Mathematical tools for kinetic equations

Perthame, Benoı̂t

doi:10.1090/s0273-0979-04-01004-3

Cited by 113 publications

(57 citation statements)

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“…We first present the kinetic model in a first subsection, with the existence theory and, in second and third subsections, we explain the derivation of the above macroscopic models. We refer to [30,56] for a general mathematical theory for kinetic equations. …”

Section: Kinetic Model Of Chemotaxis and Asymptotic Theorymentioning

confidence: 99%

PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

Perthame

2004

Appl Math

116

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Modeling the movement of cells (bacteria, amoeba) is a long standing subject and Partial Differential equations have been used several times. The most classical and successful system was proposed by Patlak [55] and Keller & Segel [39] and is formed of parabolic or elliptic equations coupled through a drift term. This model exhibits a very deep mathematical structure because smooth solutions exist for small initial norm (in the appropriate space) and blow-up for large norms. This reflects experiments on bacteria like Escherichia Coli or amoeba like Dictyostelium Discoïdeum exhibiting pointwise concentrations.For human endothelial cells, several experiments show the formation of networks that can be interpreted as the initiation of angiogenesis. To recover such patterns a hydrodynamical model seems better adapted.The two systems can be unified by a kinetic approach that was proposed for Escherichia Coli, based on more precise experiments showing a movement by 'jump and tumble'. This nonlinear kinetic model is interesting by itself and the existence theory is not complete. It is also interesting from a scaling point of view; in a diffusion limit one recovers the Keller-Segel model and in a hydrodynamical limit one recovers the model proposed for human endothelial cells.We also mention the mathematical interest of a analyzing another degenerate parabolic system (exhibiting different properties) proposed to describe the angiogenesis phenomena i.e. the formation of capillary blood vessels. 1

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Section: Kinetic Model Of Chemotaxis and Asymptotic Theorymentioning

confidence: 99%

PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

Perthame

2004

Appl Math

116

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“…that the process is close to it's invariant measure parameterized by the solution of a macroscopic PDE, in direct analogy to such ideas in kinetic theory, cf. [40], [45].…”

Section: Introduction To Hydrodynamical Limits and Spin Systemsmentioning

confidence: 99%

Stochastic hydrodynamical limits of particle systems

Katsoulakis¹,

Szepessy²

2006

Communications in Mathematical Sciences

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Abstract. Even small noise can have substantial influence on the dynamics of differential equations, e.g. for nucleation/coarsening and interface dynamics in phase transformations. The aim of this work is to establish accurate models for the noise in macroscopic differential equations, related to phase transformations/reactions, derived from more fundamental microscopic master equations.For this purpose the mathematical paradigm of the dynamic Ising model is considered in the relatively tractable case of stochastic spin flip dynamics and long range spin/spin interactions. More specifically, this paper shows that localized spatial averages, with width , of solutions to such Ising systems with long range interaction of range O(1), are approximated with error O( + (γ/ ) 2d ) in distribution by a solution of an Ito stochastic differential equation, with drift as in the corresponding mean field model and a small diffusion coefficient of order (γ/ ) d/2 , generating noise with spatial correlation length , where γ is the distance between neighboring spin sites on a uniform periodic lattice in R d . To determine the correct noise is subtle in the sense that there are expected values, i.e. observables, that require different noise: the expected values that can be accurately approximated by the Einstein-diffusion and the expected values that need an alternative diffusion related to large deviation theory are identified; for instance dendrite dynamics up to a bounded time needs Einstein diffusion while transition rates need a different diffusion model related to invariant measures.The elementary proofs use O((γ/ ) 2d ) consistency of the Kolmogorov-backward equations for the averaged spin and the stochastic differential equation and that the long range interaction yields smoothing, which contributes with the O( ) error. A new aspect of the derivation is that the error, based on residuals and weights, is computable and suitable for adaptive refinements and modeling.

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“…The same gain of half a derivative is observed in all dimensions for averages over balls of solutions f : R These smoothing effects were first discovered in [18] and [17] and later developed in several papers [4,7,9,10,11,12,13,15,19,20,21,22,23,26]. See [24] for a review and an extended bibliography.…”

Section: Introductionmentioning

confidence: 55%