Optimal decay rate for higher–order derivatives of solution to the 3D compressible quantum magnetohydrodynamic model

Juan Wang; Yinghui Zhang

doi:10.1515/anona-2021-0219

Open Access Published by De Gruyter February 9, 2022

Optimal decay rate for higher–order derivatives of solution to the 3D compressible quantum magnetohydrodynamic model

Juan Wang and Yinghui Zhang

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2021-0219

Abstract

We investigate optimal decay rates for higher–order spatial derivatives of strong solutions to the 3D Cauchy problem of the compressible viscous quantum magnetohydrodynamic model in the H⁵ × H⁴ × H⁴ framework, and the main novelty of this work is three–fold: First, we show that fourth order spatial derivative of the solution converges to zero at the L2-rate (1+t)-114 , which is same as one of the heat equation, and particularly faster than the L2-rate (1+t)-54 in Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and the L2-rate (1+t)-94 , in Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019]. Second, we prove that fifth–order spatial derivative of density ρ converges to zero at the L2-rate (1+t)-134 , which is same as that of the heat equation, and particularly faster than ones of Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019]. Third, we show that the high-frequency part of the fourth order spatial derivatives of the velocity u and magnetic B converge to zero at the L2-rate (1+t)-134 , which are faster than ones of themselves, and totally new as compared to Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019].

Keywords: Compressible quantum magnetohydrodynamic model; Optimal decay rate

MSC 2010: 35B40; 35Q35; 76W05

1 Introduction

In this paper, we consider optimal decay rates for higher–order spatial derivatives of strong solutions to the following 3D compressible viscous quantum magnetohydrodynamic (vQMHD) model:

(1.1) {ρt+ div (ρu)=0,(ρu)t+div (ρu⊗u)-μΔu-(μ+λ)∇divu+∇P(ρ)-ϑ22ρ∇(Δρρ)=(∇×B)×B,Bt-∇×(u×B)=-∇×(ν∇×B), div B=0,

where t ≥ 0 is time and x ∈ ℝ³ is the spatial coordinate, and the symbol ⊗ is the Kronecker tensor product. The unknown functions ρ = ρ(x, t) is the density, u = (u₁, u₂, u₃)(x, t) denotes the velocity, and B = (B₁, B₂, B₃)(x, t) represents the magnetic field. P = P(ρ) = aρ^γ (a > 0, γ ≥ 1) stands for the pressure. The constant viscosity coefficients μ and λ satisfy the physical conditions: μ > 0, 23μ+λ≥0 , and ν > 0 denotes the magnetic diffusion coefficient. The constant ϑ > 0 represents the Planck constant. The expression Δρρ is the so-called Bohm quantum potential satisfying:

2ρ∇(Δρρ)= div (ρ∇2ρ)=∇Δρ+|∇ρ|2∇ρρ2-∇ρΔρρ-∇ρ⋅∇2ρρ=∇Δρ-4 div(∇ρ⊗∇ρ).

The system (1.1) is supplemented with the following initial data

(1.2) (ρ,u,B)|t=0=(ρ0(x),u0(x),B0(x)).

Moreover, we assume that when the space variable goes to infinity, the initial perturbation satisfies

(1.3) lim|x|→∞(ρ0-1,u0,B0)(x)=0.

1.1 History of the problem

The quantum fluid model can provide many pieces of information for the particles in the semiconductor simulation, and it could be used to describe quantum semiconductors [6], weakly interacting Bose gases [8], and quantum trajectories of Bohmian mechanics [34]. The quantum magnetohydrodynamic (QMHD) model plays an important role in modeling and simulating electron transport, which was extended by Hass [10] later from a Wigner–Maxwell system. Moreover, this model could be used to describe global properties of quantum plasmas. It is worth mentioning that system (1.1) will reduce to the compressible MHD equations without quantum effects. There is a vast literature addressing the decay and other asymptotic behaviors of compressible fluid equations, we refer to previous studies [1, 2, 3, 4, 5, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 29, 30, 31, 32] and references therein.

In what follows, we only review some results closely related for simplicity. The study for decay rates of solutions to the QMHD model has attracted much attention of mathematicians. Pu and Guo [25] established the optimal decay rates of classical solutions near constant states via the spectral method in ℝ³. In [27], Pu and Xu showed that the classical solution of the vQMHD model has the following decay rate:

‖∇k(ρ-1)(t)‖H5-k+‖∇ku(t)‖H4-k+‖∇kB(t)‖H4-k≤C(1+t)-3+2k4,k=0,1.

Pu–Xu [28] employed the pure energy method developed by Guo–Wang [9] to obtain the optimal decay rates of higher-order spatial derivatives of solutions to the full hydrodynamic equations with quantum effects under the condition that the initial perturbation belongs to (HN+2∩H˙-s)×(HN+1∩H˙-s)×(HN∩H˙-s) for N ≥ 3 and s∈[0,32) . Recently, by making full use of Fourier splitting method, Xi–Pu–Guo [35] improved the work of Pu–Xu [27], and particularly they got

(1.4) ‖∇k(ρ-1)(t)‖H5-k+‖∇ku(t)‖H4-k+‖∇kB(t)‖H4-k≤C(1+t)-3+2k4,k=0,1,2,3.

For more results about the large time behavior of solutions to the quantum fluid model, readers can refer to [26, 36] and references therein.

It is clear that decay rate of k(k = 0, 1, 2, 3) order spatial derivative of solution in (1.4) is optimal in the sense that it coincides with decay rate of solution to the heat equation. However, decay rate in (1.4) implies that fourth order spatial derivative of solution converges to zero at L2-rate (1+t)-94 , which is slower than the L2-rate (1+t)-114 of solution to the heat equation. Moreover, fifth order spatial derivative of density ρ converges to zero at L2-rate (1+t)-94 , which is slower than the L2-rate (1+t)-134 of solution to the heat equation. Therefore, decay rates of fourth order spatial derivative of solution and fifth order spatial derivative of density ρ are not optimal in this sense. The main purpose of this paper is to give a clear answer to this issue. More precisely, our main results can be outlined as follows. Firstly, we show that fourth order spatial derivative of the solution converges to zero at the L2-rate (1+t)-114 . Secondly, we prove that fifth–order spatial derivative of density ρ converges to zero at the L2-rate (1+t)-134 . Thirdly, we show that the high-frequency part of the fourth order spatial derivatives of velocity u and magnetic B converge to zero at the L2-rate (1+t)-134 , which are faster than ones of themselves, and totally new as compared to Pu–Xu [27] and Xi–Pu–Guo [35].

