Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-23T05:06:57.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Global minimisers for anisotropic attractive–repulsive interactions

Part of: Parabolic equations and systems Partial differential equations

Published online by Cambridge University Press:  22 October 2019

GUNNAR KAIB ,
KYUNGKEUN KANG  and
ANGELA STEVENS
GUNNAR KAIB
Affiliation:
Ellerstraße 69, 40227 Düsseldorf, Germany, email: [email protected]
KYUNGKEUN KANG
Affiliation:
Department of Mathematics, Yonsei University, Seoul, Republic of Korea, email: [email protected]
ANGELA STEVENS
Affiliation:
Applied Mathematics, University of Münster, Einsteinstr. 62, D-48149 Münster, Germany, email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the existence of global minimisers for a class of attractive–repulsive interaction potentials that are in general not radially symmetric. The global minimisers have compact support. For potentials including degenerate power-law diffusion, the interaction potential can be unbounded from below. Further, a formal calculation indicates that for non-symmetric potentials global minimisers may neither be radial symmetric nor unique.

Keywords

Anisotropic attractive–repulsive interactionglobal minimisers

MSC classification

Primary: 35A15: Variational methods
Secondary: 35K65: Degenerate parabolic equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 5 , October 2020 , pp. 854 - 870
Check for updates
Copyright
© Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Auchmuty, J. F. G. & Beals, R. (1971) Variational solutions of some nonlinear free boundary problems. Arch. Rational Mech. Anal. 43(4), 255–271.CrossRefGoogle Scholar
Bedrossian, J. (2011) Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion. Appl. Math. Lett. 24(11), 1927–1932.CrossRefGoogle Scholar
Cañizo, J., Carrillo, J. A. & Patacchini, F. (2015) Existence of compactly supported global minimisers for the interaction energy. Arch. Ration. Mech. Anal. 217(3), 1197–1217.CrossRefGoogle Scholar
Carrillo, J. A., Delgadino, M. G. & Patachini, F. S. (2019) Existence of ground states for aggregation-diffusion equations. Anal. Appl. (Singap.) 17(3), 393–423.CrossRefGoogle Scholar
Carrillo, J. A., Figalli, A. & Patacchini, F. S. (2017) Geometry of minimizers for the interaction energy with mildly repulsive potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 34(5), 1299–1308.CrossRefGoogle Scholar
Carrillo, J. A., Hittmeir, S., Volzone, B. & Yao, Y. (2019) Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics. https://doi.org/10.1007/s00222-019-00898-xCrossRefGoogle Scholar
Carrillo, J. A., Mateu, J., Mora, M. G., Rondi, L., Scardia, L. & Verdera, J. (to appear) The ellipse law: Kirchhoff meets dislocations. Comm. Math. Phys.Google Scholar
Choksi, R., Fetecau, R. & Topaloglu, I. (2015) On minimizers of interaction functionals with competing attractive and repulsive potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(6), 1283–1305.CrossRefGoogle Scholar
Craig, K. & Topaloglu, I. Aggregation-Diffusion to Constrained Interaction: Minimizers & Gradient Flows in the Slow Diffusion Limit. Preprint, arXiv:1806.07415.Google Scholar
Kaib, G. (2016) Stationary States of an Aggregation Equation with Degenerate Diffusion and Attractive Potential. PhD thesis, University of Münster, Germany.Google Scholar
Kaib, G. (2017) Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential. SIAM J. Math. Anal. 49(1), 272–296.CrossRefGoogle Scholar
Lieb, E. H. & Loss, M. (2001) Analysis, 2nd ed. Graduate Studies in Mathematics, Vol. 14. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Lions, P.-L. (1981) Minimization problems in L 1(ℝ3). J. Funct. Anal. 41(2), 236–275.CrossRefGoogle Scholar
Lions, P.-L. (1984) The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(2), 109–145.CrossRefGoogle Scholar
Primi, I., Stevens, A. & Velázquez, J. J. L. (2009) Mass-selection in alignment models with non-deterministic effects. Comm. Part. Diff. Equ. 34(4–6), 419–456.CrossRefGoogle Scholar
Simione, R., Slepčev, D. & Topaloglu, I. (2015) Existence of ground states of nonlocal-interaction energies. J. Stat. Phys. 159(4), 972–986.CrossRefGoogle Scholar