Two‐dimensional stationary Navier–Stokes equations with 4‐cyclic symmetry

Abstract

This paper is concerned with the stationary Navier–Stokes equation in the whole plane and in the two–dimensional exterior domain invariant under the action of the cyclic group of order 4, and gives a condition on the potentials yielding the external force, and on the boundary value, sufficient for the unique existence of a small solution equivariant with respect to the aforementioned cyclic group.

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

“…Then every weak solution w (x) satisfying (C2E) such that u (x) = w (x) − v(x) satisfies (C2E) and the energy inequality, must coincide with w(x). This result implies the uniqueness of the solutions obtained in [36] as well as those in [33,34]. In particular, if w (x) is the weak solution constructed in the previous result such that u (x) = w (x) − v(x) satisfies (C2E), then w (x) coincides with w(x).…”

“…In addition to the property above on the uniqueness, the results in [14,35] on the stability under initial L 2 -perturbation with no restriction on the size holds, and we can replace the symmetry condition (D4E) by (C2E) by applying the improved Hardy's inequality. In other words, solutions in [33,34,36] have similar property on uniqueness and stability as physically reasonable solutions in the three-dimensional setting.…”

“…For the existence of critically decreasing solutions, the condition α = 0 is not necessary. The author [36] also showed that the solutions above become supercritically decreasing if F (x) satisfies a sharper decay condition and if, as well as the outflow condition α = 0, the data a(x) and F (x) are sufficiently small. We observe that the conditions (C4I) and (C4E) are independent of the choice of the coordinate axes, and includes rotationally symmetric functions naturally.…”

“…On the other handy, the author [36] proved, assuming only (C4I) on the domain Ω, and assuming only (C4E) on a(x) and f (x) = ∇F (x), where a(x) and F (x) are small in appropriate function spaces, the unique existence of small critically decreasing solutions satisfying the symmetry condition (C4E). In this work we imposed the invariance and equivariance under the cyclic group C 4 .…”

The Navier–Stokes equations on the plane with time-dependent external forces

“…As far as the author knows, only Galdi [6,7] obtained the unique existence for two-dimensional case, but the method relies on the periodicity. The purpose of this paper is to solve the problems above on the whole plane R 2 in the class of critically decreasing solutions or supercritically decreasing functions introduced by the author in [28,29]. Then we obtain the unique existence of small solutions, together with the continuous dependence on the datum, in these classes under appropriate assumptions on F(x, t).…”