\(L^{r}\)-Results of the Stationary Navier–Stokes Equations with Nonzero Velocity at Infinity

Abstract

We study the stationary motion of an incompressible Navier–Stokes fluid past obstacles in \(\mathbb {R}^{3}\), subject to the provided boundary velocity \(u_{b}\), external force \(f = \textrm{div} F\), and nonzero constant vector \(k {e_1}\) at infinity. We first prove that the existence of at least one very weak solution u in \(L^{3}(\Omega ) + L^{4}(\Omega )\) for an arbitrary large \(F \in L^{3/2}(\Omega ) + L^{2}(\Omega )\) provided that the flux of \(u_{b}\) on the boundary of each body is sufficiently small with respect to the viscosity \(\nu \). Moreover, we establish weak- and strong-regularity results for very weak solutions. Consequently, our existence and regularity results enable us to prove the existence of a weak solution satisfying \(\nabla u \in L^{r}(\Omega )\) for a given \(F \in L^{r}(\Omega )\) with \(3/2 \le r \le 2\), and a strong solution satisfying \(\nabla ^{2} u \in L^{s}(\Omega )\) for a given \(f \in L^{s}(\Omega )\) with \(1 < s \le 6/5\), respectively.