Kinetic and related macroscopic models for chemotaxis on networks

Borsche, Raul; Kall, Jochen; Klar, Axel; Pham, Tuan N.

doi:10.1142/s0218202516500299

Cited by 19 publications

(31 citation statements)

References 32 publications

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“…Remark 3. Another direct approach to obtain coupling conditions for the wave equation has been used in [8]. Using a full moment approximation of the distribution function in the case of bounded velocities, i.e.…”

Section: Boundary and Coupling Conditions For Macroscopic Equations Vmentioning

confidence: 99%

“…On the other hand, coupling conditions for kinetic equations on networks have been discussed in a much smaller number of publications, see [21,30,8]. In [8] a first attempt to derive a coupling condition for a macroscopic equation from the underlying kinetic model has been presented for the case of a kinetic equations for chemotaxis. In the present paper, we will present a more general and more accurate procedure.…”

Section: Introductionmentioning

confidence: 99%

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Kinetic Layers and Coupling Conditions for Macroscopic Equations on Networks I: The Wave Equation

Borsche¹,

Klar²

2018

SIAM J. Sci. Comput.

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We consider kinetic and associated macroscopic equations on networks. The general approach will be explained in this paper for a linear kinetic BGK model and the corresponding limit for small Knudsen number, which is the wave equation. Coupling conditions for the macroscopic equations are derived from the kinetic conditions via an asymptotic analysis near the nodes of the network. This analysis leads to the consideration of a fixpoint problem involving the coupled solutions of kinetic half-space problems. A new approximate method for the solution of kinetic half-space problems is derived and used for the determination of the coupling conditions. Numerical comparisons between the solutions of the macroscopic equation with different coupling conditions and the kinetic solution are presented for the case of tripod and more complicated networks.

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Section: Boundary and Coupling Conditions For Macroscopic Equations Vmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Kinetic Layers and Coupling Conditions for Macroscopic Equations on Networks I: The Wave Equation

Borsche¹,

Klar²

2018

SIAM J. Sci. Comput.

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“…Furthermore, the front tracking method is not able to handle networks with arbitrary many junctions which may contain circles. There are also some publications which use a kinetic approach to derive coupling conditions for the macroscopic model [7,8,9,20]. Recently, Borsche and Klar studied half-Riemann problems for scalar [8] and linear [7] equations with a kinetic approach to derive macroscopic coupling conditions.…”

Section: Introductionmentioning

confidence: 99%

Kinetic relaxation to entropy based coupling conditions for isentropic flow on networks

Holle

2020

Journal of Differential Equations

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We consider networks for isentropic gas and prove existence of weak solutions for a large class of coupling conditions. First, we construct approximate solutions by a vectorvalued BGK model with a kinetic coupling function. Introducing so-called kinetic invariant domains and using the method of compensated compactness justifies the relaxation towards the isentropic gas equations. We will prove that certain entropy flux inequalities for the kinetic coupling function remain true for the traces of the macroscopic solution. These inequalities define the macroscopic coupling condition. Our techniques are also applicable to networks with arbitrary many junctions which may possibly contain circles. We give several examples for coupling functions and prove corresponding entropy flux inequalities. We prove also new existence results for solid wall boundary conditions and pipelines with discontinuous crosssectional area.2010 Mathematics Subject Classification. 35L65, 76N15, 82C40.

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“…In order to retain the conservation of mass and the positivity of the numerical solution, Saito [31] proposed and analyzed an upwind finite element scheme for multidimensional problems; a slightly different approach was considered [32]. Let us also mention recent work [3,14,27] concerning the approximation of hyperbolic models of chemotaxis.…”

Section: Introductionmentioning

confidence: 99%

Chemotaxis on networks: Analysis and numerical approximation

Egger

Schöbel-Kröhn

2020

ESAIM: M2AN

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We consider the Keller-Segel model of chemotaxis on one-dimensional networks. Using a variational characterization of solutions, positivity preservation, conservation of mass, and energy estimates, we establish global existence of weak solutions and uniform bounds. This extends related results of Osaki and Yagi to the network context. We then analyze the discretization of the system by finite elements and an implicit time-stepping scheme. Mass lumping and upwinding are used to guarantee the positivity of the solutions on the discrete level. This allows us to deduce uniform bounds for the numerical approximations and to establish order optimal convergence of the discrete approximations to the continuous solution without artificial smoothness requirements. In addition, we prove convergence rates under reasonable assumptions. Some numerical tests are presented to illustrate the theoretical results.

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Kinetic and related macroscopic models for chemotaxis on networks

Cited by 19 publications

References 32 publications

Kinetic Layers and Coupling Conditions for Macroscopic Equations on Networks I: The Wave Equation

Kinetic Layers and Coupling Conditions for Macroscopic Equations on Networks I: The Wave Equation

Kinetic relaxation to entropy based coupling conditions for isentropic flow on networks

Chemotaxis on networks: Analysis and numerical approximation

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