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Equilibria of an anisotropic nonlocal interaction equation : analysis and numerics
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Carrillo, José A., Düring, Bertram, Kreusser, Lisa Maria and Schönlieb, Carlola-Bibiane (2021) Equilibria of an anisotropic nonlocal interaction equation : analysis and numerics. Discrete and Continuous Dynamical Systems - Series A, 41 (8). pp. 3985-4012. doi:10.3934/dcds.2021025 ISSN 1078-0947.
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WRAP-Equilibria-anisotropic-nonlocal-interaction-equation-analysis-numerics-2021.pdf - Accepted Version Embargoed item. Restricted access to Repository staff only - Requires a PDF viewer. Download (2244Kb) |
Official URL: http://dx.doi.org/10.3934/dcds.2021025
Abstract
In this paper, we study the equilibria of an anisotropic, nonlocal aggregation equation with nonlinear diffusion which does not possess a gradient flow structure. Here, the anisotropy is induced by an underlying tensor field. Anisotropic forces cannot be associated with a potential in general and stationary solutions of anisotropic aggregation equations generally cannot be regarded as minimizers of an energy functional. We derive equilibrium conditions for stationary line patterns in the setting of spatially homogeneous tensor fields. The stationary solutions can be regarded as the minimizers of a regularised energy functional depending on a scalar potential. A dimension reduction from the two- to the one-dimensional setting allows us to study the associated one-dimensional problem instead of the two-dimensional setting. We establish Γ-convergence of the regularised energy functionals as the diffusion coefficient vanishes, and prove the convergence of minimisers of the regularised energy functional to minimisers of the non-regularised energy functional. Further, we investigate properties of stationary solutions on the torus, based on known results in one spatial dimension. Finally, we prove weak convergence of a numerical scheme for the numerical solution of the anisotropic, nonlocal aggregation equation with nonlinear diffusion and any underlying tensor field, and show numerical results.
Item Type: | Journal Article | ||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Journal or Publication Title: | Discrete and Continuous Dynamical Systems - Series A | ||||||
Publisher: | American Institute of Mathematical Sciences | ||||||
ISSN: | 1078-0947 | ||||||
Official Date: | 5 February 2021 | ||||||
Dates: |
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Volume: | 41 | ||||||
Number: | 8 | ||||||
Page Range: | pp. 3985-4012 | ||||||
DOI: | 10.3934/dcds.2021025 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Re-use Statement: | This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Discrete and Continuous Dynamical Systems - Series A following peer review. The definitive publisher-authenticated version [insert complete citation information here] is available online at: http://dx.doi.org/10.3934/dcds.2021025 | ||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||
Date of first compliant deposit: | 13 January 2021 | ||||||
Related URLs: | |||||||
Open Access Version: |
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