2023 Fiscal Year Final Research Report
Well-posedness of the Cauchy problem for nonlinear dispersive equations and its algebraic structure
| Project/Area Number |
17K05316
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Research Field |
Mathematical analysis
|
|---|
| Research Institution | Chuo University (2018-2023)
Nagoya University (2017) |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2017-04-01 – 2024-03-31
|
| Keywords | 分散型方程式 / 非線形 / 初期値問題 / 適切性 / KdV方程式 / シュレディンガー方程式 / 調和解析 |
|---|
| Outline of Final Research Achievements |
1, We consider the fifth order KdV type equations and prove the unconditional well-poseedness in the Sobolev space when its index is greater than or equal to 1. It is optimal in the sense that the nonlinear terms can not be defined in the space-time distribution framework when the index is less than 1. The main idea is to employ the normal form reduction and a kind of cancellation properties to deal with the derivative losses.
2, We consider the Cauchy problem of a class of higher order Schrodinger type equations with constant coefficients. By employing the energy inequality, we show the L2 well-posedness, the parabolic smoothing and a breakdown of the persistence of regularity. We classify this class of equations into three types on the basis of their smoothing property.
|
| Free Research Field |
偏微分方程式論
|
| Academic Significance and Societal Importance of the Research Achievements |
非線形分散型方程式の研究はここ30年ほど大きく進展しているが，これまで主に扱われてきたのは非線形の特異性がそれほど強くない場合であり，非線形項に高階の微分を含むような方程式に対する結果は限られていた．本研究はこのようなこれまで扱いにくかった方程式に対する研究手法を切り開いたという意味で学術的意義が高いといえる．
|
