arXiv subjects

Björn Gustafsson

Publications and source records attributed to Björn Gustafsson.

18 recordsLinked to original sources

arXiv2025

Two dimensional potential theory with a view towards vortex motion: Energy, capacity and Green functions

The paper reviews some parts of classical potential theory with applications to two dimensional fluid dynamics, in particular vortex motion. Energy and forces within a system of point vortices are similar to those for point charges when the vortices are kept fixed, but the dynamics is different in the case of free vortices. Starting from Bernoulli's equation we derive these laws. Letting the number of vortices tend to infinity leads in the limit to considerations of capacity, harmonic measure and many other notions in potential theory. In particular various kinds of Green functions have a central role in the paper, where we make a difference between electrostatic and hydrodynamic Green function. We also consider the corresponding concepts in the case of closed Riemann surfaces provided with a metric. From a canonically defined monopole Green function we rederive much of the classical theory of harmonic and analytic forms. In the final section of the paper we return to the planar case, then reappearing in form of a symmetric Riemann surface, the Schottky double. Bergman kernels, electrostatic and hydrodynamic, come up naturally, and associated to the Green function the is a certain Robin function which is important for vortex motion and which also relates to capacity functions in classical potential theory.

math.CV↗

arXiv2023

On the interplay between vortices and harmonic flows: Hodge decomposition of Euler's equations in 2d

Let $Σ$ be a compact manifold without boundary whose first homology is nontrivial. Hodge decomposition of the incompressible Euler's equation in terms of 1-forms yields a coupled PDE-ODE system. The $L^2$-orthogonal components are a `pure' vorticity flow and a potential flow (harmonic, with the dimension of the homology). In this paper we focus on $N$ point vortices on a compact Riemann surface without boundary of genus $g$, with a metric chosen in the conformal class. The phase space has finite dimension $2N+ 2g$. We compute a surface of section for the motion of a single vortex ($N=1$) on a torus ($g=1$) with a non-flat metric, that shows typical features of non-integrable 2-dof Hamiltonians. In contradistinction, for flat tori the harmonic part is constant. Next, we turn to hyperbolic surfaces ($ g \geq 2$), having constant curvature -1, with discrete symmetries. Fixed points of involutions yield vortex crystals in the Poincaré disk. Finally we consider multiply connected planar domains. The image method due to Green and Thomson is viewed in the Schottky double. The Kirchhoff-Routh hamiltonian given in C.C. Lin's celebrated theorem is recovered by Marsden-Weinstein reduction from $2N+2g$ to $2N$. The relation between the electrostatic Green function and the hydrodynamical Green function is clarified. A number of questions are suggested.

math-ph↗

arXiv2023

Quadrature for quadrics

We make a systematic investigation of quadrature properties for quadrics, namely integration of holomorphic functions over planar domains bounded by second degree curves. A full understanding requires extending traditional settings by allowing domains which are multi-sheeted, in other words domains which really are branched covering surfaces of the Riemann sphere, and in addition usage of the spherical area measure instead of the Euclidean. The first part of the paper discusses two different points of view of real algebraic curves: traditionally they live in the real projective plane, which is non-orientable, but for their role for quadrature they have to be pushed to the Riemann sphere. The main results include clarifying a previous theorem (joint work with V.~Tkachev), which says that a branched covering map produces a domain with the required quadrature properties if and only it extends to be meromorphic on the double of the parametrizing Riemann surface. In the second half of the paper domains bounded by ellipses, hyperbolas, parabolas and their inverses are studied in detail, with emphasis on the hyperbola case, for which some of the results appear to be new.

