin and Löwner-Kufarev Equations
2.1 Basic Set Up in the Univalent Case
2.2 Dynamics and Subordination
2.3 The Polubarinova-Galin Versus the Löwner-Kufarev Equation
3 Weak Solutions and Balayage
3.1 Weak Formulation of the Polubarinova-Galin Equation
3.2 Weak Solutions in Terms of Balayage
3.3 Inverse Balayage
3.4 More General Laplacian Evolutions
3.5 Regularity of the Boundary via the Exponential Transform
3.6 The Resultant and the Elimination Function
4 Weak and Strong Solutions on Riemann Surfaces
4.1 Laplacian Growth on Manifolds
4.2 Examples
4.3 The Riemann Surface Solution Pulled Back to the Unit Disk
4.4 Compatibility Between Balayage and Covering Maps
5 Global Simply Connected Weak Solutions
5.1 Statement of Result, and Two Lemmas
5.2 Statement of Conjecture, and Partial Proofs
5.3 Discussion
6 General Structure of Rational Solutions
6.1 Introduction
6.2 Direct Approach
6.3 Approach via Quadrature Identities
7 Examples
7.1 Examples: Several Evolutions of a Cardioid
7.1.1 The Univalent Solution
7.1.2 A Non-univalent Solution of the Polubarinova-Galin Equation
7.1.3 A Non-univalent Solution of the Löwner-Kufarev Equation
7.1.4 A Solution for the Suction Case
7.2 Injection Versus Suction in a Riemann Surface Setting
8 Moment Coordinates and the String Equation
8.1 The Polubarinova-Galin Equation as a String Equation
8.2 The String Equation for Univalent Conformal Maps
8.3 Intuition and Physical Interpretation in the Non-univalent Case
8.4 An Example
8.4.1 General Case
8.4.2 First Subcase
8.4.3 Second Subcase
8.5 Moment Evolutions in Terms of Poisson Brackets
9 Hamiltonian Descriptions of General Laplacian Evolutions
9.1 Lie Derivatives and Interior Multiplication
9.2 Laplacian Evolutions
9.3 Schwarz Potentials and Generating Functions
9.4 Multitime Hamiltonians
10 The String Equation for Some Rational Functions
10.1 The String Equation on Quadrature Riemann Surfaces
10.2 The String Equation for Polynomials
10.3 Evolution of a Third Degree Polynomial with RealCoefficients
10.4 An Example by Ullemar
Glossary
References
Index