Stability for 2D Incompressible MHD Equations With Vertical Magnetic Dissipation and Vertical Velocity Damping Term

Abstract

This article primarily investigates the well-posedness and stability of the 2D MHD equations with vertical magnetic dissipation under Diophantine condition. The absence of horizontal magnetic dissipation brings significant challenges in estimating nonlinear terms. By employing frequency decomposition of the solutions and using the damping effect of the velocity field component $u_{2}$ , we succeed in driving some precise priori estimates of the solutions. Additionally, we investigate the existence and stability of solutions under Diophantine condition for other dissipation cases. Specifically, our results extend and generalize the corresponding results established in Zhai and Zheng and Zhu.

Keywords

2D MHD equations vertical magnetic dissipation diophantine condition

1. Introduction and Main Results

In this article, we study the following 2D magnetohydrodynamics (MHD) system in the domain $(x, t) \in T^{2} \times R_{+}$ ,

{\begin{cases} \partial_{t} u + u \cdot \nabla u + m u + \nabla P = b \cdot \nabla b, \\ \partial_{t} b + u \cdot \nabla b = (η_{1} \partial_{x_{1}}^{2} b + η_{2} \partial_{x_{2}}^{2} b) + b \cdot \nabla u, \\ div u = 0, div b = 0, \end{cases}

(1.1)

with the initial condition

\begin{aligned} (u, b) (x, 0) = (u_{0}, b_{0}) (x) . \end{aligned}

(1.2)

where

u = (u_{1}, u_{2})

is the velocity field,

b = (b_{1}, b_{2})

denotes the magnetic field,

p

stands for the scalar pressure of fluid.

The MHD system couples the Maxwell equations of electromagnetism with the Navier–Stokes equations of hydrodynamics. This system is a classical model to describe the macroscopic trajectory of conducting fluids such as electrolytes. The MHD system plays a significant role in numerous disciplines of research, such as space plasma physics, solar physics, astrophysics, and so on (see, e.g., Biskamp, 1993; Davidson, 2001; Priest & Forbes, 2000). Many mathematicians are deeply concerned with this fluid system, and they have made significant achievements in the study of such problems.

The well-posedness, stability and the long-time behavior of solutions of MHD equations have been studied by a large number of works. These results have now been successfully established by several authors via different approaches (see, e.g., Abidi & Zhang, 2017; Boardman et al., 2020; Cai & Lei, 2018; Cao et al., 2013; Cao & Wu, 2011; He et al., 2018; Khaleel & Salim, 2020; Lin et al., 2024; Pan et al., 2018; Ren et al., 2014; Shang & Yang, 2024; Wei & Zhang, 2017; Wu, 2003, 2011; Wu & Zhu, 2021; Wu, 2018; Wu et al., 2015; Zhao et al., 2023).

The stability of perturbation near a background magnetic field is an interesting topic. In the recent paper (Boardman et al., 2020), Nicki, Lin, and Wu investigated the stability of the 2D incompressible MHD equations with magnetic field and damping. The main ingredient to solve these problems is that the linearization of the perturbation system has the same wave structure. Recently, Wu and Zhu (2021) obtained the stability of background solutions of 3D MHD equations with mixed partial dissipation. It is natural to ask whether the horizontal diffusion MHD system without perturbation has a global smooth solution? Moreover, whether the 2D MHD system with magnetic field dissipation has a global solution?

In the recent paper (Wei & Zhang, 2017), they proved the global well-posedness for the 2D MHD equations with magnetic diffusion by observing that the $H^{1}$ norm and $L^{\infty}$ norm of the magnetic field $b$ decay exponentially in the torus. For the 3D MHD system with either magnetic dissipation or velocity dissipation, Chen et al. (2022) established the global existence of small solutions under Diophantine condition. Subsequently, Zhai et al. extended this result to the 2D case. Moreover, in Zhai (2023) and Zhao and Zhai (2021), they obtained the same result by weakening the dissipation in the system to either velocity damping or magnetic damping.

For the 2D MHD system with only vertical magnetic dissipation, Zheng and Zhu (2023) studied the global existence of small solutions for perturbation of the magnetic field near the special vector $(1, 0)$ . As is known to all, such perturbation of this specific vector do not belong to those vectors that satisfy the Diophantine condition.

Now, we give the Diophantine condition which was introduced in Chen et al. (2022). We consider the background magnetic field $n \in R^{2}$ satisfying the Diophantine condition: for any $k \in Z^{2} ∖ {0}$ ,

| n \cdot k | \geq \frac{c}{| k |^{r}} .

(1.3)

for some

c > 0

and

r > 1

A natural question then arises:

If the special perturbation in Zheng and Zhu (2023) is replaced by a more general vector perturbation satisfying the Diophantine condition, does the conclusion still hold?

It is not a trivial question, compared with the works in references (Chen et al., 2022; Zhai, 2023; Zhao & Zhai, 2021). Since the current article considers the MHD equations with vertical magnetic field dissipation under Diophantine condition, the absence of horizontal magnetic dissipation leads to significant challenges in estimating certain nonlinear terms, such as $b \cdot \nabla b$ . To address this difficulty, we are compelled to establish estimate for the $H^{r + α + 2}$ norm of $\nabla b$ . Thanks to the Diophantine condition, we observe that $n_{1} \neq 0$ , which provides a crucial pathway to derive the desired estimate.

Furthermore, due to the indefinite sign of the linear term $n \cdot u$ , it is not an easy task to control the nonlinear terms. Although we possess $H^{r + α + 3}$ estimate for $u_{2}$ , the estimate alone proves insufficient to bound $n \cdot u$ .

Fortunately, by meticulous analysis, we recognize that it suffices to control the zero-frequency component of $b_{2}$ in the $H^{r + α + 2}$ . By decomposing $b_{2}$ into its zero-frequency and non-zero-frequency components along the $x_{2}$ -direction, the non-zero-frequency contributions are managed via the Poincaré inequality. For the zero-frequency component, we use the governing equations and carefully select appropriate test functions to derive the necessary estimates.

A detailed exposition of this process can be found in Lemma 3.2 in the latter part of the article.

According to Remark 1.2 in Chen et al. (2022), we can show that the Diophantine condition is satisfied for almost all the vector fields $w$ in $R^{3}$ . However, when the components of $w$ are rational numbers or when one component of $w$ is zero, $w$ does not satisfy the Diophantine condition. Thus, we have both $n_{1}$ and $n_{2}$ not equal to $0$ .

In the present paper, we study the following 2D MHD system under the Diophantine condition:

{\begin{cases} \partial_{t} u + (u \cdot \nabla) u + m u + \nabla p = b \cdot \nabla b + n \cdot \nabla b, \\ \partial_{t} b + (u \cdot \nabla) b - (η_{1} \partial_{x_{1}}^{2} b + η_{2} \partial_{x_{2}}^{2} b) = (b \cdot \nabla) u + n \cdot \nabla u, \\ div u = div b = 0, \\ u (x, 0) = u_{0} (x), b (0, x) = b_{0} (x) . \end{cases}

(1.4)

Now, we give our main results.

