05566nam 22007215 450 991048349310332120251113175604.03-030-69863-710.1007/978-3-030-69863-8(CKB)4100000011807201(MiAaPQ)EBC6524990(Au-PeEL)EBL6524990(OCoLC)1243263853(PPN)254718949(DE-He213)978-3-030-69863-8(EXLCZ)99410000001180720120210322d2021 u| 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierLaplacian Growth on Branched Riemann Surfaces /by Björn Gustafsson, Yu-Lin Lin1st ed. 2021.Cham :Springer International Publishing :Imprint: Springer,2021.1 online resource (163 pages) illustrationsLecture Notes in Mathematics,1617-9692 ;22873-030-69862-9 Includes bibliographical references and index.Intro -- 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.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.Lecture Notes in Mathematics,1617-9692 ;2287Functions of complex variablesDifferential equationsPotential theory (Mathematics)Mathematical physicsSoft condensed matterFunctions of a Complex VariableDifferential EquationsPotential TheoryMathematical Methods in PhysicsFluidsFunctions of complex variables.Differential equations.Potential theory (Mathematics).Mathematical physics.Soft condensed matter.Functions of a Complex Variable.Differential Equations.Potential Theory.Mathematical Methods in Physics.Fluids.532.053Gustafsson Björn1947-853049Lin Yu-LinMiAaPQMiAaPQMiAaPQBOOK9910483493103321Laplacian growth on branched Riemann surfaces1904861UNINA