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As an extension of our previous work in Sun et.al (2018) [41], we develop a discontinuous Galerkin method for solving cross-diffusion systems with a formal gradient flow structure. These systems are associated with non-increasing entropy…
In this work we study global well-posedness and large time behaviour for a typical reaction--diffusion system, which include degenerate diffusion, and whose non-linearities arise from chemical reactions. We show that there is an {\it…
This paper is devoted to the multigrid convergence analysis for the linear systems arising from the conforming linear finite element discretization of the second order elliptic equations with anisotropic diffusion. The multigrid convergence…
The paper entitled "Well posedness of general cross-diffusion systems", by C. Choquet, C. Rosier, L. Rosier, J. Diff. Eq. 2021, is devoted to the mathematical analysis of the Cauchy problem for general cross-diffusion systems without any…
We study the regularity of weak solutions and the global existence of classical to cross-diffusion systems of $m$ equations on $N$-dimensional domains ($m,N\ge2$).
Bounded weak solutions are constructed for a degenerate parabolic system with a full diffusion matrix, which is a generalized version of the thin film Muskat system. Boundedness is achieved with the help of a sequence $(\mathcal{E}_n)_{n\ge…
We consider some cross diffusion systems which is inspired by models in mathematical biology/ecology, in particular the Shigesada-Kawasaki-Teramoto (SKT) model in population biology. We establish the global existence of strong solutions to…
Two-scale homogenization limits of parabolic cross-diffusion systems in a heterogeneous medium with no-flux boundary conditions are proved. The heterogeneity of the medium is reflected in the diffusion coefficients or by the perforated…
The weak-strong uniqueness for solutions to reaction-cross-diffusion systems in a bounded domain with no-flux boundary conditions is proved. The system generalizes the Shigesada-Kawasaki-Teramoto population model to an arbitrary number of…
Semi-discrete Runge-Kutta schemes for nonlinear diffusion equations of parabolic type are analyzed. Conditions are determined under which the schemes dissipate the discrete entropy locally. The dissipation property is a consequence of the…
One of the most fascinating phenomena observed in reaction-diffusion systems is the emergence of segregated solutions, i.e. population densities with disjoint supports. We analyse such a reaction cross-diffusion system. In order to prove…
We show in numerical simulations that a system of two coupled replicas of a binary mixture of hard spheres undergoes a phase transition in equilibrium at a density slightly smaller than the glass transition density for an unreplicated…
We investigate a class of systems of partial differential equations with nonlinear cross-diffusion and nonlocal interactions, which are of interest in several contexts in social sciences, finance, biology, and real world applications.…
The aim of this work is to propose a provably convergent finite volume scheme for the so-called Stefan-Maxwell model, which describes the evolution of the composition of a multi-component mixture and reads as a cross-diffusion system. The…
We establish the positivity of weak (and very weak) solutions to a class of cross diffusion systems which is inspired by models in mathematical biology/ecology, in particular the Shigesada-Kawasaki-Teramoto (SKT) model in population…
The aim of this work is to study the global existence of solutions for some coupled systems of reaction diffusion which describe the spread within a population of infectious disease. We consider a triangular matrix diffusion and we show…
A structure-preserving implicit Euler finite-element scheme for a degenerate cross-diffusion system for ion transport is analyzed. The scheme preserves the nonnegativity and upper bounds of the ion concentrations, the total relative mass,…
An implicit Euler finite-volume scheme for a degenerate cross-diffusion system describing the ion transport through biological membranes is analyzed. The strongly coupled equations for the ion concentrations include drift terms involving…
We study the existence and the rate of equilibration of weak solutions to a two-component system of non-linear diffusion-aggregation equations, with small cross diffusion effects. The aggregation term is assumed to be purely attractive, and…
The weak correlation between spatiotemporal fluctuations in nonequilibrium complex systems is shown to govern the fluctuation distribution, maximizing the conditional entropy associated with such fluctuations. The result is illustrated in…