On minimizers of interaction functionals with competing attractive and repulsive potentials
Choksi, Rustum ; Fetecau, Razvan C. ; Topaloglu, Ihsan
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 1283-1305 / Harvested from Numdam
Access to full text

We consider a family of interaction functionals consisting of power-law potentials with attractive and repulsive parts and use the concentration compactness principle to establish the existence of global minimizers. We consider various minimization classes, depending on the signs of the repulsive and attractive power exponents of the potential. In the special case of quadratic attraction and Newtonian repulsion we characterize in detail the ground state.

Publié le : 2015-01-01
DOI : https://doi.org/10.1016/j.anihpc.2014.09.004
Classification:  45J45,  92D25,  35A15,  35B36 
@article{AIHPC_2015__32_6_1283_0,
     author = {Choksi, Rustum and Fetecau, Razvan C. and Topaloglu, Ihsan},
     title = {On minimizers of interaction functionals with competing attractive and repulsive potentials},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {32},
     year = {2015},
     pages = {1283-1305},
     doi = {10.1016/j.anihpc.2014.09.004},
     mrnumber = {3425263},
     zbl = {1329.49019},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2015__32_6_1283_0}
}

Choksi, Rustum; Fetecau, Razvan C.; Topaloglu, Ihsan. On minimizers of interaction functionals with competing attractive and repulsive potentials. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) pp. 1283-1305. doi : 10.1016/j.anihpc.2014.09.004. http://gdmltest.u-ga.fr/item/AIHPC_2015__32_6_1283_0/

[1] D. Benedetto, E. Caglioti, M. Pulvirenti, A kinetic equation for granular media, Modél. Math. Anal. Numér. 31 no. 5 (1997), 615 -641 | Numdam | MR 1471181 | Zbl 0888.73006

[2] A. Bertozzi, J. Carrillo, T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity 22 no. 3 (2009), 683 -710 | MR 2480108 | Zbl 1194.35053

[3] A. Bertozzi, T. Laurent, Finite-time blow-up of solutions of an aggregation equation in 𝐑 n , Commun. Math. Phys. 274 no. 3 (2007), 717 -735 | MR 2328909 | Zbl 1132.35392

[4] M. Bodnar, J. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differ. Equ. 222 no. 2 (2006), 341 -380 | MR 2208049 | Zbl 1089.45002

[5] K. Fellner, G. Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci. 20 no. 12 (2010), 2267 -2291 | MR 2755500 | Zbl 1213.35079

[6] R.C. Fetecau, Y. Huang, T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity 24 no. 10 (2011), 2681 -2716 | MR 2834242 | Zbl 1288.92031

[7] D. Holm, V. Putkaradze, Aggregation of finite-size particles with variable mobility, Phys. Rev. Lett. 95 (2005), 226106

[8] D. Holm, V. Putkaradze, Formation of clumps and patches in selfaggregation of finite-size particles, Physica D 220 no. 2 (2006), 183 -196 | MR 2256869 | Zbl 1125.82021

[9] Y. Huang, A. Bertozzi, Self-similar blowup solutions to an aggregation equation in ℝ n , SIAM J. Appl. Math. 70 no. 7 (2010), 2582 -2603 | MR 2678052 | Zbl 1238.35013

[10] A. Leverentz, C. Topaz, A. Bernoff, Asymptotic dynamics of attractive-repulsive swarms, SIAM J. Appl. Dyn. Syst. 8 no. 3 (2009), 880 -908 | MR 2533628 | Zbl 1168.92045

[11] A. Mogilner, L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol. 38 (1999), 534 -570 | MR 1698215 | Zbl 0940.92032

[12] C. Topaz, A. Bertozzi, M. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol. 68 (2006), 1601 -1623 | MR 2257718 | Zbl 1334.92468

[13] G. Toscani, One-dimensional kinetic models of granular flows, Modél. Math. Anal. Numér. 34 no. 6 (2000), 1277 -1291 | Numdam | MR 1812737 | Zbl 0981.76098

[14] L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich , Birkhäuser Verlag, Basel (2008) | MR 2401600 | Zbl 1145.35001

[15] J.A. Carrillo, M. Difrancesco, A. Figalli, T. Laurent, D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J. 156 no. 2 (2011), 229 -271 | MR 2769217 | Zbl 1215.35045

