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Optimal decay-in-time rates of solutions to the Cauchy problem of 3D compressible magneto-micropolar fluids
Boundary Value Problems volume 2024, Article number: 33 (2024)
Abstract
This paper focuses on the long time behavior of the solutions to the Cauchy problem of the three-dimensional compressible magneto-micropolar fluids. More precisely, we aim to establish the optimal rates of temporal decay for the highest-order spatial derivatives of the global strong solutions by the method of decomposing frequency. Our result can be regarded as the further investigation of the one in (Wei, Guo and Li in J. Differ. Equ. 263:2457–2480, 2017), in which the authors only provided the optimal rates of temporal decay for the lower-order spatial derivatives of the perturbations of both the velocity and the micro-rotational velocity.
1 Introduction
The three-dimensional (3D) motion of compressible magneto-micropolar fluids can be described as the following system [19]:
Here the unknowns \(\rho (x,t)>0\), \(\mathbf{u}(x,t)\), \(\boldsymbol{\omega}(x,t)\), and \(\mathbf{B}(x,t)\) mean the density, the velocity, the micro-rotational velocity, and the magnetic field, respectively. The pressure \(P(\rho )\) is a smooth and strictly increasing scalar function. The positive constant ν is the so-called dynamics micro-rotation viscosity. Both the shear and bulk viscosity coefficients of fluids are represented by the parameters μ and λ satisfying that \(\mu >0\) and \(2\mu +3\lambda -4\nu \geqslant 0 \). Besides, the angular viscosity coefficients \(\mu ^{\prime} \) and \(\lambda ^{\prime} \) fulfill the conditions \(\mu ^{\prime}>0 \) and \(2\mu ^{\prime}+3\lambda ^{\prime}\geqslant 0 \). The magnetic diffusivity coefficient is denoted by σ.
While the magnetic field is absent, i.e., \(\mathbf{B}=0\), (1.1) reduces to the micropolar system. Huang, Kong, and Lian [9, 10] obtained the exponential stability of the generalized spherically symmetric solutions. For the Cauchy problem of the 3D compressible viscous micropolar system, Liu and Zhang [13] derived the optimal rates of decay-in-time of the global strong solutions with the smallness of initial perturbation in \(H^{N}(\mathbb{R}^{3})\cap{L^{1}(\mathbb{R}^{3})}\) with \(N\geqslant 4\). Furthermore, Tong, Pan, and Tan [18] used the method of spectrum analyzing to establish both the lower and upper decay-in-time rates for the solutions, which explicitly shows that the obtained convergence rates are optimal. Recently, Qin and Zhang [16] investigated the optimal decay rates for higher-order derivatives of solutions of the 3D compressible micropolar fluids system. In particular, the authors showed that the highest-order spatial derivatives of both the perturbation density and the perturbation velocity converge to zero with the decay rate \((1+t)^{-(\frac{3}{4}+\frac{N}{2})}\) for the \(L^{2}(\mathbb{R}^{3})\)-norm.
Due to the strong nonlinearity and interactions among the physical quantities, it becomes more difficult to analyze the compressible magneto-micropolar system, i.e., \(\mathbf{B} \neq 0\) in system (1.1). Amirat and Hamdache [1] extended the results in [5, 12] by establishing the global existence of weak solutions with finite energy for multi-dimensional compressible magneto-micropolar equations. The blow-up criterion of strong solutions to system (1.1) with initial vacuum can be referred to [22]. By employing \(L^{p}\)–\(L^{q}\) estimates for the linearized equations and the Fourier splitting method, Wei, Guo, and Li [19] first obtained the global-in-time existence and optimal temporal decay rates of the strong solutions to the Cauchy problem of system (1.1), in which the results can be read as follows: Supposing the initial data \((\rho _{0}, \mathbf{u}_{0}, \boldsymbol{\omega}_{0}, \mathbf{B}_{0}) \in H^{N}( \mathbb{R}^{3})\cap L^{1}(\mathbb{R}^{3})\) satisfy
and
where the integer \(N \geqslant 3\) and δ is small enough, then Cauchy problem (1.1)–(1.2) admits unique global-in-time strong solutions such that
for \(l=0,1,\ldots , N-1\) and \(k=0,1,\ldots , N\). Without the \(L^{1}\)-integrability of the initial data, Jia, Tan, and Zhou [11] recently derived the temporal decay estimates of the solution in the homogeneous Sobolev and Besov spaces. For other mathematical issues of system (1.1), the interested readers can refer to [2, 4, 6, 7, 9, 10, 17, 20–24].
However, there is no available result about the long time behavior of the highest-order spatial derivatives of the perturbation \((\rho - \bar{\rho}, \mathbf{u}, \boldsymbol{\omega})\). In this paper, inspired by the new method of decomposing the frequency in [15], we establish the optimal temporal decay rates of the highest-order (i.e., Nth order) spatial derivatives of the global strong solution for the Cauchy problem (1.1)–(1.2).
Notation
Before stating the main results, we shall introduce some basic notations used frequently in the sequel. The norm of Sobolev space \(H^{k}(\mathbb{R}^{3})\) is denoted by \(\|\cdot \|_{H^{k}}\). We use \(\langle f, g \rangle \) to denote the \(L^{2}\)-inner product between the functions f and g. \(L^{p}\) represents the usual Lebesgue space \(L^{p}(\mathbb{R}^{3})\) with the norm \(\|\cdot \|_{L^{p}}\), where \(1\leqslant p\leqslant \infty \). Enabling the cut-off function \(\phi \in C_{0}^{\infty} (\mathbb{R}_{\xi}^{3})\) to satisfy \(\phi (\xi )=1\) when \(|\xi | \leqslant 1\) and \(\phi (\xi )=0\) when \(|\xi | \geqslant 2\), we define both the low-frequency and the high-frequency parts of f as follows:
where \(\mathfrak{F}(f)\) or f̂ denotes the Fourier transform of f, and \(\mathfrak{F}^{-1}\) is its inverse. The notation \(f\lesssim g\) signifies that \(f\leqslant Cg\) with \(C>0\) being a common constant that may vary from one line to another. \(f\approx g\) means that \(f\lesssim g\) and \(g\lesssim f\). For simplicity, we denote \(\|A\|_{X}+\|B\|_{X}\) by \(\|(A,B)\|_{X}\).
Now we are in a position to state our main theorem.
Theorem 1.1
Suppose \((\rho _{0} -\bar{\rho} ,\mathbf{u}_{0},\boldsymbol{\omega}_{0},\mathbf{B}_{0}) \in H^{N}(\mathbb{R}^{3})\) with any given integer \(N\geqslant 3\). There exists a constant \(\delta >0\) such that if
then Cauchy problem (1.1)–(1.2) admits a unique global-in-time solution \((\rho ,\mathbf{u},\boldsymbol{\omega},\mathbf{B})\) satisfying
Moreover, if \((\rho _{0} -\bar{\rho} ,\mathbf{u}_{0},\boldsymbol{\omega}_{0},\mathbf{B}_{0}) \in L^{1}(\mathbb{R}^{3})\), then the following temporal decay estimates hold:
and
Remark 1.1
The temporal decay rate in (1.7) is optimal in the sense that it is the same as the one of linear solutions shown in Lemma 2.1. In addition, the estimate (1.8) implies that the \(L^{2}\)-norms of any Nth order derivatives of the micro-rotational velocity approach zero along with a rate of decay-in-time \((1 + t)^{(\frac{5}{4}+\frac{N}{2})}\). This convergence rate is quicker than the ones of both the density and the velocity.
Since the global-in-time existence and the a priori energy estimate (1.6) of the solution have been proved in [19], it suffices to establish both the temporal decay estimates (1.7) and (1.8). The remarkable thing is that, thanks to the formula (A.10), the high-frequency parts of solutions exhibit exponential decay-in-time by using an energy method, see the proof of Lemma 2.5. However, such property can not be expected for the lower frequency components. Therefore, we will first derive the decay-in-time estimates of the low-frequency parts in Sect. 2.1, and then the decay estimates of the high-frequency parts in Sect. 2.2. Based to the decay estimates as mentioned earlier and the formula (A.8), we further arrive at (1.7) (see (2.61) for the detailed derivation). Finally, we derive the faster decay-in-time (1.8) for the high-frequency parts and the micro-rotational velocity by a finer energy method in Sect. 3.