1.2 Notation

In this paper, we use H^k(ℝ³) to denote the usual Sobolev spaces with norm ‖ · ‖_H^k. Generally, we use L^p, 1 ≤ p ≤ ∞ to denote the usual L^p (ℝ³) spaces with norm ‖ · ‖_L^p. The notation a ≲ b means that a ≤ Cb, where the universal constant C > 0 may be from line to line but independent of time t. Similarly, the notation a ≳ b means that a ≥ Cb for a universal positive constant which is independent of time t. We define

∇ku=∂xαui α=k, i=1,2,3, u=(u1,u2,u3).

For a radial function ϕ∈C0∞(ℝξ3) such that ϕ(ξ) = 1 when |ξ | ≤ 1 and ϕ(ξ ) = 0 when |ξ | ≥ 2, we define the low–frequency part and the high–frequency part of f as follows

fl=𝔉-1[ϕ(ξ)f^], and fh=𝔉-1[(1-ϕ(ξ))f^].

1.3 Main results

Before stating our main results, let us recall the following results obtained in previous studies [28, 35], which will be used in this paper frequently.

Theorem 1.1

(see [28, 35]) Suppose that the initial data (ρ₀ − 1, u₀, B₀) ∈ H⁵ × H⁴ × H⁴. There exists a small constant δ > 0 such that if

(1.5) ‖ρ0-1‖H5+‖u0‖H4+‖B0‖H4≤δ,

then the solution (ρ, u, B) of (1.1)–(1.3) satisfy for all T ≥ 0

(1.6) ‖(ρ-1,u,B)(t)‖H42+‖ϑ∇ρ(t)‖H42+∫0t‖∇(u,B,ϑρ)(s)‖H42ds≤C(‖ρ0-1‖H52+‖u0‖H42+‖B0‖H42).

Moreover, provided that ‖;(ρ₀ − 1, u₀, B₀)‖_L¹ is finite additionally, then the solution (ρ, u, B) satisfy

(1.7) ‖∇k(ρ-1)(t)‖H5-k+‖∇ku(t)‖H4-k+‖∇kB(t)‖H4-k≤C(1+t)-3+2k4,

with k = 0, 1, 2, 3. Here, the positive constant C is independent of time.

Our main purpose in this paper is to establish the upper optimal decay rates for fourth order spatial derivative of the solution and fifth order spatial derivatives of the density ρ which are the same as those of the heat equation. Our main results are stated in the following theorems:

Theorem 1.2

Under all the assumptions in Theorem 1.1, the global solution (ρ, u, B) has the following time decay rate for all t ≥ T

(1.8) ‖∇4(ρ-1)(t)‖H1+‖∇4u(t)‖L2+‖∇4B(t)‖L2≲(1+t)-114.

Here T is a positive large time.

Theorem 1.3

Under all the assumptions in Theorem 1.1, the global solution (ρ, u, B) has the following time decay rate for all t ≥ T

(1.9) ‖∇5(ρ-1)(t)‖L2+‖∇4uh(t)‖L2+‖∇4Bh(t)‖L2≲(1+t)-134.

Here T is a positive large time.

Remark 1.4

It is interesting to make a comparison between Theorem 1.2 and Theorem 1.1, where the authors derived decay rate of fourth order spatial derivative of solution is the same as one of third order spatial derivative. We show that fourth order spatial derivative of the solution converges to zero at the L2-rate (1+t)-114 , which is same as that of the heat equation, and particularly faster than the L2-rate (1+t)-54 in Pu–Xu [27] and the L2-rate (1+t)-94 in Xi–Pu–Guo [35]. Therefore, our decay rate is optimal, and improve the results of [28, 35].

Remark 1.5

Compared to the main results (1.7) in [28, 35], decay rate in (1.9) is totally new and gives optimal decay rate of fifth order spatial derivative of density ϱ, which is same as that of the heat equation. In addition, it also gives optimal decay rates of the high-frequency part of fourth order spatial derivatives of the velocity u and magnetic B, which are faster than ones of themselves.

Now, let’s sketch the strategy of proving Theorem 1.2–Theorem 1.3 and explain some main difficulties and techniques involved in the process. Roughly speaking, our methods mainly involve low–frequency and high–frequency decomposition technique and delicate energy estimates. Our strategy can be outlined as follows.

For the proof of Theorem 1.2, we hope to establish the optimal decay rate for fourth order spatial derivatives of solution for the compressible viscous quantum magnetohydrodynamic equations (1.1). For the convenience of calculation, we linearize the system (1.1) around the equilibrium state, and then rewrite the linearized system in terms of the variables ϱ, u, B, which is stated in (2.1). We only need to make full use of the benefit of low-frequency and high-frequency decomposition to deal with high-frequency part of ∇⁴(ϱ, u, B). The proof mainly involves the following three steps. First, we derive high–frequency L² energy estimate of fourth order spatial derivative of the solution to the compressible vQMHD model:

12ddt∫ℝ3|∇4ϱh|2+|∇4uh|2+|∇4Bh|2+ϑ24|∇5ϱh|2dx+(2μ+λ)∫ℝ3|∇5uh|2dx+ν∫ℝ3|∇5Bh|2dx≲(1+t)-112+(1+t)-2(‖∇4ϱh‖H12+‖∇5Bh‖L22)+(1+t)-32(‖∇5uh‖L22+‖∇6ϱh‖L22).

Second, by noticing that the above energy inequality only involves dissipative estimates of u^h and B^h. In order to explore the dissipative estimate of ϱ^h, the main idea here is to employ the new interactive energy functional to get

ddt∫ℝ3∇4uh∇5ϱhdx+12‖∇5ϱh‖L22+ϑ24‖∇6ϱh‖L22≲(1+t)-112+‖∇5uh‖L22+(1+t)-32(‖∇6ϱh‖L22+‖∇5uh‖L22).

Last, we choose two sufficiently large positive constants D₀ and T₁, and define the temporal energy functional

ℰ(t)=D0(‖∇4(ϱ,u,B)h‖L22+ϑ22‖∇5ϱh‖L22)+∫ℝ3∇4uh∇5ϱhdx,

for t ≥ 0, where it is mentioned that ℰ(t) is equivalent to ‖∇4ϱh‖H12+‖∇4(u,B)h‖L22 . Combining the above estimates, we can obtain

ddtℰ(t)+‖∇4ϱh(t)‖H12+‖∇4uh(t)‖L22+‖∇4Bh(t)‖L22≲(1+t)-112,

which together with Gronwall’s argument and low-frequency decay rate gives rise to (1.8), and thus completes the proof of Theorem 1.2.

