Mathematical studies on some models arising in chemotaxis and magnetohydrodynamic turbulence

Abstract

This thesis is mainly focused on the theoretical studies on some models arising in chemotaxis and magnetohydrodynamic turbulence. The main results of this thesis consist of the following three parts. 1. A quasilinear parabolic volume-filling chemotaxis model with critical sensitivity in two dimensions is considered. In this study, a threshold number is explicitly found such that the solution exists globally with uniform-in-time bound or blows up if the initial cell mass is less than or greater than this number. Furthermore we determine the blowup time is infinite under certain conditions on the decay rate of the chemotactic sensitivity. 2. We consider the initial-boundary value problem of the attraction-repulsion Keller-Segel (ARKS) chemotaxis model describing the quorum effect in chemotaxis and the aggregation of Microglia in the central nervous system in Alzhemer' disease. First, we study the asymptotic behavior of solutions to the ARKS chemotaxis model in one dimension, where we obtain the uniform-in-time boundedness of solutions and prove that the model possesses a global attractor. For a special case where the attractive and repulsive chemical signals have the same degradation rate, we show that the solution converges to a stationary solution algebraically as time tends to infinity if the attraction dominates. In two dimensional spaces, we show that if the repulsion dominates over attraction, then the global classical solutions exist with uniform-in-time bound for large initial data. Moreover we present a Lyapunov function at the first time for the irreducible three-component attraction-repulsion chemotaxis model which plays a central role to obtain our results. 3. We establish the asymptotic nonlinear stability of solutions to the Cauchy problem of a strongly coupled Burgers system arising in magnetohydrodynamic (MHD) turbulence. We show that, as time tends to infinity, the solutions of the Cauchy problem converge to constant states or rarefaction waves with large initial data, or viscous shock waves with arbitrarily large amplitude, where the precise asymptotic behavior depends on the relationship between the left and right end states of the initial value. Our results confirm the existence of shock waves (or turbulence) numerically found in the literature.