Singular Weak Solutions Near Boundaries in a Half-space Away from Localized Force for the Stokes and Navier-Stokes Equations

Abstract

We prove that there exists a weak solution of the Stokes system with a non-zero external force and no-slip boundary conditions in a half-space of dimension three or higher such that its normal derivatives are unbounded near the boundary. A localized, divergence-free singular force causes, via a non-local effect, singular behavior of normal derivatives of the solution near the boundary, although this boundary is away from the support of the external force. The constructed solution is a weak solution with finite global energy, and it (can be compared to the one in Seregin and S̆verák (Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 385 (2010), Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 41, 200–205, 236; J. Math. Sci. 178, no. 3, 353–356 (2011)), which is a form of shear flow with only locally finite energy. A similar construction is performed) for the Navier-Stokes equations as well.