2 Decay-in-time of highest-order spatial derivatives
This section is served to establish the optimal temporal decay rate for the Nth order spatial derivatives of the solution \((\rho , \mathbf{u}, \mathbf{B})\) to Cauchy problem (1.1)–(1.2). Making use of the formulas
and then letting \(n=\rho -1\), we can rewrite system (1.1)–(1.2) into the perturbation form:
Here the positive constant \(\gamma = P'(1)\), and the nonhomogeneous source terms \(S_{i}\) (\(i=1\), 2, 3, 4) are defined by
with the nonlinear functions
By the a priori assumption
with some sufficiently small constant δ and Sobolev’s inequality of \(H^{2}\hookrightarrow L^{\infty}\), we obtain
This implies that
and for any given \(k \geqslant 1\) and \(l \geqslant 0\),
Define an increasing energy functional
Then the following proposition leads us to estimate (1.7) in Theorem 1.1.
Proposition 2.1
Under the assumptions in Theorem 1.1, it holds that
where \(\mathbf{U}_{0}=(n_{0},\mathbf{u}_{0},\boldsymbol{\omega}_{0},\mathbf{B}_{0})\).
Afterwards, our dedication turns towards demonstrating the validity of Proposition 2.1.
2.1 Decay estimates on the low-frequency part
Before establishing the temporal decay estimates on the low-frequency part of the solutions \((n,\mathbf{u},\boldsymbol{\omega},\mathbf{B})\), we shall recall some decay results for the linearized system of (2.1).
Lemma 2.1
([19])
Let \(\widetilde{\mathbf{U}}=(\widetilde{n},\widetilde{\mathbf{u}}, \widetilde{\boldsymbol{\omega}},\widetilde{\mathbf{B}})\) be the global solution to the Cauchy problem of the linearized system of (2.1). Then the following time decay properties hold for any integer \(m\geqslant 0\):
and
where \(1 \leqslant p \leqslant 2\leqslant q \leqslant \infty \).
Based on Lemma 2.1, we now present the time decay rate of the \(L^{2}\)-norm for the low-frequency part of the Nth order derivatives of the solution to the nonlinear system (2.1) as follows.
Lemma 2.2
Under the assumptions in Theorem 1.1, the solution \(\mathbf{U}:=(n,\mathbf{u},\boldsymbol{\omega},\mathbf{B})\) to the Cauchy problem of nonlinear system (2.1) satisfies the following decay estimate:
Proof
From Lemma 2.1, Duhamel’s principle, Plancherel’s theorem, and Hausdorff–Young’s inequality, it holds that
where \(\mathbf{S}=(S_{1},S_{2},S_{3},S_{4})^{T}\). By the method of decomposing the frequency, the last two terms on the right-hand side of (2.11) can be treated as follows. With the help of estimates (1.3)–(1.5), (2.4), (2.5), Lemmas A.1–A.2, and Lemma A.4, we can deduce that
Similarly, using estimates (1.3)–(1.5) and definition (2.6) of \(M(t)\), we get
Inserting (2.12) and (2.13) into (2.11) gives estimate (2.10). □
2.2 Decay estimates on the high-frequency part
To prove Proposition 2.1, we also need to establish the temporal decay estimates on the higher-frequency parts of the highest-order derivatives of the global solutions. The following lemma first provides the energy dissipation for \(\nabla ^{N +1}(\mathbf{u}^{h}, \boldsymbol{\omega}^{h}, \mathbf{B}^{h})\).
Lemma 2.3
With the assumptions in Theorem 1.1, it holds that
Proof
Performing the operator \(\mathfrak{F}^{-1}[(1-\phi (\xi ))\mathfrak{F}(\cdot )]\) onto system (2.1) yields
By taking the \(L^{2}\) inner product of \(\nabla ^{N} \text{(2.15)}_{1}\)–\(\nabla ^{N} \text{(2.15)}_{4}\) with \(\nabla ^{N}n^{h}\), \(\nabla ^{N}\mathbf{u}^{h}\), \(\nabla ^{N}\boldsymbol{\omega}^{h}\), and \(\nabla ^{N}\mathbf{B}^{h}\), respectively, we further obtain
Summing up the identities \(\gamma \times \text{(2.16)}_{1}\), (2.16)2, (2.16)3, and (2.16)4 leads to
We next deal with \(J_{i}\) \((i=1, 2, \ldots , 5)\) terms by terms.
-
(1)
Term \(J_{1}\). Following Hölder’s inequality and Young’s inequality, it holds that
$$ J_{1} = 4\nu \bigl\langle \nabla ^{N} \boldsymbol{\omega }^{h},\nabla ^{N} \nabla \times \mathbf{u}^{h} \bigr\rangle \leqslant 4\nu \bigl\Vert \nabla ^{N} \boldsymbol{\omega}^{h} \bigr\Vert _{L^{2}}^{2} +\nu \bigl\Vert \nabla ^{N} \nabla \times \mathbf{u}^{h} \bigr\Vert _{L^{2}}^{2}. $$(2.18) -
(2)
Term \(J_{2}\). To estimate the second term \(J_{2}\), we reformulate it as follows:
$$\begin{aligned} J_{2} & = \bigl\langle \nabla ^{N} S_{1}^{h}, \nabla ^{N} n^{h} \bigr\rangle \\ & =- \bigl\langle \nabla ^{N} (n\operatorname{div}\mathbf{u}+ \mathbf{u}\cdot \nabla n),\nabla ^{N} n^{h} \bigr\rangle \\ &=- \bigl\langle \nabla ^{N} (n\operatorname{div} \mathbf{u})^{h}, \nabla ^{N} n^{h} \bigr\rangle - \bigl\langle \nabla ^{N} ( \mathbf{u}\cdot \nabla n)^{h},\nabla ^{N} n^{h} \bigr\rangle \\ & =\mathcal{K}_{1}+\mathcal{K}_{2}. \end{aligned}$$(2.19)By Lemmas A.2, A.5, Hölder’s inequality, and Young’s inequality, one has
$$\begin{aligned} \vert \mathcal{K}_{1} \vert &\lesssim \bigl\Vert \nabla ^{N}(n\operatorname{div} \mathbf{u})^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N}(n\operatorname{div}\mathbf{u}) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N}\operatorname{div} \mathbf{u} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \Vert \operatorname{div} \mathbf{u} \Vert _{L^{\infty}} \bigr) \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N}\operatorname{div} \mathbf{u} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \Vert \nabla \operatorname{div}\mathbf{u} \Vert _{H^{1}}\bigr) \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl( \bigl\Vert \nabla ^{N}n \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert ^{2}_{L^{2}} \bigr). \end{aligned}$$(2.20)In addition, from the formula \(f=f^{h}+f^{l}\), it holds that
$$\begin{aligned} \mathcal{K}_{2} & =- \bigl\langle \nabla ^{N}(\mathbf{u} \cdot \nabla n)^{h}, \nabla ^{N} n^{h} \bigr\rangle \\ &=- \bigl\langle \nabla ^{N} (\mathbf{u}\cdot \nabla n)-\nabla ^{N} ( \mathbf{u}\cdot \nabla n)^{l},\nabla ^{N} n^{h} \bigr\rangle \\ &=- \bigl\langle \nabla ^{N}\bigl(\mathbf{u}\cdot \nabla n^{h}\bigr)+\nabla ^{N}\bigl( \mathbf{u}\cdot \nabla n^{l}\bigr)-\nabla ^{N}(\mathbf{u}\cdot \nabla n)^{l}, \nabla ^{N} n^{h} \bigr\rangle \\ &=- \bigl\langle \nabla ^{N} \bigl(\mathbf{u}\cdot \nabla n^{h}\bigr),\nabla ^{N} n^{h} \bigr\rangle - \bigl\langle \nabla ^{N}\bigl(\mathbf{u}\cdot \nabla n^{l}\bigr),\nabla ^{N} n^{h} \bigr\rangle + \bigl\langle \nabla ^{N} (\mathbf{u}\cdot \nabla n)^{l}, \nabla ^{N} n^{h} \bigr\rangle \\ &=\mathcal{K}_{2,1}+\mathcal{K}_{2,2}+ \mathcal{K}_{2,3}. \end{aligned}$$(2.21)Employing Lemma A.3 yields
