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26
Globalintime weak measure solutions and finitetime aggregation for nonlocal interaction equations
"... Abstract. 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 pheno ..."
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Cited by 21 (9 self)
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Abstract. 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. 1.
Asymptotic Flocking Dynamics for the kinetic CuckerSmale model
, 2009
"... Abstract. In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting ..."
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Cited by 19 (4 self)
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Abstract. In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting from a Boltzmanntype equation. The largetime behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [16] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution.
SELFSIMILAR BLOWUP SOLUTIONS TO AN AGGREGATION EQUATION
"... Abstract. We present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut = ∇ · (u∇K ∗ u), where the kernel K(x) = x. The dynamics of the blowup exhibits selfsimilar behavior in which zero mass concentrates at the core at the blowup time. Computations ..."
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Cited by 14 (8 self)
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Abstract. We present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut = ∇ · (u∇K ∗ u), where the kernel K(x) = x. The dynamics of the blowup exhibits selfsimilar behavior in which zero mass concentrates at the core at the blowup time. Computations are performed in R n for n between 2 and 10 using a method based on characteristics. In all cases studied, the selfsimilarity exhibits second kind (anomalous) scaling. Key words. aggregation equation, selfsimilarity solution of the second kind, blowup AMS subject classifications. 74H15, 74H35, 76M55, 82B24
The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels
 Chinese Annals of Mathematics, Series B
, 2009
"... with mildly singular interaction kernels ..."
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 8 (4 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.
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 monoto ..."
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Cited by 7 (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.
Asymptotics of blowup solutions for the aggregation equation
, 2011
"... We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = ∇ · (u∇K ∗ u) in R n, for homogeneous potentials K = x  γ, γ> 0. For γ> 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δring. We develop an a ..."
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Cited by 5 (2 self)
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We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = ∇ · (u∇K ∗ u) in R n, for homogeneous potentials K = x  γ, γ> 0. For γ> 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δring. We develop an asymptotic theory for the approach to this singular solution. For γ < 2, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all γ in this range, including additional asymptotic behavior in the limits γ → 0 + and γ → 2 −.
A generalized BirkhoffRott Equation for 2D Active Scalar Problems
, 2011
"... In this paper we derive evolution equations for the 2D active scalar problem when the solution is supported on 1D curve(s). These equations are a generalization of the BirkhoffRott equation when vorticity is the active scalar. The formulation is Lagrangian and they are valid for nonlocal kernels K ..."
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Cited by 5 (4 self)
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In this paper we derive evolution equations for the 2D active scalar problem when the solution is supported on 1D curve(s). These equations are a generalization of the BirkhoffRott equation when vorticity is the active scalar. The formulation is Lagrangian and they are valid for nonlocal kernels K that may include both a gradient and an incompressible term. We develop a numerical method for implementing the model which achieves second order convergence in space and fourth order in time. We verify the model by simulating classic active scalar problems such as the vortex sheet problem (in the case of inviscid, incompressible flow) and the collapse of delta ring solutions (in the case of pure aggregation), finding excellent agreement. We then study two examples with kernels of mixed type i.e., kernels that contain both incompressible and gradient flows. The first example is a vortex density model which arises in superfluids. We analyze the effect of the added gradient component on the KelvinHelmholtz instability. In the second example, we examine a nonlocal biological swarming model and study the dynamics of density rings which exhibit complicated milling behavior.
Stability and clustering of selfsimilar solutions of aggregation equations
, 2012
"... In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρt = ∇ · (ρ∇K ∗ ρ) in Rd, d ≥ 2, where K(r) = r γ /γ with γ> 2. It was previously observed 1 that radially symmetric solutions are attracted to a selfsimilar collapsing ..."
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Cited by 3 (3 self)
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In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρt = ∇ · (ρ∇K ∗ ρ) in Rd, d ≥ 2, where K(r) = r γ /γ with γ> 2. It was previously observed 1 that radially symmetric solutions are attracted to a selfsimilar collapsing shell profile in infinite time for γ> 2. In this paper we compute the stability of the similarity solution and show that the collapsing shell solution is stable for 2 < γ < 4. For γ> 4, we show that the shell solution is always unstable and destabilizes into clusters that form a simplex which we observe to be the long time attractor. We then classify the stability of these simplex solutions and prove that two dimensional (in)stability implies n dimensional (in)stability.
A study of blowups in the KellerSegel model of chemotaxis
"... We study the KellerSegel model of chemotaxis and develop a composite particlegrid numerical method with adaptive time stepping which allows us to accurately resolve singular solutions. The numerical findings (in two dimensions) are then compared with analytical predictions regarding formation and ..."
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Cited by 1 (0 self)
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We study the KellerSegel model of chemotaxis and develop a composite particlegrid numerical method with adaptive time stepping which allows us to accurately resolve singular solutions. The numerical findings (in two dimensions) are then compared with analytical predictions regarding formation and interaction of singularities obtained via analysis of the stochastic differential equations associated with the KellerSegel model. Keywords: KellerSegel, McKeanVlasov, chemotaxis, blowups, aggregation.