For the proof of Theorem 1.3, we hope to establish the fifth order spatial derivatives of the density ρ. Our main idea is to find the optimal decay rate at low–frequency and high–frequency of the fifth order spatial derivative of density, and then use the properties of high-frequency and low-frequency decomposition to obtain the optimal decay rates of the fifth order spatial derivative of density. For the low-frequency part ‖∇⁵ ϱ^l‖_L², by using the equation (2.1)₁, Theorem 1.2, Duhamel’s principle, Plancherel theorem, Hölder’s inequality and Hausdorff–Young’s inequality, we have

∥∇5ϱl(t)∥L2≲ (1+t)-134∥(ϱ,u,B)(0)∥L1+∫0t2(1+t-τ)-134∥G(τ)∥L1dτ+∫t2t(1+t-τ)-54∥|ξ|4G^l(τ)∥L∞dτ.

After some detailed calculations, we can get

(1.10) ‖∇5ϱl‖L2≲(1+t)-134.

For the high-frequency part ‖∇⁵ ϱ^l‖_L², it is easy to find that the calculation in this part is similar to that of section 2, we only need to replace the decay rate of fourth order space derivative from (1+t)-94 to (1+t)-114 . We finally arrive at

(1.11) ‖∇4ϱh(t)‖H1+‖∇4uh(t)‖L2+‖∇4Bh(t)‖L2≲(1+t)-134.

Combining the estimates (1.10) and (1.11), we conclude the proof of Theorem 1.3.

2 Proof of Theorem 1.2

In this section, we will devote ourselves to proving Theorem 1.2. We suppose that all the conditions of Theorem 1.1 are in force in this and next section. Without loss of generality, we assume that P′(1) = 1. Let us setting ϱ = ρ − 1, we can rewrite the system (1.1) in the perturbation form as

(2.1) {ϱt+ div u=G1,ut-μΔu-(μ+λ)∇ div u+∇ϱ-ϑ24∇Δϱ=G2,Bt-νΔB=G3.

Here the nonlinear source terms G₁, G₂ and G₃ are given by

G1=-ϱ div u-u⋅∇ϱ,

G2=-u⋅∇u+ϱ(ϱ+1)(μΔu+(μ+λ)∇ div u)-(P′(ϱ+1)ϱ+1-1)∇ϱ-ϑ24ϱϱ+1∇Δϱ +ϑ24(|∇ϱ|2∇ϱ(1+ϱ)3-∇ϱΔϱ(1+ϱ)2-∇ϱ⋅∇2ϱ(1+ϱ)2)+11+ϱ((∇×B)×B),G3=∇×(u×B).

The initial data are given as

(2.2) (ϱ,u,B)(x,t)|t=0=(ϱ0,u0,B0)(x)→(0,0,0) as |x|→∞.

First, we recall the L² time decay rates on the linearized system for the first two equations (2.1).

Lemma 2.1

(see [27]) Let s ≥ 0 be an integer. Assume that (ϱ, u) is the solution of the linearized system for the first two equations in (2.1) with initial data ϱ ∈ H^s+1 ∩ L¹, u₀ ∈ H^s ∩ L¹. Then, for any t ≥ 0, it holds that

‖ϱ(t)‖L2≤C(1+t)-34(‖(ϱ,u0)‖L1+‖(ϱ0,u0)‖L2),

(2.3) ‖∇k+1ϱ(t)‖L2≤C(1+t)-34-k+12(‖(ϱ0,u0)‖L1+‖(∇k+1ϱ,∇ku0)‖L2),

(2.4) ‖∇ku(t)‖L2≤C(1+t)-34-k2(‖(ϱ0,u0)‖L1+‖(∇k+1ϱ,∇ku0)‖L2),

for 0 ≤ k ≤ s.

Next, we recall the L² time decay rate on the linearized system for the third equation in (2.1).

Lemma 2.2

(see [33]) Let s ≥ 0 be an integer. Assume that B is the solution of the linearized system for the third equation in (2.1) with initial data B₀ ∈ H^s ∩ L¹. Then, for any t ≥ 0, it holds that

(2.5) ‖∇kB‖L2≤C(1+t)-34-k2‖B0‖L1,

for 0 ≤ k ≤ s.

Finally, we deduce the L² time decay rate on the nonlinear system (2.1).

Lemma 2.3

Assume that the assumptions of Theorem 1.1 are in force, the solution (ϱ, u, B) of the nonlinear system (2.1) satisfies the following decay estimate:

(2.6) ‖∇4(ϱl,ul,Bl)(t)‖L2≲(1+t)-114.

Proof

By virtue of the equation (2.1), Lemma 2.1, Lemma 2.2, Duhamel’s principle, Plancherel theorem, Hölder’s inequality and Hausdorff–Young’s inequality, we have

(2.7) ∥∇4(ϱl,ul,Bl)(t)∥L2≲ (1+t)-114∥(ϱ,u,B)(0)∥L1+∫0t2(1+t-τ)-114∥G(τ)∥L1dτ+∫t2t(1+t-τ)-54∥|ξ|3G^l(τ)∥L∞dτ.

Next, we shall estimate the second term and the third term on the right-hand side of (2.7). To begin with, by virtue of Lemma 4.2, we can bound the term by

(2.8) ‖G(τ)‖L1≲‖ div (ϱu)(τ)‖L1+‖u⋅∇u(τ)‖L1+‖ϱϱ+1[μΔu+(μ+λ)∇ div u](τ)‖L1 +‖[P′(ϱ+1)ϱ+1-P′(1)]∇ϱ(τ)‖L1+‖ϑ24ϱϱ+1∇Δϱ(τ)‖L1 +‖ϑ24(|∇ϱ|2∇ϱ(1+ϱ)3-∇ϱΔϱ(1+ϱ)2-∇ϱ⋅∇2ϱ(1+ϱ)2)(τ)‖L1+‖11+ϱ((∇×B)×B)(τ)‖L1+‖∇×(u×B)(τ)‖L1≲‖(ϱ,u)(τ)‖L2‖∇(ϱ,u)(τ)‖L2+‖ϱ(τ)‖L2‖∇2u(τ)‖L2+‖ϱ(τ)‖L2‖∇3ϱ(τ)‖L2 +‖∇ϱ(τ)‖L2‖∇2ϱ(τ)‖L2+‖B(τ)‖L2‖∇2B(τ)‖L2+‖(u,B)(τ)‖L2‖∇(u,B)(τ)‖L2≲(1+t)-32,

