Convergent Lagrangian Discretization for Drift-Diffusion with Nonlocal Aggregation

Matthes, Daniel; Söllner, Benjamin

doi:10.1007/978-3-319-49262-9_12

Cited by 8 publications

(15 citation statements)

References 15 publications

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“…The main purpose of this paper is to use a deterministic (ODEs) particle approximation method to construct solutions to a fairly wide class of aggregation-diffusion equations with nonlinear mobility, which are largely used in several contexts in population biology (see the Introduction). We stress that the presence of nonlinear mobility is new compared to several results available in the literature (see for instance [44]). This issue, together with the presence of the nonlocal transport term, suggests us to adapt to our case the strategy developed in [29] for scalar conservation laws.…”

Section: Discussionmentioning

confidence: 57%

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Solutions to aggregation–diffusion equations with nonlinear mobility constructed via a deterministic particle approximation

Fagioli

Radici

2018

Math. Models Methods Appl. Sci.

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We investigate the existence of weak type solutions for a class of aggregation-diffusion PDEs with nonlinear mobility obtained as large particle limit of a suitable nonlocal version of the follow-the-leader scheme, which is interpreted as the discrete Lagrangian approximation of the target continuity equation. We restrict the analysis to bounded, nonnegative initial data with bounded variation and away from vacuum, supported in a closed interval with zero-velocity boundary conditions. The main novelties of this work concern the presence of a nonlinear mobility term and the non strict monotonicity of the diffusion function. As a consequence, our result applies also to strongly degenerate diffusion equations. The results are complemented with some numerical simulations.

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Section: Discussionmentioning

confidence: 57%

“…In a later work [44], Matthes and Söllner tackle the problem of particle approximation for aggregation-diffusion through the discretization of the JKO. In that paper they prove convergence of the approximating particle sequence to the weak solutions of the pseudo-inverse equation.…”

Section: Introductionmentioning

confidence: 99%

Solutions to aggregation–diffusion equations with nonlinear mobility constructed via a deterministic particle approximation

Fagioli

Radici

2018

Math. Models Methods Appl. Sci.

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“…where λ is a fixed diffusion coefficient. Let us observe, that, accordingly to the results in [25,28,35], it is possible to generalize the diffusion law (5) to more general non linear expressions. Indeed, the quantity…”

mentioning

confidence: 92%

“…Once here, if U(t, w) were the inverse of W(t, z) and u(t, w) were ∂ w U(t, w) then, inspired by the results in [19,25], see also [34,35], u would be a weak solutions of the following aggregation-diffusion PDE…”

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confidence: 99%

Opinion formation systems via deterministic particles approximation

Fagioli¹,

Radici²

2021

KRM

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We propose an ODE-based derivation for a generalized class of opinion formation models either for single and multiple species (followers, leaders, trolls). The approach is purely deterministic and the evolution of the single opinion is determined by the competition between two mechanisms: the opinion diffusion and the compromise process. Such deterministic approach allows to recover in the limit an aggregation/(nonlinear)diffusion system of PDEs for the macroscopic opinion densities.

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“…In order to handle the aforementioned dynamics mathematical models composed by nonlinear aggregation/diffusion/transport equations were introduced [5,17,21,24,25,28] and deeply studied in recent years adopting different techniques and investigating possible modeling extensions (see e.g. [2,4,7,11,16,18,23] and references therein). The presence of a nonlinear mobility term in the equation may help to improve the ability of the models to catch more sophisticated phenomena.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformation

Ansini¹,

Fagioli²

2019

Preprint

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We prove an existence and uniqueness result for solutions to nonlinear diffusion equations with degenerate mobility posed on a bounded interval for a certain density u. In case of fast-decay mobilities, namely mobilities functions under a Osgood integrability condition, a suitable coordinate transformation is introduced and a new nonlinear diffusion equation with linear mobility is obtained. We observe that the coordinate transformation induces a mass-preserving scaling on the density and the nonlinearity, described by the original nonlinear mobility, is included in the diffusive process. We show that the rescaled density ρ is the unique weak solution to the nonlinear diffusion equation with linear mobility. Moreover, the results obtained for the density ρ allow us to motivate the aforementioned change of variable and to state the results in terms of the original density u without prescribing any boundary conditions.

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Convergent Lagrangian Discretization for Drift-Diffusion with Nonlocal Aggregation

Cited by 8 publications

References 15 publications

Solutions to aggregation–diffusion equations with nonlinear mobility constructed via a deterministic particle approximation

Solutions to aggregation–diffusion equations with nonlinear mobility constructed via a deterministic particle approximation

Opinion formation systems via deterministic particles approximation

Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformation

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