Determining wavenumbers for the incompressible Hall-magneto-hydrodynamics

TL;DR

This paper uses Littlewood-Paley theory to define determining wavenumbers for the Hall-MHD system, demonstrating that strong solutions exhibit almost finite dimensional long-term behavior based on bounded average wavenumbers.

Contribution

It introduces a new method to identify determining wavenumbers for Hall-MHD using Littlewood-Paley theory, linking boundedness to long-term solution behavior.

Findings

Long-term behavior of strong solutions is almost finite dimensional.

Determining wavenumbers are bounded in certain average senses.

Method provides a new perspective on Hall-MHD dynamics.

Abstract

Using Littlewood-Paley theory, one formulates the determining wavenumbers for the Hall-MHD system, defined for each individual solution $(u,b)$. It is shown that the long time behaviour of strong solutions is almost finite dimensional as the wavenumbers are bounded in certain average senses.

Equations

$$u_t+(u\cdot \nabla )u-(b\cdot \nabla )b+\nabla p=\nu \Delta u+f,$$

$$b_t+(u\cdot \nabla )b-(b\cdot \nabla )u+\eta \nabla \times ((\nabla \times b)\times b)=\mu \Delta b,$$

$$\nabla \cdot u=0,\nabla \cdot b=0,t\in {\mathbb{R}}^{+},x\in {\mathbb{T}}^3.$$

$${\Lambda }_u(t)=:$$

$${\Lambda }_b(t)=:$$

$$\frac{1}{|{\mathbb{T}}^3|}{\int }_{{\mathbb{T}}^3}u(t,x)\mathrm{d}x=\frac{1}{|{\mathbb{T}}^3|}{\int }_{{\mathbb{T}}^3}v(t,x)\mathrm{d}x=0\text{ and }\frac{1}{|{\mathbb{T}}^3|}{\int }_{{\mathbb{T}}^3}b(t,x)-h(t,x)\mathrm{d}x=0.$$

$$\big{(}u_{\leq Q_{u,v}(t)}(t),\ b_{\leq Q_{b,h}(t)}(t)\big{)}=\big{(}v_{\leq Q_{u,v}(t)}(t),h_{\leq Q_{b,h}(t)}(t)\big{)},\ \forall t>0,$$

$$\lim_{t\to\infty}\big{(}\|u(t)-v(t)\|_{L^{2}}+\|b(t)-h(t)\|_{L^{2}}\big{)}=0.$$

$$(u,b)\in L^{\infty}\big{(}0,T;(L^{2}({\mathbb{T}}^{3}))^{2}\big{)}\cap L^{2}\big{(}0,T;(H^{1}({\mathbb{T}}^{3}))^{2}\big{)}.$$

$$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\big{(}\|u\|_{L^{2}}^{2}+\|b\|_{L^{2}}^{2}\big{)}\leq-\nu\|\nabla u\|_{L^{2}}^{2}-\mu\|\nabla b\|_{L^{2}}^{2}.$$

$$\underset{t\nearrow T^{\ast }}{\mathrm{limsup}}(\Vert u(t){\Vert }_{H^s}^2+\Vert b(t){\Vert }_{H^s}^2)=\infty ,$$

$$\Vert u{\Vert }_{L^q(0,T^{\ast };L^p({\mathbb{T}}^3))}+\Vert \nabla b{\Vert }_{L^{\gamma }(0,T^{\ast };L^{\beta }({\mathbb{T}}^3))}=\infty ,$$

$$\frac{3}{p}+\frac{2}{q}\le 1,\frac{3}{\beta }+\frac{2}{\gamma }\le 1,\text{ with }p,\beta \in (3,\infty ].$$

$$\chi (\xi )=\{\begin{array}{l}1,\text{for }|\xi |\le \frac{3}{4}\\ 0,\text{for }|\xi |\ge 1,\end{array}$$

$${\Delta }_{q}u=u_q=:\sum _{k\in {\mathbb{Z}}^n}{\phi }_q(k)\hat{u}(k)e^{i2\pi k\cdot x},$$

$$u=\sum _{q=-1}^{\infty }{u}_q.$$

$$u_{\le Q}:=\sum _{q=-1}^Q{u}_q,u_{(P,Q]}:=\sum _{q=P+1}^Q{u}_q,{\tilde{u}}_q:=\sum _{|p-q|\le 1}{u}_p.$$

$$\|u\|_{H^{s}}=\Big{(}\sum_{q\geq-1}\lambda_{q}^{2s}\|u_{q}\|_{2}^{2}\Big{)}^{\frac{1}{2}}.$$

$$\Vert u_q{\Vert }_r\lesssim {\lambda }_q^{n(\frac{1}{r}-\frac{1}{s})}\Vert u_q{\Vert }_s.$$

