A note on the Liouville type theorem for the smooth solutions of the stationary Hall-MHD system

The main result of this work is to study the Liouville type theorem for the stationary Hall-MHD system on $\mathbb{R}^{3}$. Specificaly, we show that if $(u,B)$ is a smooth solutions to Hall-MHD equations satisfying $(u,B) \in L^\frac{9}{2} \mathbb{R}^{3}$, then we have $u=B=0$. This improves a recent result of Chae et al. [ 2 ] and Zujin et al. [ 14 ] .

In their famous paper [2], Chae-Degond-Liu proved (Theorem 2.5, p. 558) (see also [14]) the following Liouville-type theorem for the smooth solutions of (1.1) : ) be a smooth solution of the stationary Hall-MHD system (1.1) such that (iii) the (weak and then by classical) solution (u, B) : R 3 → R 3 is of finite energy in the sense that ∫ Then, u = B = 0.
The purpose of this note is to get rid of hypothesis (ii) and (iii) in theorem 1.1.More precisely, we shall prove the following result.
) be a smooth solution of the Hall-MHD equations (1.1) such that Remark 1.1.As mentioned in [3], if we set B = 0 in the Hall-MHD system, the above theorem reduces to the well-known Galdi result [8] for the Navier-Stokes equations (see Theorem X.9.5, pp.729-730).

Proof of Theorem 1.2
In order to prove our main result, we introduce some basic identifies in the fluid dynamic.
Remark 2.1.Based on ∇.B = 0 and Lemma 2.1, we get where I is the identical matrix.
We are now in a position to the proof of our main result.
be a smooth solution of the Hall-MHD equations (1.1) satisfies We shall first estimate the pressure in (1.1) 1 .Taking the divergence of (1.1) 1 and using the identity (2.1), we have from which we have the representation formula of the pressure, using the Riesz transforms in R 3 : Using (2.2) and Calderòn-Zygmund estimate, one has that For τ > 0, let φ τ be a real nonincreasing smooth function defined in R 3 such that and satisfying for some positive constant C independent of x ∈ R 3 .Multiplying (1.1) 1 by uφ τ and (1.1) 2 by Bφ τ , respectively, integrating by parts over R 3 and taking into acount (1.1) 3 , add the result together, we obtain ∫ where we have used the fact In the following, we will estimate all the terms on the right-hand side of (2.4).For the first integral A 1 , Hölder's inequality yields → 0 as τ → +∞.
Analogously to A 1 , an application of the Hölder inequality shows that → 0 as τ → +∞.
Similar to the treatment of A 3 , A 4 can be estimated as Finally, calculating A 5 + A 6 we obtain 2 )dx ) → 0 as τ → +∞.
Here we have used the Cauchy inequality.Consequently, letting τ → +∞ in (2.4), we obtain This completes the proof of Theorem 1.2.