Theorem 1.1

Assume that $n$ satisfies the Diophantine condition (1.3) and $m u = (0, u_{2})^{T}, (η_{1}, η_{2}) = (0, 1)$ . Let $α > 0, β > 0$ be two arbitrarily fixed constants. Suppose $N \geq (2 α + 3) r + α + 2 β + 5$ with $r > 1$ , and $(u_{0}, b_{0}) \in H^{N} (T^{2})$ with

\int_{T^{2}} u_{0} d x = \int_{T^{2}} b_{0} d x = 0.

(1.5)

If there exists a small constant

ε

such that

\begin{aligned} ‖ u_{0} ‖_{H^{N}} + ‖ b_{0} ‖_{H^{N}} \leq ε, \end{aligned}

then system (1.4) admits a global solution

(u, b) \in C ([0, \infty); H^{N})

. Moreover, for any

t \geq 0

and

r + α + 3 \leq γ \leq N

, there holds

\begin{aligned} ‖ u (t) ‖_{H^{γ}} + ‖ b (t) ‖_{H^{γ}} \leq C (1 + t)^{- \frac{(N - γ) (β + 1)}{N - r - α - 3}} . \end{aligned}

Theorem 1.2

Assume that $n$ satisfies the Diophantine condition (1.3) and $m u = (u_{1}, 0)^{T}, (η_{1}, η_{2}) = (1, 0)$ . Let $α > 0, β > 0$ be two arbitrarily fixed constants. Suppose $N \geq (2 α + 3) r + α + 2 β + 5$ with $r > 1$ , and $(u_{0}, b_{0}) \in H^{N} (T^{2})$ with

\int_{T^{2}} u_{0} d x = \int_{T^{2}} b_{0} d x = 0.

If there exists a small positive constant

ε

such that

‖ u_{0} ‖_{H^{N}} + ‖ b_{0} ‖_{H^{N}} \leq ε,

then system (1.4) admits a global solution

(u, b) \in C ([0, \infty); H^{N})

. Moreover, for any

t \geq 0

and

r + α + 3 \leq γ \leq N

, there holds

‖ u (t) ‖_{H^{γ}} + ‖ b (t) ‖_{H^{γ}} \leq C (1 + t)^{- \frac{(N - γ) (β + 1)}{N - r - α - 3}} .

Remark 1.1

If we choose $m = 0,$ and the second equation of system (1.4) has mixed dissipation

\begin{aligned} (\begin{array}{cc} \partial_{2}^{2} & 0 \\ 0 & \partial_{1}^{2} \end{array}) (b_{1}, b_{2})^{T}, \end{aligned}

then we still obtain the same results as Theorem 1.1 under the Diophantine condition and (1.5). On the other hand, if the first equation of system (1.4) has mixed dissipation

\begin{aligned} (\begin{array}{cc} \partial_{2}^{2} & 0 \\ 0 & \partial_{1}^{2} \end{array}) (u_{1}, u_{2})^{T}, \end{aligned}

we also obtain the same results as Theorem 1.1 under (1.5) and Diophantine condition.

The proof of the remark is trivial, and we only need the results of Zhao and Zhai (2021) and the following inequality,

‖ b_{1} ‖_{L^{2}} \leq ‖ \partial_{2} b_{1} ‖_{L^{2}} .

The above inequality makes use of the fact that

\int_{T} b_{1} d x_{2} = 0

. In terms of

\int_{T^{2}} b d x = 0

, we also have

\begin{aligned} ‖ b_{2} ‖_{L^{2}} \leq & ‖ \partial_{1} b_{2} ‖_{L^{2}} + ‖ \partial_{2} b_{2} ‖_{L^{2}} \\ \leq & ‖ \partial_{1} b_{2} ‖_{L^{2}} + ‖ \partial_{1} b_{1} ‖_{L^{2}} \\ \leq & ‖ \partial_{1} b_{2} ‖_{L^{2}} + ‖ \partial_{21} b_{1} ‖_{L^{2}} . \end{aligned}

The specific proof process is not provided here.

Remark 1.2

Consider the following system:

{\begin{cases} \partial_{t} u + (u \cdot \nabla) u + \nabla p - (μ_{1} \partial_{x_{1}}^{2} u + μ_{2} \partial_{x_{2}}^{2} u) = b \cdot \nabla b + n \cdot \nabla b, \\ \partial_{t} b + (u \cdot \nabla) b + m b = (b \cdot \nabla) u + n \cdot \nabla u, \\ div u = div b = 0, \\ u (x, 0) = u_{0} (x), b (0, x) = b_{0} (x) . \end{cases}

(1.6)

Assume that

n

satisfies the Diophantine condition (1.3) and

m b = (0, b_{2})^{T}, (μ_{1}, μ_{2}) = (0, 1)

, or

m b = (b_{1}, 0)^{T}, (μ_{1}, μ_{2}) = (1, 0)

, then Theorem 1.1 holds also for problem (1.6) under condition (1.5).

The proof of Remark 1.2 parallels that of Theorem 1.1 and 1.2, therefore, we omit it directly.

The article is organized as follows. In Section 2 , we give some standard energy estimates. In Section 3 , we give some key lemmas, then we give the proof of Theorem 1.1.

2. Energy Estimate

This section is devoted to some prior estimates for solutions of system (1.4). Firstly, we introduce some key inequalities and lemmas.

Poincaré inequality on $b_{1}$ and $u_{1}$ .

Since we have the initial assumption that $\int_{T^{2}} b_{0} d x = 0$ , it is easy to prove that

\int_{T^{2}} b d x = 0.

In terms of the divergence-free condition of

b

, it is easy to obtain that

\int_{T} b_{1} (x_{1}, x_{2}) d x_{2} = 0.

Thus, we have

‖ b_{1} ‖_{L^{2}} \leq C ‖ \partial_{2} b_{1} ‖_{L^{2}}, ‖ b_{2} ‖_{L^{2}} \leq C ‖ \nabla b_{2} ‖_{L^{2}} .

Moreover, by a similar proof, we also obtain

‖ u_{1} ‖_{L^{2}} \leq C ‖ \partial_{2} u_{1} ‖_{L^{2}}, ‖ u_{2} ‖_{L^{2}} \leq C ‖ \nabla u_{2} ‖_{L^{2}} .

Lemma 2.1
Let $n \in R^{2}$ satisfy the Diophantine condition, then it holds that for any $s \in R$ ,
$‖ f ‖_{H^{s}} \leq C ‖ n \cdot \nabla f ‖_{H^{s + r}},$
provided $f \in H^{s + r} (T^{2})$ with $\int_{T^{2}} f d x = 0$ .
Proof.
We can prove it by the Plancherel formula. For the detailed proof process, see reference (Chen et al., 2022; Zhai, 2023).

Recall a classical lemma.
Lemma 2.2
Let $k \geq 1$ , it holds that,
$‖ f g ‖_{H^{k}} \leq C (‖ f ‖_{L^{\infty}} ‖ g ‖_{H^{k}} + ‖ g ‖_{L^{\infty}} ‖ f ‖_{H^{k}}) .$

Now, we give the basic energy estimates of equations (1.4).

$∙$ $L^{2}$ -estimate.