$$uv=\sum _{p,q\ge -1}{u}_p{v}_q.$$

$$uv=\sum _{q\ge -1}{u}_{\le q-2}{v}_q+\sum _{q\ge -1}{u}_q{v}_{\le q-2}+\sum _{q\ge -1}{\tilde{u}}_q{v}_q,$$

$$[{\Delta }_q,u_{\le p-2}\cdot \nabla ]v_p={\Delta }_q(u_{\le p-2}\cdot \nabla v_p)-u_{\le p-2}\cdot \nabla {\Delta }_q{v}_p,$$

$$[{\Delta }_q,b_{\le p-2}\times \nabla \times ]h_p={\Delta }_q(b_{\le p-2}\times (\nabla \times h_p))-b_{\le p-2}\times (\nabla \times {\Delta }_q{h}_p).$$

$$\Vert [{\Delta }_q,u_{\le p-2}\cdot \nabla ]v_p{\Vert }_{r_1}\lesssim \Vert v_p{\Vert }_{r_2}\sum _{p^{\prime }\le p-2}{\lambda }_{p^{\prime }}\Vert u_{p^{\prime }}{\Vert }_{r_3}.$$

$$\Vert [{\Delta }_q,b_{\le p-2}\times \nabla \times ]h_p{\Vert }_r\lesssim \Vert h_p{\Vert }_r\sum _{p^{\prime }\le p-2}{\lambda }_{p^{\prime }}\Vert b_{p^{\prime }}{\Vert }_{\infty }.$$

$$b_t+\eta \nabla \times ((\nabla \times b)\times b)=\mu \Delta b,$$

$$\nabla \cdot b=0.$$

$$b_{\le Q_{b,h}(t)}(t)=h_{\le Q_{b,h}(t)}(t),\forall t>0,$$

$$\underset{t\to \infty }{\lim }\Vert b(t)-h(t){\Vert }_{L^2}=0,\forall s>0.$$

$$m_t-\mu \Delta m=-\eta \nabla \times ((\nabla \times m)\times h)-\eta \nabla \times ((\nabla \times b)\times m).$$

$$\begin{array}{rl}& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\sum _{q\ge -1}{\lambda }_q^{2s}\Vert m_q{\Vert }_2^2+\mu \sum _{q\ge -1}{\lambda }_q^{2s+2}\Vert m_q{\Vert }_2^2\\ =& \eta \sum _{q\ge -1}{\lambda }_q^{2s}{\int }_{{\mathbb{T}}^3}{\Delta }_q((\nabla \times m)\times h)\cdot (\nabla \times m_q)\mathrm{d}x\\ & +\eta \sum _{q\ge -1}{\lambda }_q^{2s}{\int }_{{\mathbb{T}}^3}{\Delta }_q((\nabla \times b)\times m)\cdot (\nabla \times m_q)\mathrm{d}x\\ =:& I+J.\end{array}$$

$$\begin{array}{rl}I=& \eta \sum _{q\ge -1}\sum _{|p-q|\le 2}{\lambda }_q^{2s}{\int }_{{\mathbb{T}}^3}{\Delta }_q((\nabla \times m_p)\times h_{\le p-2})\cdot (\nabla \times m_q)\mathrm{d}x\\ & +\eta \sum _{q\ge -1}\sum _{|p-q|\le 2}{\lambda }_q^{2s}{\int }_{{\mathbb{T}}^3}{\Delta }_q((\nabla \times m_{\le p-2})\times h_p)\cdot (\nabla \times m_q)\mathrm{d}x\\ & +\eta \sum _{q\ge -1}\sum _{p\ge q-2}{\lambda }_q^{2s}{\int }_{{\mathbb{T}}^3}{\Delta }_q((\nabla \times {\tilde{m}}_p)\times h_p)\cdot (\nabla \times m_q)\mathrm{d}x\\ =:& I_1+I_2+I_3;\end{array}$$

Full text

KEY WORDS: Hall-MHD system; determining modes

CLASSIFICATION CODE: 35Q35, 35Q85, 37L30

The work of the authors was partially supported by NSF Grant DMS–1108864.

1. Introduction

This paper deals with the finite dimensionality of solutions to the incompressible Hall-magneto-hydrodynamics (Hall-MHD) system, written as follows

[TABLE]

The above system describes the evolution of a system consisting of a magnetic field $b,$ electrons and ions, whose collective motion under $b$ can be approximated as an electrically conducting fluid with velocity field $u.$ The focus here is primarily the visco-resistive case, corresponding to positive fluid viscosity $\nu$ and magnetic resistivity $\mu.$ The external forcing term $f$, which one assumes to have zero mean, renders system (1.1)-(1.3) inhomogeneous. The Hall-MHD system is derived using generalized Ohm’s law which takes into account the effect of the electric current on the Lorentz force, neglected in the derivation of the MHD equations. The resulting extra Hall term $\eta(\nabla\times((\nabla\times b)\times b)),$ distinguishing system (1.1)-(1.3) from the conventional MHD system, becomes significant in the case of large magnetic shear. The coefficient $\eta$ here is proportional to the ion skin length.