$$\begin{aligned} \vert \mathcal{K}_{2,1} \vert \lesssim{}& \bigl\vert \bigl\langle \mathbf{u}\cdot \nabla \nabla ^{N} n^{h},\nabla ^{N} n^{h} \bigr\rangle \bigr\vert + \bigl\vert \bigl\langle \bigl[\nabla ^{N},\mathbf{u}\bigr] \nabla n^{h},\nabla ^{N} n^{h} \bigr\rangle \bigr\vert \\ \lesssim{}& \frac{1}{2} \bigl\vert \bigl\langle \operatorname{div} \mathbf{u}, \bigl\vert \nabla ^{N} n^{h} \bigr\vert ^{2} \bigr\rangle \bigr\vert + \bigl\Vert \bigl[\nabla ^{N}, \mathbf{u}\bigr]\nabla n^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \Vert \nabla \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}}^{2} \\ &{}+ \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{6}} \bigl\Vert \nabla n^{h} \bigr\Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \Vert \nabla \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}}^{2} \\ &{} + \bigl( \bigl\Vert \nabla ^{2} \mathbf{u} \bigr\Vert _{H^{1}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}} \bigl\Vert \nabla n^{h} \bigr\Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \delta \bigl( \bigl\Vert \nabla ^{N}n \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert ^{2}_{L^{2}} \bigr). \end{aligned}$$(2.22)With the help of Lemma A.6, we get
$$\begin{aligned} \vert \mathcal{K}_{2,2} \vert &\lesssim \bigl\Vert \nabla ^{N} \bigl(\mathbf{u}\cdot \nabla n^{l} \bigr) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}n^{l} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{6}} \bigl\Vert \nabla n^{l} \bigr\Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}} \Vert \nabla n \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl( \bigl\Vert \nabla ^{N}n \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert ^{2}_{L^{2}} \bigr). \end{aligned}$$(2.23)It can be obtained in a similar way that
$$\begin{aligned} \vert \mathcal{K}_{2,3} \vert &\lesssim \bigl\Vert \nabla ^{N} (\mathbf{u}\cdot \nabla n)^{l} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}(\mathbf{u}\cdot \nabla n) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N-1}\mathbf{u} \bigr\Vert _{L^{6}} \Vert \nabla n \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{2}} \Vert \nabla n \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl( \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned}$$(2.24)Gathering estimates (2.22)–(2.24), we can derive from (2.21) that
$$ \vert \mathcal{K}_{2} \vert \lesssim \delta \bigl\Vert \bigl(\nabla ^{N} n,\nabla ^{N} \mathbf{u}, \nabla ^{N+1}\mathbf{u}\bigr) \bigr\Vert _{L^{2}}^{2}. $$(2.25)Furthermore, inserting (2.20) and (2.25) into (2.19), we arrive at
$$ \vert J_{2} \vert \lesssim \delta \bigl\Vert \bigl(\nabla ^{N}n,\nabla ^{N}\mathbf{u},\nabla ^{N+1} \mathbf{u}\bigr) \bigr\Vert _{L^{2}}^{2}. $$(2.26) -
(3)
Terms \(J_{3}\) and \(J_{4}\). Following similar lines as in (2.19), the third term \(J_{3}\) can be rewritten as follows:
$$\begin{aligned} J_{3} = {}&{-} \bigl\langle \nabla ^{N} (\mathbf{u}\cdot \nabla \mathbf{u})^{h}, \nabla ^{N} \mathbf{u}^{h} \bigr\rangle \\ &{}+ \bigl\langle \nabla ^{N-1} \bigl\{ f(n)\bigl[(\mu +\nu )\Delta \mathbf{u}+( \mu +\lambda -\nu )\nabla \operatorname{div}\mathbf{u}\bigr]\bigr\} ^{h},\nabla ^{N} \operatorname{div} \mathbf{u}^{h} \bigr\rangle \\ &{}-2\nu \bigl\langle \nabla ^{N} \bigl[f(n) (\nabla \times \boldsymbol{\omega})\bigr]^{h}, \nabla ^{N}\mathbf{u}^{h} \bigr\rangle + \bigl\langle \nabla ^{N-1} \bigl[h(n) \nabla n \bigr]^{h},\nabla ^{N} \operatorname{div} \mathbf{u}^{h} \bigr\rangle \\ &{}- \biggl\langle \nabla ^{N} \biggl\{ g(n)\biggl[\mathbf{B}\cdot \nabla \mathbf{B}- \frac{1}{2}\nabla \bigl( \vert B \vert ^{2}\bigr)\biggr]\biggr\} ^{h},\nabla ^{N} \mathbf{u}^{h} \biggr\rangle \\ :={}& \mathcal{K}_{3} +\mathcal{K}_{4} + \mathcal{K}_{5} +\mathcal{K}_{6} +\mathcal{K}_{7}. \end{aligned}$$(2.27)From Lemmas A.2 and A.6, it holds for \(\mathcal{K}_{3}\) that
$$\begin{aligned} \vert \mathcal{K}_{3} \vert & \lesssim \bigl\Vert \nabla ^{N} (\mathbf{u}\cdot \nabla \mathbf{u})^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N} (\mathbf{u}\cdot \nabla \mathbf{u}) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{u} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}} \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$(2.28)Using Lemmas A.1, A.2, A.6 and Hausdorff–Young’s inequality, we infer that
$$\begin{aligned} \vert \mathcal{K}_{4} \vert & \lesssim \bigl\Vert \nabla ^{N-1}\bigl\{ f(n)\bigl[(\mu +\nu )\Delta \mathbf{u}+(\mu +\lambda -\nu )\nabla \operatorname{div}\mathbf{u}\bigr]\bigr\} ^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \operatorname{div} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}\bigl\{ f(n)\bigl[(\mu +\nu ) \Delta \mathbf{u}+(\mu + \lambda -\nu )\nabla \operatorname{div}\mathbf{u}\bigr] \bigr\} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}n \bigr\Vert _{L^{6}} \Vert \Delta \mathbf{u} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \Vert \Delta \mathbf{u} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl( \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{N}n \bigr\Vert ^{2}_{L^{2}} \bigr). \end{aligned}$$(2.29)Applying a similar argument used for \(\mathcal{K}_{3}\), we have
$$\begin{aligned} \vert \mathcal{K}_{5} \vert & \lesssim \bigl\Vert \nabla ^{N} \bigl[f(n)\nabla \times \boldsymbol{\omega}\bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N} \bigl[f(n)\nabla \times \boldsymbol{\omega}\bigr] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}\boldsymbol{\omega} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} \Vert \nabla \boldsymbol{\omega} \Vert _{L^{\infty}} \bigr) \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N+1}\boldsymbol{\omega} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{2}\boldsymbol{\omega} \bigr\Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl\Vert \bigl(\nabla ^{N+1}\mathbf{u}, \nabla ^{N} n, \nabla ^{N+1} \boldsymbol{\omega}\bigr) \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$(2.30)Besides, the following hold:
$$\begin{aligned} \vert \mathcal{K}_{6} \vert & \lesssim \bigl\Vert \nabla ^{N-1}\bigl[h(n)\nabla n\bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1}\mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}\bigl[h(n)\nabla n\bigr] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N-1} n \bigr\Vert _{L^{6}} \Vert \nabla n \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} \Vert \nabla n \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl( \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}}^{2} + \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}}^{2} \bigr) \end{aligned}$$(2.31)and
$$\begin{aligned} \vert \mathcal{K}_{7} \vert \lesssim {}& \biggl\Vert \nabla ^{N} \biggl\{ g(n)\biggl[\mathbf{B}\cdot \nabla \mathbf{B}- \frac{1}{2}\nabla \bigl( \vert \mathbf{B} \vert ^{2} \bigr)\biggr]\biggr\} ^{h} \biggr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ \lesssim {}& \biggl\Vert \nabla ^{N} \biggl\{ g(n)\biggl[\mathbf{B} \cdot \nabla \mathbf{B}- \frac{1}{2}\nabla \bigl( \vert \mathbf{B} \vert ^{2}\bigr)\biggr]\biggr\} \biggr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ \lesssim{} & \biggl\Vert \nabla ^{N} \biggl[\mathbf{B}\cdot \nabla \mathbf{B}- \frac{1}{2}\nabla \bigl( \vert \mathbf{B} \vert ^{2}\bigr)\biggr] \biggr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ \lesssim{} & \bigl( \Vert \mathbf{B} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1} \mathbf{B} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N} \mathbf{B} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{B} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ \lesssim {}& \delta \bigl\Vert \bigl(\nabla ^{N+1}\mathbf{B},\nabla ^{N+1}\mathbf{u}\bigr) \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$(2.32)Inserting (2.28)–(2.32) into (2.27) gets that