and

(2.9) ‖|ξ|3G^l(τ)‖L∞≲‖∇3Gl(τ)‖L1≲‖∇3( div (ϱu))(τ)‖L1+‖∇3(u⋅∇u)(τ)‖L1 +‖∇2(ϱϱ+1[μΔu+(μ+λ)∇ div u])(τ)‖L1+‖∇3([P′(ϱ+1)ϱ+1-P′(1)]∇ϱ)(τ)‖L1 +‖∇2(ϑ24ϱϱ+1∇Δϱ)(τ)‖L1+‖∇3(ϑ24(|∇ϱ|2∇ϱ(1+ϱ)3-∇ϱΔϱ(1+ϱ)2-∇ϱ⋅∇2ϱ(1+ϱ)2))(τ)‖L1 +‖∇3(11+ϱ((∇×B)×B))(τ)‖L1+‖∇3(∇×(u×B)(τ))‖L1≲‖(ϱ,u)(τ)‖L2‖∇4(ϱ,u)(τ)‖L2+‖∇2u(τ)‖L22+‖ϱ(τ)‖L2‖∇5ϱ(τ)‖L2 +‖∇ϱ(τ)‖L2‖∇5ϱ(τ)‖L2+‖∇ϱ(τ)‖L2‖∇4ϱ(τ)‖L2+‖∇2ϱ(τ)‖L2‖∇3ϱ(τ)‖L2 +‖∇B(τ)‖L2‖∇3B(τ)‖L2+‖(u,B)(τ)‖L2‖∇4(u,B)(τ)‖L2≲(1+t)-3.

Substituting (2.8)–(2.9) into (2.7) gives (2.6).

Proof of Theorem 1.2

The first step is to establish the energy estimate for the fourth order spatial derivative of solution. Taking

〈𝔉-1(1-ϕ(ξ))∇4(2.1)1,∇4ρh〉+〈𝔉-1(1-ϕ(ξ))∇4(2.1)2,∇4uh〉+〈𝔉-1(1-ϕ(ξ))∇4(2.1)3,∇4Bh〉,

and using integration by parts, we can obtain

(2.10) 12ddt∫ℝ3|∇4ϱh|2+|∇4uh|2+|∇4Bh|2+ϑ24|∇5ϱh|2dx+(2μ+λ)∫ℝ3|∇5uh|2dx+ν∫ℝ3|∇5Bh|2dx=〈∇4G1h,∇4ϱh〉+〈∇4G2h,∇4uh〉+〈∇4G3h,∇4Bh〉+h24〈∇5G1h,∇5ϱh〉.

The right-hand side of the above equation can be estimated as follows. Firstly, we have

(2.11) 〈∇4G1h,∇4ϱh〉=-〈∇4(ϱ div u+u⋅∇ϱ)h,∇4ϱh〉=-〈∇4(ϱ div u)h,∇4ϱh〉-〈∇4(u⋅∇ϱ)h,∇4ϱh〉:=I1+I2.

For term I₁, by using Hölder’s inequality, Lemma 4.2, Lemma 4.3, Lemma 4.1, Young’s inequality and Sobolev interpolation theorem, we arrive at

(2.12) |I1|≲‖∇3(ϱ div u)h‖L2‖∇5ϱh‖L2≲‖∇3(ϱ div u)‖L2‖∇5ϱh‖L2≲(‖ϱ‖L∞‖∇4u‖L2+‖∇u‖L3‖∇3ϱ‖L6)‖∇5ϱh‖L2≲(‖∇ϱ‖L212‖∇2ϱ‖L212‖∇4u‖L2+‖∇u‖L212‖∇2ϱ‖L212‖∇4ϱ‖L2)‖∇5ϱh‖L2≲(1+t)-154‖∇5ϱh‖L2≲(1+t)-112+(1+t)-2‖∇5ϱh‖L22.

For term I₂, we can rewrite it as follows

(2.13) I2=〈-∇4(u⋅∇ϱ)h,∇4ϱh〉=〈-∇4(u⋅∇ϱ)+∇4(u⋅∇ϱ)l,∇4ϱh〉=〈-∇4(u⋅∇ϱh)-∇4(u⋅∇ϱl)+∇4(u⋅∇ϱ)l,∇4ϱh〉=〈-∇4(u⋅∇ϱh),∇4ϱh〉+〈-∇4(u⋅∇ϱl),∇4ϱh〉+〈∇4(u⋅∇ϱ)l,∇4ϱh〉:=I2,1+I2,2+I2,3.

By routine checking, one may show that

∇4(u⋅∇ϱh)=u⋅∇5ϱh+4∇u⋅∇4ϱh+6∇2u⋅∇3ϱh+4∇3u⋅∇2ϱh+4∇4u⋅∇ϱh.

The integration by part yields directly

∫ℝ3u⋅∇(∇4ϱh)⋅∇4ϱhdx=-12∫ℝ3 div u|∇4ϱh|2dx,

and hence, we show that

(2.14) |〈 div u∇4ϱh,∇4ϱh〉|≲‖ div u‖L∞‖∇4ϱh‖L22≲‖ div u‖L∞‖∇4ϱh‖L22≲‖∇2u‖L212‖∇3u‖L212‖∇4ϱh‖L22≲(1+t)-174‖∇4ϱh‖L2≲(1+t)-112+(1+t)-3‖∇4ϱh‖L22.

Similar to the estimate (2.14), we have

(2.15) |〈∇u⋅∇4ϱh,∇4ϱh〉|≲(1+t)-112+(1+t)-3‖∇4ϱh‖L22,

and

(2.16) |〈-6∇2u⋅∇3ϱh,∇4ϱh〉|≲‖∇2u‖L3‖∇3ϱh‖L6‖∇4ϱh‖L2≲‖∇2u‖L3‖∇3ϱ‖L6‖∇4ϱh‖L2≲‖∇2u‖L212‖∇3u‖L212‖∇4ϱ‖L2‖∇4ϱh‖L2≲(1+t)-174‖∇4ϱh‖L2≲(1+t)-112+(1+t)-3‖∇4ϱh‖L22,

(2.17) |〈-4∇3u⋅∇2ϱh,∇4ϱh〉|≲‖∇2ϱh‖L3‖∇3u‖L6‖∇4ϱh‖L2≲‖∇2ϱ‖L3‖∇3u‖L6‖∇4ϱh‖L2≲‖∇2ϱ‖L212‖∇3ϱ‖L212‖∇4u‖L2‖∇4ϱh‖L2≲(1+t)-174‖∇4ϱh‖L2≲(1+t)-112+(1+t)-3‖∇4ϱh‖L22,

(2.18) |〈-∇4u⋅∇ϱh,∇4ϱh〉|≲‖∇ϱh‖L∞‖∇4u‖L2‖∇4ϱh‖L2≲‖∇2ϱ‖L212‖∇3ϱ‖L212‖∇4u‖L2‖∇4ϱh‖L2≲(1+t)-174‖∇4ϱh‖L2≲(1+t)-112+(1+t)-3‖∇4ϱh‖L22,

and hence, it follows that

(2.19) |I2,1|≲(1+t)-112+(1+t)-3‖∇4ϱh‖L22.

