Well-posedness For The Fractional Keller-segel Equations And Its Coupled Equations

This dissertion investigates the well-posedness for the fractional Keller-Segel(KS)equations derived from biology and the blowup of local smooth solutions for the incompressible Navier-Stokes-Poisson-Nernst-Planck(NSPNP)equations derived from electro-hydrodynamics.The fractional KS equations generalize the classical biological KS model,and describe the chemotaxis of biological cells to chemical stimulation in the growth environment.The incompressible NSPNP equations are strongly coupled systems by the incompressible NavierStokes(NS)equations in hydromechanics and the Poisson-Nernst-Planck(PNP)equations in electrodynamics,modeling the drift,diffusion and convection phenomena of charged particle in an isothermal,incompressible and viscous fluids.The dissertion consists of five parts:In Chapter 1,Introduction.We mainly give the research significance,background and the development of the related topics and introduce the main concent of the present dissertion.In Chapter 2,Preliminaries.We mainly introduce the basic theoretical knowledge and related lemma that need to be possessed in the process of proof in this dissertion,so as to derive the following.In Chapter 3,we are concerned with the well-posedness for the fractional KS equations.Based on the Bony paraproduct decomposition,contraction mapping theory and the properties of Besov spaces,we give the local well-posedness and the global well-posedness with small initial data of these equations for 1 < ? ? 2,generalize the previous results(H.Chen,etc.,2019).In particular,we also establish the global well-posedness with small initial data in homogeneous Besov spaces of these equations for ? = 1,this conclusion has not been discussed in the article(H.Chen,etc.,2019)and can be regarded as an innovative result based on it.In Chapter 4,we study the blowup of local smooth solutions for the incompressible NSPNP equations.Based on the Littlewood-Paley decomposition theory,energy inequality,interpolation inequality and embedding theorem,the BKM's blowup criterion in terms of horizontal component of velocity field in homogeneous Triebel-Lizorkin spaces is established for local smooth solutions,and generalize the previous results(Dong and Zhang 2010;Zhao and Bai 2016).In particular,this blowup criterion reveals that the horizontal component of velocity field is more important than the density functions of charged particles in the blowup theory of the equations.In Chapter 5,we summarize the main research conclusions of this dissertion and make a prospect for future research.