$$ \vert J_{3} \vert \lesssim \delta \bigl\Vert \bigl(\nabla ^{N} n,\nabla ^{N+1}\boldsymbol{\omega}, \nabla ^{N+1}\mathbf{u},\nabla ^{N+1}\mathbf{B}\bigr) \bigr\Vert _{L^{2}}^{2}. $$(2.33)The estimate on term \(J_{4}\) can be obtained from a similar argument used for \(J_{3}\), and it is presented as follows:
$$ \vert J_{4} \vert \lesssim \delta \bigl\Vert \bigl(\nabla ^{N} n, \nabla ^{N+1}\mathbf{u}, \nabla ^{N+1}\boldsymbol{\omega}\bigr) \bigr\Vert _{L^{2}}^{2}. $$(2.34) -
(4)
Term \(J_{5}\). Recall that
$$\begin{aligned} J_{5} ={}& \bigl\langle \nabla ^{N} \bigl[(\mathbf{B} \cdot \nabla ) \mathbf{u}\bigr]^{h}, \nabla ^{N} \mathbf{B}^{h} \bigr\rangle - \bigl\langle \nabla ^{N} \bigl[(\mathbf{u}\cdot \nabla )\mathbf{B}\bigr]^{h}, \nabla ^{N} \mathbf{B}^{h} \bigr\rangle \\ & {} - \bigl\langle \nabla ^{N}\bigl[\mathbf{B}( \operatorname{div} \mathbf{u})\bigr]^{h}, \nabla ^{N} \mathbf{B}^{h} \bigr\rangle \\ :={}&\mathcal{K}_{8} +\mathcal{K}_{9} + \mathcal{K}_{10}. \end{aligned}$$(2.35)By Lemmas A.2 and A.6, we have
$$\begin{aligned} \vert \mathcal{K}_{8} \vert & \lesssim \bigl\Vert \nabla ^{N} \bigl[(\mathbf{B}\cdot \nabla ) \mathbf{u}\bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N} \bigl[(\mathbf{B}\cdot \nabla ) \mathbf{u}\bigr] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1}\mathbf{B} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{B} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N}\mathbf{B} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{u} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{B} \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl( \bigl\Vert \nabla ^{N+1}\mathbf{B} \bigr\Vert _{L^{2}}^{2} + \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert _{L^{2}}^{2} \bigr) \end{aligned}$$(2.36)and
$$\begin{aligned} \vert \mathcal{K}_{9} \vert & \lesssim \bigl\Vert \nabla ^{N} \bigl[(\mathbf{u}\cdot \nabla ) \mathbf{B}\bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N} \bigl[(\mathbf{u}\cdot \nabla ) \mathbf{B}\bigr] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1}\mathbf{B} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}\mathbf{B} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{B} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{B} \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl( \bigl\Vert \nabla ^{N+1}\mathbf{B} \bigr\Vert _{L^{2}}^{2} + \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned}$$(2.37)It holds for term \(\mathcal{K}_{10}\) that
$$\begin{aligned} \vert \mathcal{K}_{10} \vert & \lesssim \bigl\Vert \nabla ^{N} \bigl[\mathbf{B}( \operatorname{div}\mathbf{u}) \bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N} [\mathbf{B}\operatorname{div} \mathbf{u}] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1} \mathbf{B} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{B} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N}\mathbf{B} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{u} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{B} \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl( \bigl\Vert \nabla ^{N+1}\mathbf{B} \bigr\Vert _{L^{2}}^{2} + \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned}$$(2.38)Inserting (2.36)–(2.38) into (2.35) yields that
$$ \vert J_{5} \vert \lesssim \delta \bigl\Vert \bigl(\nabla ^{N+1}\mathbf{u}, \nabla ^{N+1} \mathbf{B} \bigr) \bigr\Vert _{L^{2}}^{2}. $$(2.39)
Inserting estimates (2.18), (2.26), (2.33), (2.34), and (2.39) into (2.17), we can obtain the desired estimate (2.14). □
Next, we turn to provide the dissipation estimate for \(n^{h}\).
Lemma 2.4
With the assumptions in Theorem 1.1, it holds that
Proof
Performing the operator \(\mathfrak{F}^{-1}[(1-\phi (\xi ))\mathfrak{F}(\cdot )]\) to \(\nabla ^{N-1}\text{(2.1)}_{2}\) and taking the \(L^{2}\) inner product of both the resulting equation and \(\nabla ^{N} n^{h}\), we get
-
(1)
Terms \(J_{6}\), \(J_{7}\), and \(J_{8}\). Using Hölder’s inequality and Hausdorff–Young’s inequality, we can estimate for term \(J_{i}\) \((6 \leqslant i \leqslant 8)\) as follows:
$$\begin{aligned}& \vert J_{6} \vert \leqslant \frac{3(\mu +\nu )^{2}}{2\gamma} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}}^{2} +\frac{\gamma}{6} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}}^{2}, \end{aligned}$$(2.42)$$\begin{aligned}& \vert J_{7} \vert \leqslant \frac{3(\mu +\lambda -\nu )^{2}}{2\gamma} \bigl\Vert \nabla ^{N} \operatorname{div}\mathbf{u}^{h} \bigr\Vert _{L^{2}}^{2}+\frac{\gamma}{6} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}}^{2}, \end{aligned}$$(2.43)and
$$ \vert J_{8} \vert \leqslant \frac{6\nu ^{2}}{\gamma} \bigl\Vert \nabla ^{N} \boldsymbol{\omega}^{h} \bigr\Vert _{L^{2}}^{2} + \frac{\gamma}{6} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}}^{2}. $$(2.44) -
(2)
Term \(J_{9}\). For term \(J_{9}\), we get
$$\begin{aligned} J_{9} & =- \bigl\langle \nabla ^{N}(n \operatorname{div}\mathbf{u})^{h}, \nabla ^{N-1} \mathbf{u}^{h} \bigr\rangle - \bigl\langle \nabla ^{N}( \mathbf{u}\cdot \nabla n)^{h},\nabla ^{N-1} \mathbf{u}^{h} \bigr\rangle \\ &= :\mathcal{K}_{11} +\mathcal{K}_{12}. \end{aligned}$$(2.45)By using Hölder’s inequality, Young’s inequality, Lemmas A.1, A.2, and A.6, we arrive at
$$\begin{aligned} \vert \mathcal{K}_{11} \vert &\lesssim \bigl\Vert \nabla ^{N} (n\operatorname{div} \mathbf{u})^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N-1}\mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N} (n\operatorname{div}\mathbf{u}) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} \Vert \nabla \mathbf{u} \Vert _{L^{\infty}} \bigr) \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl( \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}}^{2} + \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned}$$(2.46)In the same way, we can get
$$\begin{aligned} \vert \mathcal{K}_{12} \vert & \lesssim \bigl\Vert \nabla ^{N-1}(\mathbf{u}\cdot \nabla n)^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}\mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}(\mathbf{u}\cdot \nabla n) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}\mathbf{u} \bigr\Vert _{L^{6}} \Vert \nabla n \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N}\mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl( \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}}^{2} + \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned}$$(2.47)Substituting (2.46) and (2.47) into (2.45) leads to
