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72
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
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.
Dimensionality of local minimizers of the interaction energy
 Arch. Rational Mech. Anal
"... Abstract. In this work we consider local minimizers (in the topology of transport distances) of the interaction energy associated to a repulsiveattractive potential. We show how the dimensionality of the support of local minimizers is related to the repulsive strength of the potential at the origin ..."
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Cited by 29 (9 self)
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Abstract. In this work we consider local minimizers (in the topology of transport distances) of the interaction energy associated to a repulsiveattractive potential. We show how the dimensionality of the support of local minimizers is related to the repulsive strength of the potential at the origin. 1.
EXISTENCE AND UNIQUENESS OF SOLUTIONS TO AN AGGREGATION EQUATION WITH DEGENERATE DIFFUSION
"... Abstract. We present an energymethodsbased proof of the existence and uniqueness of solutions of a nonlocal aggregation equation with degenerate diffusion. The equation we study is relevant to models of biological aggregation. ..."
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Cited by 19 (3 self)
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Abstract. We present an energymethodsbased proof of the existence and uniqueness of solutions of a nonlocal aggregation equation with degenerate diffusion. The equation we study is relevant to models of biological aggregation.
AGGREGATION AND SPREADING VIA THE NEWTONIAN POTENTIAL: THE DYNAMICS OF PATCH SOLUTIONS
, 2011
"... This paper considers the multidimensional active scalar problem of motion of a function ρ(x, t) by a velocity field obtained by v = −∇N ∗ρ, where N is the Newtonian potential. We prove ..."
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Cited by 19 (2 self)
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This paper considers the multidimensional active scalar problem of motion of a function ρ(x, t) by a velocity field obtained by v = −∇N ∗ρ, where N is the Newtonian potential. We prove
Stability of stationary states of nonlocal equations with singular interaction potentials
 Math. Comput. Modelling
, 2011
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Characterization of radially symmetric finite time blowup in multidimensional aggregation equations
, 2011
"... This paper studies the transport of a mass µ in R d, d ≥ 2, by a flow field v = −∇K ∗µ. We focus on kernels K = x  α /α for 2 − d ≤ α < 2. For this range we prove the existence for all time of radially symmetric measure solutions that are monotone decreasing as a function of the radius. The mon ..."
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Cited by 17 (6 self)
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This paper studies the transport of a mass µ in R d, d ≥ 2, by a flow field v = −∇K ∗µ. We focus on kernels K = x  α /α for 2 − d ≤ α < 2. For this range we prove the existence for all time of radially symmetric measure solutions that are monotone decreasing as a function of the radius. The monotonicity is preserved for all time, in contrast to the case α> 2 where radially symmetric solutions are known to lose monotonicity. In the case of the Newtonian potential (α = 2 − d) we show that under the assumption of radial symmetry the equation can be transformed into the inviscid Burgers equation on a half line. It follows that there exists a unique classical solution for all time in the case of monotone data, and a solution defined by a choice of a jump condition in the case of general radially symmetric data. In the case 2 − d < α < 2 and at the critical exponent p we exhibit initial data in L p for which the solution immediately develops a Dirac mass singularity. This extends recent work on the local illposedness of solutions at the critical exponent.
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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