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.
Highlights
Let Ω be an exterior domain in the plane R2 with C2+γ-boundary Γ with some γ ∈ (0, 1)
We are concerned with the following stationary Navier–Stokes equation in Ω:
Throug⎛hout this pap⎞er2 we assume that the external force f (x) is given by the formula f (x) = ∇ · F (x) = ⎝
Summary
Let Ω be an exterior domain in the plane R2 with C2+γ-boundary Γ with some γ ∈ (0, 1). In the author’s knowledge, there are few results on the solutions decaying sufficiently so that its stability for the nonstationary problem under initial perturbation is assumed, and the uniqueness in these classes is obtained in this class for exterior domains is obtained only by [28] without the exterior force For these problems, the author [33] showed the existence, together with the uniqueness in the small, of the solution of the stationary Navier–Stokes equation on the whole plane under the assumption that the small external force f (x) = ∇ · F (x) decays like |x|−2 as |x| → ∞ and satisfies the condition f (x⊥) = f (x) ⊥; namely, f1(x⊥) = −f2(x), f2(x⊥) = f1(x),.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