Firstly, a standard energy estimate gives
$\frac{1}{2} \frac{d}{d t} (‖ u ‖_{L^{2}}^{2} + ‖ b ‖_{L^{2}}^{2}) + ‖ \partial_{2} b ‖_{L^{2}}^{2} + ‖ m u ‖_{L^{2}}^{2} = 0,$
where we have used the cancellations,
$\begin{aligned} ⟨ u \cdot \nabla u, u ⟩ = ⟨ u \cdot \nabla b, b ⟩ = 0, \end{aligned}$

$\begin{aligned} ⟨ b \cdot \nabla b, u ⟩ + ⟨ b \cdot \nabla u, b ⟩ = 0, \end{aligned}$

$\begin{aligned} ⟨ n \cdot \nabla b, u ⟩ + ⟨ n \cdot \nabla u, b ⟩ = 0. \end{aligned}$

$∙$ $H^{s}$ -estimate.

Denote $Λ = \sqrt{- Δ}$ , for any $s > 0$ , applying $Λ^{s}$ on both sides of system (1.4), we have
${\begin{cases} \partial_{t} Λ^{s} u + (u \cdot \nabla Λ^{s}) u + m Λ^{s} u + Λ^{s} \nabla p = b \cdot \nabla Λ^{s} b + Λ^{s} (n \cdot \nabla b) - [Λ^{s}, u \cdot \nabla] u + [Λ^{s}, b \cdot \nabla] b, \\ \partial_{t} Λ^{s} b + (u \cdot \nabla Λ^{s}) b - \partial_{x_{2}}^{2} Λ^{s} b = (b \cdot \nabla Λ^{s}) u + Λ^{s} (n \cdot \nabla u) + [Λ^{s}, b \cdot \nabla] u - [Λ^{s}, u \cdot \nabla] b . \end{cases}$
Taking $L^{2}$ -inner product with $Λ^{s} u, Λ^{s} b$ respectively gives
$\begin{aligned} \frac{1}{2} \frac{d}{d t} (‖ Λ^{s} u ‖_{L^{2}}^{2} + ‖ Λ^{s} b ‖_{L^{2}}^{2}) + (m Λ^{s} u, Λ^{s} u) + ‖ \partial_{x_{2}} Λ^{s} b ‖_{L^{2}}^{2} \\ = - ⟨ [Λ^{s}, u \cdot \nabla] u, Λ^{s} u ⟩ + ⟨ [Λ^{s}, b \cdot \nabla] b, Λ^{s} u ⟩ - ⟨ [Λ^{s}, u \cdot \nabla] b, Λ^{s} b ⟩ + ⟨ [Λ^{s}, b \cdot \nabla] u, Λ^{s} b ⟩, \end{aligned}$
where we have used
$\begin{aligned} ⟨ u \cdot \nabla Λ^{s} u, Λ^{s} u ⟩ = ⟨ Λ^{s} \nabla p, Λ^{s} u ⟩ = ⟨ u \cdot \nabla Λ^{s} b, Λ^{s} b ⟩ = 0, \end{aligned}$

$\begin{aligned} ⟨ b \cdot \nabla Λ^{s} b, Λ^{s} u ⟩ + ⟨ b \cdot \nabla Λ^{s} u, Λ^{s} b ⟩ = 0, \end{aligned}$

$\begin{aligned} ⟨ Λ^{s} (n \cdot \nabla b)), Λ^{s} u ⟩ + ⟨ Λ^{s} (n \cdot \nabla u)), Λ^{s} b ⟩ = 0. \end{aligned}$
Recall a classical commutator’s estimate in Kato (1990):
$‖ [Λ^{s}, f \cdot \nabla] g ‖_{L^{2}} ≲ (‖ \nabla f ‖_{L^{\infty}} ‖ Λ^{s} g ‖_{L^{2}} + ‖ Λ^{s} f ‖_{L^{2}} ‖ \nabla g ‖_{L^{\infty}}) for~any s > 0.$
Thus we obtain
$\begin{aligned} ‖ [Λ^{s}, u \cdot \nabla] u ‖_{L^{2}} + ‖ [Λ^{s}, b \cdot \nabla] b ‖_{L^{2}} \leq C (‖ \nabla u ‖_{L^{\infty}} ‖ Λ^{s} u ‖_{L^{2}} + ‖ \nabla b ‖_{L^{\infty}} ‖ Λ^{s} b ‖_{L^{2}}), \end{aligned}$

$\begin{aligned} ‖ [Λ^{s}, u \cdot \nabla] b ‖_{L^{2}} + ‖ [Λ^{s}, b \cdot \nabla] u ‖_{L^{2}} \leq C (‖ \nabla u ‖_{L^{\infty}} ‖ Λ^{s} b ‖_{L^{2}} + ‖ Λ^{s} u ‖_{L^{2}} ‖ \nabla b ‖_{L^{\infty}}) . \end{aligned}$
It follows from the above inequalities that
$\begin{aligned} \frac{1}{2} \frac{d}{d t} (‖ Λ^{s} u ‖_{L^{2}}^{2} + ‖ Λ^{s} b ‖_{L^{2}}^{2}) + (m Λ^{s} u, Λ^{s} u) + ‖ \partial_{x_{2}} Λ^{s} b ‖_{L^{2}}^{2} \\ \leq C (‖ \nabla u ‖_{L^{\infty}} + ‖ \nabla b ‖_{L^{\infty}}) (‖ Λ^{s} u ‖_{L^{2}}^{2} + ‖ Λ^{s} b ‖_{L^{2}}^{2}) . \end{aligned}$
(2.1)
3. Some Key Lemmas

The following lemma plays a key role in controlling the nonlinear term (i.e., $u \cdot \nabla u$ ). The lemma relies heavily on the special structure of system (1.4) and Diophantine condition. It will be used to obtain the time decay property of solutions.

