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Geodesic Loewner paths with varying boundary conditions

McDonald, R; (2020) Geodesic Loewner paths with varying boundary conditions. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 10.1098/rspa.2020.0466. (In press). Green open access

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Abstract

Equations of the Loewner class subject to non-constant boundary conditions along the real axis are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian growth in which the slits represent thin fingers growing in a diffusion field. A single finger follows a curved path determined by the forcing function appearing in Loewner’s equation. This function is found by solving an ordinary differential equation whose terms depend on curvature properties of the streamlines of the diffusive field in the conformally mapped ‘mathematical’ plane. The effect of boundary conditions specifying either piecewise constant values of the field variable along the real axis, or a dipole placed on the real axis, reveal a range of behaviours for the growing slit. These include regions along the real axis from which no slit growth is possible, regions where paths grow to infinity, or regions where paths curve back toward the real axis terminating in finite time. Symmetric pairs of paths subject to the piecewise constant boundary condition along the real axis are also computed, demonstrating that paths which grow to infinity evolve asymptotically toward an angle of bifurcation of π/5.

Type: Article
Title: Geodesic Loewner paths with varying boundary conditions
Open access status: An open access version is available from UCL Discovery
DOI: 10.1098/rspa.2020.0466
Publisher version: http://dx.doi.org/10.1098/rspa.2020.0466
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Loewner equation · Laplacian growth · free boundary problems
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10111869
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