Classical and Stochastic Laplacian Growth

Table of Contents

1. Frontmatter

2. Chapter 1. Introduction and Background

   Björn Gustafsson, Razvan Teodorescu, Alexander Vasil’ev

   Abstract

   A fluid is a substance which continues to change shape as long as there are shear stresses (dependent on the velocity of deformation) present. If the force F acts over an area A, then the ratio between the tangential component of F and A gives a shear stress across the liquid. The liquid’s response to this applied shear stress is to flow. In contrast, a solid body undergoes a definite displacement or breaks completely when subject to a shear stress. Viscous stresses are linked to the velocity of deformation. In the simplest model, this relation is just linear, and a fluid possessing this property is known as a Newtonian fluid. The constant of proportionality between the viscous stress and the deformation velocity is known as the coefficient of viscosity and it is an intrinsic property of a fluid.

3. Chapter 2. Rational and Other Explicit Strong Solutions

   Björn Gustafsson, Razvan Teodorescu, Alexander Vasil’ev

   Abstract

   In this chapter we shall construct several explicit solutions to the Hele-Shaw problem, more precisely, to the Polubarinova–Galin equation, starting with the classical ones of Polubarinova-Kochina [438], [439], Galin [199] and Saffman, Taylor [488], [489]. Some properties of polynomial and rational solutions will be discussed, and it will be proved that the property of the conformal map to the fluid domain of being a polynomial or a rational function is preserved under the time evolution. The same is true also when logarithmic singularities are allowed. From these properties one easily deduces local existence and uniqueness of solutions within such classes.

4. Chapter 3. Weak Solutions and Related Topics

   Björn Gustafsson, Razvan Teodorescu, Alexander Vasil’ev

   Abstract

   In the previous chapter we discussed strong solutions, which for their definition require smooth boundaries and smooth dependence on t. This section is devoted to weak solutions, more precisely variational inequality weak solutions, which are closely related to potential theory (they can be viewed as instances of partial balayage) and also to quadrature domains and related topics.

5. Chapter 4. Geometric Properties

   Björn Gustafsson, Razvan Teodorescu, Alexander Vasil’ev

   Abstract

   In this chapter we discuss geometric properties of general Hele-Shaw flows. Special classes of univalent functions that admit explicit geometric interpretations are considered. In particular, we ask the following question: which geometrical properties are preserved during the time evolution of the moving boundary? We also discuss the geometry of weak solutions.

6. Chapter 5. Capacities and Isoperimetric Inequalities

   Björn Gustafsson, Razvan Teodorescu, Alexander Vasil’ev

   Abstract

   This is the solution of what is sometimes known as Dido’s problem because of the story that Queen Dido of Tyre bargained for some land bounded on one side by the (straight) Mediterranean coast and agreed to pay a fixed sum for as much land as could be enclosed by a bull’s hide. Both statements can be expressed in a more algebraic form which indeed underlines the fact that they are equivalent.

7. Chapter 6. Laplacian Growth and Random Matrix Theory

   Björn Gustafsson, Razvan Teodorescu, Alexander Vasil’ev

   Abstract

   The link between Laplacian growth and stochastic processes in the complex plane was discovered rather unexpectedly [581, 551], through their common relation to the multi-particle wavefunction description of the Quantum Hall Effect, in the single-Landau level approximation. As pointed out in [551], the classical Laplacian growth and its stochastic variant based on the normal random matrix theory (NRMT) can be identified to the dispersionless limit of a certain integrable hierarchy and its dispersionful versions, respectively. We discuss this formulation of the relation between the two models in the last chapters of this book; in the present chapter, we only mention the works where this relationship was already implicit, although not recognized as such.

8. Chapter 7. Integrability and Moments

   Björn Gustafsson, Razvan Teodorescu, Alexander Vasil’ev

   Abstract

   Laplacian growth is a special kind of domain variation. Following [370] we introduce the notation ∇(a) for the directional derivative corresponding to Hele-Shaw injection at the point a.

9. Chapter 8. Shape Evolution and Integrability

   Björn Gustafsson, Razvan Teodorescu, Alexander Vasil’ev

   Abstract

   The Hele-Shaw (strong or classical) advancing evolution in the plane is an example of evolution in the infinite-dimensional manifold of smooth shapes. It is a typical ‘field’ problem, i.e., given an initial shape, the further evolution is well defined at least for a short time. By shape we understand a simple closed curve in the complex plane dividing it into two simply connected domains. The study of 2D shapes is one of the central problems in the field of applied sciences. A program of such study and its importance was summarized by Mumford at ICM 2002 in Beijing [397]. The harmonic (Richardson’s) moments of the Hele-Shaw evolution (or of the Laplacian growth) are conserved (see (1.34)) under this evolution and serve as the evolution parameters or generalized times. The infinite number of evolution parameters constitutes the infinite number of degrees of freedom of the system, and clearly suggests to apply field theory methods as a natural tool of study, which logically lead to integrable systems, the dispersionless Toda hierarchies, in particular.

10. Chapter 9. Stochastic Löwner and Löwner–Kufarev Evolution

    Björn Gustafsson, Razvan Teodorescu, Alexander Vasil’ev

    Abstract

    This chapter we dedicate to the stochastic counterpart of the Löwner–Kufarev theory first recalling that one of the last (but definitely not least) contributions to this growing theory was the description by Oded Schramm in 1999–2000 [518], of the stochastic Löwner evolution (SLE), also known as the Schramm–Löwner evolution. The SLE is a conformally invariant stochastic process; more precisely, it is a family of random planar curves generated by solving Löwner’s differential equation with the Brownian motion as a driving term.

11. Backmatter