[16] J. Haile, Molecular Dynamics Simulation: Elementary Methods, John Wiley and Sons, Inc., New York (1992)

[17] D. Balagué, J. Carrillo, T. Laurent, G. Raoul, Nonlocal interactions by repulsive-attractive potentials: radial ins/stability, Physica D 260 (2013), 5 -25 | MR 3143991 | Zbl 1286.35038

[18] R.C. Fetecau, Y. Huang, Equilibria of biological aggregations with nonlocal repulsive-attractive interactions, Physica D 260 (2013), 49 -64 | MR 3143993 | Zbl 1286.35017

[19] T. Kolokolnikov, H. Sun, D. Uminsky, A. Bertozzi, A theory of complex patterns arising from 2D particle interactions, Phys. Rev. E, Rapid Commun. 84 (2011), 015203(R)

[20] J. Von Brecht, D. Uminsky, T. Kolokolnikov, A. Bertozzi, Predicting pattern formation in particle interactions, Math. Models Methods Appl. Sci. 22 no. 1 (2012), 1140002 | MR 2974182 | Zbl 1252.35056

[21] D. Balagué, J. Carrillo, T. Laurent, G. Raoul, Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal. 209 (2013), 1055 -1088 | MR 3067832 | Zbl 1311.49053

[22] M. Bodnar, J. Velasquez, Derivation of macroscopic equations for individual cell-based models: a formal approach, Math. Models Methods Appl. Sci. 28 (2005), 1757 -1779 | MR 2166611 | Zbl 1074.60101

[23] J. Carrillo, M. Chipot, Y. Huang, On global minimizers of repulsive-attractive power-law interaction energies, preprint. | MR 3268056

[24] J. Carrillo, M. Delgadino, A. Mellet, Regularity of local minimizers of the interaction energy via obstacle problems, preprint. | MR 3488544

[25] J. Cañizo, J. Carrillo, F. Patacchini, Existence of compactly supported global minimizers for the interaction energy, preprint. | MR 3356997

[26] R. Simione, D. Slepčev, I. Topaloglu, Existence of minimizers of nonlocal interaction energies, preprint.

[27] J. Bedrossian, Intermediate asymptotics for critical and supercritical aggregation equations and Patlak–Keller–Segel models, Commun. Math. Sci. 9 no. 4 (2011), 1143 -1161 | MR 2901821 | Zbl 1282.35053

[28] J. Bedrossian, N. Rodríguez, A. Bertozzi, Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion, Nonlinearity 24 (2011), 1683 -1714 | MR 2793895 | Zbl 1222.49038

[29] A. Bertozzi, D. Slepčev, Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion, Commun. Pure Appl. Anal. 9 no. 6 (2010), 1617 -1637 | MR 2684052 | Zbl 1213.35266

[30] A. Blanchet, J. Carrillo, P. Laurençot, Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differ. Equ. 35 no. 2 (2009), 133 -168 | MR 2481820 | Zbl 1172.35035

[31] M. Burger, V. Capasso, D. Morale, On an aggregation model with long and short range interactions, Nonlinear Anal., Real World Appl. 8 no. 3 (2007), 939 -958 | MR 2307761 | Zbl 1188.92040

[32] M. Burger, M. Difrancesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media 3–4 (2008), 749 -785 | MR 2448940 | Zbl 1171.35328

[33] M. Burger, M. Difrancesco, M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Commun. Math. Sci. 3 (2013), 709 -738 | MR 3061139 | Zbl 1282.35203

[34] M. Burger, R.C. Fetecau, Y. Huang, Stationary states and asymptotic behaviour of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst. 13 (2014), 397 -424 | MR 3176313 | Zbl 1302.35386

[35] J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, Appl. Math. Lett. 24 no. 11 (2011), 1927 -1932 | MR 2812240 | Zbl 1236.49090

[36] J. Auchmuty, Existence of axisymmetric equilibrium figures, Arch. Ration. Mech. Anal. 65 no. 3 (1977), 249 -261 | MR 446076 | Zbl 0366.76083

[37] J. Auchmuty, R. Beals, Variational solutions of some nonlinear free boundary problems, Arch. Ration. Mech. Anal. 43 (1971), 255 -271 | MR 337260 | Zbl 0225.49013

[38] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1 no. 2 (1984), 109 -145 | Numdam | MR 778970 | Zbl 0541.49009

