Ground States of a Two Phase Model with Cross and Self Attractive Interactions
Abstract
We consider a variational model for two interacting species (or phases), subject to cross and self attractive forces. We show existence and several qualitative properties of minimizers. Depending on the strengths of the forces, different behaviors are possible: phase mixing or phase separation with nested or disjoint phases. In the case of Coulomb interaction forces, we characterize the ground state configurations.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
L. Ambrosio and A. Braides, Functionals defined on partitions in sets of finite perimeter. II, J. Math. Pures Appl., 69 (1990), pp. 307--333.
2.
M. Bonacini and R. Cristoferi, Local and global minimality results for a nonlocal isoperimetric problem on $\R^N$, SIAM J. Math. Anal., 46 (2014), pp. 2310--2349.
3.
M. Bonacini, H. Knüpfer, and M. Röger, Optimal distribution of oppositely charged phases: Perfect screening and other properties, SIAM J. Math. Anal., 48 (2016), pp. 1128--1154.
4.
A. Burchard, R. Choksi, I. Topaloglu, Nonlocal shape optimization via interactions of attractive and repulsive potentials, preprint 2016.
5.
J. A. Can͂izo, J. A. Carrillo, and F. S. Patacchini, Existence of compactly supported global minimisers for the interaction energy, Arch. Ration. Mech. Anal., 217 (2015), pp. 1197--1217.
6.
E. A. Carlen and F. Maggi, Stability for the Brunn-Minkowski and Riesz Rearrangement Inequalities, with Applications to Gaussian Concentration and Finite Range Non-local Isoperimetry, preprint 2015.
7.
E. A. Carlen, M. Carvalho, R. Esposito, J. L. Lebowitz, and R. Marra, Free energy minimizers for a two-species model with segregation and liquid-vapor transition, Nonlinearity, 16 (2003), pp. 1075--1105.
8.
J. A. Carrillo, M. Chipot, and Y. Huang, On global minimizers of repulsive-attractive power-law interaction energies, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372, (2014).
9.
J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, and D. Slepc̆ev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), pp. 229--271.
10.
R. Choksi, R.C. Fetecau, and I. Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Ann. Inst. H. Poincaré Anal. Non Linéalre, 32 (2015), pp. 1283--1305.
11.
E. Cristiani, B. Piccoli, and A. Tosin, Multiscale Modeling of Pedestrian Dynamics: Modeling, Simulation and Applications, Springer, Berlin, 2014.
12.
A. Di Castro, M. Novaga, B. Ruffini, and E. Valdinoci, Nonlocal quantitative isoperimetric inequalities, Calc. Var. Partial Differential Equations, 54 (2015), pp. 2421--2464.
13.
M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species, Nonlinearity, 26 (2013), pp. 2777--2808.
14.
C. Escudero, F. Maciá, and J.J.L. Velázquez, Two-species-coagulation approach to consensus by group level interactions, Phys. Rev. E, 82 (2010), 016113.
15.
A. Figalli, N. Fusco, F. Maggi, V. Millot, and M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies, Comm. Math. Phys., 336 (2015), pp. 441--507.
16.
E. H. Lieb and M. Loss, Analysis, Grad. Stud. Math. 14, AMS, Providence, RI, 2001.
17.
V. Julin, Isoperimetric problem with a Coulomb repulsive term, Indiana Univ. Math. J., 63 (2014), pp. 77--89.
18.
P. Kevrevidis, A. G. Stefanov, and H. Xu, Traveling waves for the mass in mass model of granular chains, Lett. Math. Phys., 106 (2016), pp. 1067--1088.
19.
H. Knüpfer and C. Muratov, On an isoperimetric problem with a competing non-local term. I. The planar case, Commun. Pure Appl. Math., 66 (2013), pp. 1129--1162.
20.
H. Knüpfer and C. Muratov, On an isoperimetric problem with a competing non-local term. II. The general case, Commun. Pure Appl. Math., 67 (2014), pp. 1974--1994.
21.
T. Kolokolnikov, Y. Huang, and M. Pavlovski, Singular patterns for an aggregation model with a confining potential, Phys. D, 260 (2013), pp. 65--76.
22.
T. Kolokolnikov, H. Sun, D. Uminsky, and A. L. Bertozzi, A theory of complex patterns arising from 2D particle interactions, Phys. Rev. E, 84 (2011), 015203
23.
