The reaction–cross‐diffusion models for tissue growth

2021 ◽  
Author(s):  
Zehra Pinar
Keyword(s):  
Reaction Cross ◽  
Tissue Growth ◽  
Diffusion Models ◽  
Cross Diffusion

Segregation effects and gap formation in cross-diffusion models

2020 ◽  
Vol 22 (2) ◽  
pp. 175-203
Author(s):  
Martin Burger ◽  
José Carrillo ◽  
Jan-Frederik Pietschmann ◽  
Markus Schmidtchen
Keyword(s):  
Diffusion Models ◽  
Gap Formation ◽  
Cross Diffusion

Cross-diffusion models: Analytic and multiscale problems

2018 ◽  
Vol 28 (11) ◽  
pp. 2097-2102 ◽  
Author(s):  
N. Bellomo ◽  
Y. Tao ◽  
M. Winkler
Keyword(s):  
Diffusion Models ◽  
Special Issue ◽  
Multiscale Problems ◽  
Cross Diffusion ◽  
Research Perspectives ◽  
Concise Description

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.

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

2020 ◽  
Vol 30 (06) ◽  
pp. 1033-1039
Author(s):  
N. Bellomo ◽  
Y. Tao ◽  
M. Winkler
Keyword(s):  
Pattern Formation ◽  
Qualitative Analysis ◽  
Diffusion Models ◽  
Research Topic ◽  
Complex Environments ◽  
Special Issue ◽  
Cross Diffusion ◽  
Research Perspectives ◽  
Concise Description

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.

scholarly journals Splitting Schemes and Segregation in Reaction Cross-Diffusion Systems

2018 ◽  
Vol 50 (5) ◽  
pp. 5695-5718 ◽  
Author(s):  
J. A. Carrillo ◽  
S. Fagioli ◽  
F. Santambrogio ◽  
M. Schmidtchen
Keyword(s):  
Reaction Cross ◽  
Splitting Schemes ◽  
Cross Diffusion ◽  
Diffusion Systems

scholarly journals Weak–strong uniqueness of renormalized solutions to reaction–cross-diffusion systems

2019 ◽  
Vol 29 (02) ◽  
pp. 237-270 ◽  
Author(s):  
Xiuqing Chen ◽  
Ansgar Jüngel
Keyword(s):  
Strong Solution ◽  
Gradient Flow ◽  
Growth Conditions ◽  
Reaction Cross ◽  
Strong Uniqueness ◽  
Renormalized Solutions ◽  
Renormalized Solution ◽  
Cross Diffusion ◽  
Diffusion Systems ◽  
Flux Boundary Conditions

The weak–strong uniqueness for renormalized 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 species. The diffusion matrix is neither symmetric nor positive definite, but the system possesses a formal gradient-flow or entropy structure. No growth conditions on the source terms are imposed. It is shown that any renormalized solution coincides with a strong solution with the same initial data, as long as the strong solution exists. The proof is based on the evolution of the relative entropy modified by suitable cutoff functions.

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