The Hall-MHD system has a wide range of applications including modelling solar winds, designing magnetic confinement devices for fusion reactors and interpreting the origin of the geomagnetic field. It is believed to be an essential model for magnetic reconnection, an intriguing phenomenon frequently observed in space plasmas. Over the past decade, the Hall-MHD system has received more attentions from the mathematical community. Acheritogaray, Degond, Frouvelle and Liu [1] rigorously derived the system and established global existence of weak solutions on periodic domains. Chae, Degond and Liu [5] proved existence of global weak solutions in ${\mathbb{R}}^{3}$ as well as that of local smooth solutions. Chae and Lee [6] obtained blow-up criteria and small data global strong solutions. Evidences of ill-posedness can be found in [9, 18, 36]. As for the properties of solutions, temporal decay estimates in energy spaces are due to Chae and Schonbek [12]. For more mathematical results on the Hall-MHD system, e.g., well-posedness results and regularity criteria, please see [2, 15, 16, 17, 21, 22, 35, 39, 44, 45, 46, 47, 48, 49].

In the case of $b\equiv 0,$ system (1.1)-(1.3) reduces to the Navier-Stokes equations (NSE), for which the finite dimensional behaviour of solutions has been extensively studied. As alluded in Kolmogorov’s 1941 phenomenological theory [38], a turbulent flow should have a finite number of degrees of freedom. The first mathematical result in this direction, due to Foiaş and Prodi [26], stated that the higher Fourier modes of a solution to the 2D NSE are controlled by the lower modes asymptotically as time goes to infinity. More precisely, if a certain finite number of Fourier modes of a solution share the same long time behaviour with those of another solution, then the remaining infinitely many Fourier modes of the two solutions also exhibit the same long time behaviour. Thus, the notion of “determining modes” arises naturally. For the 2D NSE, estimates of the number of the determining modes were obtained by Foiaş, Manley, Temam and Treve [25] in terms of the Grashof number, and later improved by Jones and Titi [37], whereas Constantin, Foiaş, Manley and Temam [13] estimated the number of determining modes for the 3D NSE assuming the uniform boundedness of solutions in $H^{1}$. For more details concerning the study of finite dimensionality of the NSE flow, readers are referred to [14, 23, 24, 27, 28, 29, 43].

Motivated by the work of Cheskidov, Dai and Kavlie [11] where a time-dependent determining wavenumber was introduced to estimate the number of determining modes for weak solutions to the 3D NSE in an average sense, this paper aims to adapt the idea therein to the study of the Hall-MHD system. In particular, as finite dimensionality of the closely related MHD system has been investigated by Eden and Libin in [20], one is curious if such results can also be obtained for the Hall-MHD system, which differs nontrivially from the MHD system in many aspects, as illustrated in [6, 9, 36].

One introduces the determining wavenumbers for an individual weak solution $(u,b)$ to system (1.1)-(1.3). Let $\kappa:=\min\{\mu,\nu,\eta^{-1}\mu\},$ $r\in(2,3)$ and $\delta>1.$ Let $c_{r}$ be a constant depending only on $r.$ The determining wavenumbers corresponding to $u$ and $b$ are defined as follows.

[TABLE]

where $\lambda_{q}=2^{q}$ and $u_{q}=\Delta_{q}u,$ the $q$-th Littlewood-Paley projection of $u,$ to be defined in Section 2. One notices that in both definitions the conditions on the high modes resemble the ones in the definitions of the dissipation wavenumbers found in [10, 15]. It is noteworthy that the dissipation wavenumber, which separates the dissipation range from the inertial range of turbulent flows, has been utilized to establish improved regularity criteria for various fluid models in [10, 12, 15, 19]. In this paper, the following theorem shall be proved.

Theorem 1.1**.**

Let $(u,b)$ and $(v,h)$ be two weak solutions to system (1.1)-(1.3) such that for all $t\geq 0,$

[TABLE]

Let $\Lambda_{u,v}(t):=\max\{\Lambda_{u}(t),\Lambda_{v}(t)\}$ and $\Lambda_{b,h}(t):=\max\{\Lambda_{b}(t),\Lambda_{h}(t)\}$ with $\Lambda_{u}(t),$ $\Lambda_{v}(t),$ $\Lambda_{b}(t)$ and $\Lambda_{h}(t)$ defined as in Definition 1.4-1.5. Let $Q_{u,v}(t)$ and $Q_{b,h}(t)$ be such that $\Lambda_{u,v}(t)=\lambda_{Q_{u,v}(t)}\text{ and }\Lambda_{b,h}(t)=\lambda_{Q_{b,h}(t)}.$ If