N. S. Landkof, Foundations of Modern Potential Theory, Grundlehren Math. Wisse. 180, Springer-Verlag, Heidelberg, 1972.
24.
H. Levine, E. Ben-Jacob, I. Cohen, and W.-J. Rappel, Swarming patterns in Microorganisms: Some new modeling results, Proceedings of IEEE CDC, 2006, pp. 5073--5077.
25.
T. Liu, M. L. K. Langston, D. Li, J. M. Pigga, C. Pichon, A M. Todea, and A. Müller, Self-recognition among different polyprotic macroions during assembly processes in dilute solution, Science, 331 (2011), pp. 1590--1592.
26.
J. Lu and F. Otto, Nonexistence of minimizers for Thomas-Fermi-Dirac-von Weizsäcker model, Commun. Pure Appl. Math., 67 (2014), pp. 1605--1617.
27.
A. Mackey, T. Kolokolnikov, and A. L. Bertozzi, Two-species particle aggregation and stability of co-dimension one solutions, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), pp. 1411--1436.
28.
A. Magni and M. Novaga, A note on non lower semicontinuous perimeter functionals on partitions, Netw. Heterog. Media, 11 (2016), pp. 501--508.
29.
B. Maury, A. Roudneff-Chupin, F. Santambrogio, and J. Venel, Handling congestion in crowd motion models, Netw. Heterog. Media, 6 (2011), pp. 485--519.
30.
X. Ren and J. Wei, A double bubble assembly as a new phase of a ternary inhibitory system, Arch. Ration. Mech. Anal., 215 (2015), pp. 967--1034.
31.
F. Riesz, Sur une inégalité intégrale, J. Lond. Math. Soc., 5 (1930), pp. 162--168.
32.
R. Simione, D. Slepcev, and I. Topaloglu, Existence of ground states of nonlocal-interaction energies, J. Stat. Phys., 159 (2015), pp. 972--986.
33.
J. D. van der Waals and I. Verhandelingen, Kon. Akad. Wet. Amsterdam 20, 1880; II Théorie moléculaire d'une substance composée de deux matiéres différentes, Arch. Néerl., 24 (1891).
Information & Authors
Information
Published In
SIAM Journal on Mathematical Analysis
Pages: 3412 - 3443
DOI: 10.1137/15M1033976
ISSN (online): 1095-7154
Copyright
© 2016, Society for Industrial and Applied Mathematics.
History
Submitted: 5 August 2015
Accepted: 1 July 2016
Published online: 22 September 2016
Keywords
MSC codes
Authors
Funding Information
SFB Transregio: 109
John von Neumann Professorship of Tum
University of Pisa: PRA-2015-0017
DAAD
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- An exterior optimal transport problemCalculus of Variations and Partial Differential Equations, Vol. 64, No. 2 | 6 January 2025
- Ground states for a nonlocal isoperimetric problem with two interacting phasesApplicable Analysis, Vol. 103, No. 8 | 2 September 2023
- Segregation and domain formation in non-local multi-species aggregation equationsPhysica D: Nonlinear Phenomena, Vol. 456 | 1 Dec 2023
- On the Ternary Ohta–Kawasaki Free Energy and Its One-dimensional Global MinimizersJournal of Nonlinear Science, Vol. 32, No. 5 | 28 June 2022
- Existence of Ground States Solutions for a Nonlocal Isoperimetric Problem with Two PhasesAdvances in Applied Mathematics, Vol. 11, No. 05 | 1 Jan 2022
- Attractive Riesz potentials acting on hard spheresNonlinearity, Vol. 34, No. 1 | 20 January 2021
- Multiple patterns formation for an aggregation/diffusion predator-prey systemNetworks & Heterogeneous Media, Vol. 16, No. 3 | 1 Jan 2021
- Sorting Phenomena in a Mathematical Model For Two Mutually Attracting/Repelling SpeciesSIAM Journal on Mathematical Analysis, Vol. 50, No. 3 | 19 June 2018
- The isoperimetric problem for nonlocal perimetersDiscrete & Continuous Dynamical Systems - S, Vol. 11, No. 3 | 1 Jan 2018
- On a cross-diffusion model for multiple species with nonlocal interaction and size exclusionNonlinear Analysis, Vol. 159 | 1 Aug 2017
- Equilibria for an Aggregation Model with Two SpeciesSIAM Journal on Applied Dynamical Systems, Vol. 16, No. 4 | 21 December 2017
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].