math.CV↗

arXiv2022

Vortex pairs and dipoles on closed surfaces

We set up general equations of motion for point vortex systems on closed Riemannian surfaces, allowing for the case that the sum of vorticities is not zero and there hence must be counter-vorticity present. The dynamics of global circulations which is coupled to the dynamics of the vortices is carefully taken into account. Much emphasis is put to the study of vortex pairs, having the Kimura conjecture in focus. This says that vortex pairs move, in the dipole limit, along geodesic curves, and proofs for it have previously been given by S.~Boatto and J.~Koiller by using Gaussian geodesic coordinates. In the present paper we reach the same conclusion by following a slightly different route, leading directly to the geodesic equation with a reparametrized time variable. In a final section we explain how vortex motion in planar domains can be seen as a special case of vortex motion on closed surfaces, and in two appendices we give some necessary background on affine and projective connections.

math-ph↗

arXiv2020

Hadamard's variational formula and the energy-momentum tensor

The Hadamard variational formula for the Green function is formulated in terms of a polarized energy-momentum tensor and a strain tensor. This is elaborated in a general setting of subdomains of a Riemannian manifold in arbitrary dimension and linked to the way the energy-momentum tensor in general field theory appears as a result of varying the metric tensor in a Lagrangian function.

math-ph↗

arXiv2018

Vortex motion and geometric function theory: the role of connections

We formulate the equations for point vortex dynamics on a closed two dimensional Riemann manifold in the language of affine and other kinds of connections. The speed of a vortex is then expressed in terms of the difference between an affine connection derived from the coordinate Robin function and the Levi-Civita connection associated to the Riemannian metric. A Hamiltonian formulation of the same dynamics is also given. The relevant Hamiltonian function consists of two main terms. One of the terms is the well-known quadratic form based on a matrix whose entries are Green and Robin functions, while the other term describes the energy contribution from those circulating flows which are not implicit in the Green functions. One main issue of the paper is the clarification of the somewhat intricate exchanges of energy between these two terms of the Hamiltonian.

math-ph↗

arXiv2018

A field theoretic operator model and Cowen-Douglas class

In resonance to a recent geometric framework proposed by Douglas and Yang, a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space is developed. By taking advantage of the refined existing theory of the principal function of a hyponormal operator we transfer the whole action outside the spectrum, on the resolvent of the underlying operator, localized at a distinguished vector. The whole construction turns out to rely on an elementary algebra body involving analytic multipliers and Cauchy transforms. A natural field theory interpretation of the resulting resolvent functional model is proposed.

math.FA↗

arXiv2018

The string equation for non-univalent functions

For conformal maps defined in the unit disk one can define a certain Poisson bracket that involves the harmonic moments of the image domain. When this bracket is applied to the conformal map itself together with its conformally reflected map the result is identically one. This is called the string equation, and it is closely connected to the governing equation, the Polubarinova-Galin equation, for the evolution of a Hele-Shaw blob of a viscous fluid (or, by another name, Laplacian growth). In the present paper we investigate to what extent the string equation makes sense and holds for non-univalent analytic functions. We give positive answers in two cases: for polynomials and for a special class of rational functions.

math.CV↗

arXiv2017

Line bundles defined by the Schwarz function

Cauchy and exponential transforms are characterized, and constructed, as canonical holomorphic sections of certain line bundles on the Riemann sphere defined in terms of the Schwarz function. A well known natural connection between Schwarz reflection and line bundles defined on the Schottky double of a planar domain is briefly discussed in the same context.

math.CV↗

arXiv2016

Partial Balayage on Riemannian Manifolds

A general theory of partial balayage on Riemannian manifolds is developed, with emphasis on compact manifolds. Partial balayage is an operation of sweeping measures, or charge distributions, to a prescribed density, and it is closely related to (construction of) quadrature domains for subharmonic functions, growth processes such as Laplacian growth and to weighted equilibrium distributions. Several examples are given in the paper, as well as some specific results. For instance, it is proved that, in two dimensions, harmonic and geodesic balls are the same if and only if the Gaussian curvature of the manifold is constant.

