April 2019, 12(2): 303-322. doi: 10.3934/krm.2019013

Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion

Department of Information Engineering, Computer Science, and Mathematics, University of L'Aquila, Via Vetoio (Coppito), 67100, L'Aquila, Italy

Received  October 2017 Revised  August 2018 Published  November 2018

In this paper we consider a one-dimensional nonlocal interaction equation with quadratic porous-medium type diffusion in which the interaction kernels are attractive, nonnegative, and integrable on the real line. Earlier results in the literature have shown existence of nontrivial steady states if the $L^1$ norm of the kernel $G$ is larger than the diffusion constant $\varepsilon$. In this paper we aim at showing that this equation exhibits a 'multiple' behavior, in that solutions can either converge to the nontrivial steady states or decay to zero for large times. We prove the former situation holds in case the initial conditions are concentrated enough and 'close' to the steady state in the $∞$-Wasserstein distance. Moreover, we prove that solutions decay to zero for large times in the diffusion-dominated regime $\varepsilon≥ \|G\|_{L^1}$. Finally, we show two partial results suggesting that the large-time decay also holds in the complementary regime $\varepsilon < \|G\|_{L^1}$ for initial data with large enough second moment. We use numerical simulations both to validate our local asymptotic stability result and to support our conjecture on the large time decay.

Citation: Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic & Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005.

[2]

J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, Applied Mathematics Letters, 24 (2011), 1927-1932. doi: 10.1016/j.aml.2011.05.022.

[3]

J. Bedrossian, Large mass global solutions for a class of L1-critical nonlocal aggregation equations and parabolic-elliptic Patlak-Keller-Segel models, arXiv: 1403.4124.

[4]

F. BerthelinD. Chiron and M. Ribot, Stationary solutions with vacuum for a one-dimensional chemotaxis model with non-linear pressure, Commun. Math. Sci., 14 (2015), 147-186. doi: 10.4310/CMS.2016.v14.n1.a6.

[5]

A. L. Bertozzi and J. Brandman, Finite-time blow-up of L infinity weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 45-65. doi: 10.4310/CMS.2010.v8.n1.a4.

[6]

A. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in Rn, Comm. Math. Phys., 274 (2007), 717-735. doi: 10.1007/s00220-007-0288-1.

[7]

A. L. BertozziT. Laurent and J. Rosado, Lp theory for the multidimensional aggregation equation, Commun. Pure Appl. Math., 64 (2011), 45-83. doi: 10.1002/cpa.20334.

[8]

A. BlanchetJ. A. Carrillo and P. Laurencot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Cal. Var. Partial Differential Equations, 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7.

[9]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 2006 (2006), No. 44, 32 pp.

[10]

M. Bodnar and J. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differ. Equations, 222 (2006), 341-380. doi: 10.1016/j.jde.2005.07.025.

[11]

S. BoiV. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species polyergus rufescens, Nonlinear Anal, Real World Appl., 1 (2000), 163-176. doi: 10.1016/S0362-546X(99)00399-5.

[12]

M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Networks and Heterogeneous Media, 3 (2008), 749-785. doi: 10.3934/nhm.2008.3.749.

[13]

M. BurgerM. Di Francesco and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Commun. Math. Sci., 11 (2012), 709-738. doi: 10.4310/CMS.2013.v11.n3.a3.

[14]

M. BurgerR. C. Fetecau and Y. Huang, Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst., 13 (2014), 397-424. doi: 10.1137/130923786.

[15]

V. CalvezJ. A. Carrillo and F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Analysis TMA, 159 (2017), 85-128. doi: 10.1016/j.na.2017.03.008.

[16]

J. A. CarrilloD. Castorina and B. Volzone, Ground states for diffusion dominated free energies with logarithmic interaction, SIAM J. Math. Anal., 47 (2015), 1-25. doi: 10.1137/140951588.

[17]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233-258. doi: 10.4208/cicp.160214.010814a.

[18]

J. A. CarrilloM. Gualdani and G. Toscani, Finite speed of propagation in porous media by mass transportation methods, Comptes Rendus Mathematique, 338 (2004), 815-818. doi: 10.1016/j.crma.2004.03.025.

[19]

J. A. CarrilloR. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19 (2003), 971-1018. doi: 10.4171/RMI/376.

[20]

J. A. Carrillo and G. Toscani, Wasserstein metric and large-time asymptotics of nonlinear diffusion equations, New Trends in Mathematical Physics, (2004), 234-244.

[21]

R. ChoksiR. C. Fetecau and I. Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 32 (2015), 1283-1305. doi: 10.1016/j.anihpc.2014.09.004.

[22]

K. Fellner and G. Raoul, Stability of stationary states of non-local interaction equations, Mathematical and Computer Modelling, 53 (2011), 1436-1450. doi: 10.1016/j.mcm.2010.03.021.

[23]

L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for onedimensional filtration equations, SIAM Journal on Numerical Analysis, 43 (2006), 2590-2606. doi: 10.1137/040608672.