[TABLE]

then

[TABLE]

Remark 1.2*.*

Due to the Galilean invariance of the fluid equation, it suffices to assume that $u$ and $v$ are zero-mean solutions. Yet, in general the magnetic fields do not have zero means, so in the above theorem it is assumed that the space-average of $b$ is the same as that of $h.$

2. Preliminaries

2.1. Notation

Throughout the paper, $A\lesssim B$ denotes an estimate of the form $A\leq CB$ with some absolute constant $C$. Given a tempered distribution $u,$ one denotes by $\mathcal{F}u=\hat{u}$ and $\mathcal{F}^{-1}u=\check{u}$ the Fourier transform and the inverse Fourier transform of $u,$ respectively. For simplicity, the $L^{p}$-norm $\|\cdot\|_{L^{p}}$ is sometimes written as $\|\cdot\|_{p}$ , while $H^{s}$ denotes the $L^{2}$-based Sobolev spaces.

2.2. Well-posedness results for system (1.1)-(1.3)

In order to discuss the determining modes of the solutions, one had better first clarify the notions and existence of the solutions. Relevant to this paper are the following well-posedness results. From [1], it is known that in ${\mathbb{T}}^{3},$ the Leray-Hopf type weak solution to system (1.1)-(1.3) exists globally in time, just as in the case of the NSE.

Theorem 2.1** (Leray-Hopf type weak solution).**

Let the initial data $u_{0},b_{0}\in L^{2}({\mathbb{T}}^{3}).$ There exists a global weak solution $(u,b)$ to system (1.1)-(1.3) satisfying

[TABLE]

In addition, the following energy inequality holds -

[TABLE]

In [5], local existence of strong solutions was proven. Furthermore, for small initial data, the existence of strong solutions is global.

Theorem 2.2** (Strong solution).**

Let $s>\frac{5}{2}$ be an integer and $u_{0},b_{0}\in H^{s}({\mathbb{T}}^{3})$ with $\nabla\cdot u_{0}=\nabla\cdot b_{0}=0.$ Then:

i) The initial value $(u_{0},b_{0})$ generates a local-in-time classical solution (u,b)\in L^{\infty}\big{(}0,T;(H^{s}({\mathbb{T}}^{3})^{2})\big{)} with $T=T(\|u_{0}\|_{H^{s}},\|b_{0}\|_{H^{s}}).$

ii) There exists a constant $\epsilon=\epsilon(\nu,s)$ such that $(u_{0},b_{0})$ generates a global classical solution (u,b)\in L^{\infty}\big{(}0,\infty;(H^{s}({\mathbb{T}}^{3})^{2})\big{)}, provided that $\|u_{0}\|_{H^{s}}+\|b_{0}\|_{H^{s}}<\epsilon.$

To demonstrate the finite dimensional behaviour of the solutions, the following regularity criterion, found in [6], is needed.

Theorem 2.3** (Prodi-Serrin type regularity criterion).**

Let $s>\frac{5}{2}$ be an integer and $u_{0},b_{0}\in H^{s}({\mathbb{T}}^{3})$ with $\nabla\cdot u_{0}=\nabla\cdot b_{0}=0.$ Then for the first blow-up time $T^{*}<\infty$ of the classical solution to system (1.1)- (1.3), it holds that

[TABLE]

if and only if

[TABLE]

where $p,q,\beta$ and $\gamma$ satisfy the relation

[TABLE]

2.3. Littlewood-Paley decomposition

This section is a brief introduction to the Littlewood-Paley theory, a fundamental tool used throughout the paper. One starts with introducing a family of functions with annular support, $\{\varphi_{q}(\xi)\}_{q=-1}^{\infty}$, which forms a dyadic partition of unity in the frequency domain. Let $\lambda_{q}=2^{q},q\in\mathbb{Z}.$ One chooses a radial function $\chi\in C^{\infty}_{0}({\mathbb{R}}^{n})$ satisfying

[TABLE]

and define $\varphi(x)=\chi(\frac{\xi}{2})-\chi(\xi)$ and $\varphi_{q}(\xi)=\begin{cases}\varphi(\lambda_{q}^{-1}\xi),\text{for }q\geq 0,\\ \chi(\xi),\ \text{for }q=-1.\end{cases}$

For a vector field $u\in\mathcal{S}^{{}^{\prime}}({\mathbb{T}}^{n}),$ one defines the Littlewood-Paley projections as

[TABLE]

where $\hat{u}(k)$ is the $k$-th Fourier coefficient of $u.$ In particular, $\hat{u}(0)=u_{-1}.$ Thus, at least in the distributional sense $u$ can be identified as a sum of its Littlewood-Paley projections

[TABLE]

One also introduces the following notations, which appear throughout the paper,

[TABLE]

The $L^{2}$-based Sobolev spaces can thus be characterized via Littlewood-Paley projections -

[TABLE]

In addition, the following Bernstein’s inequality shall be used extensively.

