Weak solutions of the stationary Navier-Stokes equations for a viscous incompressible fluid past an obstacle

Consider the stationary Navier-Stokes equations in an exterior domain Ω ⊂ ℝ³ with smooth boundary. For every prescribed constant vector u_∞ ≠ 0 and every external force f ∈ Ḣ₂^-1(Ω), Leray (J. Math. Pures. Appl., 9:1-82, 1933) constructed a weak solution u with ∇u ∈ L₂(Ω) and u-u_∞ ∈ L₆(Ω). Here Ḣ₂^-1(Ω) denotes the dual space of the homogeneous Sobolev space Ḣ¹₂(Ω). We prove that the weak solution u fulfills the additional regularity property u-u_∞ ∈ L₄(Ω) and u_∞ · ∇u Ḣ₂^-(Ω) without any restriction on f except for f ∈ Ḣ₂^-1(Ω). As a consequence, it turns out that every weak solution necessarily satisfies the generalized energy equality. Moreover, we obtain a sharp a priori estimate and uniqueness result for weak solutions assuming only that {double pipe}f{double pipe}_Ḣ-12(Ω) and {pipe} u_∞{pipe} are suitably small. Our results give final affirmative answers to open questions left by Leray (J. Math. Pures. Appl., 9:1-82, 1933) about energy equality and uniqueness of weak solutions. Finally we investigate the convergence of weak solutions as u_∞→0 in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case u_∞≠0.