2020 Fiscal Year Annual Research Report
New perspectives on space-time estimates for dispersive equations
| Project/Area Number |
19H01796
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| Research Institution | Saitama University |
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Principal Investigator |
BEZ NEAL 埼玉大学, 理工学研究科, 准教授 (30729843)
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| Co-Investigator(Kenkyū-buntansha) |
杉本 充 名古屋大学, 多元数理科学研究科, 教授 (60196756)
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| Project Period (FY) |
2019-04-01 – 2023-03-31
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| Keywords | Strichartz estimate / Fermions / Pointwise convergence |
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| Outline of Annual Research Achievements |
Research this year has continued on the main goal of achieving a full and systematic theory of Strichartz estimates for orthonormal systems. Two projects in this direction now have been completed. The first significantly extends work of Frank-Sabin and established several new results in the case of the wave equation, Klein-Gordon equation, and the fractional Schrodinger equation. Secondly, a project concerning Carleson's problem for fermions has been completed. In this paper, we initiate the study of the pointwise convergence problem for infinitely many fermions, and extend a classical single-particle result of Kenig-Ponce-Vega in one dimension. These project raise further questions which shall be addressed later in the project.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Progress on the main goals of the research project related to Strichartz estimates for orthonormal systems is proceeding well, and during its development some related, but unexpected, new directions of research have emerged. This includes the version of Carleson's problem for fermions and this has given fresh impetus to the project.
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| Strategy for Future Research Activity |
It is planned to continue to strive to find a full and systematic theory of Strichartz estimates for orthonormal systems. Despite the progress made on this on this project up to now, to some extent, the theory still remains at the level of specific cases. A major goal is to find new ideas which will allow for a more abstract theory. Furthermore, it is planned to extend our results on Carleson's problem for fermions in several respects, including the effect of higher regularity on the initial data.
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