Nonlinear Kinetic System Research Articles

Hypocoercivity and Uniform Regularity for the Vlasov--Poisson--Fokker--Planck System with Uncertainty and Multiple Scales

We study the Vlasov--Poisson--Fokker--Planck system with uncertainty and multiple scales. Here the uncertainty, modeled by random variables, enters the solution through initial data, while the multiple scales lead the system to its high-field or parabolic regimes. With the help of proper Lyapunov-type inequalities, under some mild conditions on the initial data, the regularity of the solution in the random space, as well as exponential decay of the solution to the global Maxwellian, are established under Sobolev norms, which are uniform in terms of the scaling parameters. These are the first hypocoercivity results for a nonlinear kinetic system with random input, which are important for the understanding of the sensitivity of the system under random perturbations, and for the establishment of spectral convergence of popular numerical methods for uncertainty quantification based on (spectrally accurate) polynomial chaos expansions.

Nonlinear hybrid kinetic system of microbial bioconversion in fed-batch culture

In this paper, the nonlinear kinetic system of fed-batch fermentation is investigated in the process of glycerol bio-dissimilation to 1,3-propanediol by Klebsiella pneumoniae. Taking into account the discontinuity of the process of adding glycerol and alkali and kinetics of microbial bioconversion in fed-batch culture, we present a nonlinear hybrid kinetic system of fed-batch fermentation in terms of the kinetic system of batch culture and continuous culture, which is well suitable for the factual fed-batch fermentation. Then we discuss the existence, uniqueness and boundedness of solutions to the hybrid system as well as the relative Lipschitz continuity and differentiability of solutions with respect to the initial-state-control pairs. Finally a numerical example is employed to carry out numerical simulation to the hybrid system. The results of numerical simulation reveal that the errors between experimental and simulated values by means of the hybrid system of fed-batch culture are less than those using the previous impulsive system.

Selection Theorem for Systems with Inheritance

The problem of finite-dimensional asymptotics of infinite-dimensional dynamic systems is studied. A non-linear kinetic system with conservation of supports for distributions has generically finite-dimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. This conservation of support has a biological interpretation: inheritance. The finite-dimensional asymptotics demonstrates effects of natural selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become increasingly narrow over time and move increasingly slowly. It is possible that these peaks do not tend to fixed positions, and the path covered tends to infinity as t goes to infinity. The drift equations for peak motion are obtained. Various types of distribution stability are studied: internal stability (stability with respect to perturbations that do not extend the support), external stability or uninvadability (stability with respect to strongly small perturbations that extend the support), and stable realizability (stability with respect to small shifts and extensions of the density peaks). Models of self-synchronization of cell division are studied, as an example of selection in systems with additional symmetry. Appropriate construction of the notion of typicalness in infinite-dimensional space is discussed, and the notion of completely thin sets is introduced. Key words: Dynamics; Attractor; Evolution; Entropy; Natural selection