Lemma 2.4**.**

Let $n$ be the space dimension and let $s\geq r\geq 1,$ then

[TABLE]

**Proof: **See [3].

$\Box$

2.4. Bony’s paraproduct and commutator estimates

The product of two distributions $u$ and $v$ can be formally written as

[TABLE]

Using Bony’s paradifferential calculus, one has the following paraproduct decomposition

[TABLE]

which distinguishes three parts in the product $uv.$

To facilitate the estimations, one introduces the following commutators for the convection or inertial terms and the Hall term, respectively

[TABLE]

[TABLE]

In the upcoming sections, it shall be seen that the commutators, along with the divergence free conditions, reveal certain cancellations within the nonlinear interactions. The commutator (2.6) enjoys the following estimate, proven in [3].

Lemma 2.5**.**

For $\frac{1}{r_{1}}=\frac{1}{r_{2}}+\frac{1}{r_{3}},$ the following inequality is true -

[TABLE]

The commutator (2.7) satisfies an analogous estimate, as shown in [15].

Lemma 2.6**.**

Given that $\nabla\cdot b_{\leq p-2}=0,$ the following inequality holds -

[TABLE]

More detailed study of the aforementioned harmonic analysis tools and their applications can be found in the work of Bahouri, Chemin and Danchin [3].

3. Analysis of a reduced system

To analyze the complete Hall-MHD system, one starts by considering the fluid-free version of system (1.1)-(1.3), written as follows -

[TABLE]

The above system is named electron magneto-hydrodynamic (EMHD) equations as it describes the situation where the ions in the Hall-MHD setting are too heavy to move, leaving only the electrons in motion. As the small-scale limit of the Hall-MHD system, the EMHD equations can be used as a toy model to better understand the Hall term. In [1], the existence of weak solutions to (3.8)-(3.9) on periodic domains, analogous to that of the complete Hall-MHD system, was shown. For more studies concerning the EMHD equations, readers may consult [30, 33, 41].

In the following passages, $b$ and $h$ are two weak solutions to system (3.8)-(3.9). One aims to prove the following analogue of Theorem 1.1.

Theorem 3.1**.**

Let $\Lambda_{b,h}(t):=\max\{\Lambda_{b}(t),\Lambda_{h}(t)\}$ with $\Lambda_{b}(t)$ and $\Lambda_{h}(t)$ defined as in definition (1.5). Let $Q_{b,h}(t)$ be such that $\Lambda_{b,h}(t)=\lambda_{Q_{b,h}(t)}.$ If

[TABLE]

then

[TABLE]

**Proof: **Straightforward calculations show that $m:=b-h$ satisfies

[TABLE]

Multiplying the above equation by $\lambda_{q}^{2s}\Delta_{q}^{2}m,$ integrating by parts and summing over $q$ lead to the following identity.

[TABLE]

One further decomposes the terms $I$ and $J$ using Bony’s paraproduct.

[TABLE]

[TABLE]

One then proceeds to estimate the terms $I_{1},I_{2},I_{3}$ and $J_{1},J_{2},J_{3}.$ As for $I_{1},$ one rewrites it using the commutator notation (2.6) and notices that $I_{12}$ in the following expression vanishes.

[TABLE]

Taking into account that $m_{\leq Q_{b,h}}=0,$ one splits $I_{11}$ by the wavenumber.

[TABLE]

By Lemma 2.5, Hölder’s inequality, Definition 1.5, Young’s inequalities, one estimates $I_{111}$ as follows.

[TABLE]

One estimates $I_{112}$ using Lemma 2.5, Hölder’s inequality, Definition 1.5, Young’s and Jensen’s inequalities.

[TABLE]

For $p,q\in\mathbb{Z}$ satisfying $|p-q|\leq 2,$ it is true that $|h_{\leq q-2}-h_{\leq p-2}|\leq\sum_{i=0}^{3}|h_{q-i}|.$ Since $m_{q}=0,\forall q\leq Q_{b,h},$ the following generic bound is true -

[TABLE]

The sum is then split by the wavenumber $Q_{b,h}.$

[TABLE]

$I_{131}$ is estimated as follows.

[TABLE]

$I_{132}$ is estimated with Hölder’s inequality, Definition 1.5 and Young’s inequality.

[TABLE]

As $m_{\leq Q_{b,h}}=0,$ it is perceivable that $I_{2}$ consists of only high frequency parts and can be written as follows.

[TABLE]

Let $\delta>s.$ Hölder’s, inequality, Definition 1.5, Young’s and Jensen’s inequalities lead to

[TABLE]

$I_{3}$ is split into three terms as follows.

