5,218 research outputs found
SPLITTING SCHEMES & SEGREGATION IN REACTION-(CROSS-)DIFFUSION SYSTEMS
International audienceOne 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 existence of weak solutions for a wide class of initial data without restriction about their supports or their positivity, we propose a variational splitting scheme combining ODEs with methods from optimal transport. In addition, this approach allows us to prove conservation of segregation for initially segregated data even in the presence of vacuum
Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues
Multiphase mechanical models are now commonly used to describe living tissues
including tumour growth. The specific model we study here consists of two
equations of mixed parabolic and hyperbolic type which extend the standard
compressible porous medium equation, including cross-reaction terms. We study
the incompressible limit, when the pressure becomes stiff, which generates a
free boundary problem. We establish the complementarity relation and also a
segregation result. Several major mathematical difficulties arise in the two
species case. Firstly, the system structure makes comparison principles fail.
Secondly, segregation and internal layers limit the regularity available on
some quantities to BV. Thirdly, the Aronson-B{\'e}nilan estimates cannot be
established in our context. We are lead, as it is classical, to add correction
terms. This procedure requires technical manipulations based on BV estimates
only valid in one space dimension. Another novelty is to establish an L1
version in place of the standard upper bound
Pattern formation driven by cross--diffusion in a 2D domain
In this work we investigate the process of pattern formation in a two
dimensional domain for a reaction-diffusion system with nonlinear diffusion
terms and the competitive Lotka-Volterra kinetics. The linear stability
analysis shows that cross-diffusion, through Turing bifurcation, is the key
mechanism for the formation of spatial patterns. We show that the bifurcation
can be regular, degenerate non-resonant and resonant. We use multiple scales
expansions to derive the amplitude equations appropriate for each case and show
that the system supports patterns like rolls, squares, mixed-mode patterns,
supersquares, hexagonal patterns
A variable nonlinear splitting algorithm for reaction diffusion systems with self- and cross-diffusion
Self- and cross-diffusion are important nonlinear spatial derivative terms that are included into biological models of predator-prey interactions. Self-diffusion models overcrowding effects, while cross-diffusion incorporates the response of one species in light of the concentration of another. In this paper, a novel nonlinear operator splitting method is presented that directly incorporates both self- and cross-diffusion into a computational efficient design. The numerical analysis guarantees the accuracy and demonstrates appropriate criteria for stability. Numerical experiments display its efficiency and accurac
A variable nonlinear splitting algorithm for reaction diffusion systems with self- and cross-diffusion
Self- and cross-diffusion are important nonlinear spatial derivative terms that are included into biological models of predator-prey interactions. Self-diffusion models overcrowding effects, while cross-diffusion incorporates the response of one species in light of the concentration of another. In this paper, a novel nonlinear operator splitting method is presented that directly incorporates both self- and cross-diffusion into a computational efficient design. The numerical analysis guarantees the accuracy and demonstrates appropriate criteria for stability. Numerical experiments display its efficiency and accurac
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