Studies On Fluid Equations And Interpolation Problems In Harmonic Analysis

Navier-Stokes equations and magneto-hydrodynamic equations are the hot topics in modern mathematics,and they are the important application area of harmonic analysis.In particular,Fefferman proposed that Navier-Stokes equations problem is one of millennium problem,which shows its position in harmonic analysis and differential equations.This dissertation mainly studies the uniform analytic solution of the fractional incompressible Navier-Stokes equations in Fourier-Herz spaces and the well-posedness of magneto-hydrodynamic equations in new iteration space which is established by mutually independent microlocal information.If classical Fourier transform is used in the study of uniform analyticity,then the skills in the latter are entirely atypical,such as the technique of wavelet.The study of interpolation is one of the sources of harmonic analysis,which leads to the development of harmonic analysis,operator theory,approximation theory and differential equations.For the real interpolation space of Besov space,since some interpolation spaces will fall out of Besov spaces,Peetre suggested to give the concrete expressions of these interpolation spaces.On this issue,we consider the real interpolation of Besov space for the key index p.The wavelet developed by Meyer,Daubechies,etc.has influenced the development of modern mathematics.For the whole thesis,wavelet is our key skill.The main results of this doctoral thesis are as follows.1.We present the existence of the uniform analytic solution of the Cauchy problem for fractional incompressible Navier-Stokes equations in critical FourierHerz spaces.The main strategies are to prove that the existence of the uniform analytic solution is equivalent to the boundedness of convolution inequality on Herz space.2.In the new iteration space constructed by parametric Meyer wavelets,we establish the well-posedness of the Cauchy problem for incompressible magnetohydrodynamic equations.The velocity field and magnetic field are expanded according to parametric wavelets at specific thresholds.The evolution of these two types of fields are described by the relationship of parameterized flag microlocal quantities over binary time intervals.3.We study the real interpolation spaces of homogeneous Besov space Bps,q for the key index p.Firstly,we get some relationship between(Bp0s,q,Bp1s,q)?,r and Besov-Lorentz spaces,which depending on whether q?r or q?r.Furthermore,applying Meyer wavelets,we obtain a precise description of(Bp0s,q,Bp1s,q)?,r.