Abstract
This paper is concerned with the stationary Navier–Stokes equation in two-dimensional exterior domains with external forces and inhomogeneous boundary conditions, and shows the existence of weak solutions. This solution enjoys a new energy inequality, provided the total flux is bounded by an absolute constant. It is also shown that, under the symmetry condition, the weak solutions tend to 0 at infinity. This paper also provides two criteria for the uniqueness of weak solutions under the assumption on the existence of one small solution which vanishes at infinity. In these criteria the aforementioned energy inequality plays a crucial role.
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The author expresses his sincere gratitude to Professor G. P. Galdi and the referees for valuable comments and suggestions.
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Communicated by Y. Giga
Dedicated to Professor Reinhard Farwig on the occasion of his sixtieth birthday.
Partly supported by the International Research Training Group (IGK 1529) on Mathematical Fluid Dynamics funded by DFG and JSPS and associated with TU Darmstadt, Waseda University and the University of Tokyo, and by Grant-in-Aid for Scientific Research (C) 17K05339, Ministry of Education, Culture, Sports, Science and Technology, Japan.
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Yamazaki, M. Existence and Uniqueness of Weak Solutions to the Two-Dimensional Stationary Navier–Stokes Exterior Problem. J. Math. Fluid Mech. 20, 2019–2051 (2018). https://doi.org/10.1007/s00021-018-0397-y
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DOI: https://doi.org/10.1007/s00021-018-0397-y