| Harmonic analysis, which has been developing for nearly two hundred years, is a very active branch in mathematics, and its methods are used in almost any other branch of mathematics. In particular, its application to partial differential equation,which is concerned by more and more mathematicians, is a very active branch in mathematics research at present. For example,The Calderon-Zygmund operators of the third era and T(1),T(b) theorem offer the theoretic base of potential method to solve a class of elliptic boundary problems in non smooth domains[46]; On the other hand, the Lp-Lq estimates and space-time estimates of the linear evolution equations offer the nonlinear evolution equations new work spaces ,which is established by the estimates of oscillatory integral and potential. We can see a list of reference in the homepage of T. Tao(http://www.math.ucla.edu/tao); In addition, Littlewood-Paley theory and the real theory of Hardy spaces in harmonic analysis, turned out to be a simple and powerful tool in solving wave map equation, Navier-Stkoes equation and Euler equation [18, 27, 47, 48, 49, 58, 59].In this paper we prepare to consider several problems in the partial differential equations by the techniques of harmonic analysis. It is divided into four chapters: In Chapterl,we study the regularity of weak solutions to the Navier-stokes equations; In Chapter2, we study the Cauchy problem for the modified Navier-Stokes equations; In Chapters, we study the global well-posedness for a generalized KdV equation; In the last chapter, we study the global well-posedness of a semilinear Schrodinger equation. In the following, we will introduce the main content of every chapter.Chaperl We consider the initial value problem for the Navier-stokes equationwhere is the velocity field evaluated at the point x Rn and at time is the pressure field. and are given as the external force and the initial velocity respectively. (0.0.16) decribes the motion of a viscous incompressible fluid.Suppose that a is in L2. J. Leray and E.Hopf[43, 33] construct a global weak solution . It is well known that the solution is unique and becomes regular in two dimensions; see for instance,R. Temam [63]. In dimensions n> 3, however, the question of regularity as well as uniqueness of weak solutions has remained open. There are two ways to develop the regularity theory for (NS). One is to give a regularity criterion on weak solutions and the other is to study a better partial regularity of weak solutions. For the latter, L. Caffarelli, R. Kohn, L. Nirenberg[11] proved that the Hausdorff measure of the singular set is zero, which is the best result atpresent. J. Serrin[52, 53] is the pioneer in the former research, later on, Fabes, Jones and Riviere[28], Sohr[54], Giga[30], Struwe[56] and Takahashi[57] extended Serrin's regularity criterion: weak solution in the classare necessarily regular. H. Beirao da Veiga[1] further extended their result , and proved that if , then u is regular.A natural question is: can we find a wider regularity class of weak solutions? The main purpose of this chapter is to study this problem, and find a wider regularity class. Its main idea is originated from the work of T. Tao[60. 61] about the regularity of the wave map equation in the critical Sobolev space. The key point of our proof is the Littlewood-Paley trichotomy decomposition of the nonlinear term (u .u) in the Navier-Stokes equation. Our main results are:Chapter2 We consider the Cauchy problem for the modified Navier-Stokes equationwhere u = (u1 (t, x), un(t, x)) is the velocity field evaluated at the point x Rn and at time t € (0, T), p = p(t. x} is the pressure field, is the initial velocity, u>0 is the viscous coefficient.In this chapter, we will focus on establishing the existence and uniqueness of strong solutions. In two dimension, S. Tourville[64] gave the systemic research. Based on the Lp*Lt estimates of the semigroup S(t)(see section2.2) and the method of T. Kato [34], we obtain... |
