Energy equality for the multi-dimensional nonhomogeneous incompressible Hall-MHD equations in a bounded domain

: This paper focuses on the energy equality for weak solutions of the nonhomogeneous incompressible Hall-magnetohydrodynamics equations in a bounded domain Ω ⊂ R n ( n (cid:62) 2). By exploiting the special structure of the nonlinear terms and using the coarea formula, we obtain some su ﬃ cient conditions for the regularity of weak solutions to ensure that the energy equality is valid. For the special case n = 3, p = q = 2, our results are consistent with the corresponding results obtained by Kang-Deng-Zhou in [Results Appl. Math. 12 : 100178, 2021]. Additionally, we establish the su ﬃ cient conditions concerning ∇ u and ∇ b , instead of u and b .

When b = 0, systems (1.1)-(1.4) reduce to the incompressible Navier-Stokes system. For the energy equality of the incompressible Navier-Stokes system, the Lions-Shinbrot type criterion on the velocity was obtained by Lions [21], Shinbrot [22], Da Veiga and Yang [23] and Yu [24]. Later, Yu [25] extended Shinbrot's result to the bounded domain, with an additional Besov regularity imposed on the velocity, which is essential to deal with the boundary effects, and Nguyen et al. [26] handled the boundary effects without requiring extra conditions of velocity field u near the boundary.
For the energy equality of the compressible Navier-Stokes equations, Yu [27] proved the energy equality holds true if u ∈ L p t L q x , ρ is bounded, and √ ρ ∈ L ∞ (0, T ; H 1 (Ω)). Later on, Chen et al. [28] extended the result of [27] to the bounded domain by performing global mollification. In addition, it is worth mentioning that Berselli and Chiodaroli [29], Liang [30] and Wang and Ye [31] derived the energy equality criteria in terms of the velocity and its gradient.
Inspired by the works [16,26,31], we provide sufficient conditions on the regularity of solutions for systems (1.1)-(1.4) to ensure the energy equality holds. Compared with the results of [16], we obtain the sufficient conditions concerning ∇u and ∇b, rather than u and b, to guarantee that the energy equality is valid.
On the other hand, when n = 3, we get the following corollary by exploiting the Gagliardo-Nirenberg inequality.
Let Ω ⊂ R 3 be a bounded domain with C 2 boundary ∂Ω and (ρ, u, b, P) be a weak solution of systems (1.1)-(1.4) with initial data (1.5) and Dirichlet boundary conditions (1.6). Assume that one of the following conditions is satisfied: (1.16) Then, the energy equality (1.8) holds.
Remark 1.3. The conditions (1.16) show that we can get the regularity involving ∇u and ∇b, rather than u and b, to ensure that the energy equality (1.8) is valid.
Due to |z| 1, we have On the other hand, in view of Leibniz's formula, we conclude that Taking the norm for both sides of the above formula, we deduce where we have used the fact that |z| 1, θ ∈ [0, 1], and the constant C 1 does not rely on θ, z, ε, f and g.
For the second term of the right hand side of (2.6), one has where z = y−x ε . Similar to (2.7), we arrive at Combining (2.11) and (2.13), we get Taking into account the boundary terms, we need to use the following coarea formula for 0 < κ 1 < κ 2 established in [26]: We define the projection mapping as T (x) := x ∂Ω .
Therefore, In this part, we first give the proof of Theorem 1.1. Unlike the periodic region, we are concerned with the boundary terms produced by using integration by parts. Then, making use of the Gagliardo-Nirenberg inequality, we prove Corollary 1.1 by the results of Theorem 1.1.

Estimate of G. Taking advantage of integration by parts and free divergence condition (3.4) implies
, where the superscript "bdr" in G bdr 2 indicates that the term includes a boundary layer, and it is clear that G 4 + F 2 = 0. Thus, we only consider the terms G 1 , G bdr 2 and G 3 . First of all, for G 1 and G 3 , by exploiting integration by parts, we have  Similarly, we deduce ρ L ∞ (0,T ;L ∞ (Ω)) u L p (0,T ;W 1,q (Ω)) .
Under the assumption (1.13), by letting ε 3 → 0, we obtain lim sup which finishes the proof of the first case of (1.15).

C.
Thus, by Theorem 1.1, we get that (1.15) could ensure the energy equality (1.8).
In what follows, we give the proof of the first case of (1.16). According to Theorem 1.1, for 1 < p, q 3, one knows that the conditions ∇u, ∇b ∈ L p (0, T ; L q (Ω)) and u, b ∈ L 2p p−1 (0, T ; L 2q q−1 (Ω)) can ensure that energy equality (1.8) is valid. Therefore, to prove the first case of (1.16), we need to show that the conditions ∇u, ∇b ∈ L p (0, T ; L q (Ω)) can yield u, b ∈ L 2p p−1 (0, T ; L 2q q−1 (Ω)). To this end, for 9 5 q 3, by virtue of the Gagliardo-Nirenberg inequality, we know that

C.
Then, we complete the proof of the first case of (1.16).
Next, we treat the remaining case of (1.16). For 3 2 < q < 9 5 , it follows from the Gagliardo-Nirenberg inequality that

C.
Then, from Theorem 1.1, we know that (1.16) could guarantee the energy equality (1.8), and the proof of Corollary 1.1 is finished.

Conclusions
This paper is dedicated to the energy equality of nonhomogeneous incompressible Hall-MHD equations in a bounded domain Ω ⊂ R n (n 2). Through the special structure of the nonlinear terms, and using the coarea formula, we get some types of regularity conditions to guarantee that the energy equality is valid. It is worth noting that among them are the regularity conditions concerning ∇u and ∇b, rather than u and b.