Lemma 3.1
Assume that
$sup_{t \in [0, T]} (‖ u ‖_{H^{N}} + ‖ b ‖_{H^{N}}) \leq δ,$
for some $0 < δ < 1$ . Then there holds
$\begin{aligned} ‖ n \cdot \nabla u ‖_{H^{r + α + 2}}^{2} - \sum_{0 \leq s \leq r + α + 2} \frac{d}{d t} ⟨ Λ^{s} b, Λ^{s} (n \cdot \nabla u) ⟩ \\ \leq C (‖ \partial_{2} b ‖_{H^{r + α + 2}}^{2} + ‖ \partial_{1} b_{2} ‖_{H^{r + α + 2}}^{2}) + C ‖ m u ‖_{H^{r + α + 2}}^{2} . \end{aligned}$
(3.1)
Proof.
Applying $Λ^{s} (0 \leq s \leq r + α + 2)$ to the first equation of (1.4), and multiplying it by $Λ^{s} (n \cdot \nabla u)$ , and then integrating over $T^{3}$ , we obtain
$\begin{aligned} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} & = ⟨ Λ^{s} \partial_{t} b, Λ^{s} (n \cdot \nabla u) ⟩ + ⟨ Λ^{s} (u \cdot \nabla b), Λ^{s} (n \cdot \nabla u) ⟩ \\ - ⟨ Λ^{s} \partial_{x_{2}}^{2} b, Λ^{s} (n \cdot \nabla u) ⟩ - ⟨ Λ^{s} (b \cdot \nabla u), Λ^{s} (n \cdot \nabla u) ⟩ \\ := I_{1} + I_{2} + I_{3} + I_{4} . \end{aligned}$
By the product law, Hölder inequality and Young’s inequality, we have
$\begin{aligned} I_{3} & \leq C ‖ Λ^{s} \partial_{x_{2}}^{2} b ‖_{L^{2}} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}} \\ \leq \frac{1}{10} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} + C ‖ \partial_{2}^{2} b ‖_{H^{s}}^{2} \\ \leq \frac{1}{10} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} + C ‖ \partial_{2} b ‖_{H^{s + 1}}^{2}, \end{aligned}$
and
$\begin{aligned} I_{2} & = ⟨ Λ^{s} (u \cdot \nabla b), Λ^{s} (n \cdot \nabla u) ⟩ \\ \leq C ‖ Λ^{s} (u \cdot \nabla b) ‖_{L^{2}} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}} \\ \leq C (‖ u ‖_{L^{\infty}} ‖ \nabla b ‖_{H^{s}} + ‖ \nabla b ‖_{L^{\infty}} ‖ u ‖_{H^{s}}) ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}} \\ \leq C ‖ u ‖_{L^{\infty}}^{2} ‖ \nabla b ‖_{H^{s}}^{2} + ‖ \nabla b ‖_{L^{\infty}}^{2} ‖ u ‖_{H^{s}}^{2} + \frac{1}{10} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} \\ \leq \frac{1}{10} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} + C δ^{2} ‖ (n \cdot \nabla u) ‖_{H^{r + α + 2}}^{2} + δ^{2} ‖ \partial_{2} b ‖_{H^{r + α + 3}} + δ^{2} ‖ \partial_{1} b_{2} ‖_{L^{\infty}}^{2}, \end{aligned}$
and
$\begin{aligned} I_{4} & = ⟨ Λ^{s} (b \cdot \nabla u), Λ^{s} (n \cdot \nabla u) ⟩ \\ \leq C ‖ Λ^{s} (b \cdot \nabla u) ‖_{L^{2}} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}} \\ \leq C (‖ b ‖_{L^{\infty}} ‖ \nabla u ‖_{H^{s}} + ‖ \nabla u ‖_{L^{\infty}} ‖ b ‖_{H^{s}}) ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}} \\ \leq C ‖ b ‖_{L^{\infty}}^{2} ‖ \nabla u ‖_{H^{s}}^{2} + ‖ \nabla u ‖_{L^{\infty}}^{2} ‖ b ‖_{H^{s}}^{2} + \frac{1}{10} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} \\ \leq \frac{1}{10} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} + C δ^{2} ‖ (n \cdot \nabla u) ‖_{H^{r + α + 2}}^{2} + δ^{2} ‖ \partial_{2} b ‖_{H^{r + α + 3}} + δ^{2} ‖ \partial_{1} b_{2} ‖_{L^{\infty}}^{2} . \end{aligned}$
Now, we deal with the term $I_{1}$ , thanks to the second equality in system (1.4), we have
$\begin{aligned} I_{1} & = \frac{d}{d t} ⟨ Λ^{s} b, Λ^{s} (n \cdot \nabla u) ⟩ - ⟨ Λ^{s} b, Λ^{s} (n \cdot \nabla \partial_{t} u) ⟩ \\ = \frac{d}{d t} ⟨ Λ^{s} b, Λ^{s} (n \cdot \nabla u) ⟩ + ⟨ Λ^{s} (n \cdot \nabla b), Λ^{s} \partial_{t} u ⟩ \\ = \frac{d}{d t} ⟨ Λ^{s} b, Λ^{s} (n \cdot \nabla u) ⟩ + ⟨ Λ^{s} (n \cdot \nabla b), Λ^{s} (n \cdot \nabla b) ⟩ \\ + ⟨ Λ^{s} (n \cdot \nabla b), Λ^{s} (b \cdot \nabla b) ⟩ - ⟨ Λ^{s} (n \cdot \nabla b), Λ^{s} (u \cdot \nabla u) ⟩ - ⟨ Λ^{s} (m u), Λ^{s} (n \cdot \nabla b) ⟩, \end{aligned}$
where we have used the cancellation
$⟨ Λ^{s} (n \cdot \nabla b), Λ^{s} \nabla p ⟩ = 0.$
In terms of Hölder inequality, we obtain
$\begin{aligned} ⟨ Λ^{s} (m u), Λ^{s} (n \cdot \nabla b) ⟩ & \leq C ‖ Λ^{s} (m u) ‖_{L^{2}} ‖ Λ^{s} (n \cdot \nabla b) ‖_{L^{2}} \\ \leq ε (‖ \partial_{2} b ‖_{H^{r + α + 2}}^{2} + ‖ \partial_{1} b_{2} ‖_{H^{r + α + 2}}^{2}) + C ‖ Λ^{s} (m u) ‖_{L^{2}}^{2}, \end{aligned}$
and
$\begin{aligned} ⟨ Λ^{s} (n \cdot \nabla b), Λ^{s} (n \cdot \nabla b) ⟩ & \leq ‖ Λ^{s} (n \cdot \nabla b) ‖_{L^{2}} ‖ Λ^{s} (n \cdot \nabla b) ‖_{L^{2}} \\ \leq {(‖ \partial_{2} b ‖_{H^{r + α + 2}} + ‖ \partial_{1} b_{2} ‖_{H^{r + α + 2}})}^{2} \\ \leq ‖ \partial_{2} b ‖_{H^{r + α + 2}}^{2} + ‖ \partial_{1} b_{2} ‖_{H^{r + α + 2}}^{2} . \end{aligned}$
Furthermore, we have
$\begin{aligned} ⟨ Λ^{s} (b \cdot \nabla b), Λ^{s} (n \cdot \nabla b) ⟩ & \leq ‖ Λ^{s} (b \cdot \nabla b) ‖_{L^{2}} ‖ Λ^{s} (n \cdot \nabla b) ‖_{L^{2}} \\ \leq ‖ Λ^{s} (n \cdot \nabla b) ‖_{L^{2}} (‖ b ‖_{L^{\infty}} ‖ ‖ \nabla b ‖_{H^{s}} + ‖ \nabla b ‖_{L^{\infty}} ‖ b ‖_{H^{s}}) \\ \leq ‖ b ‖_{H^{N}} (‖ \partial_{2} b ‖_{H^{r + α + 2}} + ‖ \partial_{1} b_{2} ‖_{H^{r + α + 2}})^{2} \\ \leq ‖ \partial_{2} b ‖_{H^{r + α + 2}}^{2} + ‖ \partial_{1} b_{2} ‖_{H^{r + α + 2}}^{2}, \end{aligned}$
where we have used the Sobolev embedding, $\int_{T^{2}} b d x = 0$ and the Poincaré inequality. Moreover, one has
$\begin{aligned} ⟨ Λ^{s} (n \cdot \nabla b), Λ^{s} (u \cdot \nabla u) ⟩ & \leq ‖ Λ^{s} (n \cdot \nabla b) ‖_{L^{2}} ‖ Λ^{s} (u \cdot \nabla u) ‖_{L^{2}} \\ \leq ‖ Λ^{s} (n \cdot \nabla b) ‖_{L^{2}} (‖ u ‖_{L^{\infty}} ‖ ‖ \nabla u ‖_{H^{s}} + ‖ \nabla u ‖_{L^{\infty}} ‖ u ‖_{H^{s}}) \\ \leq ‖ \nabla u ‖_{H^{s}} ‖ Λ^{s} (n \cdot \nabla b) ‖_{L^{2}}^{2} + ‖ \nabla u ‖_{H^{s}} ‖ u ‖_{L^{\infty}}^{2} + ‖ u ‖_{H^{s}} ‖ Λ^{s} (n \cdot \nabla b) ‖_{L^{2}}^{2} \\ + ‖ u ‖_{H^{s}} ‖ \nabla u ‖_{L^{\infty}}^{2} \\ \leq δ ‖ \partial_{2} b ‖_{H^{r + α + 2}}^{2} + δ ‖ \partial_{1} b_{2} ‖_{H^{r + α + 2}}^{2} + \frac{1}{10} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} . \end{aligned}$
Collecting all above estimates, we have
$\begin{aligned} I_{1} & = ⟨ Λ^{s} \partial_{t} b, Λ^{s} (n \cdot \nabla u) ⟩ \\ \leq \frac{d}{d t} ⟨ Λ^{s} b, Λ^{s} (n \cdot \nabla u) ⟩ + C ‖ \partial_{2} b ‖_{H^{r + α + 2}}^{2} + C ‖ \partial_{1} b_{2} ‖_{H^{r + α + 2}}^{2} + \frac{1}{10} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} \\ + C ‖ m u ‖_{H^{r + α + 2}}^{2} . \end{aligned}$
Therefore, we obtain the inequality (3.1), and complete the proof of the Lemma 3.1.

