Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion

By Jacob Bedrossian, N. Rodr'iguez, Andrea L. Bertozzi. 2010

147 citations recorded by the source corpus.

Recently, there has been a wide interest in the study of aggregation equations and Patlak–Keller–Segel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the well-posedness theory of these models. We prove local well-posedness 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 well-posed. 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.

Subfields: Cancer Research, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/18675d9116c43966d69fedf87eb54e2d91d880de

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Citations in the corpus

Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data (2012, methodology): “This kind of two step bootstrap approach is common when applying similar methods [2,3,10,13,14,26,46,51].”
On minimizers of interaction functionals with competing attractive and repulsive potentials (2015, unknown): “[7, 8, 14, 17, 20, 21, 22, 23]).”
Equilibria of homogeneous functionals in the fair-competition regime (2016, background): “We refer to [3] for a discussion with more general interaction potentials in the aggregation-dominated regime.”
On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher (2011, background): “There exist many variants of the presented models with prevention of overcrowding [35, 16] or with non-linear chemo-sensitivity [3, 44], etc.”