Research on the Navier-Stokes equations by interpolation spaces and perturbation theory.

用泛函分析的方法研究了描述流体运动的N-S方程在外部区域的定常解的存在性、唯一性和稳定性，对于n维空间，前人在这方面的工作主要采用函数空间L^n。在1997年4月到1998年3月期间，我们证明了当空间维为3时，函数空间L^3中的解只在非常有限的情况下存在，因此我们必须考虑一个更大的空间L^3，*>作为存在解的函数空间。此外，对于全空间中的N-S方程，我们给出了严格大于L^&lt；n，*&gt；的Morrey空间中存在唯一小驻定解的外力充分条件。我们进一步证明了在含有非Radon测度分布的函数空间中，上述驻点解在小初始扰动下是稳定的。我们在1998年4月至1999年3月期间研究了n&lt；大于或等于&gt；3的空间中的Navier-Stokes方程的外问题。然后证明了当外力为函数空间L^&lt；n/2，*&gt；中的势的一阶导数时，函数空间中存在唯一的小定常解。我们还证明了在函数空间L^&lt；n，*&gt；中，上述定常解在小初始扰动下是稳定的。这一结果适用于更一般的外力，而不是以前用势论方法得到的结果所能处理的外力，也适用于以前用泛函分析方法无法处理的三维情况。

We studied the existence, the uniqueness and the stability of stationary solutions of the Navier-Stokes equations in exterior domains, which is regarded as a model describing the motion of fluids, by functional analytic methods.For n-dimensional spaces, the function space L^n is mainly employed in previous works on this direction. We showed during the period from April, 1997 to March, 1998 that, when the space dimension is 3, solutions in the function space L^3 exist only in very limited cases, and hence we must consider a some-what larger space L^<3, *> as the function space in which solutions exist. Moreover, for the Navier-Stokes equation in the whole space, we gave a condition on the external forces sufficient for the unique existence of small stationary solutions belonging to the Morrey spaces, which is strictly larger than the space L^<n, *>. We furthermore showed that the stationay solutions above are stable under small initial perturbation in function spaces which contain distributions other than Radon measures.We studied the exterior problem of the Navier-Stokes equations in the space for n<greater than or equal>3 during the period from April, 1998 to March, 1999. We then showed the unique existence of small stationary solutions in the function space under the condition that the external forces are given as the first order derivatives of potentials small in the function space L^<n/2, *>. We also showed that the stationary solutions above are stable under small initial perturbations in the function space L^<n, *>. This result is applicable to external forces more general than those which can be treated by previous results obtained by potential theoretical methods, and is applicable to the 3-dimensional case which could not be treated by functional analytic methods before.