$$ \vert J_{9} \vert \lesssim \delta \bigl\Vert \bigl(\nabla ^{N} n,\nabla ^{N} \mathbf{u}, \nabla ^{N+1}\mathbf{u}\bigr) \bigr\Vert _{L^{2}}^{2}. $$(2.48) -
(3)
Term \(J_{10}\). For term \(J_{10}\), we directly have
$$\begin{aligned} J_{10} = {}& {-} \bigl\langle \nabla ^{N-1}(\mathbf{u}\cdot \nabla \mathbf{u})^{h},\nabla ^{N}n^{h} \bigr\rangle \\ &{}- \bigl\langle \nabla ^{N-1}\bigl\{ f(n)\bigl[(\mu +\nu )\Delta \mathbf{u}+( \mu +\lambda -\nu )\nabla \operatorname{div}\mathbf{u}\bigr]\bigr\} ^{h},\nabla ^{N}n^{h} \bigr\rangle \\ &{}-2\nu \bigl\langle \nabla ^{N-1}\bigl[f(n) (\nabla \times \boldsymbol{\omega})\bigr]^{h}, \nabla ^{N}n^{h} \bigr\rangle - \bigl\langle \nabla ^{N-1}\bigl[h(n) \nabla n \bigr]^{h},\nabla ^{N}n^{h} \bigr\rangle \\ &{}- \biggl\langle \nabla ^{N-1}\biggl\{ g(n)\biggl[\mathbf{B}\cdot \nabla \mathbf{B}- \frac{1}{2}\nabla \bigl( \vert B \vert ^{2}\bigr)\biggr]\biggr\} ^{h},\nabla ^{N}n^{h} \biggr\rangle \\ :={} & \mathcal{K}_{13} +\mathcal{K}_{14} + \mathcal{K}_{15} + \mathcal{K}_{16} + \mathcal{K}_{17}. \end{aligned}$$(2.49)With the help of Young’s inequality and Hölder’s inequality, we obtain
$$\begin{aligned} \vert \mathcal{K}_{13} \vert &\lesssim \bigl\Vert \nabla ^{N-1}(\mathbf{u}\cdot \nabla \mathbf{u})^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N} (\mathbf{u}\cdot \nabla \mathbf{u}) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{u} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl( \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}}^{2} + \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert _{L^{2}}^{2} \bigr) \end{aligned}$$(2.50)and
$$\begin{aligned} \vert \mathcal{K}_{14} \vert &\lesssim \bigl\Vert \nabla ^{N-1}\bigl\{ f(n)\bigl[(\mu +\nu )\Delta \mathbf{u}+(\mu +\lambda -\nu )\nabla \operatorname{div}\mathbf{u}\bigr]\bigr\} ^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}\mathbf{u} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N-1}n \bigr\Vert _{L^{6}} \Vert \Delta \mathbf{u} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl( \bigl\Vert \nabla ^{N}n \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert ^{2}_{L^{2}} \bigr). \end{aligned}$$(2.51)Similarly, we get
$$\begin{aligned} \vert \mathcal{K}_{15} \vert &\lesssim \bigl\Vert \nabla ^{N-1}\bigl[f(n) (\nabla \times \boldsymbol{\omega})\bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}\bigl[f(n) (\nabla \times \boldsymbol{\omega})\bigr] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert n \Vert _{L^{3}} \bigl\Vert \nabla ^{N}\boldsymbol{\omega} \bigr\Vert _{L^{6}}+ \bigl\Vert \nabla ^{N-1}n \bigr\Vert _{L^{6}} \Vert \nabla \boldsymbol{\omega} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl( \bigl\Vert \nabla ^{N}n \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{N+1} \boldsymbol{\omega} \bigr\Vert ^{2}_{L^{2}} \bigr) \end{aligned}$$(2.52)and
$$\begin{aligned} \vert \mathcal{K}_{16} \vert & \lesssim \bigl\Vert \nabla ^{N-1}\bigl[h(n)\nabla n\bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N}\bigl[h(n)\nabla n\bigr] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N-1}n \bigr\Vert _{L^{6}} \Vert \nabla n \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \\ &\lesssim \delta \bigl\Vert \nabla ^{N} n \bigr\Vert ^{2}_{L^{2}}. \end{aligned}$$(2.53)For term \(\mathcal{K}_{17}\), we have
$$\begin{aligned} \vert \mathcal{K}_{17} \vert \lesssim & \biggl\Vert \nabla ^{N-1}\biggl\{ g(n)\biggl[\mathbf{B}\cdot \nabla \mathbf{B}- \frac{1}{2}\nabla \bigl( \vert \mathbf{B} \vert ^{2} \bigr)\biggr]\biggr\} ^{h} \biggr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ \lesssim & \biggl\Vert \nabla ^{N}\biggl\{ g(n)\biggl[\mathbf{B} \cdot \nabla \mathbf{B}- \frac{1}{2}\nabla \bigl( \vert \mathbf{B} \vert ^{2}\bigr)\biggr]\biggr\} \biggr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ \lesssim & \bigl( \Vert \mathbf{B} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}\mathbf{B} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N}\mathbf{B} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{B} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \\ \lesssim & \delta \bigl\Vert \bigl(\nabla ^{N} n, \nabla ^{N+1}\mathbf{B}\bigr) \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$(2.54)Inserting (2.50)–(2.54) into (2.49) gets
$$\begin{aligned} \vert J_{10} \vert \lesssim \bigl\Vert \bigl(\nabla ^{N} n, \nabla ^{N+1} \mathbf{u}, \nabla ^{N+1} \boldsymbol{\omega}, \nabla ^{N+1} \mathbf{B}\bigr) \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$(2.55)
Gathering (2.42)–(2.44), (2.48), and (2.55), we derive estimate (2.40) from (2.41). □
Based on the above two lemmas, we now present the temporal decay rate for the high-frequency of the Nth order derivatives of the solution.
Lemma 2.5
With the assumptions in Theorem 1.1, it holds that
Proof
By estimates (2.14), (2.40), Lemma A.6, and the smallness of δ, we have
where
with some large enough positive constant \(D_{1}\). Recalling
thus, by the formula (A.10), Gronwall’s inequality, Lemmas 2.2 and A.7, we can get
Due to the relation
estimate (2.56) can be easily obtained from (2.60). □
Proof of Proposition 2.1
Now we are in a position to prove Proposition 2.1. With the help of Lemmas 2.2, 2.5 and the frequency decompositions, we deduce that
From the definition of \(M(t)\), it holds that
By the smallness of δ, we further obtain
Thus, we complete the proof of Proposition 2.1, which immediately implies the decay rate (1.7) in Theorem 1.1. □
3 Derivation of the decay-in-time of high-frequency parts
In this section, we further derive the optimal temporal rate of decay-in-time for the Nth order spatial derivatives of the solution ω. To this end, we shall first establish the decay estimate on \(\|\nabla ^{N}\boldsymbol{\omega}^{l}\|_{L^{2}}\), which is presented as follows.
Lemma 3.1
With the assumptions in Theorem 1.1, the lower frequency part of solution ω to the Cauchy problem of system (2.1) satisfies that
Proof
Using estimate (2.9), Duhamel’s principle, Plancherel theorem, and Hausdorff–Young’s inequality, we have
Employing a similar argument used for (2.12), we easily have
On the other hand, from the decay rates (1.3)–(1.5), it holds for \(\|\nabla ^{N-1} S_{3}^{l} (\tau )\|_{L^{1}}\) that
Inserting (3.3) and (3.4) into (3.2), we arrive at the decay rate (3.1). □
We next establish the temporal decay estimates on \(\|\nabla ^{N}(n^{h},\mathbf{u}^{h},\boldsymbol{\omega}^{h},\mathbf{B}^{h})\|_{L^{2}}\).
Lemma 3.2
With the assumptions in Theorem 1.1, it holds that
Proof
Due to the decay rate (1.7) and the fine structures of the decomposition of both low and high frequencies, we next deal with the nonlinear terms \(J_{i}\) \((2 \leqslant i \leqslant 5)\) on the right-hand side of (2.17) by a manner that is different from the one used in the proof of Lemma 2.5.
