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Research of the Navier-Stokes exterior problem by using dual semigmups and the Lorentz spaces

利用对偶半映射和洛伦兹空间研究纳维-斯托克斯外问题

基本信息

批准号：
13640157
负责人：
YAMAZAKI Masao
金额：
$ 2.43万
依托单位：
Waseda University (2002-2004)Hitotsubashi University (2001)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

项目摘要

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.

在与Yoshihiro柴田的合作工作中，我们利用Lorentz空间和真实的插值之间的对偶性，得到了一类Navier-Stokes方程在n ≥ 3维外区域上存在唯一定常解的外力与时间无关的充分条件.我们的类是所谓的物理上合理的解决方案的自然推广，我们的充分条件给出了一个统一的观点，在无限远的情况下，零速度和非零速度在无穷远的情况下。我们证明了在弱Ln空间中，在小的初始扰动下，上述时间演化定态解在弱Ln空间中的稳定性。Ln空间中，并证明了上述小性可以一致地取为定常解在无穷远处的速度，进而利用次线性算子的真实的插值，将这些结果推广到含时外力情形，得到了相应的时间周期解或概周期解存在唯一性的充分条件.在弱Ln空间中，我们证明了这些解在外力扰动和解的无穷远速度一致初值条件下的稳定性.另一方面，作为将上述结果推广到一般无界区域的准备，我们推广了层区域中Stokes方程边值问题的Lp-理论，在与Takayuki Abe的合作工作中，对高阶Sobolev空间和Besov空间，得到了边值问题解唯一存在的外力的一个充分条件.特别地，我们证明了在p=无穷大的情况下解的唯一性是不成立的，并且Poiffille流可以被描述为外力和边值为零的解.