Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

and Background.- Explicit Strong Solutions.- Weak Solutions and Balayage.- Geometric Properties.- Capacities and Isoperimetric Inequalities.- General Evolution Equations.- Hele-Shaw Evolution and Strings.

http://inis.jinr.ru/sl/P_Physics/PC_Classical%20physics/PCfm_Fluid%20mechanics/Gustafsson,%20Vasiliev.%20Conformal%20and%20potential%20analysis%20in%20Hele-Shaw%20cells%20(CUP%20draft,%202004)(189s).pdf

Conformal and Potential Analysis in Hele-Shaw Cells

We consider the time-evolving displacement of a viscous fluid by another fluid of negligible viscosity in a Hele-Shaw cell, either in a channel or a radial geometry, for idealized boundary conditions developed by McLean & Saffman. The interfacial evolution is conveniently described by a time-dependent conformal map z($\zeta $, t) that maps a unit circle (or a semicircle) in the $\zeta $ plane into the viscous fluid flow region in the physical z-plane. Our paper is concerned with the singularities of the analytically continued z($\zeta $, t) in $|\zeta|$ > 1, which, on approaching $|\zeta|$ = 1, correspond to localized distortions of the actual interface. For zero surface tension, we extend earlier results to show that for any initial condition, each singularity, initially present in $|\zeta|$ > 1, continually approaches $|\zeta|$ = 1, the boundary of the physical domain, without any change in the singularity form. However, depending on the singularity type, it may or may not impinge on $|\zeta|$ = 1 in finite time. Under some assumptions, we give analytical evidence to suggest that the ill-posed initial value problem in the physical domain $|\zeta|\leq $ 1 can be imbedded in a well-posed problem in $|\zeta|\geq $ 1. We present a numerical scheme to calculate such solutions. For each initial singularity of a certain type, which in the absence of surface tension would have merely moved to a new location $\zeta \_{\text{s}}$ (t) at time t from an initial $\zeta \_{\text{s}}$(0), we find an instantaneous transformation of the singularity structure for non-zero surface tension $\beta $; however, for 0 < $\beta \ll $ 1, surface tension effects are limited to a small `inner' neighbourhood of $\zeta \_{\text{s}}$(t) when t $\ll \beta ^{-1}$. Outside the inner region, but for $|\zeta -\zeta \_{\text{s}}$(t)$|\ll $ 1, the singular behaviour of the zero surface tension solution z$\_{0}$ is reflected in z($\zeta $, t). On the other hand, for each initial zero of z$\_{\zeta}$, which for $\beta $ = 0 remains a zero of z$\_{0\zeta}$ at a location $\zeta \_{0}$(t) that is generally different from $\zeta \_{0}$(0), surface tension spawns new singularities that move away from $\zeta \_{0}$(t) and approach the physical domain $|\zeta|$ = 1. We find that even for 0 < $\beta \ll $ 1, it is possible for z - z$\_{0}$ = O(1) or larger in some neighbourhood where z$\_{0\zeta}$ is neither singular nor zero. Our findings imply that for a small enough $\beta $, the evolution of a Hele-Shaw interface is very sensitive to prescribed initial conditions in the physical domain.

Evolution of Hele Shaw interface for small surface tension

We analyze Loewner traces driven by functions asymptotic to K\sqrt{1-t}. We
prove a stability result when K is not 4 and show that K=4 can lead to non
locally connected hulls. As a consequence, we obtain a driving term \lambda(t)
so that the hulls driven by K\lambda(t) are generated by a continuous curve for
all K > 0 with K not equal to 4 but not when K = 4, so that the space of
driving terms with continuous traces is not convex. As a byproduct, we obtain
an explicit construction of the traces driven by K\sqrt{1-t} and a conceptual
proof of the corresponding results of Kager, Nienhuis and Kadanoff,
math-ph/0309006

Collisions and spirals of Loewner traces

This paper is a historical overview of the development of the topic now commonly known as Laplacian Growth, from the original Hele-Shaw experiment to the modern treatment based on integrable systems.

/pdf/from-the-hele-shaw-experiment-to-integrable-systems-a-4kccwbrzt3.pdf

From the Hele-Shaw Experiment to Integrable Systems: A Historical Overview

We consider the Lowner differential equation generating univalent maps of the unit disk (or of the upper half-plane) onto itself minus a single slit. We prove that the circular slits, tangent to the real axis are generated by Holder continuous driving terms with exponent 1/3 in the Lowner equation. Singular solutions are described, and the critical value of the norm of driving terms generating quasisymmetric slits in the disk is obtained.

/pdf/singular-and-tangent-slit-solutions-to-the-lowner-equation-o4qu9643ps.pdf

Singular and tangent slit solutions to the Lowner equation

We study one-parameter expanding evolution families of simply connected domains in the complex plane described by infinite systems of evolution parameters. These evolution parameters in some cases admit Hamiltonian formulation and lead to integrable systems. One example of such parameters is complex moments for the Laplacian growth that form a Whitham-Toda integrable hierarchy. Another example we deal with is related to expanding coefficient bodies for conformal maps given by Lowner subordination chains. The coefficients' bodies are proved to form a Liouville partially integrable Hamiltonian system for each fixed index and the first integrals are obtained. We also discuss the contact structure of this system.

Univalent Functions and Integrable Systems

We study one-parameter expanding evolution families of simply connected domains in the complex plane described by infinite systems of evolution parameters. These evolution parameters in some cases admit Hamiltonian formulation and lead to integrable systems. One example of such parameters is complex moments for the Laplacian growth that form a Whitham-Toda integrable hierarchy. Another example we deal with is related to expanding coefficient bodies for conformal maps given by Lowner subordination chains. The coefficients bodies are proved to form a Liouville partially integrable Hamiltonian system for each fixed index and the first integrals are obtained. We also discuss the contact structure of this system.

Univalent functions and integrable systems

One of the folklore questions in the theory of free boundary problems is the lifetime of the starlike dynamics in a Hele-Shaw cell. We prove precisely that, starting with a starlike analytic phase domain Omega(0), the Hele-Shaw chain of subordinating domains Omega( t), Omega(0) = Omega(0), exists for an infinite time under injection at the point of starlikeness.

/pdf/infinite-lifetime-for-the-starlike-dynamics-in-hele-shaw-4hq8hoqepc.pdf

Infinite lifetime for the starlike dynamics in Hele-Shaw cells

We consider coefficient bodies $\mathcal M_n$ for univalent functions. Based on the L\"owner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then $\mathcal M_n$ are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracket-generating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system and calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case $\mathcal M_3$.

Dmitri Prokhorov

Papers

Singular and tangent slit solutions to the Lowner equation

Univalent Functions and Integrable Systems

Univalent functions and integrable systems

Infinite lifetime for the starlike dynamics in Hele-Shaw cells

Sub-Riemannian geometry of the coefficients of univalent functions