[TABLE]

Invoking Definition 1.5 and applying Hölder’s, Young’s and Jensen’s inequalities, one can estimate $I_{31},I_{32}$ and $I_{33}$ as follows.

[TABLE]

[TABLE]

[TABLE]

Thus, the estimation for $I$ is completed.

$J_{1},J_{2}$ and $J_{3}$ remain to be estimated. One can write $J_{1},$ whose low frequency parts vanish due to $m_{\leq Q_{b,h}}=0$, as

[TABLE]

Recalling Definition 1.5, one can estimate $J_{1}$ using Hölder’s, Young’s and Jensen’s inequalities, provided that $\delta>s+1.$

[TABLE]

$J_{2}$ can be partitioned into two terms with $Q_{b,h}$.

[TABLE]

To estimate $J_{21},$ one applies Hölder’s and Young’s inequalities.

[TABLE]

For $J_{22},$ Hölder’s inequality, Definition 1.5, Young’s and Jensen’s inequalities yield

[TABLE]

Taking advantage of $m_{\leq Q_{b,h}}=0,$ one write $J_{3}$ as

[TABLE]

which can then be estimated as follows.

[TABLE]

Let $c_{r}=1-(2\mu)^{-1}.$ Assembling all the estimates above leads to

[TABLE]

Setting $s=0,$ one sees that $\frac{\mathrm{d}}{\mathrm{d}t}\|m\|_{L^{2}}^{2}\lesssim-\lambda_{0}^{2}\|m\|_{L^{2}}^{2}.$ The desired result then follows from Grönwall’s inequality.

$\Box$

4. Proof of Theorem 1.1

Let $(u,b)$ and $(v,h)$ be two weak solutions to system (1.1)-(1.3). Straightforward calculations show that the difference $(w,m):=(u-v,b-h)$ satisfies the following system of equations.

[TABLE]

Utilizing the wavenumbers, one shall eventually prove that $(w,m)$ satisfies the following inequality

[TABLE]

which leads to theorem (1.1).

To this end, one considers a frequency-localized version of system (4.10) in energy spaces. Multiplying the equations by $\lambda_{q}^{2s}\Delta_{q}^{2}w$ and $\lambda_{q}^{2s}\Delta_{q}^{2}m$ respectively, integrating by parts and summing over $q,$ one obtains

[TABLE]

and

[TABLE]

The tasks are then to control the terms $A$–$J.$

4.1. Estimation of A

The estimates for $A$ fall into the same line as those in [11]. Bony’s decomposition leads to the following -

[TABLE]

Using Definition 1.4, one then separate the lower and higher modes of $A_{1}.$

[TABLE]

To control the lower modes, one uses Definition 1.4, Lemma 2.4, Hölder’s and Young’s inequalities.

[TABLE]

The higher modes are estimated as follows.

[TABLE]

It follows from the condition $w_{\leq Q_{u,v}}=0$ that

[TABLE]

Recalling Definition 1.4, one then estimates $A_{2}$ using Hölder’s and Young’s inequalities.

[TABLE]

Separating lower and higher modes of $A_{3}$ with the wavenumber $Q_{u,v}$ results in

[TABLE]

One has no difficulty in controlling the few lower modes.

[TABLE]

The higher modes are estimated using Definition 1.4, Hölder’s, Young’s and Jensen’s inequalities.

[TABLE]

4.2. Estimation of B

As a result of Bony’s paraproduct decomposition

[TABLE]

Since $w_{\leq Q_{u,v}}=0,$ $B_{1}$ consists of only higher modes.

[TABLE]

Let $1-\frac{3}{r}<0.$ One can estimate $B_{1}$ using Definition 1.4, Hölder’s, Young’s and Jensen’s inequalities.

[TABLE]

Splitting $B_{2}$ with the wavenumber $Q_{u,v}$ results in

[TABLE]

The estimate for the lower modes $|B_{21}|+|B_{22}|$ are as follows.

[TABLE]

The estimate for $B_{23}$ follows from Definition 1.4 and Hölder’s inequality.

[TABLE]

Similar to previous terms, $|B_{3}|$ is bounded above by the estimates for the lower modes and for the higher modes.

[TABLE]

The term $B_{31},$ consisting of scarce lower modes, can be controlled with ease.

[TABLE]

As a result of Definition 1.4, Hölder’s, Young’s and Jensen’s inequalities, $B_{32}$ can be estimated as follows.

[TABLE]

4.3. Estimation of C

Bony’s paraproduct decomposition yields

[TABLE]

Moreover, one rewrites $C_{1}$ using the commutator as

[TABLE]

As will be seen later, $C_{12}$ cancels a part of the term $G.$

Taking into account that $m_{\leq Q_{b,h}}=0,$ one split $C_{11}$ using the wavenumber $Q_{b,h}.$

[TABLE]

By Definition 1.5, Hölder’s and Young’s inequalities, the following estimate holds.

