1.1 Model and synopsis of related studies

This article focuses on the following compressible magnetohydrodynamic (MHD) system without shear viscosity, given by

(1.1)

where $\mathbb T^2$ denotes the two-dimensional periodic torus. Here, $\rho =\rho (t,x)$ , $ \mathbf {u} = \mathbf {u} (t,x),$ and $\mathbf {B}= \mathbf {B} (t,x)$ represent the fluid density, velocity field, and the magnetic field, respectively. The pressure $P=P(\rho )$ is a function of the density, which is assumed to be smooth and strictly increasing. The constant $\nu>0$ is the bulk viscosity coefficient, and $\sigma>0$ is the electrical conductivity of the field.

The MHD system (1.1) possesses a special equilibrium state characterized by a constant density, a zero velocity field, and a background magnetic field. This article aims to conduct a perturbation analysis around this state, investigating fundamental questions, such as global existence, stability, and large-time asymptotic behavior. Our study is motivated by two primary factors. First, we seek to provide a rigorous mathematical explanation for the experimentally observed stabilizing effect of a background magnetic field (see, e.g., [Reference Alexakis1–Reference Bardos, Sulem and Sulem3, Reference Chen, Zhang and Zhou5, Reference Gallet, Berhanu and Mordant14, Reference Gallet and Doering15, Reference Zhai44]). Second, we address the inherent mathematical challenges caused by the absence of shear viscosity in the system. The lack of uniform ellipticity, when only bulk viscosity is present, significantly complicates the control of nonlinear terms and demands novel analytical approaches.

We outline some physical background for the system (1.1). The dynamics of electrically conducting fluids interacting with magnetic fields are governed by the MHD equations. Compressible MHD models serve as the fundamental framework for theoretically describing turbulence in the solar wind [Reference Tandberg-Hanssen and Emslie35]. Since observed fluctuations often involve density variations, plasma compressibility effects must be considered in the theory. In standard MHD theory, the momentum equation includes both shear viscosity and volume viscosity, namely,

(1.2) $$ \begin{align} \left\{\begin{aligned} &\partial_t \rho + {\mathrm{div}} (\rho \mathbf{u}) =0, \\ &\rho\partial_t \mathbf{u}+ \rho\mathbf{u}\cdot\nabla \mathbf{u}-\mu\Delta \mathbf{u} - \nu \nabla {\mathrm{div}} \mathbf{u}+\nabla P=(\nabla\times\mathbf{B})\times\mathbf{B},\\ &\partial_t \mathbf{B}-\sigma\Delta \mathbf{B}+\mathbf{u}\cdot\nabla\mathbf{B}-\mathbf{B}\cdot\nabla\mathbf{u}+\mathbf{B} \text{ div}\, \mathbf{u}=0,\\ &{\mathrm{div}} \mathbf{B} =0,\\ &(\rho, \mathbf{u} ,\mathbf{B})|_{t=0}=(\rho_0, \mathbf{u} _0,\mathbf{B}_0), \end{aligned}\right. \end{align} $$

where $\mu \ge 0$ is the shear viscosity coefficient. In general, shear viscosity coefficient $\mu $ in the system (1.2) plays a central role in the study of fluid dynamics. Ultimately, it is the source of all forms of loss and drag and gives rise to the instabilities which lead to flow turbulence. As a result, its molecular interpretation, variation with temperature and pressure, and numerical values can be found in nearly every reference on fluid mechanics. However, several fluids, including common diatomic gases, are seen to have bulk viscosities which are hundreds or thousands of times larger than their shear viscosities. Therefore, significant attention has been drawn to these fluids with bulk viscosities that are large compared to their shear viscosities, even in the case of zero shear viscosity, that is, system (1.1). As Landau and Lifshitz noted in [Reference Landau and Lifshitz24], “if the relaxation time of these processes is long, significant energy dissipation occurs when the fluid is compressed or expanded,” leading to a relatively large volume viscosity coefficient and a much smaller shear viscosity coefficient. This observation has been echoed in more recent studies [Reference Cramer8, Reference Graves and Argrow17], which highlight that “several fluids, including common diatomic gases, have bulk viscosities that are hundreds or thousands of times larger than their shear viscosities.” For more discussions about the system (1.1) and related models, we refer the reader to the papers [Reference Besse and Cheverry4, Reference Ewing, Goodwin, McGlashan and Trusler12, Reference Herzfeld and Litovitz18] and the references therein.