For the term I_2,2, we have

(2.20) |I2,2|≲‖∇4(u⋅∇ϱl)‖L2‖∇4ϱh‖L2≲(‖u‖L∞‖∇5ϱl‖L2+‖∇ϱl‖L∞‖∇4u‖L2)‖∇4ϱh‖L2≲(‖∇u‖L212‖∇2u‖L212‖∇4ϱ‖L2+‖∇2ϱl‖L212‖∇3ϱl‖L212‖∇4u‖L2)‖∇4ϱh‖L2≲(‖∇u‖L212‖∇2u‖L212‖∇4ϱ‖L2+‖∇ϱ‖L212‖∇2ϱ‖L212‖∇4u‖L2)‖∇4ϱh‖L2≲(1+t)-154‖∇4ϱh‖L2≲(1+t)-112+(1+t)-2‖∇4ϱh‖L22.

For the term I_2,3, we get

(2.21) |I2,3|≲‖∇4(u⋅∇ϱ)l‖L2‖∇4ϱh‖L2≲‖∇3(u⋅∇ϱ)‖L2‖∇4ϱh‖L2≲(‖u‖L∞‖∇4ϱ‖L2+‖∇ϱ‖L3‖∇3u‖L6)‖∇4ϱh‖L2≲(‖∇u‖L212‖∇2u‖L212‖∇4ϱ‖L2+‖∇ϱ‖L212‖∇2ϱ‖L212‖∇4u‖L2)‖∇4ϱh‖L2≲(1+t)-154‖∇4ϱh‖L2≲(1+t)-112+(1+t)-2‖∇4ϱh‖L22.

Summing up estimates (2.19)–(2.21), we arrive at

(2.22) |I2|≲(1+t)-112+(1+t)-2‖∇4ϱh‖H12.

Applying equation (2.1), it holds that

(2.23) 〈∇4G2h,∇4uh〉=〈-∇4(u⋅∇u)h,∇4uh〉+〈∇4{ϱ(ϱ+1)[μΔv+(μ+λ)∇ div u]}h,∇4uh〉+〈-∇4{[P′(ϱ+1)ϱ+1-P′(1)]∇ϱ}h,∇4uh〉+〈-∇4(ϑ24ϱϱ+1∇Δϱ)h,∇4uh〉+〈∇4[ϑ24(|∇ϱ|2∇ϱ(1+ϱ)3-∇ϱΔϱ(1+ϱ)2-∇ϱ⋅∇2ϱ(1+ϱ)2)]h,∇4uh〉+〈∇4[11+ϱ((∇×B)×B)]h,∇4uh〉:=∑j=38Ij.

The first term on the right-hand side of the above equation can be estimated as follows

(2.24) |I3|≲‖∇3(u⋅∇u)h‖L2‖∇5uh‖L2≲‖∇3(u⋅∇u)‖L2‖∇5uh‖L2≲(‖u‖L∞‖∇4u‖L2+‖∇3u‖L6‖∇u‖L3)‖∇5uh‖L2≲‖∇u‖L212‖∇2u‖L212‖∇4u‖L2‖∇5uh‖L2≲(1+t)-154‖∇5uh‖L2≲(1+t)-112+(1+t)-2‖∇5uh‖L22.

Similarly, using the integration by parts, the terms I₄ ∼ I₈ can be estimated as follows

(2.25) |I4|≲‖∇3{ϱ(ϱ+1)[μΔu+(μ+λ)∇ div u]}h‖L2‖∇5uh‖L2≲‖∇3{ϱ(ϱ+1)[μΔu+(μ+λ)∇ div u]}‖L2‖∇5uh‖L2≲(‖ϱ‖L∞‖∇5u‖L2+‖∇2u‖L3‖∇3ϱ‖L6)‖∇5uh‖L2≲(‖∇ϱ‖L212‖∇2ϱ‖L212‖∇5u‖L2+‖∇2u‖L212‖∇3u‖L212‖∇4ϱ‖L2)‖∇5uh‖L2≲(1+t)-154‖∇5uh‖L2+(1+t)-174‖∇5uh‖L2≲(1+t)-112+(1+t)-2‖∇5uh‖L22.

(2.26) |I5|≲‖∇3{[P′(ϱ+1)ϱ+1-P′(1)]∇ϱ}h‖L2‖∇5uh‖L2≲‖∇3{[P′(ϱ+1)ϱ+1-P′(1)]∇ϱ}‖L2‖∇5uh‖L2≲(‖ϱ‖L∞‖∇4ϱ‖L2+‖∇ϱ‖L3‖∇3ϱ‖L6)‖∇5uh‖L2≲‖∇ϱ‖L212‖∇2ϱ‖L212‖∇4ϱ‖L2‖∇5uh‖L2≲(1+t)-154‖∇5uh‖L2≲(1+t)-112+(1+t)-2‖∇5uh‖L22.

For term I₆, we can rewrite it as follows

(2.27) I6=〈-∇4(ϑ24ϱϱ+1∇Δϱ)h,∇4uh〉=〈-∇4(ϑ24ϱϱ+1∇Δϱ)+∇4(ϑ24ϱϱ+1∇Δϱ)l,∇4uh〉=〈-∇4(ϑ24ϱϱ+1∇Δϱh)-∇4(ϑ24ϱϱ+1∇Δϱl)+∇4(ϑ24ϱϱ+1∇Δϱ)l,∇4uh〉:=I6,1+I6,2+I6,3.

Similar to the proof of (2.19)–(2.21), for I_6,1, I_6,2 and I_6,3, it holds that

|I6,1|≲‖∇3(ϑ24ϱϱ+1∇Δϱh)‖L2‖∇5uh‖L2≲(‖ϱ‖L∞‖∇6ϱh‖L2+‖∇3ϱh‖L6‖∇3ϱ‖L3)‖∇5uh‖L2≲(‖∇ϱ‖L212‖∇2ϱ‖L212‖∇6ϱh‖L2+‖∇3ϱ‖L212‖∇4ϱ‖L212‖∇4ϱ‖L2)‖∇5uh‖L2≲(1+t)-32‖∇6ϱh‖L2‖∇5uh‖L2+(1+t)-92‖∇5uh‖L2≲(1+t)-112+(1+t)-32(‖∇6ϱh‖L22+‖∇5uh‖L22).

|I6,2|≲‖∇3(ϑ24ϱϱ+1∇Δϱl)‖L2‖∇5uh‖L2≲(‖ϱ‖L∞‖∇6ϱl‖L2+‖∇3ϱl‖L6‖∇3ϱ‖L3)‖∇5uh‖L2≲(‖ϱ‖L∞‖∇5ϱ‖L2+‖∇3ϱl‖L6‖∇3ϱ‖L3)‖∇5uh‖L2≲(‖∇ϱ‖L212‖∇2ϱ‖L212‖∇5ϱ‖L2+‖∇3ϱ‖L212‖∇4ϱ‖L212‖∇4ϱ‖L2)‖∇5uh‖L2≲(1+t)-154‖∇5uh‖L2+(1+t)-92‖∇5uh‖L2≲(1+t)-112+(1+t)-2‖∇5uh‖L22.

