Well-Posedness and Smoothing Effect for Nonlinear Dispersive Equations

where α is a real constant with 2α/3 ̸∈ Z and T > 0. In (1), all the parameters are normalized except for α. Equation (1) appears as a mathematical model for nonlinear pulse propagation phenomena in various fields of physics, especially in nonlinear optics (see [54], [27] and [1]). So far, equation (1) without the third order derivative, that is, the cubic NLS equation has attracted much mathematical and physical interest. Recently, as the ultra-short pulse has become important in the photonic crystal fiber, an increasing attention among theoretical and experimental physicists in nonlinear optics has been paid to the role of the third order dispersion in equation (1). From a viewpoint of the PDE theory, the well-posedness issue of the Cauchy problem for nonlinear evolution equations such as (1)-(2) is one of the most fundamental problems. The Cauchy problem is said to be locally (resp. globally) well-posed if the following three properties hold: (i) local (resp. global) existence of solutions, (ii) uniqueness of solutions, (iii) continuous dependence of solutions on initial data. We refer to the local and the global well-posedness as (LWP) and (GWP), respectively. The Caucy problem is said to be ill-posed if it is not well-posed. Many mathematicians have been studying what is the largest space where the Cauchy problem of a nonlinear evolution equation at hand is well-posed. Especially, there has been a great progress in nonlinear dispersive equations for the last two decades. In this note, we take equation (1) as an example to explain recent results obtained by the author in collaboration with Nobu Kishimoto (RIMS, Kyoto University), Tomoyuki Miyaji (Meiji Inst. Adv. Stud. Math. Sci., Meiji University), Tadahiro Oh (The University of Edinburgh) and Nikolay Tzvetkov (University of Cergy-Pontoise). First, in Section 2, we consider solving the Cauchy problem (1)-(2) in H for s < 0, elements of which may not be functions but distributions. When we work with the space consisting of distributions, the problem we immediately meet with is how the nonlinear term can make sense, because the product of distributions is not necessarily well-defined. In 1993, Bourgain [4] presented the so-called Forier restriction method with the Fourier restriction norm. The Fourier restriction norm and its variants succeeded in capturing specific features of nonlinear oscillations and have been applied to many nonlinear evolution equations describing nonlinear wave phenomena (see, e.g., [4], [13], [18], [22]-[26], [32], [34]-[36], [40], [42], [44]-[49], [57]-[60]). The Fourier restriction method also led a new mathematical insight into the nonlinear interaction