LEADER 04200nam 2200553 450 001 996466561903316 005 20231110232131.0 010 $a3-030-69863-7 035 $a(CKB)4100000011807201 035 $a(MiAaPQ)EBC6524990 035 $a(Au-PeEL)EBL6524990 035 $a(OCoLC)1243263853 035 $a(PPN)254718949 035 $a(EXLCZ)994100000011807201 100 $a20211014d2021 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aLaplacian growth on branched Riemann surfaces /$fBjo? Gustafsson and Yu-Lin Lin 210 1$aCham, Switzerland :$cSpringer,$d[2021] 210 4$d©2021 215 $a1 online resource (163 pages) $cillustrations 225 1 $aLecture Notes in Mathematics ;$vv.2287 311 $a3-030-69862-9 320 $aIncludes bibliographical references and index. 327 $aIntro -- 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 Lo?wner-Kufarev Equations -- 2.1 Basic Set Up in the Univalent Case -- 2.2 Dynamics and Subordination -- 2.3 The Polubarinova-Galin Versus the Lo?wner-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 Lo?wner-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. 327 $a8.4 An Example -- 8.4.1 General Case -- 8.4.2 First Subcase -- 8.4.3 Second Subcase -- 8.5 Moment Evolutions in Terms of Poisson Brackets -- 9 Hamiltonian Descriptions of General Laplacian Evolutions -- 9.1 Lie Derivatives and Interior Multiplication -- 9.2 Laplacian Evolutions -- 9.3 Schwarz Potentials and Generating Functions -- 9.4 Multitime Hamiltonians -- 10 The String Equation for Some Rational Functions -- 10.1 The String Equation on Quadrature Riemann Surfaces -- 10.2 The String Equation for Polynomials -- 10.3 Evolution of a Third Degree Polynomial with RealCoefficients -- 10.4 An Example by Ullemar -- Glossary -- References -- Index. 410 0$aLecture Notes in Mathematics 606 $aFluid dynamics 606 $aGeometric function theory 606 $aDinàmica de fluids$2thub 606 $aTeoria geomètrica de funcions$2thub 608 $aLlibres electṛnics$2thub 615 0$aFluid dynamics. 615 0$aGeometric function theory. 615 7$aDinàmica de fluids 615 7$aTeoria geomètrica de funcions 676 $a532.053 700 $aGustafsson$b Bjo?rn$f1947-$0853049 702 $aLin$b Yu-Lin 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996466561903316 996 $aLaplacian growth on branched Riemann surfaces$91904861 997 $aUNISA