非线性化学反应扩散方程组重整化解的平衡收敛性
Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction-diffusion systems
摘要：
研究了非线性反应扩散方程组重正化解到平衡点的收敛性。所考虑的反应扩散系统来自具有质量作用动力学的化学反应网络，并满足复平衡条件。通过应用所谓的熵的方法，我们表明，如果系统没有边界平衡，那么任何重正化的解决方案指数收敛到复杂的平衡的速率，可以计算明确的有限维不等式。这个不等式是通过矛盾论证证明的，因此不是明确的。一个明确的证明方法，但是，提供了一个特定的应用程序建模可逆的酶反应，利用特定的结构的守恒定律。我们的方法也是有用的研究具有边界平衡的系统的平衡趋势。更确切地说，为了证明具有边界平衡点的系统收敛于平衡点，我们建立了一个充分条件，即沿系统轨迹的一个修正的有限维不等式。通过假设这个条件，这大致意味着系统产生太多的熵，以保持接近边界平衡无限的时间，熵方法显示指数收敛到平衡的重正化解决方案的复杂平衡系统的边界平衡。
英文摘要：
The convergence to equilibrium for renormalised solutions to nonlinear reaction-diffusion systems is studied. The considered reaction-diffusion systems arise from chemical reaction networks with mass action kinetics and satisfy the complex balanced condition. By applying the so-called entropy method, we show that if the system does not have boundary equilibria, then any renormalised solution converges exponentially to the complex balanced equilibrium with a rate, which can be computed explicitly up to a finite dimensional inequality. This inequality is proven via a contradiction argument and thus not explicitly. An explicit method of proof, however, is provided for a specific application modelling a reversible enzyme reaction by exploiting the specific structure of the conservation laws. Our approach is also useful to study the trend to equilibrium for systems possessing boundary equilibria. More precisely, to show the convergence to equilibrium for systems with boundary equilibria, we establish a sufficient condition in terms of a modified finite dimensional inequality along trajectories of the system. By assuming this condition, which roughly means that the system produces too much entropy to stay close to a boundary equilibrium for infinite time, the entropy method shows exponential convergence to equilibrium for renormalised solutions to complex balanced systems with boundary equilibria.
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