[TABLE]

As a result of Definition 1.5, Hölder’s and Young’s inequalities, the following estimate for $C_{113}$ is true.

[TABLE]

$|C_{13}|$ is bounded above by two terms as follows.

[TABLE]

The estimate for $C_{131}$ is as follows.

[TABLE]

$C_{132}$ enjoys the following estimate, thanks to Definition 1.5.

[TABLE]

Since $\nabla m_{\leq Q_{b,h}}=0,$ the lower modes of $C_{2}$ vanish and it can be seen that

[TABLE]

which is estimated using Hölder’s, Young’s and Jensen’s inequalities as

[TABLE]

One splits $C_{3}$ into lower and higher modes.

[TABLE]

$C_{31},$ made up from the scarce lower modes, is estimated as follows.

[TABLE]

One recalls Definition 1.5 and applies Hölder’s, Young’s and Jensen’s inequalities to bound $C_{32}.$

[TABLE]

4.4. Estimation of D

Bony’s paraproduct decomposition yields

[TABLE]

Utilizing the wavenumber $Q_{b,h},$ one splits $D_{1}$ into three terms.

[TABLE]

One can estimate $|D_{11}|+|D_{12}|$ without difficulties.

[TABLE]

By Definition 1.5, Hölder’s, Young’s and Jensen’s inequalities, one has

[TABLE]

It turns out that $D_{2}$ consists of only higher modes, as $m_{\leq Q_{b,h}}=0.$

[TABLE]

Using Hölder’s Young’s and Jensen’s inequalities, one estimates $D_{2}.$

[TABLE]

One split $D_{3}$ into the lower modes, which are rather few, and the higher modes, which are the majority.

[TABLE]

$D_{31}$ satisfies the following estimate.

[TABLE]

The estimate for $D_{32}$ follows from Definition 1.5, Hölder’s, Young’s and Jensen’s inequalities.

[TABLE]

4.5. Estimation of E

One decomposes $E$ using Bony’s paraproduct.

[TABLE]

Utilizing the wavenumber $Q_{u,v},$ $E_{1}$ is split into two.

[TABLE]

By Definition 1.4, Hölder’s, Young’s and Jensen’s inequalities, $E_{11}$ and $E_{12}$ are estimated in the following ways.

[TABLE]

[TABLE]

By the commutator notation, one has

[TABLE]

where $E_{22}$ vanishes as $\nabla\cdot v_{\leq q-2}=0.$

Splitting $E_{21}$ by the wavenumber $Q_{u,v},$ one has

[TABLE]

Using Definition 1.4, Lemma 2.5, Hölder’s and Young’s inequalities, one can estimate $E_{211}.$

[TABLE]

The term $E_{212}$ can be estimated in a similar fashion.

[TABLE]

The estimate for $E_{213}$ follows from Definition 1.4, Lemma 2.5, Hölder’s and Young’s inequalities.

[TABLE]

Explicitly writing out $(v_{\leq p-2}-v_{\leq q-2})$ leads to

[TABLE]

The estimate for $E_{231}$ is as follows.

[TABLE]

By Definition 1.4, Hölder’s and Young’s inequalities, $E_{232}$ can be estimated.

[TABLE]

One separates the lower and higher modes of $E_{3}$ using the wavenumber $Q_{u,v}.$

[TABLE]

With the help of Definition 1.4, Hölder’s, Young’s and Jensen’s inequalities, the terms $E_{31}$ and $E_{32}$ can be under control.

[TABLE]

[TABLE]

4.6. Estimation of F

By Bony’s paraproduct decomposition, one has

[TABLE]

Using the fact that $m_{\leq Q_{b,h}}=0,$ one splits $F_{1}$ into two terms.

[TABLE]

To estimate $F_{11},$ one uses Definition 1.5, Hölder’s and Young’s inequalities.

[TABLE]

By Definition 1.5, Hölder’s and Young’s inequalities, $F_{12}$ satisfies the following.

[TABLE]

$F_{2}$ is split into lower and higher modes based on the wavenumber $Q_{b,h}$ as well as the fact that $m_{\leq Q_{b,h}}=0.$

[TABLE]

It follows from Definition 1.5, Hölder’s, Young’s and Jensen’s inequalities that

[TABLE]

One estimates $F_{22}$ with the help of Hölder’s, Young’s and Jensen’s inequalities.

[TABLE]

As $m_{\leq Q_{b,h}}=0,$ one splits $F_{3}$ into two terms.

[TABLE]

The estimate for $F_{31}$ is as follows.

