Rigorous Derivation of Population Cross-Diffusion Systems from Moderately Interacting Particle Systems

Chen, Li; Daus, Esther S.; Holzinger, Alexandra; Jüngel, Ansgar

doi:10.1007/s00332-021-09747-9

Cited by 16 publications

(23 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A ij = 0 for j = i, see [38,13]. It is worth to mention that a stochastic approach (based on moderate limits) to derive cross-diffusion systems from particle systems is done in the recent work [9].…”

Section: Further Perspectivesmentioning

confidence: 99%

Porous medium equation as limit of nonlocal interaction

Burger¹,

Esposito²

2022

Preprint

View full text Add to dashboard Cite

This paper studies the convergence of solutions of a nonlocal interaction equation to the solution of the quadratic porous medium equation in the limit of a localising interaction kernel. The analysis is carried out at the level of the (nonlocal) partial differential equations and we use the gradient flow structure of the equations to derive bounds on energy, second order moments, and logarithmic entropy. The dissipation of the latter yields sufficient regularity to obtain compactness results and pass to the limit in the localised convolutions. The strategy we propose relies on a discretisation scheme which could be slightly modified in order to extend our result to PDEs with no gradient flow structure. Our analysis allows to treat the case of limiting weak solutions of the non-viscous porous medium equation at relevant low regularity, assuming the initial value to have finite energy and entropy. However, the latter excludes particle solutions of the nonlocal interaction equation.

show abstract

Section: Further Perspectivesmentioning

confidence: 99%

Porous medium equation as limit of nonlocal interaction

Burger¹,

Esposito²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…, and the conclusion follows as soon as ε N satisfies (46). Recent applications of these results can be found in [73] and [115]. The reference [73] presents a generalisation of [203] to a multi-species system with non globally Lipschitz interactions.…”

Section: Theorem 32 ([31]mentioning

confidence: 96%

Propagation of chaos: A review of models, methods and applications. Ⅱ. Applications

Chaintron

Diez

2022

KRM

View full text Add to dashboard Cite

<p style='text-indent:20px;'>The notion of propagation of chaos for large systems of interacting particles originates in statistical physics and has recently become a central notion in many areas of applied mathematics. The present review describes old and new methods as well as several important results in the field. The models considered include the McKean-Vlasov diffusion, the mean-field jump models and the Boltzmann models. The first part of this review is an introduction to modelling aspects of stochastic particle systems and to the notion of propagation of chaos. The second part presents concrete applications and a more detailed study of some of the important models in the field.</p>

show abstract

“…The weak formulation for the SKT system without self-diffusion is weaker than that one with self-diffusion, since we have only the gradient regularity ∇ u i ∈ L 1 (O), and A ij ( u) may be nonintegrable. However, system (1) can be written in Laplacian form according to (8), which allows for the "very weak" formulation stated in Theorem 4. The condition on γ if d = 2 is needed to prove the fractional time regularity for the approximative solutions.…”

Section: Given Two Quadratic Matricesmentioning

confidence: 99%

“…The deterministic analog of ( 1)-( 3) generalizes the two-species model of [37] to an arbitrary number of species. The deterministic model can be derived rigorously from nonlocal population systems [19,35], stochastic interacting particle systems [8], and finite-state jump Markov models [2,13]. The original system in [37] also contains a deterministic environmental potential and Lotka-Volterra terms, which are neglected here for simplicity.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Global martingale solutions for stochastic Shigesada-Kawasaki-Teramoto population models

Braukhoff¹,

Huber²,

Jüngel³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

The existence of global nonnegative martingale solutions to cross-diffusion systems of Shigesada-Kawasaki-Teramoto type with multiplicative noise is proven. The model describes the stochastic segregation dynamics of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. The diffusion matrix is generally neither symmetric nor positive semidefinite, which excludes standard methods for evolution equations. Instead, the existence proof is based on the entropy structure of the model, a novel regularization of the entropy variable, higher-order moment estimates, and fractional time regularity. The regularization technique is generic and is applied to the population system with self-diffusion in any space dimension and without self-diffusion in two space dimensions.

show abstract

Rigorous Derivation of Population Cross-Diffusion Systems from Moderately Interacting Particle Systems

Cited by 16 publications

References 28 publications

Porous medium equation as limit of nonlocal interaction

Porous medium equation as limit of nonlocal interaction

Propagation of chaos: A review of models, methods and applications. Ⅱ. Applications

Global martingale solutions for stochastic Shigesada-Kawasaki-Teramoto population models

Contact Info

Product

Resources

About