TY - JOUR

T1 - Existence and Uniqueness of Weak Solutions to the Two-Dimensional Stationary Navier–Stokes Exterior Problem

AU - Yamazaki, Masao

N1 - Funding Information:
The author expresses his sincere gratitude to Professor G. P. Galdi and the referees for valuable comments and suggestions. 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. The author declares that he has no conflict of interest.
Publisher Copyright:
© 2018, The Author(s).

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

KW - Energy inequality

KW - Exterior problem

KW - Stationary Navier–Stokes equations

KW - Weak-strong uniqueness

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U2 - 10.1007/s00021-018-0397-y

DO - 10.1007/s00021-018-0397-y

M3 - Article

AN - SCOPUS:85056756357

SN - 1422-6928

VL - 20

SP - 2019

EP - 2051

JO - Journal of Mathematical Fluid Mechanics

JF - Journal of Mathematical Fluid Mechanics

IS - 4

ER -