-
(1)
Term \(J_{2}\). For term \(\mathcal{K}_{1}\), one has
$$\begin{aligned} \vert \mathcal{K}_{1} \vert \lesssim{}& \bigl\Vert \nabla ^{N} (n\operatorname{div} \mathbf{u})^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \bigl\Vert \nabla ^{N}(n\operatorname{div}\mathbf{u}) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N}\operatorname{div}\bigl( \mathbf{u}^{l} + \mathbf{u}^{h}\bigr) \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \Vert \operatorname{div} \mathbf{u} \Vert _{L^{\infty}} \bigr) \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \bigl( \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N} \operatorname{div} \mathbf{u}^{l} \bigr\Vert _{L^{2}} + \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N} \operatorname{div}\mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &{}+ \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} \Vert \nabla \operatorname{div}\mathbf{u} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \bigl((1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} +\delta \bigl\Vert \nabla ^{N} \operatorname{div}\mathbf{u}^{h} \bigr\Vert _{L^{2}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}}) \\ \lesssim{}& (1+t)^{-\frac{5+2N}{2}} + \bigl(\delta +(1+t)^{-\frac{3}{2}} \bigr) \bigl( \bigl\Vert \nabla ^{N}\operatorname{div}\mathbf{u}^{h} \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert ^{2}_{L^{2}} \bigr). \end{aligned}$$(3.6)For term \(\mathcal{K}_{2,1}\), we can derive that
$$\begin{aligned} \vert \mathcal{K}_{2,1} \vert \lesssim{}& \bigl\vert \bigl\langle \mathbf{u} \cdot \nabla \nabla ^{N} n^{h}, \nabla ^{N} n^{h} \bigr\rangle \bigr\vert + \bigl\vert \bigl\langle \bigl[\nabla ^{N}, \mathbf{u}\bigr]\nabla n^{h}, \nabla ^{N} n^{h} \bigr\rangle \bigr\vert \\ \lesssim{}& \frac{1}{2} \bigl\vert \bigl\langle \operatorname{div} \mathbf{u}, \bigl\vert \nabla ^{N} n^{h} \bigr\vert ^{2} \bigr\rangle \bigr\vert + \bigl\Vert \bigl[\nabla ^{N},\mathbf{u}\bigr]\nabla n^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \Vert \nabla \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}}^{2} \\ &{} + \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{2}} \bigl\Vert \nabla n^{h} \bigr\Vert _{L^{\infty}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \delta \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}}^{2} +(1+t)^{- \frac{7}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& (1+t)^{-\frac{5+2N}{2}} + \bigl((1+t)^{-\frac{5}{2}}+\delta \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$(3.7)Similarly, we also have
$$\begin{aligned} \vert \mathcal{K}_{2,2} \vert & \lesssim \bigl\Vert \nabla ^{N} \bigl(\mathbf{u}\cdot \nabla n^{l}\bigr) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}n^{l} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{2}} \bigl\Vert \nabla n^{l} \bigr\Vert _{L^{\infty}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{2} n \bigr\Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}} +(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert ^{2}_{L^{2}} \end{aligned}$$(3.8)and
$$\begin{aligned} \vert \mathcal{K}_{2,3} \vert & \lesssim \bigl\Vert \nabla ^{N} (\mathbf{u}\cdot \nabla n)^{l} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}(\mathbf{u}\cdot \nabla n) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N-1}\mathbf{u} \bigr\Vert _{L^{6}} \Vert \nabla n \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{2}} \Vert \nabla n \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}}+(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert ^{2}_{L^{2}}. \end{aligned}$$(3.9)From estimates (3.7)–(3.9), it holds that
$$\begin{aligned} \vert \mathcal{K}_{2} \vert \lesssim (1+t)^{-\frac{5+2N}{2}} + \bigl((1+t)^{- \frac{3}{2}} +\delta \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert ^{2}_{L^{2}}. \end{aligned}$$(3.10)Inserting (3.6) and (3.10) into (2.19) yields that
$$\begin{aligned} \vert J_{2} \vert & \lesssim \vert \mathcal{K}_{1} \vert + \vert \mathcal{K}_{2} \vert \\ & \lesssim (1+t)^{- \frac{5+2N}{2}} + \bigl((1+t)^{-\frac{3}{2}}+\delta \bigr) \bigl( \bigl\Vert \nabla ^{N} \operatorname{div}\mathbf{u}^{h} \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert ^{2}_{L^{2}} \bigr). \end{aligned}$$(3.11) -
(2)
Terms \(J_{3}\) and \(J_{4}\). For term \(J_{3}\), integrating by parts, we get
$$\begin{aligned} J_{3} ={}& \bigl\langle \nabla ^{N-1}(\mathbf{u}\cdot \nabla \mathbf{u})^{h},\nabla ^{N}\operatorname{div} \mathbf{u}^{h} \bigr\rangle \\ & {}+ \bigl\langle \nabla ^{N-1}\bigl\{ f(n)\bigl[(\mu +\nu )\Delta \mathbf{u}+( \mu +\lambda -\nu )\nabla \operatorname{div}\mathbf{u}\bigr]\bigr\} ^{h},\nabla ^{N} \operatorname{div} \mathbf{u}^{h} \bigr\rangle \\ & {}+ 2\nu \bigl\langle \nabla ^{N-1}\bigl[f(n) (\nabla \times \boldsymbol{\omega})\bigr]^{h}, \nabla ^{N}\operatorname{div} \mathbf{u}^{h} \bigr\rangle + \bigl\langle \nabla ^{N-1} \bigl[h(n)\nabla n\bigr]^{h},\nabla ^{N} \operatorname{div} \mathbf{u}^{h} \bigr\rangle \\ & {}+ \biggl\langle \nabla ^{N-1}\biggl\{ g(n)\biggl[B\cdot \nabla B- \frac{1}{2} \nabla \bigl( \vert B \vert ^{2}\bigr) \biggr]\biggr\} ^{h},\nabla ^{N}\operatorname{div} \mathbf{u}^{h} \biggr\rangle \\ :={}& \mathcal{K}'_{3} +\mathcal{K}'_{4} +\mathcal{K}'_{5} + \mathcal{K}'_{6} +\mathcal{K}'_{7}. \end{aligned}$$(3.12)It holds for \(\mathcal{K}'_{3}\) that
$$\begin{aligned} \bigl\vert \mathcal{K}'_{3} \bigr\vert & \lesssim \bigl\Vert \nabla ^{N-1}(\mathbf{u}\cdot \nabla \mathbf{u})^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1}\mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}(\mathbf{u}\cdot \nabla \mathbf{u}) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}\mathbf{u} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{u} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigl\Vert \nabla ^{N}\mathbf{u} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N}\mathbf{u} \bigr\Vert _{L^{2}} \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N+1}\mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}}+(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert ^{2}_{L^{2}}. \end{aligned}$$(3.13)For \(\mathcal{K}'_{4}\), we have
$$\begin{aligned} \bigl\vert \mathcal{K}'_{4} \bigr\vert \lesssim{}& \bigl\Vert \nabla ^{N-1}\bigl\{ f(n)\bigl[(\mu +\nu )\Delta \mathbf{u}+(\mu +\lambda -\nu )\nabla \operatorname{div}\mathbf{u}\bigr]\bigr\} ^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}\operatorname{div}\mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \bigl\Vert \nabla ^{N-1}\bigl\{ f(n)\bigl[(\mu +\nu ) \Delta \mathbf{u}+(\mu + \lambda -\nu )\nabla \operatorname{div}\mathbf{u}\bigr] \bigr\} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}\bigl(\mathbf{u}^{l}+ \mathbf{u}^{h} \bigr) \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}n \bigr\Vert _{L^{6}} \Vert \Delta \mathbf{u} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \bigl( \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N+1}\mathbf{u}^{l} \bigr\Vert _{L^{2}} + \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \Vert \Delta \mathbf{u} \Vert _{H^{1}} \bigr) \\ &{}\times \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \bigl((1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} +\delta \bigl\Vert \nabla ^{N+1}\mathbf{u}^{h} \bigr\Vert _{L^{2}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& (1+t)^{-\frac{5+2N}{2}} + \bigl(\delta +(1+t)^{-\frac{3}{2}} \bigr) \bigl\Vert \nabla ^{N+1}\mathbf{u}^{h} \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$(3.14)Similarly, we deduce
