Global renormalized solutions to reaction-cross-diffusion systems

Ansgar J\"ungel; Xiuqing Chen

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Global renormalized solutions to reaction-cross-diffusion systems

Xiuqing Chen, Ansgar Jüngel

Analysis of PDEs · Submitted 4 Nov 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The global-in-time existence of renormalized solutions to reaction-cross-diffu-sion systems for an arbitrary number of variables in bounded domains with no-flux boundary conditions is proved.

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The global-in-time existence of renormalized solutions to reaction-cross-diffu-sion systems for an arbitrary number of variables in bounded domains with no-flux boundary conditions is proved. The cross-diffusion part describes the segregation of population species and is a generalization of the Shigesada-Kawasaki-Teramoto model. The diffusion matrix is not diagonal and generally neither symmetric nor positive semi-definite, but the system possesses a formal gradient-flow or entropy structure. The reaction part includes reversible reactions of mass-action kinetics and does not obey any growth condition. The existence result generalizes both the condition on the reaction part required in the boundedness-by-entropy method and the proof of J. Fischer for reaction-diffusion systems with diagonal diffusion matrices.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The global-in-time existence of renormalized solutions to reaction-cross-diffu-sion systems for an arbitrary number of variables in bounded domains with no-flux boundary conditions is proved.	result 0.91	0.94	0.96	asserted 1.00
s1	The cross-diffusion part describes the segregation of population species and is a generalization of the Shigesada-Kawasaki-Teramoto model.	background 0.93	0.15	0.03	asserted 1.00
s2	The diffusion matrix is not diagonal and generally neither symmetric nor positive semi-definite, but the system possesses a formal gradient-flow or entropy structure.	background 0.72	0.39	0.03	asserted 0.99
s3	The reaction part includes reversible reactions of mass-action kinetics and does not obey any growth condition.	contribution 0.55	0.46	0.03	asserted 1.00
s4	The existence result generalizes both the condition on the reaction part required in the boundedness-by-entropy method and the proof of J. Fischer for reaction-diffusion systems with diagonal diffusion matrices.	contribution 0.55	0.81	0.10	asserted 1.00

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Original abstract

The global-in-time existence of renormalized solutions to reaction-cross-diffu-sion systems for an arbitrary number of variables in bounded domains with no-flux boundary conditions is proved. The cross-diffusion part describes the segregation of population species and is a generalization of the Shigesada-Kawasaki-Teramoto model. The diffusion matrix is not diagonal and generally neither symmetric nor positive semi-definite, but the system possesses a formal gradient-flow or entropy structure. The reaction part includes reversible reactions of mass-action kinetics and does not obey any growth condition. The existence result generalizes both the condition on the reaction part required in the boundedness-by-entropy method and the proof of J. Fischer for reaction-diffusion systems with diagonal diffusion matrices.

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2 arXiv papers cite this in 9 sentences

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This paper is a continuation of our work [ref] , in which we proved the global existence of renormalized solutions to a class of reaction-cross-diffusion systems describing the evolution of population species.

Weak-strong uniqueness of renormalized solutions to reaction-cross-diffusion systems · Introduction · 0.92

Uses its method or tool 2 papers

Then we can compute [equation] By choosing $ψ= 1$ , or more precisely a smooth version of $1$ with compact support in $[0,T-δ]$ then let $δ→0$ (see [ref] for more details) in the definition of the renormalized solutions, we get [equation] Our first goal now is to pass to the limit $η→0$ in [ref].

Trend to equilibrium of renormalized solutions to reaction-cross-diffusion systems · Proof of the main result · 0.89
The proof of the lemma is the same as in step 1 of the proof of Lemma 11 in [ref] .

Weak-strong uniqueness of renormalized solutions to reaction-cross-diffusion systems · Some auxiliary results · 0.99

Compares with it 1 paper

Also note that in [ref] the detailed balance diffusion condition was $π_i a_ij = π_j a_ji$ for some positive constants $π_i>0$.

Trend to equilibrium of renormalized solutions to reaction-cross-diffusion systems · Introduction · 0.86

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In this paper, we combine the entropy method used in [ref] and the concept of renormalized solutions of [ref] .

Global existence analysis of cross-diffusion population systems for multiple species · Introduction · 0.97

Cites as background 1 paper

Let us briefly comment on conditions [ref] and [ref]; for details, we refer to [ref] .

Global existence analysis of cross-diffusion population systems for multiple species · Introduction · 0.95

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