2015 Fiscal Year Final Research Report
Stochastic Processes and Statistical Phenomena behind Partial Differential Equaitons
| Project/Area Number |
23340030
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | Meiji University (2012-2015)
Osaka University (2011) |
|---|
Principal Investigator |
NAWA Hayato 明治大学, 理工学部, 教授 (90218066)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
SAKAJO Takashi 京都大学, 大学院理学研究科, 教授 (10303603)
YOSHIDA Nobuo 名古屋大学, 大学院多元数理科学研究科, 教授 (40240303)
FUKUIZUMI Reika 東北大学, 大学院情報科学研究科, 准教授 (00374182)
Matsumoto Takeshi 京都大学, 大学院理学研究科, 助教 (20346076)
|
|---|
| Co-Investigator(Renkei-kenkyūsha) |
AKAHORI Takafumi 静岡大学, 大学院工学研究科, 准教授 (90437187)
KIKUCHI Hiroaki 津田塾大学, 学芸学部, 講師 (00612277)
|
|---|
| Research Collaborator |
Gadi Fibich Tel Aviv University, 教授
Anne de Bouard Ecole Polytechnique, CNRS, director
OHKITANI Koji The University of Sheffield, 教授
|
|---|
| Project Period (FY) |
2011-04-01 – 2016-03-31
|
| Keywords | 非線形偏微分方程式 / 確率微分方程式 / 乱流 / 非圧縮性オイラー方程式 / 散逸的弱解 / 非線形シュレーディンガー方程式 / 爆発解 / 重対数法則 |
|---|
| Outline of Final Research Achievements |
Revisiting Kolmogorov's scaling laws and Onsager's conjecture, we made an assessment of their mathematical relevance from the view point of stochastic processes. Then we employed the Karman-Howarth-Monin relation as the energy dissipation rate to propose a new mathematical model of turbulence in the light of dissipative weak solutions of the incompressible Euler equations of which our sample space of turbulence consists. Besides, we conducted a numerical computation to verify the existence of a Gibbs measure on our sample space. We also investigated the blowup problem for the pseudo-conformally invariant nonlinear Schroedinger equations simultaneously. We established the loglog-law on the blowup rate for a class of blowup solutions by means of Nelson diffusions. Through out our project, we learned the importance of the use of our idea and method to be enhanced and to investigate other type of nonlinear PDEs, which led us to a new KAKENHI project continuing this attempt further.
|
| Free Research Field |
非線形偏微分方程式，応用確率過程論，変分法
|
