The Navier-Stokes equation with distributions as initial data and application to self-similar solutions

In this paper we construct new function spaces of the Besov type from the Morrey spaces, and show that the Navier-Stokes equation and semilinear heat equations have unique time-global solutions with a bound for small initial data, which are not necessarily Radon measures, in some of the above function spaces. Thus we obtain some self-similar solutions of these equations provided that the initial data is a homogeneous function. We also constructed a local version of these function spaces, and give conditions on the initial data in these spaces sufficient for the unique existence of time-local solutions of the above equations with a bound near t = 0. Next, as a generalization of the above result on the Navier-Stokes equation, we consider the stationary Navier-Stokes equation with an external force, and give a condition on the external force sufficient for the existence of small solutions belonging to appropriate Morrey spaces. Further, the stability of the above stationary solution, in Morrey spaces larger than or equal to the original Morrey spaces, is shown under an appropriate condition. The stability of the above stationary solution in new function spaces introduced above is also verified. Also in this case, we obtain some self-similar solutions provided that the external force and the initial data are homogeneous. Part of the results of this paper is proved in [KYI] and [KY2].