Here, it should be pointed out that system (1.1) does not fit into conventional mathematical frameworks so that its well-posedness has not been fully clarified. Before we state our main results, we briefly review some of the existing results related to the system (1.2).

• • When $\mu>0$ and $\sigma>0$ , system (1.2) represents the compressible viscous resistive MHD equations. Due to their extensive physical applications, complexity, and the rich phenomena they model, these equations have garnered significant attention. The compressible MHD system with viscosity and resistance has been the subject of substantial research, focusing on the global existence of weak solutions [Reference Ducomet and Feireisl11, Reference Hu and Wang21], local and global well-posedness of strong solutions with vacuum [Reference Hong, Hou, Peng and Zhu20, Reference Li, Xu and Zhang26], and large-time behavior [Reference Wu, Zhang and Zou37], among others.

• • When $\mu>0$ and $\sigma =0$ , system (1.2) reduces to the compressible viscous non-resistive MHD equations. Wu and Wu [Reference Wu and Wu38] achieved the global well-posedness of these equations in ${\mathbb {R}}^2$ for strong solutions using a systematic approach. Tan and Wang [Reference Tan and Wang34] established the global existence of smooth solutions in a 3D compressible barotropic viscous non-resistive MHD system in a horizontally infinite flat layer. Wu and Zhu [Reference Wu and Zhu40] pioneered the investigation of this system on bounded domains using pure energy estimates. Dong et al. [Reference Dong, Wu and Zhai10] proved the global existence of strong solutions for the $2\frac 12$ -D compressible non-resistive MHD equations with small initial data. Recently, Wu and Zhai [Reference Wu and Zhai39] demonstrated the global well-posedness of these equations in $ \mathbb {T} ^3$ under the condition that the initial magnetic field is sufficiently close to an equilibrium state. For weak solutions, Li and Sun [Reference Li and Sun27] established the existence of global weak solutions for the 2D non-resistive compressible MHD equations. This result was extended by [Reference Liu and Zhang28] to include a density-dependent viscosity coefficient and a non-monotone pressure law.

• • When $\mathbf {B} = \mathbf {0}$ and $\mu =\nu =0$ , system (1.2) reduces to the isentropic compressible Euler equations which have been shown to form finite-time singularities from smooth initial data, such as shocks and cusps in all dimensions (1D, 2D, and 3D), see, for example, [Reference Christodoulou6, Reference Christodoulou7, Reference Luk and Speck29, Reference Merle, Raphael, Rodnianski and Szeftel30, Reference Sideris33, Reference Xin41, Reference Yin43]. In the compressible regime of fluids, the presence of the shear viscosity prevents the formation of shocks near the equilibrium. As the shear viscosity vanishes, the shock formation is expected even for small initial perturbations. However, in electrically conducting fluids, the volume viscosity is expected to play a similar role in prohibiting the shock formation.

• • When $\mathbf {B} = \mathbf {0}$ and $\mu>0$ , $\nu>0$ , system (1.2) reduces to the isentropic compressible Navier–Stokes equations which have been extensively investigated (see, e.g., [Reference Feireisl13, Reference Hoff19, Reference Li, Wang and Xin25, Reference Xin41, Reference Xin and Yan42]).

