Optimal critical mass in the two dimensional Keller–Segel model in R2

By Jean Dolbeault, Benoit Perthame. 2004

304 citations recorded by the source corpus.

Abstract The Keller–Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that, in two space dimensions, for small initial mass there is global existence of classical solutions and for large initial mass blow-up occurs. In this Note we complete this picture and give an explicit value for the critical mass when the system is set in the whole space. To cite this article: J. Dolbeault, B. Perthame, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Subfields: Mathematical Physics, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/d3861171cacdd7692fe3cdbc08e93857f63970e9

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Citations in the corpus

The two-dimensional Keller-Segel model after blow-up (2009, result): “First results in this direction have been obtained in [12] and recently completed in [ 8 ] and [2].”
Delayed blow‐up for chemotaxis models with local sensing (2020, background): “he microscopic local sensing models. Based on the properties of the entropy, which is bounded below only for the case of subcritical mass (which is related to logarithmic functional inequalities, cf. [10]), it is not surprising that there is a similar threshold behaviour for (1.1a), (1.1b), in parti”
Equilibria of homogeneous functionals in the fair-competition regime (2016, methodology): “The two-dimensional logarithmic case (m = 1, k = 0) is the so-called Keller–Segel system in its simplest formulation [35,36,41,33,31,9,43].”
Critical space for the parabolic-parabolic Keller–Segel model in Rd (2006, unknown): “In dimension d = 2, L 1 ( R d ) is the critical space: for M below the critical mass 8 π/χ there are global weak solutions, for M above the critical mass there is blow-up in ﬁnite time [6,2,13,1].”
Functional inequalities, thick tails and asymptotics for the critical mass Patlak–Keller–Segel model (2010, background): “dt FPKS[ρ(t)] = − Z R2 ρ(t,x) ∇ δFPKS[ρ(t)] δρ (x) 2 dx. (1.3) In particular, along such solutions, t→ FPKS[ρ(t)] is monotone non–increasing. The key to exploiting this monotonicity, as discovered in [17], is the sharp logarithmic Hardy–Littlewood–Sobolev (Log HLS) inequality [2, 9]: BCCJuly22,2013 ”
Global existence for chemotaxis with finite sampling radius (2006, background): “For two-dimensional domains, global existence depends on a threshold: when the initial mass lies below the threshold solutions exist globally, while above the threshold solutions blow up in finite time (see references in Horstmann [12], or the new book of Suzuki [32], and Dolbeault and Perthame [7] ”
Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion (2010, methodology): “We extend the free energy methods of [13, 15, 27] to obtain the sharp critical mass for a class of models with general nonlinear diffusion and inhomogeneous kernels.”
Blow-up in multidimensional aggregation equations with mildly singular interaction kernels (2009, methodology): “shares some features with the classical Patlak-Keller-Segel model for chemotaxis [22, 34] without diusion, see [12, 8, 9, 18 ] for the state of the art in this problem.”
Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ2 (2007, background): “Perthame announced in [22] that if χM < 8π there is global existence of solution for the PatlakKeller-Segel system (1) in a weak sense.”
PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic (2004, methodology): “We also refer to [21] for a proof of the optimal critical mass in two dimensions.”
Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data (2012, methodology): “Using the sharp logarithmic Hardy–Littlewood– Sobolev inequality (see for example [18]) this implies the global existence of any weak solution which has finite initial free energy provided that the total mass is less than 8π [13,30].”
On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher (2011, background): “The gap between the 4/CGNS and 8π was not filled before [31] when the link with the logarithmic Hardy-Littlewood-Sobolev inequality was made: Let f be a non-negative function in L1(R2) such that f log f and f log(1 + |x|2) belong to L1(R2).”