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51
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
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.
The PatlakKellerSegel model and its variations: properties of solutions via maximum principle. arXiv:1102.0092
, 2011
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ON THE PARABOLICELLIPTIC PATLAKKELLERSEGEL SYSTEM IN DIMENSION 2 AND HIGHER
"... Abstract. This review is dedicated to recent results on the 2d parabolicelliptic PatlakKellerSegel model, and on its variant in higher dimensions where the diffusion is of critical porous medium type. Both of these models have a critical mass Mc such that the solutions exist globally in time if t ..."
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Cited by 10 (2 self)
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Abstract. This review is dedicated to recent results on the 2d parabolicelliptic PatlakKellerSegel model, and on its variant in higher dimensions where the diffusion is of critical porous medium type. Both of these models have a critical mass Mc such that the solutions exist globally in time if the mass is less than Mc and above which there are solutions which blowup in finite time. The main tools, in particular the free energy, and the idea of the methods are set out. A number of open questions are also stated. hal00620500, version 1 7 Sep 2011 1. Biological background Chemotaxis is defined as a move of an organism along a chemical concentration gradient. Bacteria can produce this chemoattractant themselves, creating thus a longrange nonlocal interaction between them. We are interested in a very simplified model of aggregation at the scale of cells by chemotaxis: myxamoebaes or bacterias experience a random walk to spread in the space and find food. But in starvation conditions, the dictyostelium discoideum emit a chemical signal: cyclic adenosine monophosphate (cAMP). They move towards a higher concentration of cAMP. Their behaviour is thus the result of a competition between
The parabolicparabolic KellerSegel system with critical diffusion as a gradient flow
 in R d , d ≥ 3, Comm. Partial Differential Equations
"... Abstract. It is known that, for the parabolicelliptic KellerSegel system with critical porousmedium diffusion in dimension Rd, d ≥ 3 (also referred to as the quasilinear SmoluchowskiPoisson equation), there is a critical value of the chemotactic sensitivity (measuring in some sense the strength ..."
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Cited by 9 (1 self)
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Abstract. It is known that, for the parabolicelliptic KellerSegel system with critical porousmedium diffusion in dimension Rd, d ≥ 3 (also referred to as the quasilinear SmoluchowskiPoisson equation), there is a critical value of the chemotactic sensitivity (measuring in some sense the strength of the drift term) above which there are solutions blowing up in finite time and below which all solutions are global in time. This global existence result is shown to remain true for the parabolicparabolic KellerSegel system with critical porousmedium type diffusion in dimension Rd, d ≥ 3, when the chemotactic sensitivity is below the same critical value. The solution is constructed by using a minimising scheme involving the KantorovichWasserstein metric for the first component and the L2norm for the second component. The cornerstone of the proof is the derivation of additional estimates which relies on a generalisation to a nonmonotone functional of a method due to Matthes, McCann, & Savare ́ (2009). 1.
Uniqueness for KellerSegeltype chemotaxis models
 Discrete Contin. Dyn. Syst
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Global existence and finite time blowup for critical PatlakKellerSegel models with inhomogeneous diffusion
, 2011
"... The L1critical parabolicelliptic PatlakKellerSegel system is a classical model of chemotactic aggregation in microorganisms wellknown to have critical mass phenomena [10, 8]. In this paper we study this critical mass phenomenon in the context of PatlakKellerSegel models with spatially varyin ..."
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Cited by 6 (1 self)
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The L1critical parabolicelliptic PatlakKellerSegel system is a classical model of chemotactic aggregation in microorganisms wellknown to have critical mass phenomena [10, 8]. In this paper we study this critical mass phenomenon in the context of PatlakKellerSegel models with spatially varying diffusivity and decay rate of the chemoattractant. The primary tool for the proof of global existence below the critical mass is the use of pseudodifferential operators to precisely evaluate the leading order quadratic portion of the potential energy (interaction energy). Under the assumption of radial symmetry, blowup is proved above critical mass using a maximumprinciple type argument based on comparing the mass distribution of solutions to a barrier consisting of the unique stationary solutions of the scaleinvariant PKS. Although effective where standard Virial methods do not apply, this method seems to be dependent on the assumption of radial symmetry. For technical reasons we work in dimensions three and higher where L1critical variants of the PKS have porous mediatype nonlinear diffusion on the organism density, resulting in finite speed of propagation and simplified functional inequalities. 1
Existence, uniqueness and Lipschitz dependence for PatlakKellerSegel and NavierStokes in R2 with measurevalued initial data. arXiv:1205.1551
, 2012
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Sparse Stabilization and Control of the CuckerSmale Model
, 2013
"... From a mathematical point of view selforganization can be described as patterns to which certain dynamical systems modeling social dynamics tend spontaneously to be attracted. In this paper we explore situations beyond selforganization, in particular how to externally control such dynamical system ..."
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Cited by 4 (4 self)
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From a mathematical point of view selforganization can be described as patterns to which certain dynamical systems modeling social dynamics tend spontaneously to be attracted. In this paper we explore situations beyond selforganization, in particular how to externally control such dynamical systems in order to eventually enforce pattern formation also in those situations where this wished phenomenon does not result from spontaneous convergence. Our focus is on dynamical systems of CuckerSmale type, modeling consensus emergence, and we question the existence of stabilization and optimal control strategies which require the minimal amount of external intervention for nevertheless inducing consensus in a group of interacting agents. First we follow a greedy approach, by designing instantaneous feedback controls with two different sparsity properties: componentwise sparsity, meaning that the controls have at most one nonzero component at every instant of time and their implementation is based on a variational criterion involving ℓ1norm penalization terms; time sparsity, meaning that the number of switchings is bounded on every compact interval of time, and such controls are realized by means of a sampleandhold procedure. Controls sharing these two sparsity features are very realistic and convenient for practical issues. Moreover we show
Multidimensional degenerate KellerSegel system with critical diffusion exponent 2n/(n + 2
 SIAM J. Math. Anal
, 2012
"... Abstract. This paper deals with a degenerate diffusion Patlak–Keller–Segel system in n ≥ 3 dimension. The main difference between the current work and many other recent studies on the same model is that we study the diffusion exponent m = 2n/(n + 2), which is smaller than the usual exponent m ∗ = 2 ..."
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Cited by 4 (3 self)
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Abstract. This paper deals with a degenerate diffusion Patlak–Keller–Segel system in n ≥ 3 dimension. The main difference between the current work and many other recent studies on the same model is that we study the diffusion exponent m = 2n/(n + 2), which is smaller than the usual exponent m ∗ = 2 − 2/n used in other studies. With the exponent m = 2n/(n + 2), the associated free energy is conformal invariant, and there is a family of stationary solutions Uλ,x0(x) = C ( λ λ2+x−x02) n+2 2 ∀λ> 0, x0 ∈ Rn. For radially symmetric solutions, we prove that if the initial data are strictly below Uλ,0(x) for some λ, then the solution vanishes in L 1 loc as t → ∞; if the initial data are strictly above Uλ,0(x) for some λ, then the solution either blows up at a finite time or has a mass concentration at r = 0 as time goes to infinity. For general initial data, we prove that there is a global weak solution provided that the Lm norm of initial density is less than a universal constant, and the weak solution vanishes as time goes to infinity. We also prove a finite time blowup of the solution if the Lm norm for initial data is larger than the Lm norm of Uλ,x0(x), which is constant independent of λ and x0, and the free energy of initial data is smaller than that of Uλ,x0(x).