When the shear viscosity coefficient $\mu =0$ , it remains a challenging open problem whether system (1.1) admits global classical solutions for the general smooth initial data. The term $\nu \nabla \text {div}\, $ provides only partial ellipticity, making the classical dissipation mechanisms for velocity inapplicable. To the best of our knowledge, there are few mathematical results for these fluids. This article focuses on the global existence and stability of solutions to system (1.1) near the steady-state $ (\bar {\rho }, 0, \bar {\mathbf {B}}) $ with $\bar {\rho }>0$ being a constant and $\bar {\mathbf {B}}$ a constant vector field in a periodic domain.

1.2 Main result

For simplicity, we take the background magnetic field to be $\bar {\mathbf {B}}= \mathbf {e}_2=(0,1)^{\top }$ . We restrict the initial perturbations to lie in certain symmetry classes, a choice partly motivated by [Reference Pan, Zhou and Zhu32, Reference Zhou and Zhu45]. Without loss of generality, we set

$$ \begin{align*} \left|\mathbb{T}^2\right|=(2\pi)^2\stackrel{\mathrm{def}}{=}\alpha. \end{align*} $$

The initial data are assumed to satisfy

(1.3) $$ \begin{align} \int_{\mathbb{T}^2}\rho_0\,dx=\bar{\rho},\quad\int_{\mathbb{T} ^2}\rho_0 \mathbf{u} _0\,dx=0, \quad\text{and}\quad \int_{\mathbb{T} ^2} \mathbf{B}_{0,2}(x)\ dx =\alpha\neq0, \end{align} $$

together with the symmetry conditions:

(1.4) $$ \begin{align} \rho_0(x),\quad \mathbf{u} _{0,1}(x), \quad \mathbf{B}_{0,2}(x)\quad &\text{are even periodic with respect to } x_2, \end{align} $$

(1.5) $$ \begin{align} \mathbf{u} _{0,2}(x),\quad \mathbf{B}_{0,1}(x) \quad &\text{are odd periodic with respect to } x_2. \end{align} $$

One may verify that the conditions (1.3)–(1.5) are preserved under the time evolution, so that for all $t \geq 0$ ,

(1.6) $$ \begin{align} \int_{\mathbb{T}^2}\rho\,dx=\bar{\rho},\quad\int_{\mathbb{T} ^2}\rho \mathbf{u} \,dx=0, \quad\text{and}\quad \int_{\mathbb{T} ^2} \mathbf{B}_{2}(x)\ dx =\alpha\neq0, \end{align} $$

and

(1.7) $$ \begin{align} \rho(x),\quad \mathbf{u} _{1}(x), \quad \mathbf{B}_{2}(x)\quad &\text{are even periodic with respect to } x_2, \end{align} $$

(1.8) $$ \begin{align} \mathbf{u} _{2}(x),\quad \mathbf{B}_{1}(x) \quad &\text{are odd periodic with respect to } x_2. \end{align} $$

We now state our main result for the 2D MHD system (1.1).

Theorem 1.1 Let $\nu> 0$ and $\sigma> 0$ in (1.1). Let ${N}\ge 12$ be an integer. Consider the system (1.1) with initial data $(\rho _0, \mathbf {u} _0, \mathbf {B} _0)$ satisfying (1.3)–(1.5), and, for some constant $c_0>0$ ,

$$ \begin{align*} \rho_0-\bar{\rho}\in H^{N}( \mathbb{T} ^2),\quad c_0\le\rho_0\le c_0^{-1},\quad \mathbf{u} _0\in H^{N}( \mathbb{T} ^2), \quad \mathbf{B} _0-\mathbf{e}_2\in H^{N}( \mathbb{T} ^2). \end{align*} $$

Then there exists a small constant $\varepsilon $ such that, if

$$ \begin{align*} \left\lVert \rho_0-\bar{\rho} \right\rVert_{H^{N}}+\left\lVert \mathbf{u} _0 \right\rVert_{H^{N}}+\left\lVert \mathbf{B} _0-\mathbf{e}_2 \right\rVert_{H^{N}}\le\varepsilon, \end{align*} $$

then the system (1.1) admits a unique global solution $(\rho -\bar {\rho }, \mathbf {u} , \mathbf {B} -\mathbf {e}_2)\in C([0,\infty ); H^{N}( \mathbb {T} ^2))$ . Moreover, for any $ [\frac {N}{2}] +4\le \beta \le N $ and $t\ge 0$ , the solution satisfies the decay estimate

