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.
| Original language | English |
|---|---|
| Pages (from-to) | 3985-4012 |
| Number of pages | 28 |
| Journal | Discrete and Continuous Dynamical Systems- Series A |
| Volume | 41 |
| Issue number | 8 |
| Early online date | 1 Jan 2021 |
| DOIs | |
| Publication status | Published - 1 Aug 2021 |
Bibliographical note
Funding Information:Acknowledgments. JAC was supported the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 883363) and by the EPSRC through grant number EP/P031587/1. BD has been supported by the Leverhulme Trust research project grant ‘Novel discretizations for higher-order nonlinear PDE’ (RPG-2015-69). LMK was supported by the European Union Horizon 2020 research and innovation programmes under the Marie Skodowska-Curie grant agreement No. 777826 NoMADS and No. 691070 CHiPS, the EPSRC grant Nr. EP/L016516/1, the German Academic Scholarship Foundation (Studienstiftung des Deutschen Volkes), the Cantab Capital Institute for the Mathematics of Information and Magdalene College, Cambridge (Nevile Research Fellowship). CBS acknowledges support from the Leverhulme Trust (Breaking the non-convexity barrier, and Unveiling the Invisible), the Philip Leverhulme Prize, the EPSRC grant Nr. EP/M00483X/1, the EPSRC Centre Nr. EP/N014588/1, the European Union Horizon 2020 research and innovation programmes under the Marie Skodowska-Curie grant agreement No. 777826 NoMADS and No. 691070 CHiPS, the Cantab Capital Institute for the Mathematics of Information and the Alan Turing Institute.
Publisher Copyright:
© 2021 American Institute of Mathematical Sciences. All rights reserved.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.
Keywords
- Finite volume methods
- Nonlinear diffusion
- Nonlocal aggregation
- Pattern formation
- Stationary states
ASJC Scopus subject areas
- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics