"On the long time behaviour of irreversible enzyme systems"Einav, AmitReaction-Diffusion systems appear naturally in many biological and chemical phenomena with underlying chemical reactions. A commonly used method to investigate such systems is to explore their entropies, which are physically motivated Lyaponuv functionals. An extremely prolific approach to the study of the long time behaviour of these (and many other) systems is the so-called Entropy method, where one attempts to find a functional inequality connecting the aforementioned entropies and their production, which is the absolute rate of the time change of these entropies. This approach is frequently used in the investigation of reversible reaction-diffusion systems where one is guaranteed a strictly positive equilibrium state, which is essential to being able to use most of the natural entropies of such systems. The method can’t be directly applied, however, in situations where some substances that are involved in the reactions disappear over time – a common feature in irreversible systems. In our talk, based on recent work with Marcel Braukhoff and Bao Quoc Tang, we will present a new approach to circumvent the problems that arise when dealing with decaying substances by considering “cut off” entropies which, when combined with a decreasing mass term, give a new entropy functional for which we can apply the ideas of the entropy method. We will use this approach in the setting of a well known irreversible enzyme system to not only find explicit rates of convergence for the modification of the natural Boltzmann entropy, but also for the $L^\infty$ norm of the associated concentrations. |
« back