(1.9) $$ \begin{align} \left\lVert (\rho-\bar{\rho})(t) \right\rVert_{H^{\beta}}+\left\lVert \mathbf{u} (t) \right\rVert_{H^{\beta}}+\left\lVert \mathbf{B} (t)-\mathbf{e}_2 \right\rVert_{H^{\beta}}\le C \varepsilon (1+t)^{-\frac{3({N}-\beta)}{2\big({N}-( [\frac{N}{2}] +4)\big)}}. \end{align} $$

The symmetry conditions (1.4) and (1.5) are introduced to exploit the hidden dissipation mechanism that plays a central role in our analysis. Due to the absence of shear viscosity, the system lacks full ellipticity, and the classical dissipation mechanisms for the velocity are not available. Instead, our approach relies on a hidden dissipation mechanism arising from the coupling between the velocity and the magnetic field, through which the directional derivative $\partial _2 \mathbf {u} $ becomes effectively dissipative. The parity structure imposed by (1.4) and (1.5) allows us to apply Lemma 2.3, which yields control of $\nabla \mathbf {u} $ in terms of $\partial _2 \mathbf {u} $ and $ \text { div}\, \mathbf {u} $ . In particular, these assumptions ensure that the vorticity has zero average in the $x_2$ -direction, eliminating low-frequency obstructions and enabling the recovery of dissipation. The assumptions are not merely technical, but are closely tied to the mechanism through which the magnetic field stabilizes the system. They identify a natural class of perturbations around the equilibrium for which the hidden dissipation becomes effective.

1.3 Difficulties and scheme of the proof

We explain the main difficulties and our strategies for proving Theorem 1.1. For notational convenience, we set $\bar \rho =1$ and define

$$ \begin{align*}a\stackrel{\mathrm{def}}{=}\rho-1\quad\text{and}\quad \mathbf{H} \stackrel{\mathrm{def}}{=} \mathbf{B} -{ \mathbf{e}_2 }.\end{align*} $$

It then follows from the system (1.1) that the perturbation $(a, \mathbf {u} , \mathbf {H} )$ satisfies

(1.10) $$ \begin{align} \left\{\begin{aligned} &\partial_t a + \text{ div}\, \mathbf{u} + \text{ div}\, (a \mathbf{u} )=0,\\ &(a+1)\partial_t \mathbf{u} + (a+1) \mathbf{u} \cdot\nabla \mathbf{u} - \nu \nabla \text{ div}\, \mathbf{u} +\nabla P=\partial_2 \mathbf{H} + \mathbf{H} \cdot\nabla \mathbf{H} -\nabla \mathbf{H} _2- \mathbf{H} \nabla \mathbf{H} ,\\ &\partial_t \mathbf{H} -\sigma\Delta \mathbf{H} + \mathbf{u} \cdot\nabla \mathbf{H} =\partial_2 \mathbf{u} + \mathbf{H} \cdot\nabla \mathbf{u} - \mathbf{e}_2 \text{ div}\, \mathbf{u} - \mathbf{H} \text{ div}\, \mathbf{u} ,\\ & \text{ div}\, \mathbf{H} =0,\\ &(a, \mathbf{u} , \mathbf{H} )|_{t=0}=(a_0, \mathbf{u} _0, \mathbf{H} _0). \end{aligned}\right. \end{align} $$