[TABLE]

One uses Hölder’s, Young’s and Jensen’s inequalities to estimate $F_{32}.$

[TABLE]

4.7. Estimation of G

Using Bony’s paraproduct decomposition, one has

[TABLE]

Taking into account that $m_{\leq Q_{b,h}}=0,$ one separates lower and higher modes of $G_{1}$ by the wavenumber $Q_{b,h}.$

[TABLE]

Thanks to the fact that $q=Q_{b,h}+1$ or $Q_{b,h}+2$, one can control $G_{11}.$

[TABLE]

Using Definition 1.5, Hölder’s, Young’s and Jensen’s inequalities, one has

[TABLE]

Rewriting $G_{2}$ using the commutator notation yields

[TABLE]

One further splits $G_{21}$ into three parts by the wavenumber $Q_{b,h}.$

[TABLE]

Using Definition 1.5, Hölder’s and Young’s inequalities, one can estimate $|G_{211}|+|G_{212}|.$

[TABLE]

The estimate for $G_{213}$ is as follows.

[TABLE]

As noted before, $G_{22}$ and $C_{12}$ cancel each other.

[TABLE]

Since $m_{\leq Q_{b,h}}=0,$ $G_{23}$ consists of mostly higher modes.

[TABLE]

By Definition 1.5, Hölder’s and Young’s inequalities, one has

[TABLE]

$G_{232}$ is estimated as follows.

[TABLE]

One divides $G_{3}$ into lower and higher modes using the wavenumber $Q_{b,h}.$

[TABLE]

One can estimate $G_{31}$ in the following way.

[TABLE]

Meanwhile, by Definition 1.5, Hölder’s, Young’s and Jensen’s inequalities, it holds that

[TABLE]

4.8. Estimation of H

By Bony’s paraproduct decomposition, one has

[TABLE]

By the wavenumber $Q_{u,v},$ the term $H_{1}$ can be split into three parts.

[TABLE]

One can estimate $H_{11}$ with the help of Definition 1.4, Hölder’s and Young’s inequalities.

[TABLE]

To estimate $H_{12},$ one recalls Definition 1.4 and applies Hölder’s and Young’s inequalities.

[TABLE]

As a result of Definition 1.4, Hölder’s, Young’s and Jensen’s inequalities, one has

[TABLE]

$H_{2}$ is split into lower and higher modes.

[TABLE]

which are estimated by Definition 1.4, Hölder’s, Young’s and Jensen’s inequalities.

[TABLE]

[TABLE]

One also divides $H_{3}$ into two terms.

[TABLE]

By Definition 1.4, Hölder’s, Young’s and Jensen’s inequalities, one has

[TABLE]

For $H_{32},$ the following estimate holds.

[TABLE]

4.9. Conclusion

As the terms $I$ and $J$ are already estimated in Section 3, one sums up all the previous estimates and chooses a suitable constant $c_{r}$ to obtain

[TABLE]

One can see that \big{(}\|w\|_{L^{2}}^{2}+\|m\|_{L^{2}}^{2}\big{)} decays to [math] exponentially as $t\to\infty$ as a result of Grönwall’s inequality.

$\Box$

5. Bounds on the wavenumbers

In [11], it was shown that the time average of the determining wavenumber for a weak solution to the Navier-Stokes equations is bounded above by Kolmogorov’s dissipation wavenumber via the average energy dissipation rate $\varepsilon:=\langle\|\nabla u\|^{2}_{L^{2}}\rangle,$ where $\langle\cdot\rangle$ signifies the time average. For the 2D MHD system, it is also known that explicit dimension estimates of functional invariant sets can be given by the energy dissipation rate.

Yet, in the case of the Hall-MHD system, it seems impossible to bound the wavenumber $\Lambda_{b,h}(t)$ using the average magnetic energy dissipation rate $\langle\|\nabla b\|_{L^{2}}^{2}\rangle$. Fortunately, restricting one’s attentions to strong solutions can lead to a reasonable bound on $\Lambda_{b,h}(t)$ in an average sense. Indeed, whenever $\Lambda_{b,h}(t)>\lambda_{0}$, it must be that one of the conditions in Definition 1.5 is unfulfilled, i.e., $\|b_{Q_{b,h}(t)}\|_{\infty}>c_{r}\kappa$ or $\|b_{\leq Q_{b,h}(t)-1}\|_{\infty}>c_{r}\kappa.$

The inequality $\|b_{Q_{b,h}(t)}\|_{\infty}>c_{r}\kappa$ implies that

[TABLE]

By Lemma 2.4, one has

[TABLE]

Meanwhile, if $\|b_{\leq Q_{b,h}(t)-1}\|_{\infty}>c_{r}\kappa,$ then

[TABLE]

which, by Lemma 2.4, yields

[TABLE]

Hence, for (u,b)\in L^{\infty}\big{(}0,\infty;(H^{s}({\mathbb{T}}^{3}))^{2}\big{)}, one has, by Theorem 2.3, the following bound.

[TABLE]