Turbulence and singularity of the Einstein equation with a negative cosmological constant studied from classical turbulence theory

Research Project

Project/Area Number	16K13850
Research Category	Grant-in-Aid for Challenging Exploratory Research
Allocation Type	Multi-year Fund
Research Field	Mathematical physics/Fundamental condensed matter physics
Research Institution	Kyoto University
Principal Investigator	Matsumoto Takeshi 京都大学, 理学研究科, 助教 (20346076)
Project Period (FY)	2016-04-01 – 2018-03-31
Project Status	Completed (Fiscal Year 2017)
Budget Amount *help	¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000) Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000) Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords	古典乱流のカスケード / スケーリング則 / 偏微分方程式の爆発解 / 乱流理論 / 重力方程式
Outline of Final Research Achievements	It has been known that a certain spherical symmetric Einstein equation with a negative cosmological constant has a turbulent solution by a numerical simulation. The equation has a global conservation law. The turbulent solution successively generates small-scale activities by respecting a specific scaling law. The generation occurring in an accelerated manner is considered to reach the infinitesimally small scale in a finite time. We studied this turbulent solution and its singular behavior with the method of analyzing turbulence of the Navier-Stokes equations. In particular, we adopted the picture of the energy cascade in the Navier-Stokes turbulence to the conservative quantity of the turbulent solution. So far we were not able to explain the exponent of the power-law scaling law phenomenologicaly. Nevertheless, the cascade picture shed a new light to the small-scale generation mechanism of the turbulent solution.