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下記の通り、サイエンスフロンティア（数学専攻）を開講いたします。

講師紹介

招聘教授：In-Jee Jeong（ソウル国立大学）

講義タイトル

Takeuchi-Mizohata condition and wellposedness of degenerate dispersive systems

講義日程、場所

日程：5月29日（水）2限

　　　  30日（木）3限

              31日（金）２限 

場所：理学部E棟408（杉本）

講義内容

The goal of these lectures is to give an introduction to the initial value problem for dispersive PDEs, focusing on concrete examples such as the Schrodinger and KdV equations. We will start from the most basic dispersive equations and then proceed to more complicated systems, where two major sources of complications are: (1) potential degeneracy of the principal term (the term involving the largest number of derivatives) of the PDE and (2) presence and slow decay in space of the coefficient of the sub-principal term. These two issues are naturally intertwined, since while one may perform a spatial change of variables which normalizes the principal term coefficient, it generates a problematic sub-principal term in general. When such complications arise, wellposedness (existence and uniqueness of the solution) of the initial value problem depends in a delicate way. Interestingly, such dispersive PDEs arise in various physical models, including water waves and magneto-hydrodynamic systems.

In the first part, we begin with basic properties of linear dispersive equations, such as decay and smoothing estimates, the primary examples being linear Schrodinger and KdV equations. These estimates will lead to various wellposedness results for nonlinear dispersive equations. We first discuss these equations in the simplest case when the principal term is of constant coefficient, where the Fourier transform method is very effective. Then we introduce  wave packets, which can be considered as a replacement of Fourier transform adapted to the non constant coefficient case.

In the second part, we consider the initial value problem for the linear Schrodinger equation with a sub-principal (term involving one spatial derivative of the solution). Takeuchi and Mizohata proved a necessary and sufficient condition on the coefficient of the sub-principal term for L^2 wellposedness. This condition played a fundamental role in the subsequent development in the wellposedness theory for dispersive equations. We provide a proof of a quantitative version of Mizohata's theorem and its generalizations, which is based on construction of wave packets and use of the L^2 testing argument. We also perform a similar analysis for the KdV equation case.

In the last part, we discuss the initial value problem for nonlinear and degenerate dispersive equations, where "degenerate" means that the coefficient of the principal term vanishes at some point in space. Using wave packets, we prove a general "illposedness" result for such equations in standard Sobolev spaces, in the sense that even for infinitely smooth and compactly supported initial data, there may not exist any smooth solutions. Finally, we shall discuss a way to obtain wellposedness theory for such equations using weighted Sobolev spaces and modified energy estimates, the general principle being that the function spaces should be adapted to the "geometry" of the solution.

受け入れ教員:阿部健