The local well-posedness of (1.10) can be shown via a procedure that is now standard for compressible equations (see, e.g., [Reference Kawashima23]). The focus of the proof is on the global bound of $(a, \mathbf {u} , \mathbf {H} )$ in $H^N(\mathbb T^2)$ . We use the bootstrapping argument and start by making the ansatz that, for any $T>0$ ,

$$ \begin{align*}\sup_{t\in[0,T]}\left\lVert (a, \mathbf{u} , \mathbf{H} ) \right\rVert_{H^{N}}\le \delta \end{align*} $$

for suitably chosen $0<\delta <1$ . The main efforts are devoted to proving that, if the initial norm is taken to be sufficiently small, namely,

$$ \begin{align*}\|( a_0, \mathbf{u} _0, \mathbf{H} _0)\|_{H^N} \le \varepsilon \end{align*} $$

for sufficiently small $\varepsilon>0$ , then

(1.11) $$ \begin{align} \sup_{t\in[0,T]}\left\lVert (a, \mathbf{u} , \mathbf{H} ) \right\rVert_{H^{N}}\le \frac{\delta}{2}. \end{align} $$

It is not trivial to prove (1.11). The difficulty is due to the lack of dissipation or damping in the equations of ${\mathbb {P}} \mathbf {u} $ and a, here $\mathbb P=\mathbb I - \nabla \Delta ^{-1} \mbox {div}$ is the projection onto divergence-free vector fields. Without any stabilizing mechanism, the norms of ${\mathbb {P}} \mathbf {u} $ and a would grow in time and it would be impossible to establish (1.11). Therefore, it is crucial to exploit some potential smoothing and stabilizing effects due to the coupling and interaction of a, $ \mathbf {u} $ , and $ \mathbf {H} $ . This article is able to discover the hidden dissipation in two quantities. The first one is the directional derivative of $ \mathbf {u} $ along $ \mathbf {e}_2 $ , namely, $ \partial _2 \mathbf {u} $ , while the second one is the density a. To be more precise, we rewrite system (1.10) as

$$ \begin{align*} \left\{\begin{aligned} &\partial_t a+ \text{ div}\, \mathbf{u} =f_1,\\ &\partial_t \mathbf{u} -\nu\nabla \text{ div}\, \mathbf{u} +\nabla a+\partial_2 \mathbf{H} -\nabla \mathbf{H} _2 =f_2,\\ &\partial_t \mathbf{H} -\sigma\Delta \mathbf{H} +\partial_2 \mathbf{u} + \mathbf{e}_2 \text{ div}\, \mathbf{u} =f_3, \end{aligned}\right. \end{align*} $$

where $f_1$ , $f_2$ , and $f_3$ are essentially nonlinear terms. We outline the main steps in the proof of (1.11) and explain how the coupling and interaction lead to the dissipative in $\partial _2 \mathbf {u} $ and the damping effect of a.

The first step estimates the norm $\|( a, \mathbf {u} , \mathbf {H} )\|_{H^\ell }$ with $0\le \ell \le N$ in terms of the $L^\infty $ -norms of $a,\, \mathbf {u} ,\, \mathbf {H} \,$ and their gradients, and some low-order Sobolev norms,

(1.12) $$ \begin{align} \frac12\frac{d}{dt}\left\lVert (a, \mathbf{u} , \mathbf{H} ) \right\rVert_{H^{\ell}}^2 +\nu\left\lVert \text{div}\, \mathbf{u} \right\rVert_{H^{\ell}}^2+\sigma\left\lVert \nabla \mathbf{H} \right\rVert_{H^{\ell}}^2\le C \mathrm{Y}^{(\ell)}_{\infty}(t)\left\lVert (a, \mathbf{u} , \mathbf{H} ) \right\rVert_{H^{\ell}}^2, \end{align} $$

