2004 Fiscal Year Final Research Report Summary
Research of the Navier-Stokes exterior problem by using dual semigmups and the Lorentz spaces
| Project/Area Number |
13640157
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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 |
Basic analysis
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| Research Institution | Waseda University (2002-2004)
Hitotsubashi University (2001) |
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Principal Investigator |
YAMAZAKI Masao Waseda University, Faculty of Science and Engineering, Professor (20174659)
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| Co-Investigator(Kenkyū-buntansha) |
SHIBATA Yoshihiro Waseda University, Faculty of Science and Engineering, Professor (50114088)
TANAKA Kazunaga Waseda University, Faculty of Science and Engineering, Professor (20188288)
FUJITA Takahiko Hitotsubashi University, Graduate School of Commerce, Professor (50144316)
ISHIMURA Naoyuki Hitotsubashi University, Graduate School of Economics, Associate Professor (80212934)
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| Project Period (FY) |
2001 – 2004
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| Keywords | Navier-Stokes equation / exterior domain / stationary solution / periodic solution / stability / boundary value problem / Lorentz spaces / real interpolation |
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| Research Abstract |
In a joint work with Yoshihiro Shibata, we obtained a sufficient condition on time-independent external forces for the unique existence of a stationary solution in a certain class of the Navier-Stokes equation in exterior domains of dimension n greater than or equal to 3 by using the duality between the Lorentz spaces and real interpolation. Our class is a natural generalization of the so-called physically reasonable solutions, and our suffirient condition gives a unified view for the case with zero velocity at infinity and the case with non-zero velocity at infinity.
Next, in a joint work with Yuko Enomoto and Yoshihiro Shibata, we verified the stability in the weak-Ln space of the stationary solution above for time-evolution under small initial perturbation in the weak-Ln space, and showed that the smallness above can be taken uniformly in the velocity at infinity of the stationary solution.
Furthermore, by using real interpolation for sublinear operators, we generalized these results for time-dependent external forces, and obtained a sufficient condition for the unique existence of the corresponding time-periodic or almost periodic solutions. We also showed the stability of these solutions in weak-Ln spaces under perturbations on the external forces and initial data uniform in the velocity at infinity of the solutions.
On the other hand, as a preparation for generalized the results above for general unbounded domains, we generalized the Lp-theory on the boundary value problem for the Stokes equation in a layer domain, in a joint work with Takayuki Abe for higher-order Sobolev spaces and Besov spaces, and obtained a sufficient condition on the external forces for the unique existence of the solution of the boundary value problem. In particular, we showed that the uniqueness of the solution fails in the case p=infinity, and that the Poiseuille flow can be characterized as the solution with zero as the external forces and boundary values.
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