Volume effects in the Keller-Segel model : energy estimates preventing blow-up

By Vincent Calvez, José Antonio Carrillo. 2006

249 citations recorded by the source corpus.

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/3df291ecba6e424058ef921603ee19645f6a827a

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Citations in the corpus

Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion (2010, methodology): “In this paper we extend the free energy methods of [13, 27, 18, 14] to estimate the critical mass for a wide range of kernels and nonlinear diffusion, which include these known results.”
On the global existence of solutions to an aggregation model (2008, background): “The author delivered some a-priori estimates guaranteeing the boundedness of solutions (in dimension 2), which in turn justifles nonexistence of their flnite or inflnite time blow-up (see also [ 24 ] where the results from [9] have been improved and extended to the whole space IR2).”
On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher (2011, result): “In dimension 2, these results were obtained by [42, 17].”
Equilibria of homogeneous functionals in the fair-competition regime (2016, background): “The effect of substituting linear diffusion by non-linear diffusion with m > 1 in two dimensions and higher was described in [11,51] where it is shown that solutions exist globally in time for all values of the parameter χ > 0.”
A critical blow-up exponent in a chemotaxis system with nonlinear signal production (2018, unknown): “2) in which the second equation is replaced by either 0 = ∆v − v + u or 0 = ∆v − μ + u with μ = 1 |Ω| ∫ Ω u0, partially even more detailed results are available ([9], [8], [16]).”
A coupled chemotaxis-fluid model: Global existence (2011, background): “This leads to a nonlinear porous-medium-like diffusion instead of a linear one, see e.g. [4] [11].”
Uniqueness for Keller-Segel-type chemotaxis models (2012, background): “We finally refer to [11, 13, 14, 22] for different results concerning the existence of solutions satisfying the boundedness of the Fisher information and/or the uniform bounds of the solutions for particular choices of ε ≥ 0, α ≥ 0, and nonlinear diffusions.”
Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data (2012, methodology): “Indeed, estimates on the vertical distribution of mass have long been known to be a key controlling quantity for PKS (see for example [13,14,46]) so it is natural that such control also produces hypercontractive estimates, as already seen in [13].”
A coupled Keller–Segel–Stokes model: global existence for small initial data and blow-up delay (2012, background): “The Keller–Segel system in higher space dimensions has been investigated in [13,14], and results for the system with nonlinear diffusion to prevent blow-up can be found in [9,24].”
Existence, Uniqueness and Lipschitz Dependence for Patlak–Keller–Segel and Navier–Stokes in $${\mathbb{R}^2}$$R2 with Measure-Valued Initial Data (2012, background): “[46, 13, 14]) so it is natural that such control also produces hypercontractive estimates, as already seen in [13].”