Using the properties of low-frequency decomposition, it holds that

|I6,3|≲‖∇3(ϑ24ϱϱ+1∇Δϱ)l‖L2‖∇5uh‖L2≲‖∇2(ϑ24ϱϱ+1∇Δϱ)‖L2‖∇5uh‖L2≲(‖ϱ‖L∞‖∇5ϱ‖L2+‖∇3ϱ‖L6‖∇2ϱ‖L3)‖∇5uh‖L2≲(‖∇ϱ‖L212‖∇2ϱ‖L212‖∇5ϱ‖L2+‖∇2ϱ‖L212‖∇3ϱ‖L212‖∇4ϱ‖L2)‖∇5uh‖L2≲(1+t)-154‖∇5uh‖L2≲(1+t)-112+(1+t)-2‖∇5uh‖L22.

Putting the above estimates into (2.27) yields directly

(2.28) |I6|≲(1+t)-112+(1+t)-32(‖∇6ϱh‖L22+‖∇5uh‖L22).

For the terms I₇, I₈, we have

(2.29) |I7|≲‖∇3[ϑ24(|∇ϱ|2∇ϱ(1+ϱ)3-∇ϱΔϱ(1+ϱ)2-∇ϱ⋅∇2ϱ(1+ϱ)2)]h‖L2‖∇5uh‖L2≲‖∇3[ϑ24(|∇ϱ|2∇ϱ(1+ϱ)3-∇ϱΔϱ(1+ϱ)2-∇ϱ⋅∇2ϱ(1+ϱ)2)]‖L2‖∇5uh‖L2≲(‖∇ϱ‖L∞‖∇5ϱ‖L2+‖∇4ϱ‖L6‖∇2ϱ‖L3)‖∇5uh‖L2≲‖∇2ϱ‖L212‖∇3ϱ‖L212‖∇5ϱ‖L2‖∇5uh‖L2≲(1+t)-174‖∇5uh‖L2≲(1+t)-112+(1+t)-3‖∇5uh‖L22,

(2.30) |I8|≲‖∇3[11+ϱ((∇×B)×B)]h‖L2‖∇5uh‖L2≲‖∇3[11+ϱ((∇×B)×B)]‖L2‖∇5uh‖L2≲(‖B‖L∞‖∇4B‖L2+‖∇B‖L3‖∇3B‖L6)‖∇5uh‖L2≲‖∇B‖L212‖∇2B‖L212‖∇4B‖L2‖∇5uh‖L2≲(1+t)-154‖∇5uh‖L2≲(1+t)-112+(1+t)-2‖∇5uh‖L22.

Summing up (2.23)–(2.26), (2.28)–(2.30), we obtain

(2.31) |〈∇4G2h,∇4uh〉|≲(1+t)-112+(1+t)-32(‖∇5uh‖L22+‖∇6ϱh‖L22),

and we have

(2.32) |〈∇4G3h,∇4Bh〉|≲‖∇3(∇×(u×B))h‖L2‖∇5Bh‖L2≲‖∇3(∇×(u×B))‖L2‖∇5Bh‖L2≲(‖u‖L∞‖∇4B‖L2+‖B‖L∞‖∇4u‖L2)‖∇5Bh‖L2≲(‖∇u‖L212‖∇2u‖L212‖∇4B‖L2+‖∇B‖L212‖∇2B‖L212‖∇4u‖L2)‖∇5Bh‖L2≲(1+t)-154‖∇5Bh‖L2≲(1+t)-112+(1+t)-2‖∇5Bh‖L22.

For the term 〈 ∇5G1h , ∇⁵ ϱ^h〉, we have

(2.33) ϑ24〈∇5G1h,∇5ϱh〉=-ϑ24〈∇5(ϱ div u+u⋅∇ϱ)h,∇5ϱh〉=-ϑ24〈∇5(ϱ div u)h,∇5ϱh〉+ϑ24〈-∇5(u⋅∇ϱ)h,∇5ϱh〉:=I9+I10.

For the term I₉, by using the properties of low-frequency and high-frequency decomposition, we get

(2.34) I9=ϑ24〈-∇5(ϱ div u)h,∇5ϱh〉=ϑ24〈-∇5(ϱ div u)+∇5(ϱ div u)l,∇5ϱh〉=ϑ24〈-∇5(ϱ div uh)-∇5(ϱ div ul)+∇5(ϱ div u)l,∇5ϱh〉=I9,1+I9,2+I9,3.

For the term I_9,1, by using integration by parts, we obtain

(2.35) |I9,1|≲‖∇4(ϱ div uh)‖L2‖∇6ϱh‖L2≲(‖ϱ‖L∞‖∇5uh‖L2+‖∇uh‖L3‖∇4ϱ‖L6)‖∇6ϱh‖L2≲(‖∇ϱ‖L212‖∇2ϱ‖L212‖∇5uh‖L2+‖∇u‖L212‖∇2u‖L212‖∇5ϱ‖L2)‖∇6ϱh‖L2≲(1+t)-32‖∇5uh‖L2‖∇6ϱh‖L2+(1+t)-154‖∇6ϱh‖L2≲(1+t)-112+(1+t)-32(‖∇6ϱh‖L22+‖∇5uh‖L22).

For the term I_9,2, we have

(2.36) |I9,2|≲‖∇4(ϱ div ul)‖L2‖∇6ϱh‖L2≲(‖ϱ‖L∞‖∇5ul‖L2+‖∇ul‖L3‖∇4ϱ‖L6)‖∇6ϱh‖L2≲(‖∇ϱ‖L212‖∇2ϱ‖L212‖∇4u‖L2+‖∇u‖L212‖∇2u‖L212‖∇5ϱ‖L2)‖∇6ϱh‖L2≲(1+t)-154‖∇6ϱh‖L2≲(1+t)-112+(1+t)-2‖∇6ϱh‖L22.

For the term I_9,3, we get

(2.37) |I9,3|≲‖∇4(ϱ div u)l‖L2‖∇6ϱh‖L2≲‖∇3(ϱ div u)‖L2‖∇6ϱh‖L2≲(‖ϱ‖L∞‖∇4u‖L2+‖∇u‖L3‖∇3ϱ‖L6)‖∇6ϱh‖L2≲(‖∇ϱ‖L212‖∇2ϱ‖L212‖∇4u‖L2+‖∇u‖L212‖∇2u‖L212‖∇4ϱ‖L2)‖∇6ϱh‖L2≲(1+t)-154‖∇6ϱh‖L2≲(1+t)-112+(1+t)-2‖∇6ϱh‖L22.

