Small stable stationary solutions in Morrey spaces of the Navier-Stokes equation (Q1917583)

Recently, many authors studied the Cauchy problem for the Navier-Stokes equation in \(\mathbb{R}^n\) in the framework of Morrey spaces. For example, Giga and Miyakawa and Kato gave sufficient conditions for the unique existence of time-global solutions. For previous papers related to this problem, see the references of the authors [Commun. Partial Differ. Equations 19, No. 5-6, 959-1014 (1994; Zbl 0803.35068)], which studied the above Cauchy problem in new function spaces larger than the corresponding Morrey spaces. However, these papers considered only the case where the external force vanishes identically or decays as \(t\to \infty\). The purpose of this paper is to generalize the results on the global solvability in the works above to the case with a stationary external force by showing the unique existence and the stability of a small stationary solution in suitable Morrey spaces under appropriate assumptions on the external force.