Well-posedness And Gevrey Regularity Of Debye-Hückel System In Fourier-Besov Space

The Debye-Hückel system originated in the late 19 th century and was originally used as a basic model to describe the diffusion of ions in electrolyte.In the 1980 s,scholars began to study it mathematically,focusing on boundary value problems,wellposedness,asymptotics and Gevery regularity.In this paper,we prove that the Debye-Hückel system and fractional Debye-Hückel system are wellposed with small initial values in Fourier-Besov space,Gevrey regularity of the mild solutions,and obtain the time decay estimates of the mild solutions.The overall structure of this paper is as follows:In the first chapter,we introduce the origin and recent research status of DebyeHückel system and fractional Debye-Hückel system respectively,analyze the results of scholars on the well-posedness and analyticity of the equations in detail,then we describe the work and conclusions of this paper,and explain the main proof methods.In the second chapter,we introduce some definitions of symbols and formulas,bony decomposition theory in Littlewood Paley theory,Besov space and Fourier-Besov space,and give some properties of functions in Fourier-Besov space.In the third chapter,we first rewrite the equation into the form of integral equation according to Duhamel’s principle,and then estimate the linear term and nonlinear term of the equation respectively.We prove the well-posedness of Debye-Hückel system in Fourier-Besov space by using Fourier localization method,Bony decomposition theory and key inequalities in functional analysis,On this basis,we obtain that the mild solution is Gevrey regular.According to the above conclusion,we can deduce that the time decay rate of the mild solution in the Fourier Besov space is about t^-δ/2.In the fourth chapter,we prove that the fractional Debye-Hückel system still has the properties of Debye-Hückel system.The proof idea is consistent with that in Chapter three,but we need to pay special attention to the treatment of parameters.Firstly,we prove its well-posedness: Under certain parameter constraints and the initial value is small enough,there must be a global time mild solution.Then we prove that the mild solution is Gevrey regular,and obtain that the time decay rate of the mild solution is about t^-δ/2β.