Cross-diffusion Models Research Articles

A class of free boundary problems describing the local-nonlocal cross-diffusion models with two species in ecology

We study a class of free boundary problems describing the local-nonlocal cross-diffusion models with two species in ecology. The first is a native species with nonlocal cross-diffusion, and the second is an invasion species with local cross-diffusion. Since the local and nonlocal cross-diffusions are taken into account, the models are described by the strongly coupled systems of integral and differential equations. In addition, the position of each free boundary is determined by the Stefan condition. By using the transformations of function and variable, various estimates and contraction mapping theorem, we show the global existence, uniqueness and regularity of solutions for the free boundary problems. Besides, the long time behavior of the solution for a special type of local-nonlocal cross-diffusion competition model is also obtained.

Existence of global solutions for cross-diffusion models in a fractional setting

Existence of global solutions for cross-diffusion models in a fractional setting

Cross-diffusion models in complex frameworks from microscopic to macroscopic

This paper deals with the micro–macro derivation of models from the underlying description provided by methods of the kinetic theory for active particles. We consider the so-called exotic models according to the definition proposed in [ N. Bellomo, N. Outada, J. Soler, Y. Tao and M. Winkler, Chemotaxis and cross diffusion models in complex environments: Modeling towards a multiscale vision, Math. Models Methods Appl. Sci. 32 (2022) 713–792]. The first part of the presentation focuses on a survey and a critical analysis of some phenomenological models known in the literature. We refer to a selection of case studies, in detail, the transport of virus models, social dynamics, and Keller–Segel in a fluid. The second part shows how a Hilbert-type approach can be developed to derive models at the macroscale from the underlying description provided by the kinetic theory of active particles. The third part deals with the derivation of macroscopic models corresponding to the selected case studies. Finally, a forward look into the future research perspectives is proposed.

Two-species nonlocal cross-diffusion models with free boundaries

A class of two-species nonlocal cross-diffusion models with free boundaries in one space dimension is investigated. In the models, the two species exist initially in (−∞,s10] and (−∞,s20], respectively, and then spread into the right space. The spreading fronts and nonlocal cross-diffusion of the species are described by the free boundaries and integral diffusion operators, respectively. By introducing some parameterized ODE problems and by applying the contraction mapping theorem and deriving some estimates, we give the global existence and uniqueness of solutions for the models. These results are applied to the nonlocal cross-diffusion prey-predator and competition models.

Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision

This paper proposes a review focused on exotic chemotaxis and cross-diffusion models in complex environments. The term exotic is used to denote the dynamics of models interacting with a time-evolving external system and, specifically, models derived with the aim of describing the dynamics of living systems. The presentation first, considers the derivation of phenomenological models of chemotaxis and cross-diffusion models with particular attention on nonlinear characteristics. Then, a variety of exotic models is presented with some hints toward the derivation of new models, by accounting for a critical analysis looking ahead to perspectives. The second part of the paper is devoted to a survey of analytical problems concerning the application of models to the study of real world dynamics. Finally, the focus shifts to research perspectives within the framework of a multiscale vision, where different paths are examined to move from the dynamics at the microscopic scale to collective behaviors at the macroscopic scale.

Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system

<p style='text-indent:20px;'>Recently, the authors proved [<xref ref-type="bibr" rid="b2">2</xref>] that the Maxwell-Stefan system with an incompressibility-like condition on the total flux can be rigorously derived from the multi-species Boltzmann equation. Similar cross-diffusion models have been widely investigated, but the particular case of a perturbative incompressible setting around a non constant equilibrium state of the mixture (needed in [<xref ref-type="bibr" rid="b2">2</xref>]) seems absent of the literature. We thus establish a quantitative perturbative Cauchy theory in Sobolev spaces for it. More precisely, by reducing the analysis of the Maxwell-Stefan system to the study of a quasilinear parabolic equation on the sole concentrations and with the use of a suitable anisotropic norm, we prove global existence and uniqueness of strong solutions and their exponential trend to equilibrium in a perturbative regime around any macroscopic equilibrium state of the mixture. As a by-product, we show that the equimolar diffusion condition naturally appears from this perturbative incompressible setting.</p>

Fast stable finite difference schemes for nonlinear cross-diffusion

The dynamics of cross-diffusion models leads to a high computational complexity for implicit difference schemes, turning them unsuitable for tasks that require results in real-time. We propose the use of two operator splitting schemes for nonlinear cross-diffusion processes in order to lower the computational load, and establish their stability properties using discrete L2 energy methods. Furthermore, by attaining a stable factorization of the system matrix as a forward-backward pass, corresponding to the Thomas algorithm for self-diffusion processes, we show that the use of implicit cross-diffusion can be competitive in terms of execution time, widening the range of viable cross-diffusion coefficients for on-the-fly applications.

Segregation effects and gap formation in cross-diffusion models

In this paper, we extend the results of [8] by proving exponential asymptotic H^1 -convergence of solutions to a one-dimensional singular heat equation with L^2 -source term that describe evolution of viscous thin liquid sheets while considered in the Lagrange coordinates. Furthermore, we extend this asymptotic convergence result to the case of a time inhomogeneous source. This study has also independent interest for the porous medium equation theory.

Chemotaxis systems in complex frameworks: Pattern formation, qualitative analysis and blowup prevention

A presentation of a special issue is delivered by this editorial paper on the derivation and related analytical problems, of cross diffusion models in complex environments. A brief introduction to motivations toward this research topic is presented in the first part. Subsequently, a concise description of the contents follows. Lastly, some ideas on possible research perspectives are proposed mainly focusing on modeling topics.