In order to estimate the norm $H^{r + α + 2}$ of $\partial_{1} b_{2}$ , we define the zero frequency and non-zero frequency of $b_{2}$ , respectively,
$b_{2, 0} = \int_{T} b_{2} d x_{2}, b_{2, \neq} = b_{2} - \int_{T} b_{2} d x_{2} .$
It is obvious that we have
$\int_{T} b_{2, \neq} d x_{2} = 0.$
Using the Poincaré inequality, we have
$‖ b_{2, \neq} ‖_{L^{2}} \leq C ‖ \partial_{2} b_{2, \neq} ‖_{L^{2}} \leq C ‖ \partial_{2} b_{2} ‖_{L^{2}} .$
Using the above lemma and the Poincaré inequality, we have the following proposition.
Proposition 3.1
Assume that
$sup_{t \in [0, T]} (‖ u ‖_{H^{N}} + ‖ b ‖_{H^{N}}) \leq δ,$
for some $0 < δ < 1$ . Then there holds
$\begin{aligned} ‖ n \cdot \nabla u ‖_{H^{r + α + 2}}^{2} - \sum_{0 \leq s \leq r + 3} \frac{d}{d t} ⟨ Λ^{s} b, Λ^{s} (n \cdot \nabla u) ⟩ \leq C (‖ \partial_{2} b ‖_{H^{r + α + 2}}^{2} + ‖ \partial_{21} b_{2} ‖_{H^{r + α + 2}}^{2}) \\ + C ‖ \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2} + C ‖ (m u) ‖_{H^{r + α + 2}}^{2} . \end{aligned}$
(3.2)

Compared with literature Chen et al. (2022) and Zhai (2023), since our magnetic field equation only contains vertical dissipation, we are unable to establish closure for the estimate of certain nonlinear term (i.e., $b \cdot \nabla b$ ). By analyzing the structure of the system, we find that it suffices to obtain the zero-mode estimate for $\partial_{1} b_{2}$ .

