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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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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
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 5 (4 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.
The PatlakKellerSegel model and its variations: properties of solutions via maximum principle. arXiv:1102.0092
, 2011
"... solutions via maximum principle ..."
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 1 (1 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
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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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
An aggregation equation with degenerate diffusion: qualitative property of solutions
, 2012
"... We study a nonlocal aggregation equation with degenerate diffusion, set in a periodic domain. This equation represents the generalization to m> 1 of the McKean–Vlasov equation where here the “diffusive ” portion of the dynamics are governed by Porous medium self–interactions. We focus primarily o ..."
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We study a nonlocal aggregation equation with degenerate diffusion, set in a periodic domain. This equation represents the generalization to m> 1 of the McKean–Vlasov equation where here the “diffusive ” portion of the dynamics are governed by Porous medium self–interactions. We focus primarily on m ∈ (1, 2] with particular emphasis on m = 2. In general, we establish regularity properties and, for small interaction, exponential decay to the uniform stationary solution. For m = 2, we obtain essentially sharp results on the rate of decay for the entire regime up to the (sharp) transitional value of the interaction parameter. 1
STATIONARY STATES OF QUADRATIC DIFFUSION EQUATIONS WITH LONGRANGE ATTRACTION
"... Abstract. We study the existence and uniqueness of nontrivial stationary solutions to a nonlocal aggregation equation with quadratic diffusion arising in many contexts in population dynamics. The equation is the Wasserstein gradient flow generated by the energy E, which is the sum of a quadratic fre ..."
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Abstract. We study the existence and uniqueness of nontrivial stationary solutions to a nonlocal aggregation equation with quadratic diffusion arising in many contexts in population dynamics. The equation is the Wasserstein gradient flow generated by the energy E, which is the sum of a quadratic free energy and the interaction energy. The interaction kernel is taken radial and attractive, nonnegative and integrable, with further technical smoothness assumptions. The existence vs. nonexistence of such solutions is ruled by a threshold phenomenon, namely nontrivial steady states exist if and only if the diffusivity constant is strictly smaller than the total mass of the interaction kernel. In the one dimensional case we prove that steady states are unique up to translations and mass constraint. The strategy is based on a strong version of the KreinRutman theorem. The steady states are symmetric with respect to their center of mass x0, compactly supported on sets of the form [x0 − L, x0 + L], C 2 on their support, strictly decreasing on (x0, x0+L). Moreover, they are global minimizers of the energy functional E. The results are complemented by numerical simulations. 1.