[39] C. Le Bris, Some results on the Thomas–Fermi–Dirac–von Weizsäcker model, Differ. Integral Equ. 6 (1993), 337 -353 | MR 1195387 | Zbl 0828.49024

[40] C. Le Bris, P.-L. Lions, From atoms to crystals: a mathematical journey, Bull. Am. Math. Soc. 42 no. 3 (2005), 291 -363 | MR 2149087 | Zbl 1136.81371

[41] E. Lieb, Thomas–Fermi and related theories of atoms and molecules, Rev. Mod. Phys. 52 (1981), 603 -641 | MR 629207 | Zbl 1049.81679

[42] J. Lu, F. Otto, Nonexistence of a minimizer for the Thomas–Fermi–Dirac–von Weizsäcker model, Commun. Pure Appl. Math. 67 no. 10 (2014), 1605 -1617 | MR 3251907 | Zbl 1301.49002

[43] R. Choksi, M. Peletier, Small volume fraction limit of the diblock copolymer problem: I. Sharp interface functional, SIAM J. Math. Anal. 42 no. 3 (2010), 1334 -1370 | MR 2653253 | Zbl 1210.49050

[44] R. Choksi, M. Peletier, Small volume fraction limit of the diblock copolymer problem: II. Diffuse interface functional, SIAM J. Math. Anal. 43 no. 2 (2011), 739 -763 | MR 2784874 | Zbl 1223.49056

[45] R. Choksi, P. Sternberg, On the first and second variations of a nonlocal isoperimetric problem, J. Reine Angew. Math. 611 (2007), 75 -108 | MR 2360604 | Zbl 1132.35029

[46] V. Julin, Isoperimetric problem with a coulombic repulsive term, preprint. | MR 3218265

[47] H. Knuepfer, C. Muratov, On an isoperimetric problem with a competing non-local term. I. The planar case, Commun. Pure Appl. Math. 66 no. 7 (2013), 1129 -1162 | MR 3055587 | Zbl 1269.49087

[48] H. Knuepfer, C. Muratov, On an isoperimetric problem with a competing non-local term. II. The general case, Commun. Pure Appl. Math. (2014), http://dx.doi.org/10.1002/cpa.21479

[49] H. Poincaré, Sur une théorème de M. Liapunoff rélatif a l'équilibre d'une masse fluide, C. R. Acad. Sci. 104 (1887), 622 -625

[50] E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math. 57 (1977), 93 -105 | MR 471785 | Zbl 0369.35022

[51] E. Lieb, M. Loss, Analysis.

[52] A. Bertozzi, J. Von Brecht, H. Sun, T. Kolokolnikov, D. Uminsky, Ring patterns and their bifurcations in a nonlocal model of biological swarms, Commun. Math. Sci. (2014), http://www.math.ucla.edu/~bertozzi/papers/bigring-final.pdf | Zbl 1331.35044

[53] K. Craig, A. Bertozzi, A blob method for the aggregation equation, preprint. | MR 3471104

[54] J.A. Carrillo, M. Difrancesco, A. Figalli, T. Laurent, D. Slepčev, Confinement in nonlocal interaction equations, Nonlinear Anal. 75 no. 2 (2012), 550 -558 | MR 2847439 | Zbl 1233.35032

[55] A. Bertozzi, T. Laurent, L. Flavien, Aggregation and spreading via the Newtonian potential: the dynamics of patch solutions, Math. Models Methods Appl. Sci. 22 no. 1 (2012), 1140005 | MR 2974185 | Zbl 1241.35153

[56] K. Fellner, G. Raoul, Stability of stationary states of non-local equations with singular interaction potentials, Math. Comput. Model. 53 no. 7–8 (2011), 1436 -1450 | MR 2782822 | Zbl 1219.35342

[57] E. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. Math. (2) 118 no. 2 (1983), 349 -374 | MR 717827 | Zbl 0527.42011

[58] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin (2000) | MR 1736116 | Zbl 0939.49001

[59] P. Billingsley, Weak Convergence of Measures: Applications in Probability, Society for Industrial and Applied Mathematics, Philadelphia, PA (1971) | MR 310933 | Zbl 0271.60009

[60] A. Bernoff, C. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst. 10 no. 1 (2011), 212 -250 | MR 2788924 | Zbl 1255.35012