Cross-diffusion models: Analytic and multiscale problems

A presentation of a special issue on the derivation of cross-diffusion models and on the related analytical problems is proposed in this note. A brief introduction to motivations and recently published literature is presented in the first part. Subsequently, a concise description of the contents of the papers published in the issue follows. Finally, some ideas on possible research perspectives are proposed.

Study of cross-diffusion induced Turing patterns in a ratio-dependent prey-predator model via amplitude equations

• Spatio-temporal prey-predator model with cross-diffusion is studied. • Effects of cross-diffusion terms are studied–analytically and numerically. • Weakly nonlinear analysis is used to find the amplitude equations. • Numerical simulations are used to see how cross-diffusion influence Turing patterns. • Complete scenario of pattern formation in model with cross-diffusion is provided. Cross-diffusion models the situation where the presence, absence or abundance of one species of population affects the movement of other species of population in the domain under consideration and vice versa. Inclusion of cross-diffusion terms makes the modeling approach more realistic and shows significant impact on the spatio-temporal pattern formation scenario. In this paper, cross-diffusion is considered in a prey-predator model with ratio-dependent functional response, in addition to self-diffusion. Weakly nonlinear analysis is used near the Turing bifurcation boundary to derive the amplitude equations. From the stability analysis of the amplitude equations, conditions for emergence of Turing patterns such as cold spot, hot spot, mixture of spots and stripes and labyrinthine are identified. The analytical results are then verified with the help of numerical simulations. Results are general in nature and can be used to study the effect of cross-diffusion on other prey predator models both analytically and numerically.

Cross-Diffusion Systems for Image Processing: II. The Nonlinear Case

In this paper the use of nonlinear cross-diffu\-sion systems to model image restoration is investigated, theoretically and numerically. In the first case, well-posedness, scale-space properties and long time behaviour are analyzed. From a numerical point of view, a computational study of the performance of the models is carried out, suggesting their diversity and potentialities to treat image filtering problems. The present paper is a continuation of a previous work of the same authors, devoted to linear cross-diffusion models. \keywords{Cross-diffusion \and Complex diffusion \and Image restoration}

From a multiscale derivation of nonlinear cross-diffusion models to Keller–Segel models in a Navier–Stokes fluid

This paper deals with a micro–macro derivation of a variety of cross-diffusion models for a large system of active particles. Some of the models at the macroscopic scale can be viewed as developments of the classical Keller–Segel model. The first part of the presentation focuses on a survey and a critical analysis of some phenomenological models known in the literature. The second part is devoted to the design of the micro–macro general framework, where methods of the kinetic theory are used to model the dynamics of the system including the case of coupling with a fluid. The third part deals with the derivation of macroscopic models from the underlying description, delivered within a general framework of the kinetic theory.

Do some chemotaxis-growth models possess Lyapunov functionals?

In spirit of a result by W. Alt from 1980 we give some sufficient criteria that guarantee the existence of Lyapunov functionals for parabolic cross-diffusion models including chemotaxis-growth models with non-diffusive chemotactic signals (resp. with non-diffusive memory).

Bifurcation Approach to Analysis of Travelling Waves in Some Taxis–Cross-Diffusion Models

An overview of recently obtained authors’ results on traveling wave solutions of some classes of PDEs is presented. The main aim is to describe all possible travelling wave solutions of the equations. The analysis was conducted using the methods of qualitative and bifurcation analysis in order to study the phase-parameter space of the corresponding wave systems of ODEs. In the first part we analyze the wave dynamic modes of populations described by the “growth - taxis - diffusion polynomial models. It is shown that “suitable nonlinear taxis can affect the wave front sets and generate non-monotone waves, such as trains and pulses, which represent the exact solutions of the model system. Parametric critical points whose neighborhood displays the full spectrum of possible model wave regimes are identified; the wave mode systematization is given in the form of bifurcation diagrams. In the second part we study a modified version of the FitzHugh-Nagumo equations, which model the spatial propagation of neuron firing. We assume that this propagation is (at least, partially) caused by the cross-diffusion connection between the potential and recovery variables. We show that the cross-diffusion version of the model, besides giving rise to the typical fast travelling wave solution exhibited in the original “diffusion FitzHugh-Nagumo equations, additionally gives rise to a slow traveling wave solution. We analyze all possible traveling wave solutions of the model and show that there exists a threshold of the cross-diffusion coefficient (for a given speed of propagation), which bounds the area where “normal impulse propagation is possible. In the third part we describe all possible wave solutions for a class of PDEs with cross-diffusion, which fall in a general class of the classical Keller-Segel models describing chemotaxis. Conditions for existence of front-impulse, impulse-front, and front-front traveling wave solutions are formulated. In particular, we show that a non-isolated singular point in the ODE wave system implies existence of free-boundary fronts.

Macroscopic Cross-Diffusion Models Derived from Spatially Discrete Continuous Time Microscopic Models

We formulate a continuous time, discrete in space model for two spatially interacting species. The spatial interaction is described in terms of a measure for the desire or ability of a population to move from one location into a neighboring site. This can depend on local densities of both populations in the current and the target site. Refining the spatial resolution and passing to a continuous in space model, one obtains a system of partial differential equations with cross diffusion terms. We show that certain cross-diffusion models that have been used in the literature to describe interacting species can be derived as special cases with our approach.

Remarks on some Lotka–Volterra type cross-diffusion models

The present article collects some results for Lotka–Volterra type models in the presence of cross-diffusion effects. It discusses the existence and the nonexistence of nonhomogeneous steady state solutions and deals with the existence of traveling wave solutions in the presence of cross-diffusion effects.

On some cross-diffusion models in population dynamics and their connections to well-posed filters in signal enhancement processes

The present note studies some cross-diffusion models in population dynamics. The well-posedness of these systems is established and Lyapunov functions are introduced. Furthermore, the given results imply the well-posedness of some filters in signal enhancement processes of one-dimensional signals.