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

Abstract

Consider the stationary Navier–Stokes equations in an exterior domain $$\varOmega \subset \mathbb{R }^3 $$ with smooth boundary. For every prescribed constant vector $$u_{\infty } \ne 0$$ and every external force $$f \in \dot{H}_2^{-1} (\varOmega )$$ , Leray (J. Math. Pures. Appl., 9:1–82, 1933) constructed a weak solution $$u $$ with $$\nabla u \in L_2 (\varOmega )$$ and $$u - u_{\infty } \in L_6(\varOmega )$$ . Here $$\dot{H}^{-1}_2 (\varOmega )$$ denotes the dual space of the homogeneous Sobolev space $$\dot{H}^1_{2}(\varOmega ) $$ . We prove that the weak solution $$u$$ fulfills the additional regularity property $$u- u_{\infty } \in L_4(\varOmega )$$ and $$u_\infty \cdot \nabla u \in \dot{H}_2^{-1} (\varOmega )$$ without any restriction on $$f$$ except for $$f \in \dot{H}_2^{-1} (\varOmega )$$ . 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 $$\Vert f\Vert _{\dot{H}^{-1}_2(\varOmega )}$$ and $$|u_{\infty }|$$ 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 $$u_{\infty } \rightarrow 0$$ in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case $$u_{\infty } \ne 0$$ .