Generalized Scale-Invariant Solutions to the Two-Dimensional Stationary Navier--Stokes Equations

Abstract

New explicit solutions to the incompressible Navier-Stokes equations in R 2 \ {0} are determined, which generalize the scale-invariant solutions found by Hamel. These new solutions are invariant under a particular combination of the scaling and rotational symmetries. They are the only solutions invariant under this new symmetry in the same way as the Hamel solutions are the only scaleinvariant solutions. While the Hamel solutions are parameterized by a discrete parameter n, the flux Φ and an angle θ 0 , the new solutions generalize the Hamel solutions by introducing an additional parameter a which produces a rotation. The new solutions decay like |x| −1 as the Hamel solutions, and exhibit spiral behavior. The new variety of asymptotes induced by the existence of these solutions further emphasizes the difficulties faced when trying to establish the asymptotic behavior of the Navier-Stokes equations in a two-dimensional exterior domain or in the whole plane.

Steady solutions of the Navier-Stokes equations in the plane

“…In the exterior of a disk, Hillairet & Wittwer (2013) proved the existence of solutions that decay like |x| −1 at infinity provided that the boundary condition on the disk is close to µe r for |µ| > √ 48. To our knowledge, these last two results together with the exact solutions found by Hamel (1917); Guillod & Wittwer (2015b) are the only ones showing the existence of solutions in two-dimensional exterior domains satisfying (1.3c) with u ∞ = 0 and a known decay rate at infinity. We now analyze the implications of the decay of the velocity on the linear and nonlinear terms and on the net force.…”

“…As shown by theorem 3.8, the compatibility condition corresponding to the net torque M can be lifted by the exact solution M e θ /r. We remark that another way of lifting this compatibility condition might be given by the small exact solutions found by Guillod & Wittwer (2015b). The other two compatibility conditions are not invariant quantities and therefore much more difficult to lift (see also chapter 5).…”

“…More precisely, when the net force is nonzero, the asymptotic behavior is given by U F , however when the net force is vanishing the asymptotic behavior seems to be much less universal. In some regime, the asymptote is given by a double wake U F + U −F so that the net force is effectively zero (see figure 1.5), in some other regime by the harmonic solution µe θ /r , and finally can also be the exact scale -invariant solution up to a rotation discussed in Guillod & Wittwer (2015b). The presence of the double wake is surprising, because intuitively on would expect that the solution should behave like the Stokes solution, i.e.…”

“…These solutions are parameterized by the flux Φ ∈ R, an angle θ 0 , and a discrete parameter n. As explained by Šverák (2011, §5), they are far from the Stokes solutions decaying like |x| −1 , so cannot be used to lift the compatibility conditions of the Stokes equations. In an attempt to obtain the correct asymptotic behavior, Guillod & Wittwer (2015b) Finally, we discuss the supercritical case, that is to say the two-dimensional Navier-Stokes equations for a nonzero net force F = 0. By assuming that the decay of the solution is homogeneous, the previous power counting argument shows that the solution cannot decay faster than |x| −1/2 .…”

Existence of the stationary Navier-Stokes flow in $\mathbb{R}^2$ around a radial flow

“…Recently, Guillod [7] showed the existence of a pair (u, F ) solving (1.1), where F is dependent on u and is constructed around an arbitrarily given small function k having zero integral and decaying faster than |x| −3 . Moreover, Guillod-Wittwer [8] found solutions to (1.1) which is scaling invariant with respect to a rotation conversion.…”