Fourier - Besov Space And Oscillation Integral And Its Application

The methods and techniques, as well as the systematic concept and theory in harmonic analysis, play an important role in solving the practicality problem in PDE. In this thesis, we use the Littlewood-Paley theory to specialize in the Fourier-Besov spaces, including the definition, the properties and the applications in generalized Navier-Stokes equations. Besides, we study the estimates and well-posedness of damped wave equation by the techniques from oscillatory integral.Now we sketch the outline of this thesis.In Chapter 1 we introduce the development and the background of Fourier-Besov spaces. We also review some results on generalized Navier-Stokes equations, unimodular Fourier multipliers, damped wave equation. In contrast to the known results, the main theorem in this thesis also state in this chapter.In Chapter 2 we focus on the definition and properties of Fourier-Besov spaces. We first review the basic knowledge of Littlewood-Paley theory and Besov spaces. Similarly with that of Besov spaces, we define the Fourier-Besov spaces by dyadic decomposition. Based on this definition, we prove some most common and useful properties of this spaces, including the equivalence, the relationship with other spaces, the embedding, the interpolation and especially the product estimate etc. All the contents in this chapter are also preliminaries for the following two chapters.In Chapter 3 and 4 we apply the Fourier-Besov spaces to the generalized Navier-Stokes equations. In Chapter 3 we prove global existence for small initial data in Fourier-Besov spaces and the long-time decay property. Especially, we prove the well-posedness in the endpoint case β= 1/2. In Chapter 4 we prove the blow-up criterion and space analyticity for solutions in these spaces. The key to prove the blow-up criterion is to solve the equation in other spaces which is continue on time. And to prove the space analyticity we use Gevrey class.In Chapter 5 we focus on the application of oscillatory in higher order damped wave equation. By the form of the fundamental solution of this equation, the kernel behave like heat operator in lower frequency and is a kind of oscillatory integral in higher frequency. So we calculate the point-wise estimates for the kernel in three parts, and obtain the Lp estimate of the fundamental solution. In the last, we apply these estimates to prove global existence for the higher order damped wave equation.