where $\mathrm {Y}^{(\ell )}_{\infty }(t)$ (defined in Lemma 3.3) is given by

$$ \begin{align*} \mathrm{Y}^{(\ell)}_{\infty}(t)\stackrel{\mathrm{def}}{=}& \left\lVert (\nabla a,\nabla \mathbf{u} ,\nabla \text{ div}\, \mathbf{u} ,\nabla \mathbf{H} ) \right\rVert_{L^\infty}+\left\lVert a \right\rVert_{H^{[\frac{\ell-1}{2}]+4}} +\left\lVert a \right\rVert_{H^{\ell-[\frac{\ell-1}{2}]}} \notag\\ &+\left\lVert \text{div}\, \mathbf{u} \right\rVert_{H^{\ell-[\frac{\ell-1}{2}]}}+\left\lVert a \right\rVert_{H^{[\frac{\ell-1}{2}]+3}}^2 +\left\lVert (\nabla a,\nabla \mathbf{u} ,\nabla \mathbf{H} ) \right\rVert_{L^\infty}^2 \notag\\ &+\|(a, \mathbf{H} )\|_{L^{\infty}}^2 +\|a\|_{L^{\infty}}^2\| \mathbf{H} \|_{L^{\infty}}^2+\| \mathbf{H} \|_{L^{\infty}}^2\|\nabla \mathbf{H} \|_{L^{\infty}}^2. \end{align*} $$

In order to obtain a global bound for $\|(a, \mathbf {u} , \mathbf {H} )\|_{H^\ell }$ , we need to control the time integral of $\mathrm {Y}^{(\ell )}_{\infty }(t)$ . Due to the lack of dissipation or damping terms in the equations of ${\mathbb {P}} \mathbf {u} $ and a, the time integrals involved in $a, \mathbf {u} $ cannot be bounded directly. Our idea is to exploit the hidden dissipation in the quantities $ \partial _2 \mathbf {u} $ and a.

The second main step is to combine the equation of $\mathbb P \mathbf {u} $ (the divergence part of $ \mathbf {u} $ ) and the equation of $ \mathbf {H} $ to establish the stabilizing effect in $ \partial _2 \mathbf {u} $ . We take advantage of the special structure in the following system:

$$ \begin{align*}\begin{cases} \partial_t \mathbb P \mathbf{u} = \partial _2 \mathbf{H} + \mathbb P f_2, \\ \partial_t \mathbf{H} -\sigma\Delta \mathbf{H} -\partial_2 \mathbf{u} + \mathbf{e}_2 \text{ div}\, \mathbf{u} =f_3, \end{cases} \end{align*} $$

and make the stabilizing effect more explicit by using a “cross-energy estimate”

$$ \begin{align*}\frac{d}{dt}\sum_{0\le s\le { [\frac{N}{2}] +3}}\int_{\mathbb{T} ^2} \Lambda ^{s} \mathbf{H} \cdot \Lambda ^{s}\partial_2 \mathbf{u} \,dx. \end{align*} $$

We obtain, as stated in Lemma 3.4, that

(1.13) $$ \begin{align} &\left\lVert \partial_2 \mathbf{u} \right\rVert_{H^{ [\frac{N}{2}] +3}}^2-\frac{d}{dt}\sum_{0\le s\le { [\frac{N}{2}] +3}}\int_{\mathbb{T} ^2}{\Lambda^s} \mathbf{H} \cdot{\Lambda^s}\partial_2 \mathbf{u} \,dx\nonumber\\ &\quad\le C\delta^2\left\lVert a \right\rVert_{H^{ [\frac{N}{2}] +4}}^2+ C\left\lVert \nabla \mathbf{H} \right\rVert_{H^{ [\frac{N}{2}] +4}}^2+C\left\lVert \text{div}\, \mathbf{u} \right\rVert_{H^{ [\frac{N}{2}] +4}}^2, \end{align} $$

which yields the time integrability of $\left \lVert \partial _2 \mathbf {u} (t) \right \rVert _{H^{ [\frac {N}{2}] +3}}^2$ . Combining with the generalized Poincaré type inequality induced by Lemma 2.3, we are then able to control the time integrability of low-order Sobolev norms of $ \mathbf {u} $ .

