The Navier--Stokes problem in a two-dimensional domain with angulate outlets to infinity

S. A. Nazarov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 28, Tome 257 (1999), pp. 207-227

Résumé

The Navier–Stokes problem in a plane domain with two angulate outlets to infinity, as usual, is supplied either by the flux condition, or by the pressure drop one. It is proven for small data that there exists a solution with the velocity field decay $O(|x|^{-1})$ as $|x|\to\infty$ (if one of the angles equals or greater than $\pi$, the additional symmetry assumptions are needed). Since the nonlinear and linear terms are asymptotically of the same power, the results are based on the complete investigation of the linearized Stokes problem in weighted spaces with detached asymptotics (angular parts in the representations are not fixed).