L®-results of the stationary Navier-Stokes flows around a rotating obstacle

In this article, we study the stationary motion of an incompressible Navier-Stokes flow past a rotating body R3 \Omega, which is also translating about the x1-axis with nonzero constant velocity -ke(1). Assume that the body rotates with a constant angular velocity omega = |omega|e(1) and the external body force is given by f - divF. Our main purpose is to prove the existence of a weak solution u satisfying del u is an element of L-r(Omega) for arbitrary large F is an element of L-r(Omega), 3/2 <= r <= 2, provided that the flux of ub is sufficiently small with respect to the viscosity nu. This is obtained by the regularity result that mainly use localization procedure. In the end, we also present an optimal existence theorem for strong and very weak solutions.