[24]

S. GottliebC. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43 (2001), 89-112. doi: 10.1137/S003614450036757X.

[25]

D. HelbingI. J. FarkasP. Molnar and T. Vicsek, Simulation of pedestrian crowds in normal and evacuation situations, Pedestrian and Evacuation Dynamics, 21 (2002), 21-58.

[26]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.

[27]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[28]

G. Kaib, Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential, SIAM J. Math. Anal., 49 (2017), 272-296. doi: 10.1137/16M1072450.

[29]

I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: Properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2012), 568-602. doi: 10.1137/110823584.

[30]

H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Archive for rational mechanics and analysis, 172 (2004), 407-428. doi: 10.1007/s00205-004-0307-8.

[31]

E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174. doi: 10.1007/BF01217684.

[32]

P. L. Lions, The concentration-compactness principle in calculus of variations. The locally compact case, part 1, Ann. Inst. Henri Poincar e, Anal. Nonlin., 1 (1984), 109-145. doi: 10.1016/S0294-1449(16)30428-0.

[33]

R. McCann, A convexity principle for interacting gases, Advances in mathematics, 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.

[34]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570. doi: 10.1007/s002850050158.

[35]

D. MoraleV. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: from individuals to populations, Journal of Mathematical Biology, 50 (2005), 49-66. doi: 10.1007/s00285-004-0279-1.

[36]

G. Russo, Deterministic diffusion of particles, Communications on Pure and Applied Mathematics, 43 (1990), 697-733. doi: 10.1002/cpa.3160430602.

[37]

L. ScardiaR. PeerlingsM. Geers and M. A. Peletier, Mechanics of dislocation pile-ups: A unification of scaling regimes, Journal of the Mechanics and Physics of Solids, 70 (2014), 42-61. doi: 10.1016/j.jmps.2014.04.014.

[38]

K. Sznajd-Weron and J. Sznajd. Opinion evolution in closed community, Opinion evolution in closed community, Int. J. Mod. Phys. C, 11 (2000), 1157-1166.

[39]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bulletin of Mathematical Biology, 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6.

[40]

G. Toscani, Kinetic and hydrodynamic models of nearly elastic granular flows, Monatsh. Math., 142 (2004), 179-192. doi: 10.1007/s00605-004-0241-8.

[41]

J. L. Vazquez, The Porous Medium Equation, Oxford Mathematical Monographs, 2007.

[42]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, AMS, Providence, RI, 2003. doi: 10.1007/b12016.

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005.

[2]

J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, Applied Mathematics Letters, 24 (2011), 1927-1932. doi: 10.1016/j.aml.2011.05.022.

[3]

J. Bedrossian, Large mass global solutions for a class of L1-critical nonlocal aggregation equations and parabolic-elliptic Patlak-Keller-Segel models, arXiv: 1403.4124.

[4]

F. BerthelinD. Chiron and M. Ribot, Stationary solutions with vacuum for a one-dimensional chemotaxis model with non-linear pressure, Commun. Math. Sci., 14 (2015), 147-186. doi: 10.4310/CMS.2016.v14.n1.a6.

[5]

A. L. Bertozzi and J. Brandman, Finite-time blow-up of L infinity weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 45-65. doi: 10.4310/CMS.2010.v8.n1.a4.

[6]

A. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in Rn, Comm. Math. Phys., 274 (2007), 717-735. doi: 10.1007/s00220-007-0288-1.

[7]

A. L. BertozziT. Laurent and J. Rosado, Lp theory for the multidimensional aggregation equation, Commun. Pure Appl. Math., 64 (2011), 45-83. doi: 10.1002/cpa.20334.

[8]

A. BlanchetJ. A. Carrillo and P. Laurencot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Cal. Var. Partial Differential Equations, 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7.

[9]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 2006 (2006), No. 44, 32 pp.

[10]

M. Bodnar and J. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differ. Equations, 222 (2006), 341-380. doi: 10.1016/j.jde.2005.07.025.

[11]

S. BoiV. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species polyergus rufescens, Nonlinear Anal, Real World Appl., 1 (2000), 163-176. doi: 10.1016/S0362-546X(99)00399-5.

[12]

M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Networks and Heterogeneous Media, 3 (2008), 749-785. doi: 10.3934/nhm.2008.3.749.

[13]

M. BurgerM. Di Francesco and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Commun. Math. Sci., 11 (2012), 709-738. doi: 10.4310/CMS.2013.v11.n3.a3.

[14]

M. BurgerR. C. Fetecau and Y. Huang, Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst., 13 (2014), 397-424. doi: 10.1137/130923786.

[15]

V. CalvezJ. A. Carrillo and F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Analysis TMA, 159 (2017), 85-128. doi: 10.1016/j.na.2017.03.008.

[16]

J. A. CarrilloD. Castorina and B. Volzone, Ground states for diffusion dominated free energies with logarithmic interaction, SIAM J. Math. Anal., 47 (2015), 1-25. doi: 10.1137/140951588.

[17]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233-258. doi: 10.4208/cicp.160214.010814a.

