Linear Stability And Asymptotic Stability Of Laminar Flow Solutions Of Two Types Of Incompressible Fluid Systems

The research content of this paper is divided into two parts.The first part: we studied the rotation between two concentric rotating cylinder effect and heat convection effect under coupling action of stability and instability phenomenon,through the Normal-Modes method,we present the linearized system in the oscillation frequency,axial wavelength and thermal gradient under the condition of stability and instability criterion,and the tiny disturbance of axisymmetric case we also are simulated.Part two: we study the exponential stability of the two dimensional incompressible MHD equations layered on the periodic region T~2= [0,1] × [0,1].The stability mechanism that makes our results possible is dissipation due to the mixed effects of background flows.To be more precise,although the magnetic potential is controlled by the transport equation,the algebraic structure of incompressible conditions can be used to show that the linearized MHD equations retain a non-local damping mechanism,and we demonstrate this stability result by combining energy estimation and decay estimation.