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54
Globalintime weak measure solutions, finitetime aggregation and confinement for nolocal interaction equations
, 2009
"... In this paper, we provide a wellposedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of ..."
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Cited by 68 (19 self)
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In this paper, we provide a wellposedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. We develop an existence theory that enables one to go beyond the blowup time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, we show the finite time total collapse of the solution onto a single point, for compactly supported initial measures. Finally, we give conditions on compensation between the attraction at large distances and local repulsion of the potentials to have globalintime confined systems for compactly supported initial data. Our approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations.
Functional inequalities, thick tails and asymptotics for the critical mass PatlakKellerSegel model
, 2011
"... We investigate the long time behavior of the critical mass PatlakKellerSegel equation. This equation has a one parameter family of steadystate solutions λ, λ> 0, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attrac ..."
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Cited by 51 (12 self)
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We investigate the long time behavior of the critical mass PatlakKellerSegel equation. This equation has a one parameter family of steadystate solutions λ, λ> 0, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attraction for them using an entropy functional Hλ coming from the critical fast diffusion equation in R 2. We construct solutions of PatlakKellerSegel equation satisfying an entropyentropy dissipation inequality for Hλ. While the entropy dissipation for Hλ is strictly positive, it turns out to be a difference of two terms, neither of which need to be small when the dissipation is small. We introduce a strategy of controlled concentration to deal with this issue, and then use the regularity obtained from the entropyentropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards λ. In the present paper, we do not provide any estimate of the rate of convergence, but we discuss how this would result from a stability result for a certain sharp GagliardoNirenbergSobolev inequality.
Stable stationary states of nonlocal interaction equations
 Math. Models Methods Appl. Sci
"... analysis, numerical simulation In this article, we are interested in the largetime behaviour of a solution to a nonlocal interaction equation, where a density of particles/individuals evolves subject to an interaction potential and an external potential. It is known that for regular interaction p ..."
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Cited by 42 (4 self)
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analysis, numerical simulation In this article, we are interested in the largetime behaviour of a solution to a nonlocal interaction equation, where a density of particles/individuals evolves subject to an interaction potential and an external potential. It is known that for regular interaction potentials, stable stationary states of this equations are generically finite sums of Dirac masses. For a finite sum of Dirac masses, we give i) a condition to be a stationary state, ii) two necessary conditions of linear stability w.r.t. shifts and reallocations of individual Dirac masses, and iii) show that these linear stability conditions implies local nonlinear stability. Finally, we show that for regular repulsive interaction potential Wε converging to a singular repulsive interaction potentialW, the Diractype stationary states ρ̄ε approximate weakly a unique stationary state ρ ̄ ∈ L∞. We illustrate our results with numerical examples. 1
A macroscopic crowd motion model of gradient flow type
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2010
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Local and Global WellPosedness for Aggregation Equations and PatlakKellerSegel Models with Degenerate Diffusion
, 2010
"... Recently, there has been a wide interest in the study of aggregation equations and PatlakKellerSegel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the wellposedness theory of these models. We prove local wellposedness on b ..."
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Cited by 34 (9 self)
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Recently, there has been a wide interest in the study of aggregation equations and PatlakKellerSegel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the wellposedness theory of these models. We prove local wellposedness on bounded domains for dimensions d ≥ 2 and in all of space for d ≥ 3, the uniqueness being a result previously not known for PKS with degenerate diffusion. We generalize the notion of criticality for PKS and show that subcritical problems are globally wellposed. For a fairly general class of problems, we prove the existence of a critical mass which sharply divides the possibility of finite time blow up and global existence. Moreover, we compute the critical mass for fully general problems and show that solutions with smaller mass exists globally. For a class of supercritical problems we prove finite time blow up is possible for initial data of arbitrary mass. 1
Nonlocal interactions by repulsiveattractive potentials: radial ins/stability
, 2011
"... Abstract. We investigate nonlocal interaction equations with repulsiveattractive radial potentials. Such equations describe the evolution of a continuum density of particles in which they repulse each other in the short range and attract each other in the long range. We prove that under some condit ..."
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Cited by 32 (10 self)
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Abstract. We investigate nonlocal interaction equations with repulsiveattractive radial potentials. Such equations describe the evolution of a continuum density of particles in which they repulse each other in the short range and attract each other in the long range. We prove that under some conditions on the potential, radially symmetric solutions converge exponentially fast in some transport distance toward a spherical shell stationary state. Otherwise we prove that it is not possible for a radially symmetric solution to converge weakly toward the spherical shell stationary state. We also investigate under which condition it is possible for a nonradially symmetric solution to converge toward a singular stationary state supported on a general hypersurface. Finally we provide a detailed analysis of the specific case of the repulsiveattractive power law potential as well as numerical results. 1.
Refined asymptotics for the subcritical KellerSegel system and related functional inequalities
, 2010
"... We analyze the rate of convergence towards selfsimilarity for the subcritical KellerSegel system in the radially symmetric twodimensional case and in the corresponding onedimensional case for logarithmic interaction. We measure convergence in Wasserstein distance. The rate of convergence towards ..."
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Cited by 22 (3 self)
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We analyze the rate of convergence towards selfsimilarity for the subcritical KellerSegel system in the radially symmetric twodimensional case and in the corresponding onedimensional case for logarithmic interaction. We measure convergence in Wasserstein distance. The rate of convergence towards selfsimilarity does not degenerate as we approach the critical case. As a byproduct, we obtain a proof of the logarithmic HardyLittlewoodSobolev inequality in the one dimensional and radially symmetric two dimensional case based on optimal transport arguments. In addition we prove that the onedimensional equation is a contraction with respect to Fourier distance in the subcritical case.
Stability of stationary states of nonlocal equations with singular interaction potentials
 Math. Comput. Modelling
, 2011
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Passing to the limit in a wasserstein gradient flow: From diffusion to reaction
 Calc. Var. Partial Differential Equations
"... Abstract. We study a singularlimit problem arising in the modelling of chemical reactions. At finite ε> 0, the system is described by a FokkerPlanck convectiondiffusion equation with a doublewell convection potential. This potential is scaled by 1/ε, and in the limit ε → 0, the solution conce ..."
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Cited by 17 (6 self)
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Abstract. We study a singularlimit problem arising in the modelling of chemical reactions. At finite ε> 0, the system is described by a FokkerPlanck convectiondiffusion equation with a doublewell convection potential. This potential is scaled by 1/ε, and in the limit ε → 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savaré, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805–1825, 2010, using the linear structure of the equation. In this paper we reprove the result by using solely the Wasserstein gradientflow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a secondorder system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradientflow structure, we prove that the sequence of rescaled solutions is precompact in an appropriate topology. We then prove a Gammaconvergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the εproblem converge to a solution of the limiting problem.
A masstransportation approach to a one dimensional fluid mechanics model with nonlocal velocity
 Adv. Math
, 2012
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