The parabolic-parabolic Keller-Segel model in R2

By Vincent Calvez, Lucilla Corrias. 2008

224 citations recorded by the source corpus.

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/61a9352a166eaa0ec358287e52ffe435b4e8dbd9

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Citations in the corpus

Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane (2012, unknown): “3 of [7] that if m(u0; R2) = 8π , then there exists a constant C0 > 0 such that E(t) ≥ −C0(1 + t) for t ∈ (0, T ).”
Global Solutions to the Coupled Chemotaxis-Fluid Equations (2010, result): “The further literature on the Keller–Segel model is indeed huge and thus out of the scope of this paper; see recent work [ 1 , 2 , 5 ] and references therein.”
Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces (2009, background): “[8] L. Corrias, B. Perthame, Asymptotic decay for the solutions of the parabolic–parabolic Keller–Segel chemotaxis in critical spaces, Math.”
A review of the mean field limits for Vlasov equations (2014, background): “Below the critical value, global solutions exist, see [44], [35], [136].”
COUPLED CHEMOTAXIS FLUID MODEL (2010, methodology): “For the elliptic-parabolic Keller-Segel model in the 2 D case, the results are summarized in [3] and for the parabolic-parabolic model recent progress has been achieved in [4].”
Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane (2014, background): “The global existence result for the mass of u below 8π has been extended to the two dimensional parabolic-parabolic system in [12, 21].”
A coupled chemotaxis-fluid model: Global existence (2011, methodology): “For more references on the general Keller–Segel system, the interested reader can refer to recent work [1,2,5].”
A coupled Keller–Segel–Stokes model: global existence for small initial data and blow-up delay (2012, methodology): “The proof works in the same way as in [10] and it uses the bound established in Lemma 3.”
Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion (2010, methodology): “If the problem is critical, it is well-known that in PKS there exists a critical mass, and solutions with larger mass can blow up in finite time [15, 34, 10, 27, 14, 19, 13, 57, 58, 18].”
Uniqueness for Keller-Segel-type chemotaxis models (2012, background): “2 except the L∞ bounds in [15] for d = 2 and the subcritical mass case m < 8π.”
On the global well-posedness theory for a class of PDE models for criminal activity (2013, background): “Similar inequalities has been extensively used for similar reasons, see for example [20].”
Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data (2012, background): “1) has also been analyzed in various contexts (see for example [6,16,43,57,62]).”
Nondegeneracy of blow-up points for the parabolic Keller–Segel system (2014, background): “On the other hand, in dimension N = 2, global existence was proved to hold for M < 4π in general bounded domains [1,14,38], and for M < 8π for radial solutions in a ball [38] or for general solutions in the whole space [6].”
LOCAL EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A PDE MODEL FOR CRIMINAL BEHAVIOR (2010, unknown): “(5) ∀ v ∈ H(Ω), γ, k ∈ Z ∪ 0, and 0 ≤ ≤ 1: ‖J v‖m+γ ≤ cmγ γ ‖v‖m , ∣∣J Dkv∣∣∞ ≤ ck N/2+γ−k ‖v‖k .”