A Study On The Dispersion Equation In Modular Space

This PH.D thesis focuses on the study of dispersive equations on modulation spaces. The research on dispersive equations has a long history and rich theoret-ical system. Recently, it is the uniform decomposition that has become of great importance on the improvement of dispersive estimates. So studying dispersive equations on modulation spaces is feasible. There are two kinds of question about the well-posedness:one is the global well-posedness with small initial data in M2,1s, and the other is the local well-posedness for generalized Mp,1 initial data. For the former, we extend the global well-posedness with small initial data and show that the global well-posedness with small initial data essentially depend on the ones with low regularity. We also obtain a strong blow up criterion. Then for the latter, we establish the perturbation theory by dividing the time interval. Moreover, we investigate the regularity of the solution in modulation spaces, that is, the solution keeps its regularity along the lifetime. We also get some scatter-ing results. Thanks to the advantage of the dispersive estimates on modulation space, one can get the wide range of index of the Strichartz estimates on mod-ulation space. We establish the Strichartz estimates on α-modulation spaces. Finally, we study the wellposedness of the generalized Schrodinger equation.We now list our main content and main results briefly in each chapter.In Chapter 1, we introduce the definitions and equivalent characterizations of some function spaces, especially the modulation spaces, α-modulations space which we study in this thesis. We show the basic properties and make a distinc-tion about the free semigroup estimate between modulation spaces and Lebesgue spaces. And we also reveal the advantage of the estimates on modulation spaces.In Chapter 2, we investigate properties of Schrodinger equation, and es-tablish the local theory, blowup criterion, perturbation theory. The scattering theory and the regularity of solutions are also considered here.In Chapter 3, we present some Strichartz estimates for dispersive equations, containing the Schrodinger equation, non-elliptic equation, and the wave equa- tion.In Chapter 4, we mainly investigate the wellposedness theory for high order Schrodinger equations, including the global wellposedness with small M2,1s initial data and that with a loss of regularity with small Mp,1s initial data.