Nonlinear degenerate cross-diffusion systems with nonlocal interaction

This paper presents new analytical results for a class of nonlinear parabolic systems of partial different equations with small cross-diffusion which have been proposed to describe the dynamics of a variety of large systems of interacting particles. Under suitable assumptions, we prove existence of classical solutions and we show exponential convergence in time to the stationary state. Furthermore, we consider the special case of one mobile and one immobile species, for which the system reduces to a nonlinear equation of Fokker-Planck type. In this framework, we improve the equilibration result obtained for the general system and we derive L ∞ -bounds for the solutions in two spatial dimensions. We conclude by illustrating the behaviour of solutions with numerical experiments in one and two spatial dimensions.1991 Mathematics Subject Classification. 35B40, 35B45, 35K51, 65N08.

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“…The asymptotic gradient flow system presented in [11] can be written in the general form of (2). In particular: (20) A(x, u) =…”

Section: 1mentioning

confidence: 99%

Trend to equilibrium for systems with small cross-diffusion

et al. 2020

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“…In this spirit, the "Evolution Variational Inequality" (E.V.I.) linked with the auxiliary gradient flow is crucial in order to obtain useful refined estimates (see for instance [34,36]). The connection between gradient flows and evolutionary PDEs of diffusion type shown in [1,42,51,53] allows us to consider the (decoupled) system…”

Section: 2mentioning

confidence: 99%

Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions

Carrillo¹,

Francesco²,

Esposito³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulative distribution functions, for which the system can be stated as a gradient flow on the Hilbert space L 2 (0, 1) 2 according to the classical theory by Brézis. For absolutely continuous initial data we construct solutions using a minimising movement scheme in the set of probability measures. In addition we show that the scheme preserves finiteness of the L m -norms for all m ∈ [1, +∞] and of the second moments. We then provide a characterisation of equilibria and prove that they are achieved (up to time subsequences) in the large time asymptotics. We conclude the paper constructing two examples of non-uniqueness of measure solutions emanating from the same (atomic) initial datum, showing that the notion of gradient flow solution is necessary to single out a unique measure solution.

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“…It is known that the thus obtained discrete gradient approximation converge to a solution of the non-linear Fokker-Planck equation (1.1) as τ tends to zero. Note, this scheme has been similarly applied to a variety of PDEs and systems of PDEs with gradient flow structure in the L 2 -Wasserstein or in a L 2 -Wasserstein-like space: non-local Fokker-Planck equations [9,11,35]; Fokker-Planck equations on manifolds [13,34]; fourth order fluid and quantum models [15,16,25]; chemotaxis systems [4,5,39]; Poisson-Nernst-Planck equations [20]; multi-component fluid systems [22]; Cahn-Hilliard equations [24]; degenerate cross-diffusion systems [29,40].…”

Section: Introductionmentioning

confidence: 99%

A BDF2-approach for the non-linear Fokker-Planck equation

Plazotta¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We prove convergence of a variational formulation of the BDF2 method applied to the nonlinear Fokker-Planck equation. Our approach is inspired by the JKO-method and exploits the differential structure of the underlying L 2 -Wasserstein space. The technique presented here extends and strengthens the results of our own recent work on the BDF2 method for general metric gradient flows in the special case of the non-linear Fokker-Planck equation: firstly, we do not require uniform semi-convexity of the augmented energy functional; secondly, we prove strong instead of merely weak convergence of the timediscrete approximations; thirdly, we directly prove without using the abstract theory of curves of maximal slope that the obtained limit curve is a weak solution of the non-linear Fokker-Planck equation.

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Nonlinear degenerate cross-diffusion systems with nonlocal interaction

Cited by 46 publications

References 51 publications

Trend to equilibrium for systems with small cross-diffusion

Trend to equilibrium for systems with small cross-diffusion

Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions

A BDF2-approach for the non-linear Fokker-Planck equation

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