On the other hand, we note that $n_{1} \neq 0$ holds due to the Diophantine condition satisfied by $n$ . In order to estimate the term $‖ \partial_{1} b_{2} ‖_{H^{r + α + 2}}^{2}$ , we need to establish the following lemma.
Lemma 3.2
Assume that
$sup_{t \in [0, T]} (‖ u ‖_{H^{N}} + ‖ b ‖_{H^{N}}) \leq δ,$
for some $0 < δ < 1$ . Then there holds
$\begin{aligned} ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2} - \frac{d}{d t} \sum_{0 \leq s \leq r + α + 2} ⟨ Λ^{s} u, Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ⟩ \\ \leq C ‖ u_{2} ‖_{H^{r + α + 3}}^{2} + δ^{2} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} + δ ‖ \partial_{2} b ‖_{H^{r + α + 3}}^{2} . \end{aligned}$
(3.3)
Proof.
Since $n_{1}$ satisfies the Diophantine condition, $n_{1} \neq 0$ . Applying $Λ^{s} (0 \leq s \leq r + α + 2)$ to the first equation of system (1.4), multiplying it by $Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0}))$ , and then integrating over $T^{2}$ ; Applying $Λ^{s} (0 \leq s \leq r + α + 2)$ to the second equation of (1.4), multiplying it by $- Λ^{s} (n_{1} \cdot \partial_{1} (u_{1, 0}, u_{2, 0}))$ , and then integrating over $T^{2}$ , adding these two results together, we obtain
$\begin{aligned} ⟨ Λ^{s} (n \cdot \nabla b), Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ⟩ - ⟨ Λ^{s} (n \cdot \nabla u), Λ^{s} (n_{1} \cdot \partial_{1} (u_{1, 0}, u_{2, 0})) ⟩ \\ = ⟨ Λ^{s} \partial_{t} u, Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ⟩ - ⟨ Λ^{s} \partial_{t} b, Λ^{s} (n_{1} \cdot \partial_{1} (u_{1, 0}, u_{2, 0})) ⟩ \\ + ⟨ Λ^{s} (u \cdot \nabla u), Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ⟩ + ⟨ Λ^{s} (\nabla p), Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ⟩ \\ + ⟨ Λ^{s} \partial_{2}^{2} b, Λ^{s} (n_{1} \cdot \partial_{1} (u_{1, 0}, u_{2, 0})) ⟩ - ⟨ Λ^{s} (u \cdot \nabla b), Λ^{s} (n_{1} \cdot \partial_{1} (u_{1, 0}, u_{2, 0})) ⟩ \\ - ⟨ Λ^{s} (b \cdot \nabla b), Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ⟩ + ⟨ Λ^{s} (b \cdot \nabla u), Λ^{s} (n_{1} \cdot \partial_{1} (u_{1, 0}, u_{2, 0})) ⟩ \\ + ⟨ Λ^{s} m u, Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ⟩ \\ := J_{1} + J_{2} + J_{3} + J_{4} + J_{5} + J_{6} + J_{7} + J_{8} + J_{9} . \end{aligned}$
Note that
$\partial_{1} b_{1, 0} = \partial_{x_{1}} \int_{T} b_{1} d x_{2} = \int_{T} \partial_{1} b_{1} d x_{2} = - \int_{T} \partial_{2} b_{2} d x_{2} = 0.$
Then
$\begin{aligned} ⟨ Λ^{s} (n \cdot \nabla b), Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ⟩ & = ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2} + ‖ n_{1} \partial_{1} b_{1, 0} ‖_{H^{r + α + 2}}^{2} \\ = ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2}, \end{aligned}$
where we have used the fact that $\partial_{2} (b_{1, 0}, b_{2, 0}) = 0$ and integration by parts. Then, using the similar properties and divergence-free condition for $u$ , one has
$\begin{aligned} ⟨ Λ^{s} (n \cdot \nabla u), Λ^{s} (n_{1} \cdot \partial_{1} (u_{1, 0}, u_{2, 0})) ⟩ & = ‖ n_{1} \partial_{1} u_{1, 0} ‖_{H^{r + α + 2}}^{2} + ‖ n_{1} \partial_{1} u_{2, 0} ‖_{H^{r + α + 2}}^{2} \\ \leq ‖ \partial_{1} u_{2} ‖_{H^{r + α + 2}}^{2}, \end{aligned}$
and
$\begin{aligned} J_{1} + J_{2} = \frac{d}{d t} ⟨ Λ^{s} u, Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ⟩ . \end{aligned}$
Note that $m u = u_{2}$ , and by Hölder inequality and Young inequality, one has
$\begin{aligned} J_{9} & = ⟨ Λ^{s} u_{2}, n_{1} Λ^{s} \partial_{1} b_{2, 0} ⟩ \\ \leq \frac{1}{10} ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2} + ‖ u_{2} ‖_{H^{r + α + 2}}^{2}, \end{aligned}$
and
$\begin{aligned} J_{3} & = ⟨ Λ^{s} (u \cdot \nabla u), Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ⟩ \\ = ‖ Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ‖_{L^{2}} (‖ u ‖_{L^{\infty}} ‖ ‖ \nabla u ‖_{H^{s}} + ‖ \nabla u ‖_{L^{\infty}} ‖ u ‖_{H^{s}}) \\ \leq \frac{1}{10} ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2} + δ^{2} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} . \end{aligned}$
Using integration by parts and divergence-free condition for $u$ and $b$ , we have
$J_{4} = ⟨ Λ^{s} (\nabla p), Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ⟩ = 0,$
and
$J_{5} = ⟨ Λ^{s} \partial_{2}^{2} b, Λ^{s} (n_{1} \cdot \partial_{1} (u_{1, 0}, u_{2, 0})) ⟩ = 0.$
Moreover, one has
$\begin{aligned} J_{6} & = - ⟨ Λ^{s} (u \cdot \nabla b), Λ^{s} (n_{1} \cdot \partial_{1} (u_{1, 0}, u_{2, 0})) ⟩ \\ \leq ‖ Λ^{s} (n_{1} \cdot \partial_{1} (u_{1, 0}, u_{2, 0})) ‖_{L^{2}} (‖ u ‖_{L^{\infty}} ‖ ‖ \nabla b ‖_{H^{s}} + ‖ \nabla b ‖_{L^{\infty}} ‖ u ‖_{H^{s}}) \\ \leq δ ‖ u_{2} ‖_{H^{r + α + 3}}^{2} + δ^{2} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} + δ ‖ \partial_{2} b ‖_{H^{r + α + 3}}^{2} + δ ‖ \partial_{1} b_{2, 0} ‖_{H^{r + α + 3}}^{2}, \end{aligned}$
where we have used the fact that $\partial_{2} (u_{1, 0}, u_{2, 0}) = 0$ and integration by parts. Furthermore, it follows from Lemma 2.2, Hölder inequality and Young inequality that
$\begin{aligned} - J_{7} & = ⟨ Λ^{s} (b \cdot \nabla b), Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ⟩ \\ \leq ‖ Λ^{s} (n_{1} \cdot \partial_{1} (b_{1, 0}, b_{2, 0})) ‖_{L^{2}} (‖ b ‖_{L^{\infty}} ‖ ‖ \nabla b ‖_{H^{s}} + ‖ \nabla b ‖_{L^{\infty}} ‖ b ‖_{H^{s}}) \\ \leq \frac{1}{10} ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2} + δ^{2} ‖ \partial_{2} b, \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2}, \end{aligned}$
and
$\begin{aligned} J_{8} & = ⟨ Λ^{s} (b \cdot \nabla u), Λ^{s} (n_{1} \cdot \partial_{1} (u_{1, 0}, u_{2, 0})) ⟩ \\ \leq ‖ Λ^{s} (n_{1} \cdot \partial_{1} (u_{1, 0}, u_{2, 0})) ‖_{L^{2}} (‖ b ‖_{L^{\infty}} ‖ ‖ \nabla u ‖_{H^{s}} + ‖ \nabla u ‖_{L^{\infty}} ‖ b ‖_{H^{s}}) \\ \leq ‖ \nabla u ‖_{H^{s}} ‖ b ‖_{L^{\infty}}^{2} + ‖ \nabla u ‖_{H^{s}} ‖ Λ^{s} (n_{1} \cdot \partial_{1} (u_{1, 0}, u_{2, 0})) ‖_{L^{2}}^{2} + δ ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} + δ ‖ \partial_{2} b ‖_{H^{r + α + 3}}^{2} \\ + δ ‖ \partial_{1} b_{2, 0} ‖_{H^{r + α + 3}}^{2} \\ \leq C ‖ u_{2} ‖_{H^{r + α + 3}}^{2} + δ ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} + δ ‖ \partial_{2} b ‖_{H^{r + α + 3}}^{2} + δ ‖ \partial_{1} b_{2, 0} ‖_{H^{r + α + 3}}^{2} . \end{aligned}$
Collecting the above all estimates, we complete the proof of Lemma 3.2.
Proof. Proof of the Theorem 1.1

For any $(u_{0}, b_{0}) \in H^{N} (T^{2})$ , the local well-posedness of system (1.4) can be proved by using a standard energy method. Thus, we can assume that there exist $T > 0$ and a unique solution $(u, b) \in C ([0, T]; H^{N})$ of system (1.4).

Now, we prove the solution is global by the bootstrap argument. We assume that

sup_{t \in [0, T]} (‖ u ‖_{H^{N}} + ‖ b ‖_{H^{N}}) \leq δ,

(3.4)

for some

0 < δ < 1

to be determined later.

