Global renormalized solutions to reaction-cross-diffusion systems with self-diffusion

Chen, Xiuqing; Jüngel, Ansgar

doi:10.1016/j.jde.2019.06.010

Cited by 16 publications

(38 citation statements)

References 32 publications

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“…By choosing ψ " 1 , or more precisely a smooth version of 1 with compact support in r0, T´δs then let δ Ñ 0 (see [5,Lemma 11] for more details) in the definition of the renormalized solutions, we get…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…Proof of Theorem 1.1. The existence of a global renormalized solution follows from [5, Theorem 1] since under the complex balanced condition the reactions satisfy (14), which is (H4) in [5] with π i " 1 and λ i "´log u i8 for all i " 1, . .…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…, n and δ ij denotes the Kronecker delta. They are assumed to satisfy (in analogy to [5]) either the weak cross-diffusion condition (4) α :" min i"1,...,n˜a ii´1 4 n ÿ i"1`? a ij´?…”

Section: Introductionmentioning

confidence: 99%

“…Let Ω be a bounded domain with smooth boundary BΩ. Assume a i0 , a ii ą 0, a ij ě 0, and let the diffusion matrix Apuq satisfy either (4) or (5). Assume that (S) satisfies the complex balanced condition (6).…”

Section: Introductionmentioning

confidence: 99%

“…This should not be confused with the detailed balance condition occuring in reactions or in even more general micro-reversible processes. Also note that in[5] the detailed balance diffusion condition was π i a ij " π j a ji for some positive constants π i ą 0. Here we choose π i " 1 for i " 1, .…”

mentioning

confidence: 99%

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Trend to equilibrium of renormalized solutions to reaction–cross-diffusion systems

Daus

Tang

2019

Applied Mathematics Letters

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The convergence to equilibrium of renormalized solutions to reaction-cross-diffusion systems in a bounded domain under no-flux boundary conditions is studied. The reactions model complex balanced chemical reaction networks coming from mass-action kinetics and thus do not obey any growth condition, while the diffusion matrix is of cross-diffusion type and hence nondiagonal and neither symmetric nor positive semi-definite, but the system admits a formal gradient-flow or entropy structure. The diffusion term generalizes the population model of Shigesada, Kawasaki and Teramoto to an arbitrary number of species. By showing that any renormalized solution satisfies the conservation of masses and a weak entropy-entropy production inequality, it can be proved under the assumption of no boundary equilibria that all renormalized solutions converge exponentially to the complex balanced equilibrium with a rate which is explicit up to a finite dimensional inequality.

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…, n and δ ij denotes the Kronecker delta. They are assumed to satisfy (in analogy to [5]) either the weak cross-diffusion condition (4) α :" min i"1,...,n˜a ii´1 4 n ÿ i"1`? a ij´?…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Trend to equilibrium of renormalized solutions to reaction–cross-diffusion systems

Daus

Tang

2019

Applied Mathematics Letters

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A brief overview of existence results and decay time estimates for a mathematical modeling of scintillating crystals

Davı́

2021

Math Methods in App Sciences

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Inorganic scintillating crystals can be modelled as continua with microstructure.For rigid and isothermal crystals, the evolution of charge carriers becomes in this way described by a reaction-diffusion-drift equation coupled with the Poisson equation of electrostatic. Here, we give a survey of the available existence and asymptotic decays results for the resulting boundary value problem, the latter being a direct estimate of the scintillation decay time. We also show how to recover various approximated models which encompass also the two most used phenomenological models for scintillators, namely, the kinetic and diffusive ones. Also for these cases, we show, whenever it is possible, which existence and asymptotic decays estimate results are known to date. KEYWORDS entropy methods, existence of solutions of PDE, exponential rate of convergence, reaction-diffusion-drift equations, scintillators MSC CLASSIFICATION 35K57; 35B40; 35B45 INTRODUCTIONA scintillator crystal is a material which converts ionizing radiations into photons in the frequency range of visible light, hence its name. It acts as a true "wavelength shifter" and in such a role is used as radiation sensor into high-energy physics, in medical imaging and in security applications. 1 The physics of scintillation, which is a complex multi-scale phenomenon (see, e.g., Vasil'ev and Getkin 2 ) can be described within a continuum approach at three scales: at a microscopic scale, the incoming energy E generates a population of charged energy carriers which moves in straight directions for few nanometer 3 and whose density N = N(E) can be found by the means of approximated solutions of the Bethe-Bloch equation. 4-7* These energy carriers wander and migrate within a greater region either generating other energy carriers or recombining with emission of photons h𝜈. In the process, some energy is lost, and a scintillator is a material in which

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Global Renormalised Solutions and Equilibration of Reaction–Diffusion Systems with Nonlinear Diffusion

et al. 2023

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The global existence of renormalised solutions and convergence to equilibrium for reaction–diffusion systems with nonlinear diffusion are investigated. The system is assumed to have quasi-positive nonlinearities 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 nonlinear diffusion. When the reaction network does not admit boundary equilibria, the complex balanced equilibrium is shown, by using the entropy method, to exponentially attract renormalised solutions in the same compatibility class. This convergence extends even to a range of nonlinear 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.

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Global renormalized solutions to reaction-cross-diffusion systems with self-diffusion

Cited by 16 publications

References 32 publications

Trend to equilibrium of renormalized solutions to reaction–cross-diffusion systems

Trend to equilibrium of renormalized solutions to reaction–cross-diffusion systems

A brief overview of existence results and decay time estimates for a mathematical modeling of scintillating crystals

Global Renormalised Solutions and Equilibration of Reaction–Diffusion Systems with Nonlinear Diffusion

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