Classical and Stochastic Laplacian Growth

Abstract

More information about this series at http://www.springer.com/series/5032 Advances in Mathematical Fluid Mechanics is a forum for the publication of high quality monographs, or collections of works, on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. Its mathematical aims and scope are similar to those of the Journal of Mathematical Fluid Mechanics. In particular, mathematical aspects of computational methods and of applications to science and engineering are welcome as an important part of the theory. So also are works in related areas of mathematics that have a direct bearing on fluid mechanics.The monographs and collections of works published here may be written in a more expository style than is usual for research journals, with the intention of reaching a wide audience. Collections of review articles will also be sought from time to time.

On integrability and exact solvability in deterministic and stochastic Laplacian growth

“…A zero surface tension Hele-Shaw free-boundary problem (see e.g. [7], [23], [68]) is a deterministic version of the Laplacian Growth. It describes time evolution of a domain Ω = Ω(t) in the z = x + iy plane, when the boundary ∂Ω of the domain is driven by the gradient of a scalar field P = P (x, y, t), often referred as "pressure" (see Fig.…”

“…The simplest class of these is usually unified by the name of Laplacian Growth, also known as the Hele-Shaw problem, which refers to dynamics of a moving front ("boundary" or "interface") between two distinct phases in two dimensions driven by a harmonic scalar field that is a potential for the growth velocity field (see e.g. [7], [68], and [23] that contains more up to date results). Such models describe for example fingering viscous flows in fluid mechanics.…”

“…The problem can be naturally regularized by taking into account the surface tension of an interface (see e.g [27]. or section 1.4.4 of[23]). Unfortunately, this generalization is no longer integrable.…”

Finite term relations for the exponential orthogonal polynomials

“…The first column of the power moments, {a j0 , j ≥ 0}, namely the "harmonic" moments, coincides with the first column of the exponential moments, {b j0 , j ≥ 0}, as is easily seen from (5.1). Hele-Shaw flow moving boundary problems, or Laplacian growth (see [22,11] for general theory), are characterized by these moments changing according to some simple law. One case is the squeezing version of Hele-Shaw flow, meaning that a viscous fluid blob, represented by the principal function g, is confined between two parallel plates and one simply squeezes the plates together.…”

The string equation for polynomials

“…For conformal maps to bounded domains proofs based on somewhat different ideas and involving explicitly the Schwarz function were given in [? ], [3]. For convenience we briefly recall the proof below.…”