math.DG↗

arXiv2014

Non-univalent solutions of the Polubarinova-Galin equation

We study non-univalent solutions of the Polubarinova-Galin equation, describing the time evolution of the conformal map from the unit disk onto a Hele-Shaw blob of fluid subject to injection at one point. In particular, we tackle the difficulties arising when the map is not even locally univalent, in which case one has to pass to weak solutions developing on a branched covering surface of the complex plane. One major concern is the construction of this Riemann surface, which is not given in advance but has to be constantly up-dated along with the solution. Once the Riemann surface is constructed the weak solution is automatically global in time, but we have had to leave open the question whether the weak solution can be kept simply connected all the time (as is necessary to connect to the Polubarinova-Galin equation). A certain crucial statement, a kind of stability statement for free boundaries, has therefore been left as a conjecture only. Another major part of the paper concerns the structure of rational solutions (as for the derivative of the mapping function). Here we have fairly complete results on the dynamics. Several examples are given.

math.CV↗

arXiv2012

Exponential transforms, resultants and moments

We give an overview of some recent developments concerning harmonic and other moments of plane domains, their relationship to the Cauchy and exponential transforms, and to the meromorphic resultant and elimination function. The paper also connects to certain topics in mathematical physics, for example domain deformations generated by harmonic gradients (Laplacian growth) and related integrable structures.

math.CV↗

arXiv2011

On the curvature of some free boundaries in higher dimensions

It is known that any subharmonic quadrature domain in two dimensions satisfies a natural inner ball condition, in other words there is a specific upper bound on the curvature of the boundary. This result directly applies to free boundaries appearing in obstacle type problems and in Hele-Shaw flow. In the present paper we make partial progress on the corresponding question in higher dimensions. Specifically, we prove the equivalence between several different ways to formulate the inner ball condition, and we compute the Brouwer degree for a geometrically important mapping related to the Schwarz potential of the boundary. The latter gives in particular a new proof in the two dimensional case.

math.AP↗

arXiv2011

On the dynamics of roots and poles for solutions of the Polubarinova-Galin equation

We study the dynamics of roots of f'(z,t), where f(z,t) is a locally univalent polynomial solution of the Polubarinova-Galin equation for the evolution of the conformal map onto a Hele-Shaw blob subject to injection at one point. We give examples of the sometimes complicated motion of roots, but show also that the asymptotic behavior is simple. More generally we allow f'(z,t) to be a rational function and give sharp estimates for the motion of poles and for the decay of the Taylor coefficients. We also prove that any global in time locally univalent solution actually has to be univalent.

math.CV↗

arXiv2011

On the exponential transform of multi-sheeted algebraic domains

We introduce multi-sheeted versions of algebraic domains and quadrature domains, allowing them to be branched covering surfaces over the Riemann sphere. The two classes of domains turn out to be the same, and the main result states that the extended exponential transform of such a domain agrees, apart from some simple factors, with the extended elimination function for a generating pair of functions. In an example we discuss the algebraic curves associated to level curves of the Neumann oval, and determine which of these give rise to multi-sheeted algebraic domains.

math.CV↗

arXiv2009

Critical points of Green's function and geometric function theory

We study questions related to critical points of the Green's function of a bounded multiply connected domain in the complex plane. The motion of critical points, their limiting positions as the pole approaches the boundary and the differential geometry of the level lines of the Green's function are main themes in the paper. A unifying role is played by various affine and projective connections and corresponding Möbius invariant differential operators. In the doubly connected case the three Eisenstein series $E_2$, $E_4$, $E_6$ are used. A specific result is that a doubly connected domain is the disjoint union of the set of critical points of the Green's function, the set of zeros of the Bergman kernel and the separating boundary limit positions for these. At the end we consider the projective properties of the prepotential associated to a second order differential operator depending canonically on the domain.

math.CV↗

arXiv2009

On the Jacobian of the harmonic moment map

In this paper we represent harmonic moments in the language of transfinite functions, that is projective limits of polynomials in infinitely many variables. We obtain also an explicit formula for the Jacobian of a generalized harmonic moment map.

math.CV↗

arXiv2006

A note on H-convergence

We give a criterion for H-convergence of conductivity matrices in terms of ordinary weak convergence of the factors in certain quotient representations of the matrices.

math.AP↗