$$\begin{aligned} \bigl\vert \mathcal{K}'_{5} \bigr\vert &\lesssim \bigl\Vert \nabla ^{N-1}\bigl[f(n)\nabla \times \boldsymbol{\omega} \bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1}\mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}\bigl[f(n)\nabla \times \boldsymbol{\omega}\bigr] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} \boldsymbol{\omega} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1} n \bigr\Vert _{L^{6}} \Vert \nabla \boldsymbol{\omega} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N}\boldsymbol{\omega} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \Vert \nabla \boldsymbol{\omega} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}} +(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}}^{2} \end{aligned}$$(3.15)and
$$\begin{aligned} \bigl\vert \mathcal{K}'_{6} \bigr\vert & \lesssim \bigl\Vert \nabla ^{N-1}\bigl[h(n)\nabla n \bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1}\mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}\bigl[h(n)\nabla n\bigr] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}n \bigr\Vert _{L^{6}} \Vert \nabla n \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \Vert \nabla n \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}} +(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$(3.16)For term \(\mathcal{K}'_{7}\), we have
$$\begin{aligned} \bigl\vert \mathcal{K}'_{7} \bigr\vert & \lesssim \biggl\Vert \nabla ^{N} \biggl\{ g(n)\biggl[\mathbf{B}\cdot \nabla \mathbf{B}-\frac{1}{2}\nabla \bigl( \vert \mathbf{B} \vert ^{2}\bigr)\biggr]\biggr\} ^{h} \biggr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \biggl\Vert \nabla ^{N} \biggl\{ g(n)\biggl[\mathbf{B} \cdot \nabla \mathbf{B}- \frac{1}{2}\nabla \bigl( \vert \mathbf{B} \vert ^{2}\bigr)\biggr]\biggr\} \biggr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \biggl\Vert \nabla ^{N} \biggl[\mathbf{B}\cdot \nabla \mathbf{B}- \frac{1}{2}\nabla \bigl( \vert \mathbf{B} \vert ^{2}\bigr)\biggr] \biggr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{B} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1} \mathbf{B} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N}\mathbf{B} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{B} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla \mathbf{B} \Vert _{H^{1}} \bigl\Vert \nabla ^{N+1} \bigl( \mathbf{B}^{l}+ \mathbf{B}^{h}\bigr) \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N+1}\bigl(\mathbf{B}^{l}+ \mathbf{B}^{h} \bigr) \bigr\Vert _{L^{2}} \Vert \nabla \mathbf{B} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl((1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}}+\delta \bigl\Vert \nabla ^{N+1}\mathbf{B}^{h} \bigr\Vert _{L^{2}} \bigr) \bigl\Vert \nabla ^{N+1}\mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}} + \bigl(\delta +(1+t)^{-\frac{3}{2}} \bigr) \bigl( \bigl\Vert \nabla ^{N+1}\mathbf{B}^{h} \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert ^{2}_{L^{2}} \bigr). \end{aligned}$$(3.17)Substituting (3.13)–(3.17) into (3.12), we can arrive at
$$ \vert J_{3} \vert \lesssim (1+t)^{-\frac{5+2N}{4}} + \bigl(\delta +(1+t)^{- \frac{3}{2}} \bigr) \bigl( \bigl\Vert \nabla ^{N+1}\mathbf{B}^{h} \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert ^{2}_{L^{2}} \bigr). $$(3.18)Employing a similar argument used for \(J_{3}\), we have
$$ \vert J_{4} \vert \lesssim (1+t)^{-\frac{5+2N}{4}} + \bigl(\delta +(1+t)^{- \frac{3}{2}} \bigr) \bigl\Vert \nabla ^{N+1}\boldsymbol{\omega}^{h} \bigr\Vert _{L^{2}}^{2}. $$(3.19) -
(3)
Term \(J_{5}\). For the last term \(J_{5}\), one has
$$\begin{aligned} J_{5} = {}& {-} \bigl\langle \nabla ^{N-1}\bigl[(\mathbf{B} \cdot \nabla ) \mathbf{u}\bigr]^{h}, \nabla ^{N} \operatorname{div}\mathbf{B}^{h} \bigr\rangle + \bigl\langle \nabla ^{N-1}\bigl[(\mathbf{u}\cdot \nabla ) \mathbf{B}\bigr]^{h}, \nabla ^{N} \operatorname{div}\mathbf{B}^{h} \bigr\rangle \\ &{} + \bigl\langle \nabla ^{N-1}\bigl[\mathbf{B}(\operatorname{div} \mathbf{u})\bigr]^{h}, \nabla ^{N} \operatorname{div} \mathbf{B}^{h} \bigr\rangle \\ =:{}& \mathcal{K}'_{8} +\mathcal{K}'_{9} +\mathcal{K}'_{10}. \end{aligned}$$(3.20)It holds for term \(\mathcal{K}'_{8}\) that
$$\begin{aligned} \bigl\vert \mathcal{K}'_{8} \bigr\vert & \lesssim \bigl\Vert \nabla ^{N-1}\bigl[(\mathbf{B}\cdot \nabla ) \mathbf{u}\bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1}\mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}\bigl[(\mathbf{B}\cdot \nabla ) \mathbf{u}\bigr] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1}\mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{B} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N}\mathbf{u} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}\mathbf{B} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{u} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla \mathbf{B} \Vert _{H^{1}} \bigl\Vert \nabla ^{N}\mathbf{u} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N}\mathbf{B} \bigr\Vert _{L^{2}} \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N+1}\mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}} +(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{B}^{h} \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$(3.21)Similarly, we have
$$\begin{aligned} \bigl\vert \mathcal{K}'_{9} \bigr\vert & \lesssim \bigl\Vert \nabla ^{N-1}\bigl[(\mathbf{u}\cdot \nabla ) \mathbf{B}\bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1}\mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}\bigl[(\mathbf{u}\cdot \nabla ) \mathbf{B}\bigr] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1}\mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N}\mathbf{B} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}\mathbf{u} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{B} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigl\Vert \nabla ^{N}\mathbf{B} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N}\mathbf{u} \bigr\Vert _{L^{2}} \Vert \nabla \mathbf{B} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N+1}\mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}} +(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{B}^{h} \bigr\Vert _{L^{2}}^{2} \end{aligned}$$(3.22)and
$$\begin{aligned} \bigl\vert \mathcal{K}'_{10} \bigr\vert & \lesssim \bigl\Vert \nabla ^{N-1}\bigl[\mathbf{B}( \operatorname{div} \mathbf{u})\bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1} \mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}[\mathbf{B}\operatorname{div} \mathbf{u}] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1} \mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{B} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}\mathbf{B} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{u} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla \mathbf{B} \Vert _{H^{1}} \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N} \mathbf{B} \bigr\Vert _{L^{2}} \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N+1}\mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{B}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}} +(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{B}^{h} \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$(3.23)Inserting (3.21)–(3.23) into (3.20) leads to
$$\begin{aligned} \vert J_{5} \vert \lesssim (1+t)^{-\frac{5+2N}{4}} +(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{B}^{h} \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$(3.24)
Then we complete the proof of Lemma 3.2 by substituting estimates (3.11), (3.18), (3.19), and (3.24) into (2.17). □
To enclose the energy estimate, it is necessary to establish the dissipation estimate for \(\nabla ^{N} n^{h}\) in a different way.
Lemma 3.3
With the assumptions in Theorem 1.1, it holds that
Proof
Now, we aim to present the estimates on the last two terms on the right-hand side of (2.41).
