Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion

Fellner K, Fischer JL, Kniely M, Tang BQ. Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion. arXiv, .

Preprint | Submitted | English
Author
Fellner, Klemens; Fischer, Julian LISTA ; Kniely, MichaelISTA ; Tang, Bao Quoc
Department
Abstract
The global existence of renormalised solutions and convergence to equilibrium for reaction-diffusion systems with non-linear diffusion are investigated. The system is assumed to have quasi-positive non-linearities and to satisfy an entropy inequality. The difficulties in establishing global renormalised solutions caused by possibly degenerate diffusion are overcome by introducing a new class of weighted truncation functions. By means of the obtained global renormalised solutions, we study the large-time behaviour of complex balanced systems arising from chemical reaction network theory with non-linear diffusion. When the reaction network does not admit boundary equilibria, the complex balanced equilibrium is shown, by using the entropy method, to exponentially attract all renormalised solutions in the same compatibility class. This convergence extends even to a range of non-linear diffusion, where global existence is an open problem, yet we are able to show that solutions to approximate systems converge exponentially to equilibrium uniformly in the regularisation parameter.
Publishing Year
Date Published
2021-09-24
Journal Title
arXiv
Acknowledgement
A major part of this work was carried out when B. Q. Tang visited the Institute of Science and Technology Austria (IST Austria). The hospitality of IST Austria is greatly acknowledged. This work is partially supported by the International Research Training Group IGDK 1754 “Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures”, funded by the German Research Council (DFG) project number 188264188/GRK1754 and the Austrian Science Fund (FWF) under grant number W 1244-N18 and NAWI Graz.
IST-REx-ID

Cite this

Fellner K, Fischer JL, Kniely M, Tang BQ. Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion. arXiv.
Fellner, K., Fischer, J. L., Kniely, M., & Tang, B. Q. (n.d.). Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion. arXiv.
Fellner, Klemens, Julian L Fischer, Michael Kniely, and Bao Quoc Tang. “Global Renormalised Solutions and Equilibration of Reaction-Diffusion Systems with Non-Linear Diffusion.” ArXiv, n.d.
K. Fellner, J. L. Fischer, M. Kniely, and B. Q. Tang, “Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion,” arXiv. .
Fellner K, Fischer JL, Kniely M, Tang BQ. Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion. arXiv, .
Fellner, Klemens, et al. “Global Renormalised Solutions and Equilibration of Reaction-Diffusion Systems with Non-Linear Diffusion.” ArXiv.
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