[18]

J. A. CarrilloM. Gualdani and G. Toscani, Finite speed of propagation in porous media by mass transportation methods, Comptes Rendus Mathematique, 338 (2004), 815-818. doi: 10.1016/j.crma.2004.03.025.

[19]

J. A. CarrilloR. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19 (2003), 971-1018. doi: 10.4171/RMI/376.

[20]

J. A. Carrillo and G. Toscani, Wasserstein metric and large-time asymptotics of nonlinear diffusion equations, New Trends in Mathematical Physics, (2004), 234-244.

[21]

R. ChoksiR. C. Fetecau and I. Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 32 (2015), 1283-1305. doi: 10.1016/j.anihpc.2014.09.004.

[22]

K. Fellner and G. Raoul, Stability of stationary states of non-local interaction equations, Mathematical and Computer Modelling, 53 (2011), 1436-1450. doi: 10.1016/j.mcm.2010.03.021.

[23]

L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for onedimensional filtration equations, SIAM Journal on Numerical Analysis, 43 (2006), 2590-2606. doi: 10.1137/040608672.

[24]

S. GottliebC. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43 (2001), 89-112. doi: 10.1137/S003614450036757X.

[25]

D. HelbingI. J. FarkasP. Molnar and T. Vicsek, Simulation of pedestrian crowds in normal and evacuation situations, Pedestrian and Evacuation Dynamics, 21 (2002), 21-58.

[26]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.

[27]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[28]

G. Kaib, Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential, SIAM J. Math. Anal., 49 (2017), 272-296. doi: 10.1137/16M1072450.

[29]

I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: Properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2012), 568-602. doi: 10.1137/110823584.

[30]

H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Archive for rational mechanics and analysis, 172 (2004), 407-428. doi: 10.1007/s00205-004-0307-8.

[31]

E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174. doi: 10.1007/BF01217684.

[32]

P. L. Lions, The concentration-compactness principle in calculus of variations. The locally compact case, part 1, Ann. Inst. Henri Poincar e, Anal. Nonlin., 1 (1984), 109-145. doi: 10.1016/S0294-1449(16)30428-0.

[33]

R. McCann, A convexity principle for interacting gases, Advances in mathematics, 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.

[34]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570. doi: 10.1007/s002850050158.

[35]

D. MoraleV. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: from individuals to populations, Journal of Mathematical Biology, 50 (2005), 49-66. doi: 10.1007/s00285-004-0279-1.

[36]

G. Russo, Deterministic diffusion of particles, Communications on Pure and Applied Mathematics, 43 (1990), 697-733. doi: 10.1002/cpa.3160430602.

[37]

L. ScardiaR. PeerlingsM. Geers and M. A. Peletier, Mechanics of dislocation pile-ups: A unification of scaling regimes, Journal of the Mechanics and Physics of Solids, 70 (2014), 42-61. doi: 10.1016/j.jmps.2014.04.014.

[38]

K. Sznajd-Weron and J. Sznajd. Opinion evolution in closed community, Opinion evolution in closed community, Int. J. Mod. Phys. C, 11 (2000), 1157-1166.

[39]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bulletin of Mathematical Biology, 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6.

[40]

G. Toscani, Kinetic and hydrodynamic models of nearly elastic granular flows, Monatsh. Math., 142 (2004), 179-192. doi: 10.1007/s00605-004-0241-8.

[41]

J. L. Vazquez, The Porous Medium Equation, Oxford Mathematical Monographs, 2007.

[42]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, AMS, Providence, RI, 2003. doi: 10.1007/b12016.

Figure 1.  Two different asymptotic behaviors of the solution depending on the initial datum
Figure 4.  Growing of the second moment of the solution to 3 in time in the regime $1/3<\varepsilon<\|G\|_{L^1}$. $\varepsilon = 0.5, $ $G(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}$, and initial datum 40 with $\delta = 0.05.$
Figure 2.  Convergence of the solution to the steady state in the aggregation-dominated regime $0<\varepsilon<\|G\|_{L^1}$. $\varepsilon = 0.002, $ $G(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}.$ (a) $\rho_{0}(x) = \frac{93}{8}(1-\frac{961}{4}x^2), $ (b) $\rho_{0}(x) = \frac{21}{8}(1-\frac{49}{4}x^2).$
Figure 3.  Decay of the solution to zero in the diffusion-dominated regime $\varepsilon\geq\|G\|_{L^{1}}$. $\varepsilon = 2, $ $G(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}, $ $\rho_{0}(x) = \frac{21}{8}(1-\frac{49}{4}x^2)$.
Figure 5.  Convergence of the solution to the steady state in the aggregation-dominated regime $0<\varepsilon<\|G\|_{L^{1}}$. $\varepsilon = 0.5, $ $G(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}.$ (a) $\rho_{0}(x) = \frac{9}{8}(1-\frac{9}{4}x^2), $ (b) $\rho_{0}(x) = \frac{105}{400}(1-\frac{1225}{10000}x^2).$
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