Putting estimates (2.35)–(2.37) into (2.34), it arrives at

(2.38) |I9|≲(1+t)-112+(1+t)-32(‖∇6ϱh‖L22+‖∇5uh‖L22).

Similarly, for the term I₁₀, we can rewrite it as follows

(2.39) I10=ϑ24〈-∇5(u⋅∇ϱ)h,∇5ϱh〉=ϑ24〈-∇5(u⋅∇ϱ)+∇5(u⋅∇ϱ)l,∇5ϱh〉=ϑ24〈-∇5(u⋅∇ϱh)-∇5(u⋅∇ϱl)+∇5(u⋅∇ϱ)l,∇5ϱh〉=I10,1+I10,2+I10,3.

Similar to the proof of (2.35), we have

(2.40) |I10,1|≲‖∇4(u⋅∇ϱh)‖L2‖∇6ϱh‖L2≲(‖u‖L∞‖∇5ϱh‖L2+‖∇ϱh‖L∞‖∇4u‖L2)‖∇6ϱh‖L2≲(‖∇u‖L212‖∇2u‖L212‖∇5ϱh‖L2+‖∇2ϱ‖L212‖∇3ϱ‖L212‖∇4u‖L2)‖∇6ϱh‖L2≲(1+t)-174‖∇6ϱh‖L2+(1+t)-154‖∇6ϱh‖L2≲(1+t)-112+(1+t)-2‖∇6ϱh‖L22,

(2.41) |I10,2|≲‖∇4(u⋅∇ϱl)‖L2‖∇6ϱh‖L2≲(‖u‖L∞‖∇5ϱl‖L2+‖∇ϱl‖L∞‖∇4u‖L2)‖∇6ϱh‖L2≲(‖∇u‖L212‖∇2u‖L212‖∇4ϱ‖L2+‖∇2ϱ‖L212‖∇3ϱ‖L212‖∇4u‖L2)‖∇6ϱh‖L2≲(1+t)-154‖∇6ϱh‖L2≲(1+t)-112+(1+t)-2‖∇6ϱh‖L22,

and

(2.42) |I10,3|≲‖∇4(u⋅∇ϱ)l‖L2‖∇6ϱh‖L2≲‖∇3(u⋅∇ϱ)‖L2‖∇6ϱh‖L2≲(‖u‖L∞‖∇4ϱ‖L2+‖∇ϱ‖L3‖∇3u‖L6)‖∇6ϱh‖L2≲(‖∇u‖L212‖∇2u‖L212‖∇4ϱ‖L2+‖∇ϱ‖L212‖∇2ϱ‖L212‖∇4u‖L2)‖∇6ϱh‖L2≲(1+t)-154‖∇6ϱh‖L2≲(1+t)-112+(1+t)-2‖∇6ϱh‖L22.

It follows from the estimates (2.40)–(2.42) and (2.39) that

(2.43) |I10|≲(1+t)-112+(1+t)-2‖∇6ϱh‖L22.

Combining with (2.12), (2.22), (2.31)–(2.32), (2.38) and (2.43), it arrives at

(2.44) 12ddt∫ℝ3|∇4ϱh|2+|∇4uh|2+|∇4Bh|2+ϑ24|∇5ϱh|2dx +(2μ+λ)∫ℝ3|∇5uh|2dx+ν∫ℝ3|∇5Bh|2dx≲(1+t)-112 +(1+t)-2(‖∇4ϱh‖H12+‖∇5Bh‖L22)+(1+t)-32(‖∇5uh‖L22+‖∇6ϱh‖L22).

In order to close the estimate, we will establish the dissipation estimate for ∇⁴ ϱ^h. Applying the operator 𝔉⁻¹(1 − ϕ(ξ )) to ∇⁴(2.1)₂, multiplying the resulting equality by ∇⁵ ϱ^h, integrating over ℝ³, we can deduce that

(2.45) ddt∫ℝ3∇4uh∇5ϱhdx+‖∇5ϱh‖L22+ϑ24‖∇6ϱh‖L22=‖∇5uh‖L22-μ〈∇5uh,∇6ϱh〉 -(μ+λ)〈∇4 div uh,∇6ϱh〉-〈∇4 div (ϱu)h,∇6uh〉+〈∇4G2h,∇5ϱh〉.

Using the Young inequality, we can obtain

(2.46) |μ〈∇5uh,∇6ϱh〉|≤μ2‖∇5uh‖L22+14‖∇6ϱh‖L22.

Similarly, we have

(2.47) |(μ+λ)〈∇4 div uh,∇6ϱh〉|≤(μ+λ)2‖∇4 div uh‖L22+14‖∇6ϱh‖L22.

For 〈∇⁴div(ϱu)^h, ∇⁵ ϱ^h〉, we obtain

(2.48) |〈∇4 div (ϱu)h,∇5uh〉|≲(1+t)-112+(1+t)-32‖∇5uh‖L22.

We notice that

(2.49) 〈∇4G2h,∇5ϱh〉=〈-∇4(u⋅∇u)h,∇5ϱh〉+〈∇4{ϱ(ϱ+1)[μΔu+(μ+λ)∇ div u]}h,∇5ϱh〉+〈-∇4{[P′(ϱ+1)ϱ+1-P′(1)]∇ϱ}h,∇5ϱh〉+〈-∇4(ϑ24ϱϱ+1∇Δϱ)h,∇5ϱh〉+〈∇4[ϑ24(|∇ϱ|2∇ϱ(1+ϱ)3-∇ϱΔϱ(1+ϱ)2-∇ϱ⋅∇2ϱ(1+ϱ)2)]h,∇5ϱh〉+〈∇4[11+ϱ((∇×B)×B)]h,∇5ϱh〉:=∑j=16Jj.

The right-hand side of the above equation can be estimated as follows:

(2.50) |J1|≲‖∇3(∇u⋅u)h‖L2‖∇6ϱh‖L2≲‖∇3(∇u⋅u)‖L2‖∇6ϱh‖L2≲(‖u‖L∞‖∇4u‖L2+‖∇3u‖L6‖∇u‖L3)‖∇6ϱh‖L2≲‖∇u‖L212‖∇2u‖L212‖∇4u‖L2‖∇6ϱh‖L2≲(1+t)-154‖∇6ϱh‖L2≲(1+t)-112+(1+t)-2‖∇6ϱh‖L22.

Similar to the proof of (2.38), we have

(2.51) |J2|≲(1+t)-112+(1+t)-32(‖∇6ϱh‖L22+‖∇5uh‖L22).

