A remark on Liouville-type theorems for the stationary Navier–Stokes equations in three space dimensions

A remark on the Liouville problem for stationary Navier-Stokes equations in Lorentz and Morrey spaces

“…where h t denotes the heat kernel, plays a very important role in the analysis on the Navier-Stokes equations (stationary and non stationary) since this is the largest space which is invariant under scaling properties of these equations (see the article [1] and the books [14] and [15] for more references). Thus, in order obtain the identity U = 0, we have supposed U ∈Ḃ −1 ∞,∞ (R 3 ) which is a condition on U less restrictive compared to those made in [13] and [18].…”

“…in [13], but in this more general space some supplementary hypothesis were needed to obtain U = 0. Indeed, in Theorem 1.2 of the article [13] it is proven that if U ∈ L 9 2 ,∞ (R 3 ) then we have the estimate R 3 | ∇ ⊗ U(x)| 2 dx ≤ c U 3 L 9 2 ,∞…”

A Liouville type theorem for axially symmetric D-solutions to steady Navier–Stokes equations

“…Recently, in [8], Kozono etc showed that if the vorticity ω decays faster than |x| − 5 3 at infinity, then D-solutions in R 3 are 0. However, to the best of our knowledge, there is no known result for the a priori decay rate (at infinity) of a general D-solution or vorticity in dimension three.…”

On the Vanishing of Some D-Solutions to the Stationary Magnetohydrodynamics System

“…The study is partly motivated by the related Liouville problem of the stationary Navier-Stokes equations, which has attracted much attention in recent years and is still far from being fully understood. See for example [1,2,3,4,5,7,8,9,12,13] and the reference therein. First, in full 3D case, the Liouville-type theorem holds provided the vanishing of u θ and h θ .…”