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Steady Navier-Stokes equations with Poiseuille and Jeffery-Hamel flows in $$\mathbb R^2$$. (English) Zbl 1427.35178
Summary: We consider a steady flow of an incompressible viscous fluid in a two-dimensional unbounded domain with unbounded boundaries. The domain has two outlets. The part of the domain upstream is a cylinder and the part of the domain downstream is a wedge. In the part of the domain upstream, the velocity is required to approach the Poiseuille flow. In the part of the domain downstream, the velocity is required to approach Jeffery-Hamel’s flow. This problem has been treated by C. J. Amick [Lect. Notes Math. 771, 1–11 (1980; Zbl 0425.35078)] and L. E. Frankel, V. A. Solonnikov and many others. Recently, the author [J. Math. Sci., Tokyo 21, No. 1, 61–77 (2014; Zbl 1329.35234)] obtained the unique solution of Jeffery-Hamel’s flows in a wedge. Therefore we reconsider this problem. In this paper, we succeed in proving the existence of such a steady flow under the restricted flux condition which depends only on the part of the domain upstream and downstream.
##### MSC:
 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 35A01 Existence problems for PDEs: global existence, local existence, non-existence
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