| Project/Area Number |
12640200
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | TOHOKU UNIVERSITY |
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Principal Investigator |
NAGASAWA Takeyuki Tohoku University, Math. Inst., Associate Professor, 大学院・理学研究科, 助教授 (70202223)
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| Co-Investigator(Kenkyū-buntansha) |
TSUTSUMI Yoshio Tohoku University, Math. Inst., Professor, 大学院・理学研究科, 教授 (10180027)
TAKAGI Izumi Tohoku University, Math. Inst., Professor, 大学院・理学研究科, 教授 (40154744)
SHIMAKURA Norio Tohoku University, Math. Inst., Professor, 大学院・理学研究科, 教授 (60025393)
FUJIIE Seturo Tohoku University, Math. Inst., Lecturer, 大学院・理学研究科, 講師 (00238536)
KOZONO Hideo Tohoku University, Math. Inst., Professor, 大学院・理学研究科, 教授 (00195728)
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| Project Period (FY) |
2000 – 2001
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| Keywords | Navier-Stokes equations / Energy identity / Energy inequality / Weak solution / Regurality / Uniqueness |
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| Research Abstract |
Though the mathematical research on the Navier-Stokes equations that describe the motion of incompressible fluid has a long history, the theory has not completed yet. In particular concerning the problems of the regularity and uniqueness of weak solutions, we have only partial answers. A weak solution satisfies equations only in some weak sense, and therefore it may have singular points. If a solution is smooth, then it preserves energy with respect to time variable. For some weak solutions we can show the non-increasing property of energy, but it is uncertain whether they preserve energy or not. It is called the "energy inequality." For weak solutions the integrability of time-derivative is unclear only from the definition. This is why we cannot show the preservation of energy. This fact suggests that time-derivative is a singular measure with respect to "time". Consequently we must consider the integral of this "singular" part to show the preservation of energy.
We know that it is possible to estimate the decrease of energy from below by use of fractional time-derivative for the weak solution constructed by the method of discrete Morse flow. This is a new estimate. The purpose of this research is to study the possibility of such a refinement for any weak solution. For precise, we clarified the following facts. We consider weak solutions as functions which map to the space of square-integrable functions. Then the decrease of energy is related to the limit of the 1/2-time-difference of solutions in the sense of Nikol'skii. If we assume that the limit is zero with or without some speed, then we can show the energy identity with an additional term which compensates the decrease of energy. Furthermore without the assumption of the existence of limit, we can show the energy identity with another additional term of different expression. The difference of expression comes from that of topology of convergence of limit of time-difference.
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