The boundedness-by-entropy principle for cross-diffusion systems

Abstract

A novel principle is presented which allows for the proof of bounded weak solutions to a class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure. The main feature of these systems is that the diffusion matrix may be generally neither symmetric nor positive semi-definite. The key idea of the principle is to employ a transformation of variables, determined by the entropy density, which is defined by the gradient-flow formulation. The transformation yields at the same time a positive semi-definite diffusion matrix, suitable gradient estimates as well as lower and/or upper bounds of the solutions. These bounds are a consequence of the transformation of variables and are obtained without the use of a maximum principle. Several classes of cross-diffusion systems are identified which can be solved by this technique. The systems are formally derived from continuous-time random walks on a lattice modeling, for instance, the motion of ions, cells, or fluid particles.

References in corpus (1)

• Improved duality estimates and applications to reaction-diffusion equations

Cited by in corpus (31)

• Global existence analysis of cross-diffusion population systems for multiple species
• Transport distances and geodesic convexity for systems of degenerate diffusion equations
• Global weak solutions to fully cross-diffusive systems with nonlinear diffusion and saturated taxis sensitivity
• Bound-Preserving Finite-Volume Schemes for Systems of Continuity Equations with Saturation
• Flow Characteristics in a Crowded Transport Model
• Existence analysis of a stationary compressible fluid model for heat-conducting and chemically reacting mixtures
• Exponential time decay of solutions to reaction-cross-diffusion systems of Maxwell-Stefan type
• Well-posedness of a cross-diffusion population model with nonlocal diffusion
• Global existence analysis of energy-reaction-diffusion systems
• An entropy structure preserving space-time formulation for cross-diffusion systems: Analysis and Galerkin discretization
• Reduced order modelling of nonlinear cross-diffusion systems
• A meeting point of entropy and bifurcations in cross-diffusion herding
• On a cross-diffusion system arising in image denosing
• Macroscopic behaviour in a two-species exclusion process via the method of matched asymptotics
• Stability estimates for systems with small cross-diffusion
• Computer-assisted proofs for some nonlinear diffusion problems
• Active Crowds
• Maximal mixed parabolic-hyperbolic regularity for the full equations of multicomponent fluid dynamics
• Swapping in lattice-based cell migration models
• A cross-diffusion system derived from a Fokker-Planck equation with partial averaging
• Global well-posedness of advective Lotka-Volterra competition systems with nonlinear diffusion
• On multi-species diffusion with size exclusion
• Stationary solutions and large time asymptotics to a cross-diffusion-Cahn-Hilliard system
• Global existence for a class of viscous systems of conservation laws
• Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain: linearized system
• Convergence of solutions of a rescaled evolution nonlocal cross-diffusion problem to its local diffusion counterpart
• Study of an entropy dissipating finite volume scheme for a nonlocal cross-diffusion system
• An economic cross-diffusion mutualistic model for cities emergence
• Evolution of dietary diversity and a starvation driven cross-diffusion system as its singular limit
• Turing instability analysis of a singular cross-diffusion problem
• A structure-preserving LDG discretization of the Fisher-Kolmogorov equation for modeling neurodegenerative diseases