Taking $s = r + α + 3$ in (2.1), and $m = (0, 1)$ , we have

\begin{aligned} \frac{1}{2} \frac{d}{d t} (‖ u ‖_{H^{r + α + 3}}^{2} + ‖ b ‖_{H^{r + α + 3}}^{2}) + ‖ u_{2} ‖_{H^{r + α + 3}}^{2} + ‖ \partial_{2} b ‖_{H^{r + α + 3}}^{2} \\ \leq C (‖ \nabla u ‖_{L^{\infty}} + ‖ \nabla b ‖_{L^{\infty}}) (‖ u ‖_{H^{r + α + 3}}^{2} + ‖ b ‖_{H^{r + α + 3}}^{2}) . \end{aligned}

(3.5)

Now, we first choose

A >> B >> 1

, and then perform the following operation:

(3.5) \times A + (3.3) \times B + (3.2)

. By this, we obtain

\begin{aligned} \frac{1}{2} \frac{d}{d t} {A (‖ u ‖_{H^{r + α + 3}}^{2} + ‖ b ‖_{H^{r + α + 3}}^{2}) - B \sum_{0 \leq s \leq r + α + 2} ⟨ Λ^{s} u, Λ^{s} (n \cdot \nabla b) ⟩ - \sum_{0 \leq s \leq r + α + 2} ⟨ Λ^{s} b, Λ^{s} (n \cdot \nabla u) ⟩} \\ + A ‖ u_{2} ‖_{H^{r + α + 3}}^{2} + A ‖ \partial_{2} b ‖_{H^{r + α + 3}}^{2} + B ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2} + ‖ n \cdot \nabla u ‖_{H^{r + α + 2}}^{2} \\ \leq C (‖ \nabla u ‖_{L^{\infty}} + ‖ \nabla b ‖_{L^{\infty}}) (‖ u ‖_{H^{r + α + 3}}^{2} + ‖ b ‖_{H^{r + α + 3}}^{2}) + C B ‖ u_{2} ‖_{H^{r + α + 2}}^{2} + B δ^{2} ‖ Λ^{s} (n \cdot \nabla u) ‖_{L^{2}}^{2} \\ + δ B ‖ \partial_{2} b ‖_{H^{r + α + 3}}^{2} + C (‖ \partial_{2} b ‖_{H^{r + α + 2}}^{2} + ‖ \partial_{21} b_{2} ‖_{H^{r + α + 2}}^{2}) + C ‖ \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2} + C ‖ u_{2} ‖_{H^{r + α + 2}}^{2} . \end{aligned}

Thus, we have

\begin{aligned} \frac{1}{2} \frac{d}{d t} {A (‖ u ‖_{H^{r + α + 3}}^{2} + ‖ b ‖_{H^{r + α + 3}}^{2}) - B \sum_{0 \leq s \leq r + α + 2} ⟨ Λ^{s} u, Λ^{s} (n \cdot \nabla b) ⟩ - \sum_{0 \leq s \leq r + α + 2} ⟨ Λ^{s} b, Λ^{s} (n \cdot \nabla u) ⟩} \\ + A ‖ u_{2} ‖_{H^{r + α + 3}}^{2} + A ‖ \partial_{2} b ‖_{H^{r + α + 3}}^{2} + B ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2} + B ‖ n_{1} \partial_{1} b_{1, 0} ‖_{H^{r + α + 2}}^{2} + ‖ n \cdot \nabla u ‖_{H^{r + α + 2}}^{2} \\ \leq C (‖ \nabla u ‖_{L^{\infty}} + ‖ \nabla b ‖_{L^{\infty}}) (‖ u ‖_{H^{r + α + 3}}^{2} + ‖ b ‖_{H^{r + α + 3}}^{2}), \end{aligned}

provided that

δ

is small enough.

Thanks to $H^{α + 1} ↪ L^{\infty} (T^{2})$ and Lemma 2.1, we get

\begin{aligned} A ‖ \nabla u ‖_{L^{\infty}} ‖ u ‖_{H^{r + α + 3}}^{2} & \leq A ‖ u ‖_{H^{a + 2}} ‖ u ‖_{H^{r + α + 3}}^{2} \leq A ‖ n \cdot \nabla u ‖_{H^{r + a + 2}} ‖ u ‖_{H^{r + α + 3}}^{2} \\ \leq \frac{1}{10} ‖ n \cdot \nabla u ‖_{H^{r + a + 2}}^{2} + C A^{2} ‖ u ‖_{H^{r + α + 3}}^{4} . \end{aligned}

Similarly, we obtain

A ‖ \nabla u ‖_{L^{\infty}} ‖ b ‖_{H^{r + α + 3}}^{2} \leq C A ‖ u ‖_{H^{a + 2}} ‖ b ‖_{H^{r + α + 3}}^{2} \leq \frac{1}{10} ‖ n \cdot \nabla u ‖_{H^{r + a + 2}}^{2} + C A^{2} ‖ b ‖_{H^{r + α + 3}}^{4},

and

\begin{aligned} A ‖ \nabla b ‖_{L^{\infty}} ‖ u ‖_{H^{r + α + 3}}^{2} & \leq C A ‖ b ‖_{H^{a + 2}} ‖ u ‖_{H^{r + α + 3}}^{2} \\ \leq \frac{1}{10} ‖ b ‖_{H^{a + 2}}^{2} + C A^{2} ‖ u ‖_{H^{r + α + 3}}^{4} \\ \leq \frac{1}{10} (‖ \partial_{2} b ‖_{H^{r + α + 3}}^{2} + ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2}) + C A^{2} ‖ u ‖_{H^{r + α + 3}}^{4}, \end{aligned}

and

\begin{aligned} A ‖ \nabla b ‖_{L^{\infty}} ‖ b ‖_{H^{r + α + 3}}^{2} & \leq C A ‖ b ‖_{H^{a + 2}} ‖ b ‖_{H^{r + α + 3}}^{2} \\ \leq \frac{1}{10} ‖ b ‖_{H^{a + 2}}^{2} + C A^{2} ‖ b ‖_{H^{r + α + 3}}^{4} \\ \leq \frac{1}{10} (‖ \partial_{2} b ‖_{H^{r + α + 3}}^{2} + ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2}) + C A^{2} ‖ b ‖_{H^{r + α + 3}}^{4} . \end{aligned}

For any

N \geq 2 r + α + 4

, by the classical interpolation inequality in the Sobolev spaces, there holds

‖ u ‖_{H^{r + α + 3}} \leq ‖ u ‖_{H^{α + 2}}^{\frac{1}{2}} ‖ u ‖_{H^{N}}^{\frac{1}{2}} \leq C δ^{\frac{1}{2}} ‖ n \cdot \nabla u ‖_{H^{r + a + 2}}^{\frac{1}{2}},

and

‖ b ‖_{H^{r + α + 3}} \leq ‖ b ‖_{H^{α + 2}}^{\frac{1}{2}} ‖ b ‖_{H^{N}}^{\frac{1}{2}} \leq δ^{\frac{1}{2}} {(‖ \partial_{2} b ‖_{H^{r + α + 3}} + ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}})}^{\frac{1}{2}} .

