Well-posedness And Partial Large Initial Value Solutions For Two Types Of Fluid Equations

As an important branch of mathematics,harmonic analysis has a great impact on many scientific research fields and practical applications.This article mainly focuses on the Navier-Stokes-Nernst-Planck-Poisson coupled system and Keller-Segel system of the following two types of fluid equations,studying their well-posedness and global solutions with partial large initial values in the Fourier-Besov space and Besov space.The main contents of this paper are as follows:In the first chapter,the research background and current research situation of Navier-Stokes-Nernst-Planck-Poisson equation and Keller-Segel equation at home and abroad are introduced respectively,and the relevant results of well-posedness and global solutions of the above two kinds of fluid equations in various spaces in recent years are comprehensively discussed,and the core content and main theorems studied in this paper are given.In the second chapter,we briefly review the homogeneous Littlewood-Paley decomposition theory,give the representation of the Fourier-Besov space and the related Chemin-Lener type space through the definition of the homogeneous Besov space,and give the Bernstein lemma,Bony decomposition,Minkowski inequality and other properties,as well as the estimates that need to be used in this paper.In the third chapter,we mainly study the well-posedness of the Navier-Stokes-Nernst-Planck-Poisson equation in the Fourier-Besov space and the global solution of the large initial value.By using Bony decomposition and Fourier localization techniques,we give the product estimation of the equation in the Fourier-Besov space,and obtain the global well-posedness by ignoring the assumption that the initial data is small enough,and prove that the global existence of the solution can be obtained only when the partial initial value is small enough.In the fourth chapter,we use Littlewood-Paley analysis to study the Cauchy problem of the parabolic Keller-Segel system in Besov space.With the help of the algebraic structure of the equation,we prove that the equation is globally well-posedness with small initial data.Furthermore,it is proved that in bounded cases,even if some initial values are large,the global solution still exists.In addition,by proving the Gevery regularity of the global solution,we also obtain the time decay rate of the global solution.