Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

By Fabio A. C. C. Chalub, Peter A. Markowich, Benoit Perthame, Christian Schmeiser. 2004

268 citations recorded by the source corpus.

Subfields: Modeling and Simulation, Molecular Biology, Oncology

View Paper: https://www.semanticscholar.org/paper/4ca66097b3fdf738031037ba35210a5af10d8bda

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Citations in the corpus

Global Existence for a Kinetic Model of Chemotaxis via Dispersion and Strichartz Estimates (2007, methodology): “Following the reasoning in Chalub et al. (2004) we estimate f as follows: f t x v ≤ Cf0 x − tv v + C ∫ t 0 t − s x − sv ds + C 3∑ j=1 fj t x v (3.6) where f1 t x v = ∫ t 0 ∫ V Ss t − s x − sv+ v f t − s x − sv v′ dv′ ds = ∫ t 0 Ss t − s x − sv+ v t − s x − sv ds (3.7a) f2 t x v = ∫ t 0 ∫ V S t − s…”
PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic (2004, background): “In case of the nonlinear model (12)–(13), it is proved ([ 13 ]) that, as " ! 0,”
MULTICELLULAR BIOLOGICAL GROWING SYSTEMS: HYPERBOLIC LIMITS TOWARDS MACROSCOPIC DESCRIPTION (2007, background): “Subsequent relevant contributions in this area are due to various authors, among others.(23,34,30,15,18,26,16) In these works biological systems are considered for which interactions do not follow classical mechanical rules, and biological activity may play a relevant role in determining the dynamic”
Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ2 (2007, methodology): “Let us finally mention that the PKS system has been derived from kinetic models of chemotaxis [52, 45, 17].”
Global Solutions to the Coupled Chemotaxis-Fluid Equations (2010, unknown): “[8, 7].”
Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions (2004, unknown): “For the mathematical analysis of the mesoscopic approach and the derivation of the macroscopic models from the mesoscopic ones, we refer to [39], [38], [43], [12], [27], [17] and the references therein.”
General relative entropy inequality: an illustration on growth models (2005, background): “and especially we consider the non-conservative and non-symmetric case as motivated by [11, 20, 8].”
MODEL HIERARCHIES FOR CELL AGGREGATION BY CHEMOTAXIS (2006, methodology): “We impose the following assumptions on the turning kernel: When Sε → S0 and ρε → ρ0 as ε→ 0, then Tε(v → v′)[Sε, ρε] = T0(v → v′)[S0, ρ0] + εT1(v → v′)[S0, ρ0] + o(ε) where the convergence Tε → T0 is made precise in reference [4].”
Mathematical tools for kinetic equations (2004, background): “For instance, in case of the nonlinear model of chemotaxis where equation (87) is coupled with (45)–(46), we can prove (see [99], [34]) that fε(t, x, ξ) → %(t, x), Sε(t, x) → S(t, x) and that the Keller-Segel model holds in the limit ∂ ∂t %(t, x) + div(χ%∇S) = div(D∇%), t ≥ 0, x ∈ R, −∆S = %, with t”
Embedding of Semigroups of Lipschitz Maps into Positive Linear Semigroups on Ordered Banach Spaces Generated by Measures (2009, background): “Examples include branching random evolution [9], kinetic chemotaxis models concerning the run-and-tumble type of movement of flagellated bacteria like E. coli, B. subtilis or V. cholerae [18, 19, 2 ] and the extension of these to amoebae like Dictyostelium discoideum [3, 11], and cell cycle models ”
A STATISTICAL MODEL OF CRIMINAL BEHAVIOR (2008, background): “Deriving continuum equations from the underlying micro-scale behavior of a system is a common procedure in mathematical biology.(33,39,19,15,14,22,4,18,3) It is no surprise, then, that Eqs.”
General entropy equations for structured population models and scattering (2004, unknown): “Here 0 kT (·) ∈ L∞(Rd ) and 0 K(x,y) ∈ L1 ∩L∞(Rd) and especially we aim to treat the non-symmetric c as motivated by [8,14,4].”
Global existence for chemotaxis with finite sampling radius (2006, background): “More details and rigorous justifications can be found in [9], [22], [5].”