As a result, we have

\begin{aligned} \frac{1}{2} \frac{d}{d t} {A (‖ u ‖_{H^{r + α + 3}}^{2} + ‖ b ‖_{H^{r + α + 3}}^{2}) - B \sum_{0 \leq s \leq r + α + 2} ⟨ Λ^{s} u, Λ^{s} (n \cdot \nabla b) ⟩ - \sum_{0 \leq s \leq r + α + 2} ⟨ Λ^{s} b, Λ^{s} (n \cdot \nabla u) ⟩} \\ + A ‖ u_{2} ‖_{H^{r + α + 3}}^{2} + A ‖ \partial_{2} b ‖_{H^{r + α + 3}}^{2} + B ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2} + ‖ n \cdot \nabla u ‖_{H^{r + α + 2}}^{2} \\ \leq \frac{1}{10} ‖ n \cdot \nabla u ‖_{H^{r + a + 2}}^{2} + \frac{1}{10} (‖ \partial_{2} b ‖_{H^{r + α + 3}}^{2} + ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2}) + C ‖ \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2} . \end{aligned}

(3.6)

Define

D (t) = A ‖ u_{2} ‖_{H^{r + α + 3}}^{2} + A ‖ \partial_{2} b ‖_{H^{r + α + 3}}^{2} + B ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}}^{2} + ‖ n \cdot \nabla u ‖_{H^{r + α + 2}}^{2},

and

E (t) = A (‖ u ‖_{H^{r + α + 3}}^{2} + ‖ b ‖_{H^{r + α + 3}}^{2}) - B \sum_{0 \leq s \leq r + α + 2} ⟨ Λ^{s} u, Λ^{s} (n \cdot \nabla b) ⟩ - \sum_{0 \leq s \leq r + α + 2} ⟨ Λ^{s} b, Λ^{s} (n \cdot \nabla u) ⟩ .

Let

A >> B >> 1

be such that

E (t) \geq ‖ u ‖_{H^{r + α + 3}}^{2} + ‖ b ‖_{H^{r + α + 3}}^{2} .

Then, choosing

δ > 0

small enough, it follows from the inequality (3.6) that

\frac{d}{d t} E (t) + \frac{1}{2} D (t) \leq 0.

For any

N \geq (2 α + 3) r + α + 2 β + 5

, one has

‖ u ‖_{H^{r + α + 3}} \leq ‖ u ‖_{H^{α + 2}}^{\frac{2 (β + 1)}{2 β + 3}} ‖ u ‖_{H^{N}}^{\frac{1}{2 β + 3}} \leq C δ^{\frac{1}{2 β + 3}} ‖ n \cdot \nabla u ‖_{H^{r + α + 2}}^{\frac{2 (β + 1)}{2 β + 3}},

which implies that

\begin{aligned} E (t) & \leq ‖ u ‖_{H^{r + α + 3}}^{2} + ‖ b ‖_{H^{r + α + 3}}^{2} \\ \leq ‖ u ‖_{H^{α + 2}}^{\frac{4 (β + 1)}{2 β + 3}} ‖ u ‖_{H^{N}}^{\frac{2}{2 β + 3}} + ‖ b ‖_{H^{α + 2}}^{\frac{4 (β + 1)}{2 β + 3}} ‖ b ‖_{H^{N}}^{\frac{2}{2 β + 3}} \\ \leq C δ^{\frac{1}{2 β + 3}} ‖ n \cdot \nabla u ‖_{H^{r + α + 2}}^{\frac{2 (β + 1)}{2 β + 3}} + C δ^{\frac{1}{2 β + 3}} {(‖ \partial_{2} b ‖_{H^{r + α + 3}} + ‖ n_{1} \partial_{1} b_{2, 0} ‖_{H^{r + α + 2}})}^{\frac{4 (β + 1)}{2 β + 3}} \\ \leq C (D (t))^{\frac{2 (β + 1)}{2 β + 3}} . \end{aligned}

Therefore, we obtain a Lyapunov-type differential inequality

\frac{d}{d t} E (t) + c (E (t))^{\frac{2 (β + 1)}{2 β + 3}} \leq 0.

Solving this inequality yields

E (t) \leq C (1 + t)^{- 2 (β + 1)} .

(3.7)

Taking

s = N

in (2.1) and the Sobolev embedding gives

\frac{d}{d t} (‖ u ‖_{H^{N}}^{2} + ‖ b ‖_{H^{N}}^{2}) + ‖ \partial_{2} b ‖_{H^{N}}^{2} + ‖ u_{2} ‖_{H^{N}}^{2} \leq C (‖ u ‖_{H^{3}} + ‖ b ‖_{H^{3}}) (‖ u ‖_{H^{N}}^{2} + ‖ b ‖_{H^{N}}^{2}) .

(3.8)

Thanks to the inequality (3.7), we have

\int_{0}^{t} (‖ u ‖_{H^{3}} + ‖ b ‖_{H^{3}}) d τ \leq C,

so, applying the Gronwall inequality to (3.8), we have

‖ u ‖_{H^{N}}^{2} + ‖ b ‖_{H^{N}}^{2} \leq C (‖ u_{0} ‖_{H^{N}}^{2} + ‖ b_{0} ‖_{H^{N}}^{2}) .

Take the initial data such that

(‖ u_{0} ‖_{H^{N}}^{2} + ‖ b_{0} ‖_{H^{N}}^{2})

is sufficiently small and satisfies

C (‖ u_{0} ‖_{H^{N}}^{2} + ‖ b_{0} ‖_{H^{N}}^{2}) \leq (δ / 2)

. Then, by a bootstrap argument, it can be deduced that the local solution can be extended to a global solution.

Moreover, by (3.7), we also establish the following decay rate

‖ u (t) ‖_{H^{r + α + 3}} + ‖ b (t) ‖_{H^{r + α + 3}} \leq C (1 + t)^{- (β + 1)} .

Thus, for any

γ > r + α + 3

, choosing

N > γ

and using the interpolation inequality

‖ f (t) ‖_{H^{γ}} \leq ‖ f (t) ‖_{H^{r + α + 3}}^{\frac{N - γ}{N - r - α - 3}} ‖ f (t) ‖_{H^{N}}^{\frac{γ - r - α - 3}{N - r - α - 3}},

we get the decay rate for the higher order energy

‖ u (t) ‖_{H^{γ}} + ‖ b (t) ‖_{H^{γ}} \leq C (1 + t)^{- \frac{(N - γ) (β + 1)}{N - r - α - 3}} .

Thus we complete the proof of Theorem 1.1.

Proof. Proof of the Theorem 1.2.

The proof of Theorem 1.2 parallels that of Theorem 1.1. We only outline the key steps in the core argument. Firstly, we need to obtain an estimate analogous to Lemma 3.1, that is, an inequality similar to (3.1). However, it should be noted that the right-hand side of this inequality now involves the norm of $‖ \partial_{2} b_{1} ‖_{H^{r + α + 2}}$ .

Secondly, we need to estimate the norm of $‖ \partial_{2} b_{1} ‖_{H^{r + α + 2}}$ . We follow the strategy in the proof of Lemma 3.2 but with a modified procedure: we decompose $b_{1}$ with respect to the $x_{1}$ -variable into its zero-frequency and non-zero frequency components, then specifically estimate the zero-mode component $‖ \partial_{2} b_{1} ‖_{H^{r + α + 2}}$ . For conciseness, we omit the technical details here.

Footnotes

Acknowledgments

Chen was supported by start-up funds for doctoral research of Anhui Normal University (No. 762350). Qin was supported by the National Natural Science Foundation of China (No. 12171486), the Science and Technology Innovation Program of Hunan Province (No. 2024RC3021), the Young Backbone Teachers Project of Hunan Province and Natural Science Foundation for Excellent Young Scholars of Hunan Province (No. 2023JJ20057). Zhu was supported by the research funds for the Central South University, China (No. 2025ZZTS0601).

Author Contributions

All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

ORCID iD

Rui Zhu

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