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

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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.