The third main step is to establish the stabilizing property of the quantity a. The idea is to make use of the interaction between the equation of $a $ and that of $\mathbb Q \mathbf {u} $ :

(1.14) $$ \begin{align} \begin{cases} \partial_t a+ \text{ div}\, \mathbf{u} =f_1,\\ \partial_t {\mathbb{Q}} \mathbf{u} -\nu\Delta {\mathbb{Q}} \mathbf{u} +\nabla (a + \mathbf{H} _2)={\mathbb{Q}} f_2, \end{cases} \end{align} $$

where $\mathbb Q= \nabla \Delta ^{-1} \mbox {div}$ projects vectors onto their gradient parts. Noting that ${\mbox {div} {\mathbb {Q}} \mathbf {u} = \text { div}\, \mathbf {u} }$ , we observe that (1.14) involves a wave structure for a and ${\mathbb {Q}} \mathbf {u} $

$$ \begin{align*}\begin{cases} \partial_{tt} a- \nu \Delta \partial_t a-\Delta a =\Delta \mathbf{H} _2, \\ \partial_{tt} {\mathbb{Q}} \mathbf{u} -\nu \Delta \partial_t {\mathbb{Q}} \mathbf{u} -\nabla \mathbf{u} =-\partial_t\Delta \mathbf{H} _2 \end{cases} \end{align*} $$

if neglecting some nonlinear terms. Alternatively, we can introduce the effective velocity by

$$ \begin{align*} \mathbf{G} \stackrel{\mathrm{def}}{=}{\mathbb{Q}} \mathbf{u} -\frac1\nu\Delta^{-1}\nabla {a}. \end{align*} $$

The effective velocity $ \mathbf {G} $ then satisfies the following system:

$$ \begin{align*} \left\{\begin{aligned} &\partial_t a +\frac1\nu{a}+ \text{ div}\, \mathbf{G} =f_1,\\ &\partial_t \mathbf{G} -\nu\Delta \mathbf{G} =\frac1\nu{\mathbb{Q}} \mathbf{u} -\nabla \mathbf{H} _2 +{\mathbb{Q}} f_2-\frac1\nu\Delta^{-1}\nabla f_1. \end{aligned}\right. \end{align*} $$

These equations clearly reveal the damping in a and the dissipation in $ \mathbf {G} $ . We then estimate the Sobolev norms $\|a\|_{H^{ [\frac {N}{2}] +4}}$ and $\| \mathbf {G} \|_{H^{ [\frac {N}{2}] +4}}$ . Combining (1.12) with $\ell = [\frac {N}{2}] +4$ and (1.13) in the previous steps, we then obtain a self-contained inequality.

The last step is to establish (1.11) and close the bootstrapping argument. The energy inequality obtained in the previous steps, together with interpolation inequalities, allows us to show that

(1.15) $$ \begin{align} \|(a, \mathbf{u} , \mathbf{H} )(t)\|_{H^{ [\frac{N}{2}] +4}} \le C\, (1+ t)^{-\frac32}, \end{align} $$

In particular, the time integral of $\mathrm {Y}^{(\ell )}_{\infty }(t)$ in (1.12) is time integrable:

$$ \begin{align*}\int_0^\infty \mathrm{Y}^{(\ell)}_{\infty}(t) \,dt \le C <\infty. \end{align*} $$

Grönwall’s inequality then yields

$$ \begin{align*}\|(a, \mathbf{u} , \mathbf{H} )(t)\|_{H^N} \le C\, \|(a_0, \mathbf{u} _0, \mathbf{H} _0)\|_{H^N}. \end{align*} $$

Taking the initial norm to be sufficiently small, we achieve (1.11). The decay rate in (1.9) is a consequence of (1.15) and an interpolation inequality.

1.4 Organization of the article

The remainder of this article is structured into three sections. Section 2 collects several auxiliary inequalities that will be instrumental in our subsequent analysis. Subsequently, Section 3 is devoted to the proof of Theorem 1.1.