| 1. |
- Gustafsson, Björn, 1947-
(författare)
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Vortex Pairs and Dipoles on Closed Surfaces
- 2022
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Ingår i: Journal of nonlinear science. - : Springer Nature. - 0938-8974 .- 1432-1467. ; 32:5
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Kavallaris, Nikos I., et al.
(författare)
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Dynamics of Shadow System of a Singular Gierer–Meinhardt System on an Evolving Domain
- 2020
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Ingår i: Journal of nonlinear science. - : Springer. - 0938-8974 .- 1432-1467. ; 31:1
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Tidskriftsartikel (refereegranskat)abstract
- The main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular Gierer–Meinhardt model on an isotropically evolving domain. In the case where the inhibitor’s response to the activator’s growth is rather weak, then the shadow system of the Gierer–Meinhardt model is reduced to a single though non-local equation whose dynamics is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blow-up results for this non-local equation, which can be interpreted as instability patterns of the shadow system. In particular, a diffusion-driven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which then is destabilised via diffusion-driven blow-up, is observed. The latter indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns. Most of the theoretical results are verified numerically, whilst the numerical approach is also used to exhibit the dynamics of the shadow system when analytical methods fail.
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