One-dimensional chemotaxis kinetic model

tabar, Mohsen Sharifi

doi:10.1007/s00030-010-0088-8

Cited by 2 publications

(5 citation statements)

References 19 publications

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“…It has been shown that the solutions to the 2D spherical symmetric kinetic model (2.9) blow up in finite time. While for the one dimensional kinetic local model (2.7), although the strict analysis is lacking [31], the numerical results strongly suggest that the solutions also blow up. In these cases, the convergence of the schemes described above would be questionable when simulations are performed on the fixed grids.…”

Section: Adaptive Grids For Solutions With Blowupmentioning

confidence: 92%

“…We numerically check the open problem of the blow up property of the solution to the one dimensional kinetic local model (2.7). As mentioned before, a theoretical prediction on this blow up is still lacking, see [31]. We consider the initial data given by (5.1).…”

Section: The 1d Local Model: Blow Up In Finite Timementioning

confidence: 99%

“…m c < M c . Moreover, the appearance of blow-up in the one dimensional case has not been clarified yet, see [31]. The second aim of this work is to investigate these questions through numerical simulation.…”

Section: Introductionmentioning

confidence: 99%

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An Asymptotic Preserving Scheme for the Diffusive Limit of Kinetic Systems for Chemotaxis

Carrillo

Yan

2013

Multiscale Model. Simul.

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In this work we numerically study the diffusive limit of run & tumble kinetic models for cell motion due to chemotaxis by means of asymptotic preserving schemes. It is well-known that the diffusive limit of these models leads to the classical Patlak-Keller-Segel macroscopic model for chemotaxis. We will show that the proposed scheme is able to accurately approximate the solutions before blow-up time for small parameter. Moreover, the numerical results indicate that the global solutions of the kinetic models stabilize for long times to steady states for all the analyzed parameter range. We also generalize these asymptotic preserving schemes to two dimensional kinetic models in the radial case. The blow-up of solutions is numerically investigated in all these cases.Hereis the density of chemoattractant, whose governing equation will be described later.A very interesting consequence is that even in one dimension, this system blows-up over this critical mass. This makes possible to numerically analyse this highly subtle phenomena. This modified Keller-Segel system was numerically solved in [3] by using optimal transportation ideas and the scheme proven to be convergent in the subcritical case.Alt in [1, 2] derived (1.4) from a transport equation similar to (1.1) via a stochastic model. Later the kinetic system (1.1)-(1.2) was formulated in [26]. Othmer and Hillen in [16,27] studied the formal diffusion limit of this system by moment expansions. In [11] Chalub et al. proved that, in three dimensions with suitable assumptions on the turning kernel T ε , the solution to the kinetic system (1.1)-(1.2) globally exists for any initial total mass. Moreover, they gave a rigorous proof of the convergence to the macroscopic limit for all time invervals of existence of the limiting Keller-Segel system. In the following work [18,19] the results are extended to more general cases.

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Section: Adaptive Grids For Solutions With Blowupmentioning

confidence: 92%

Section: The 1d Local Model: Blow Up In Finite Timementioning

confidence: 99%

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An Asymptotic Preserving Scheme for the Diffusive Limit of Kinetic Systems for Chemotaxis

Carrillo

Yan

2013

Multiscale Model. Simul.

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“…The long time behavior of the subcritical case is unclear yet. Also, theoretic proof of the blow up in the 1D case is not available [46]. The microscopic kinetic model, with interesting properties and mysterious behaviors, make it appealing to investigate the system numerically.…”

Section: Introductionmentioning

confidence: 99%

“…The long time behavior of the subcritical case is unclear yet. Also, theoretic proof of the blow up in the 1D case is not available [46].…”

mentioning

confidence: 99%

A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs

Jin¹,

Lu²,

Pareschi³

2018

Multiscale Model. Simul.

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In this paper, we develop a stochastic Asymptotic-Preserving (sAP) scheme for the kinetic chemotaxis system with random inputs, which will converge to the modified Keller-Segel model with random inputs in the diffusive regime. Based on the generalized Polynomial Chaos (gPC) approach, we design a high order stochastic Galerkin method using implicit-explicit (IMEX) Runge-Kutta (RK) time discretization with a macroscopic penalty term. The new schemes improve the parabolic CFL condition to a hyperbolic type when the mean free path is small, which shows significant efficiency especially in uncertainty quantification (UQ) with multi-scale problems. The stochastic Asymptotic-Preserving property will be shown asymptotically and verified numerically in several tests. Many other numerical tests are conducted to explore the effect of the randomness in the kinetic system, in the aim of providing more intuitions for the theoretic study of the chemotaxis models.

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One-dimensional chemotaxis kinetic model

Cited by 2 publications

References 19 publications

An Asymptotic Preserving Scheme for the Diffusive Limit of Kinetic Systems for Chemotaxis

An Asymptotic Preserving Scheme for the Diffusive Limit of Kinetic Systems for Chemotaxis

A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs

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