53 research outputs found
Resonances in Loewner equations
We prove that given a Herglotz vector field on the unit ball of
of the form with for all , its evolution family admits an associated Loewner chain,
which is normal if no real resonances occur. Hence the Loewner-Kufarev PDE
admits a solution defined for all positive times.Comment: 23 pages, corrected typos, revised argument in Proposition 5.5,
result unchange
Theorem on the existence of solutions of quasi-static moving boundary problems
Using the theory of conformal mappings, we show that two-dimensional quasi-static moving boundary problems can be described by a non-linear Löwner-Kufarev equation and a functional relation between the shape of the boundary and the velocity at the boundary. Together with the initial data, this leads to an initial value problem. Assuming that satisfies certain conditions, we prove a theorem stating that this initial value problem has a local solution in time. The proof is based on some straightforward estimates on solutions of Löwner-Kufarev equations and an iteration technique
Loewner equations on complete hyperbolic domains
We prove that, on a complete hyperbolic domain D\subset C^q, any Loewner PDE
associated with a Herglotz vector field of the form H(z,t)=A(z)+O(|z|^2), where
the eigenvalues of A have strictly negative real part, admits a solution given
by a family of univalent mappings (f_t: D\to C^q) such that the union of the
images f_t(D) is the whole C^q. If no real resonance occurs among the
eigenvalues of A, then the family (e^{At}\circ f_t) is uniformly bounded in a
neighborhood of the origin. We also give a generalization of Pommerenke's
univalence criterion on complete hyperbolic domains.Comment: 19 pages, revised exposition, improved results, added reference
Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the KdV hierarchy
We show that unstable fingering patterns of two dimensional flows of viscous
fluids with open boundary are described by a dispersionless limit of the KdV
hierarchy. In this framework, the fingering instability is linked to a known
instability leading to regularized shock solutions for nonlinear waves, in
dispersive media. The integrable structure of the flow suggests a dispersive
regularization of the finite-time singularities.Comment: Published versio
Bubble break-off in Hele-Shaw flows : Singularities and integrable structures
Bubbles of inviscid fluid surrounded by a viscous fluid in a Hele-Shaw cell
can merge and break-off. During the process of break-off, a thinning neck
pinches off to a universal self-similar singularity. We describe this process
and reveal its integrable structure: it is a solution of the dispersionless
limit of the AKNS hierarchy. The singular break-off patterns are universal, not
sensitive to details of the process and can be seen experimentally. We briefly
discuss the dispersive regularization of the Hele-Shaw problem and the
emergence of the Painlev\'e II equation at the break-off.Comment: 27 pages, 9 figures; typo correcte
Laplacian Growth and Whitham Equations of Soliton Theory
The Laplacian growth (the Hele-Shaw problem) of multi-connected domains in
the case of zero surface tension is proven to be equivalent to an integrable
systems of Whitham equations known in soliton theory. The Whitham equations
describe slowly modulated periodic solutions of integrable hierarchies of
nonlinear differential equations. Through this connection the Laplacian growth
is understood as a flow in the moduli space of Riemann surfaces.Comment: 33 pages, 7 figures, typos corrected, new references adde
Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy
We continue the analysis of the spectral curve of the normal random matrix
ensemble, introduced in an earlier paper. Evolution of the full quantum curve
is given in terms of compatibility equations of independent flows. The
semiclassical limit of these flows is expressed through canonical differential
forms of the spectral curve. We also prove that the semiclassical limit of the
evolution equations is equivalent to Whitham hierarchy.Comment: 14 page
Normal random matrix ensemble as a growth problem
In general or normal random matrix ensembles, the support of eigenvalues of
large size matrices is a planar domain (or several domains) with a sharp
boundary. This domain evolves under a change of parameters of the potential and
of the size of matrices. The boundary of the support of eigenvalues is a real
section of a complex curve. Algebro-geometrical properties of this curve encode
physical properties of random matrix ensembles. This curve can be treated as a
limit of a spectral curve which is canonically defined for models of finite
matrices. We interpret the evolution of the eigenvalue distribution as a growth
problem, and describe the growth in terms of evolution of the spectral curve.
We discuss algebro-geometrical properties of the spectral curve and describe
the wave functions (normalized characteristic polynomials) in terms of
differentials on the curve. General formulae and emergence of the spectral
curve are illustrated by three meaningful examples.Comment: 44 pages, 14 figures; contains the first part of the original file.
The second part will be submitted separatel
Abstract basins of attraction
Abstract basins appear naturally in different areas of several complex
variables. In this survey we want to describe three different topics in which
they play an important role, leading to interesting open problems
Loewner evolution driven by a stochastic boundary point
We consider evolution in the unit disk in which the sample paths are
represented by the trajectories of points evolving randomly under the
generalized Loewner equation. The driving mechanism differs from the SLE
evolution, but nevertheless solutions possess similar invariance properties.Comment: 23 pages, 6 figure
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