2014 Fiscal Year Annual Research Report
New Frontiers in Kinetic Equation Theory
| Project/Area Number |
26887008
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| Research Institution | Saitama University |
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Principal Investigator |
BEZ NEAL 埼玉大学, 研究機構研究企画推進室, 准教授 (30729843)
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| Project Period (FY) |
2014-08-29 – 2016-03-31
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| Keywords | Smoothing estimates / Dispersive equations |
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| Outline of Annual Research Achievements |
Progress has been achieved on numerous fronts, primarily by pioneering certain techniques from harmonic analysis and geometric analysis to problems in partial differential equations.
The Funk-Hecke theorem, in particular, has proved to be particularly effective. This is a result from classical harmonic analysis and has been used to make several breakthroughs on this grant. For example, a number of new results have been obtained in the context of Kato-smoothing estimates and related trace estimates, especially regarding the identification of optimal constants and extremal input functions in these estimates. Such estimates are well-known to have numerous consequences in the theory of dispersive and wave-like partial differential equations and it is anticipated that the new results obtained on this grant will find several applications in these directions.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
The original plan was to apply powerful techniques from harmonic and geometric analysis to develop the theory of kinetic equations. In addition to making progress on this goal, such techniques have also been used to solve a number of problems in closely related topics, including smoothing estimates for dispersive and wave-like partial differential equations.
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| Strategy for Future Research Activity |
In the next year of this grant, the techniques which have been utilised in the research achievements so far will be adopted to push forward the rigorous mathematical theory of the kinetic transport equation. More specifically, the Funk-Hecke theorem will be used to understand the smoothing effect of the velocity averages for the solution of the kinetic transport equation in the case of square-integrable initial data. To go beyond this and to provide a fuller theory in more general Lebesgue spaces, other techniques from harmonic analysis will be adopted.
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