-
(1)
Term \(J_{9}\). Integrating by parts, one has
$$\begin{aligned} \vert \mathcal{K}_{11} \vert & \lesssim \bigl\Vert \nabla ^{N-1}(n\operatorname{div} \mathbf{u})^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}(n\operatorname{div}\mathbf{u}) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N}\mathbf{u} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}n \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{u} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N}\mathbf{u} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}}+(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert ^{2}_{L^{2}} \end{aligned}$$(3.26)and
$$\begin{aligned} \vert \mathcal{K}_{12} \vert & \lesssim \bigl\Vert \nabla ^{N-1}(\mathbf{u}\cdot \nabla n)^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}(\mathbf{u}\cdot \nabla n) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}\mathbf{u} \bigr\Vert _{L^{6}} \Vert \nabla n \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N}\mathbf{u} \bigr\Vert _{L^{2}} \Vert \nabla n \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}}+(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert ^{2}_{L^{2}}. \end{aligned}$$(3.27)Plugging (3.26) and (3.27) into (2.45), we have
$$\begin{aligned} \vert J_{9} \vert \lesssim (1+t)^{-\frac{5+2N}{2}}+(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N+1}\mathbf{u}^{h} \bigr\Vert ^{2}_{L^{2}}. \end{aligned}$$(3.28) -
(2)
Term \(J_{10}\). In the same way, we deduce
$$\begin{aligned} \vert \mathcal{K}_{13} \vert & \lesssim \bigl\Vert \nabla ^{N-1}(\mathbf{u}\cdot \nabla \mathbf{u})^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}(\mathbf{u}\cdot \nabla \mathbf{u}) \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \mathbf{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N}\mathbf{u} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}\mathbf{u} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{u} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigl\Vert \nabla ^{N}\mathbf{u} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N} \mathbf{u} \bigr\Vert _{L^{2}} \Vert \nabla \mathbf{u} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}}+(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert ^{2}_{L^{2}} \end{aligned}$$(3.29)and
$$\begin{aligned} \vert \mathcal{K}_{14} \vert \lesssim{}& \bigl\Vert \nabla ^{N-1}\bigl\{ f(n)\bigl[(\mu +\nu ) \Delta \boldsymbol{u}+(\mu +\lambda -\nu ) \nabla \operatorname{div}\boldsymbol{u}\bigr]\bigr\} ^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N+1}\bigl(\mathbf{u}^{l}+ \mathbf{u}^{h} \bigr) \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}n \bigr\Vert _{L^{6}} \Vert \Delta \mathbf{u} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \bigl( \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N+1}\mathbf{u}^{l} \bigr\Vert _{L^{2}} + \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}} \\ &{} + \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \Vert \Delta \mathbf{u} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& \bigl((1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} +\delta \bigl\Vert \nabla ^{N+1}\mathbf{u}^{h} \bigr\Vert _{L^{2}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim{}& (1+t)^{-\frac{5+2N}{2}} + \bigl(\delta +(1+t)^{-\frac{3}{2}} \bigr) \bigl( \bigl\Vert \nabla ^{N+1}\mathbf{u}^{h} \bigr\Vert _{L^{2}}^{2} + \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned}$$(3.30)$$\begin{aligned} \vert \mathcal{K}_{15} \vert & \lesssim \bigl\Vert \nabla ^{N-1}\bigl[f(n) (\nabla \times \boldsymbol{\omega})\bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N} \bigl[f(n) (\nabla \times \boldsymbol{\omega})\bigr] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N}\boldsymbol{\omega} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}n \bigr\Vert _{L^{6}} \Vert \nabla \boldsymbol{\omega} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N}\boldsymbol{\omega} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \Vert \nabla \boldsymbol{\omega} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}} +(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}}^{2} \end{aligned}$$(3.31)and
$$\begin{aligned} \vert \mathcal{K}_{16} \vert & \lesssim \bigl\Vert \nabla ^{N-1}\bigl[h(n)\nabla n\bigr]^{h} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl\Vert \nabla ^{N-1}\bigl[h(n)\nabla n\bigr] \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert n \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}n \bigr\Vert _{L^{6}} \Vert \nabla n \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N} n \bigr\Vert _{L^{2}} \\ &\lesssim \bigl( \Vert \nabla n \Vert _{H^{1}} \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N}n \bigr\Vert _{L^{2}} \Vert \nabla n \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ &\lesssim (1+t)^{-\frac{5+2N}{2}} +(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$(3.32)For term \(\mathcal{K}_{17}\), we have
$$\begin{aligned} \vert \mathcal{K}_{17} \vert \lesssim & \biggl\Vert \nabla ^{N-1}\biggl\{ g(n)\biggl[\mathbf{B}\cdot \nabla \mathbf{B}- \frac{1}{2}\nabla \bigl( \vert \mathbf{B} \vert ^{2} \bigr)\biggr]\biggr\} ^{h} \biggr\Vert _{L^{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim & \biggl\Vert \nabla ^{N-1}\biggl\{ g(n) (\mathbf{B}\cdot \nabla \mathbf{B})-\biggl[g(n) \frac{1}{2}\nabla \bigl( \vert \mathbf{B} \vert ^{2}\bigr)\biggr]^{h}\biggr\} \biggr\Vert _{L^{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ \lesssim & \bigl( \Vert \mathbf{B} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{N}\mathbf{B} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N-1}\mathbf{B} \bigr\Vert _{L^{6}} \Vert \nabla \mathbf{B} \Vert _{L^{3}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim & \bigl( \Vert \nabla \mathbf{B} \Vert _{H^{1}} \bigl\Vert \nabla ^{N}\mathbf{B} \bigr\Vert _{L^{2}} + \bigl\Vert \nabla ^{N}\mathbf{B} \bigr\Vert _{L^{2}} \Vert \mathbf{B} \Vert _{H^{1}} \bigr) \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert _{L^{2}} \\ \lesssim & (1+t)^{-\frac{5}{4}-\frac{3}{4}-\frac{N}{2}} \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}} \\ \lesssim & (1+t)^{-\frac{5+2N}{2}} +(1+t)^{-\frac{3}{2}} \bigl\Vert \nabla ^{N} n^{h} \bigr\Vert ^{2}_{L^{2}}. \end{aligned}$$(3.33)Putting estimates (3.29)–(3.33) together, we can derive from (2.49) that
$$\begin{aligned} \vert J_{10} \vert \lesssim (1+t)^{-\frac{5+2N}{2}} + \bigl( \delta +(1+t)^{- \frac{3}{2}} \bigr) \bigl( \bigl\Vert \nabla ^{N+1} \mathbf{u}^{h} \bigr\Vert _{L^{2}}^{2} + \bigl\Vert \nabla ^{N}n^{h} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned}$$(3.34)
Plugging (2.42)–(2.44), (3.28), and (3.34) into (2.41), we can derive estimate (3.25). □
From Lemmas 3.2, 3.3 and the smallness of δ, we can obtain
where
with some large enough positive constant \(D_{2}\). Besides, it can be easily obtained that
Making use of estimates (3.35), (3.37) and Gronwall’s inequality yields
Thanks to the relations
and \(f = f^{l} +f^{h}\), the decay rate (1.8) can be derived from (3.1) and (3.38). Thus we complete the proof of Theorem 1.1.
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Funding
This work was supported by NSFC (Grant Nos. 12022102, 12231016 and 12371233) and the Natural Science Foundation of Fujian Province of China (Grant Nos. 2020J02013 and 2022J01105).
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Xinyu Cui: Writing–Original Draft; Methodology; Shengbin Fu: Writing–Review & Editing; Funding acquisition; Rui Sun: Writing - Review & Editing; Validation; Fangfang Tian.
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Appendix
Appendix
This appendix is devoted to providing some basic mathematical tools used frequently in the previous sections. The detailed proof of the following Gagliardo–Nirenberg inequality can be referred to [3, 14].
Lemma A.1
Let \(0\leqslant i, j\leqslant k\), it holds that
where \(\alpha \in [\frac{i}{k}, 1]\) and satisfies
Especially, while \(p=q=r=2\), we have
Lemma A.2
It holds that for \(k\geqslant 0\),
where \(p_{1}, p_{2}, p_{3} \in (1,+\infty )\) and
We can further deduce the following commutator estimate from Lemma A.2.
Lemma A.3
Let f and g be smooth functions belonging to \(H^{k} \cap L^{\infty}\) for any integer \(k \geqslant 1\), and then we define the commutator
It holds that
Here p, \(p_{1}\), \(p_{2}\), \(p_{3}\) are defined as in Lemma A.2.
Lemma A.4
Assume that \(\|f\|_{L^{\infty}} \leqslant 1\). Let \(F(f)\) be a smooth function of f with bounded derivatives of any order. Then, for any given integer \(k\geqslant 1\) and any given real number \(1 \leqslant p\leqslant \infty \), we have
Lemma A.5
We have, for any function \(f \in H^{2}(\mathbb{R}^{3})\),
-
(i)
\(\|f\|_{L^{r}} \lesssim \|f\|_{H^{1}}\), \(2\leqslant r \leqslant 6\),
-
(ii)
\(\|f\|_{L^{\infty}}\lesssim \|\nabla f\|_{H^{1}}\).
Lemma A.6
If \(f \in H^{k}(\mathbb{R}^{3})\), then we have
Lemma A.7
([8])
Suppose that \(b_{1}, b_{2}\in \mathbb{R}^{3}\) and \(b_{1}>0\), \(b_{2}>0\), it holds that for \(t \in \mathbb{R}_{+}\),
where \(C(b_{1}, b_{2})\) is the constant that only depends on \(b_{1}\), \(b_{2}\).
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Cui, X., Fu, S., Sun, R. et al. Optimal decay-in-time rates of solutions to the Cauchy problem of 3D compressible magneto-micropolar fluids. Bound Value Probl 2024, 33 (2024). https://doi.org/10.1186/s13661-024-01839-1
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DOI: https://doi.org/10.1186/s13661-024-01839-1