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

作者：Heck Horst; Kim Hyunseok^*; Kozono Hideo

来源：Mathematische Annalen, 2013, 356(2): 653-681.

DOI：10.1007/s00208-012-0861-6

Consider the stationary Navier-Stokes equations in an exterior domain with smooth boundary. For every prescribed constant vector and every external force , Leray (J. Math. Pures. Appl., 9:1-82, 1933) constructed a weak solution with and . Here denotes the dual space of the homogeneous Sobolev space . We prove that the weak solution fulfills the additional regularity property and without any restriction on except for . 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 and 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 in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case .

• 出版日期2013-6