For J₃, we get

(2.52) |J3|≲‖∇3{[P′(ϱ+1)ϱ+1-P′(1)]∇ϱ}h‖L2‖∇6ϱh‖L2≲‖∇3{[P′(ϱ+1)ϱ+1-P′(1)]∇ϱ}‖L2‖∇6ϱh‖L2≲(‖ϱ‖L∞‖∇4ϱ‖L2+‖∇ϱ‖L3‖∇3ϱ‖L6)‖∇6ϱh‖L2≲‖∇ϱ‖L212‖∇2ϱ‖L212‖∇4ϱ‖L2‖∇6ϱh‖L2≲(1+t)-154‖∇6ϱh‖L2≲(1+t)-112+(1+t)-2‖∇6ϱh‖L22.

For J₄, we can rewrite it as follows

(2.53) J4=ϑ24〈∇3(ϱϱ+1∇Δϱ)h,∇6ϱh〉=ϑ24〈∇3(ϱϱ+1∇Δϱ)-∇3(ϱϱ+1∇Δϱ)l,∇6ϱh〉=ϑ24〈∇3(ϱϱ+1∇Δϱh)+∇3(ϱϱ+1∇Δϱl)-∇3(ϱϱ+1∇Δϱ)l,∇6ϱh〉=J4,1+J4,2+J4,3.

For J_4,1, we obtain

(2.54) |J4,1|≲‖∇3(ϱϱ+1∇Δϱh)‖L2‖∇6ϱh‖L2≲(‖ϱ‖L∞‖∇6ϱh‖L2+‖∇3ϱh‖L6‖∇3ϱ‖L3)‖∇6ϱh‖L2≲(‖∇ϱ‖L212‖∇2ϱ‖L212‖∇6ϱh‖L2+‖∇3ϱ‖L212‖∇4ϱ‖L212‖∇4ϱh‖L2)‖∇6ϱh‖L2≲(1+t)-32‖∇6ϱh‖L2+(1+t)-92‖∇6ϱh‖L2≲(1+t)-112+(1+t)-32‖∇6ϱh‖L22.

For J_4,2, we get

(2.55) |J4,2|≲‖∇3(ϱϱ+1∇Δϱl)‖L2‖∇6ϱh‖L2≲(‖ϱ‖L∞‖∇6ϱl‖L2+‖∇3ϱ‖L6‖∇3ϱl‖L3)‖∇6ϱh‖L2≲(‖∇ϱ‖L212‖∇2ϱ‖L212‖∇5ϱ‖L2+‖∇3ϱl‖L212‖∇4ϱl‖L212‖∇4ϱ‖L2)‖∇6ϱh‖L2≲+(1+t)-154‖∇6ϱh‖L2≲(1+t)-112+(1+t)-2‖∇6ϱh‖L22.

For J_4,3, we get

(2.56) |J4,3|≲‖∇3(ϱϱ+1∇Δϱ)l‖L2‖∇6ϱh‖L2≲‖∇2(ϱϱ+1∇Δϱ)‖L2‖∇6ϱh‖L2≲(‖ϱ‖L∞‖∇5ϱ‖L2+‖∇3ϱ‖L6‖∇2ϱ‖L3)‖∇6ϱh‖L2≲(‖∇ϱ‖L212‖∇2ϱ‖L212‖∇5ϱ‖L2+‖∇2ϱ‖L212‖∇3ϱ‖L212‖∇4ϱ‖L2)‖∇6ϱh‖L2≲(1+t)-154‖∇6ϱh‖L2+(1+t)-174‖∇6ϱh‖L2≲(1+t)-112+(1+t)-2‖∇6ϱh‖L22.

The combination of (2.53)–(2.56) give rises to

(2.57) |J4|≲(1+t)-112+(1+t)-32‖∇6ϱh‖L22,

(2.58) |J5|≲‖∇3[ϑ24(|∇ϱ|2∇ϱ(1+ϱ)3-∇ϱΔϱ(1+ϱ)2-∇ϱ⋅∇2ϱ(1+ϱ)2)]h‖L2‖∇6ϱh‖L2≲‖∇3[ϑ24(|∇ϱ|2∇ϱ(1+ϱ)3-∇ϱΔϱ(1+ϱ)2-∇ϱ⋅∇2ϱ(1+ϱ)2)]‖L2‖∇6ϱh‖L2≲(‖∇ϱ‖L∞‖∇5ϱ‖L2+‖∇4ϱ‖L6‖∇2ϱ‖L3)‖∇6ϱh‖L2≲‖∇2ϱ‖L212‖∇3ϱ‖L212‖∇5ϱ‖L2‖∇6ϱh‖L2≲(1+t)-174‖∇6ϱh‖L2≲(1+t)-112+(1+t)-3‖∇6ϱh‖L22,

and

(2.59) |J6|≲‖∇3[11+ϱ((∇×B)×B)]h‖L2‖∇6ϱh‖L2≲‖∇3[11+ϱ((∇×B)×B)]‖L2‖∇6ϱh‖L2≲(‖B‖L∞‖∇4B‖L2+‖∇B‖L3‖∇3B‖L6)‖∇6ϱh‖L2≲‖∇B‖L212‖∇2B‖L212‖∇4B‖L2‖∇6ϱh‖L2≲(1+t)-154‖∇6ϱh‖L2≲(1+t)-112+(1+t)-2‖∇6ϱh‖L22.

It follows from the estimates (2.50)–(2.52) and (2.57)–(2.59) that

(2.60) |〈∇3G2h,∇4ϱh〉|≲(1+t)-112+(1+t)-32(‖∇6ϱh‖L22+‖∇5uh‖L22),

which together with (2.46)–(2.48) implies immediately

(2.61) ddt∫ℝ3∇4uh∇5ϱhdx+12‖∇5ϱh‖L22+ϑ24‖∇6ϱh‖L22≲(1+t)-112+‖∇5uh‖L22+(1+t)-32(‖∇6ϱh‖L22+‖∇5uh‖L22).

Last, we will close the estimate to prove the decay rate (1.8). Choosing sufficient large time T₁ and positive constant D₀, we define the temporary energy functional

(2.62) ℰ(t)=D0(‖∇4(ϱ,u,B)h‖L22+ϑ22‖∇5ϱh‖L22)+∫ℝ3∇4uh∇5ϱhdx,

for t ≥ 0, where it is clear that ℰ(t) is equivalent to ‖∇4ϱh‖H12+‖∇4(u,B)h‖L22 since D₀ is large enough.

Combining the estimates (2.44) with (2.61) and using Lemma 2.3, for t ≥ T₁, one may deduce that

(2.63) ddtℰ(t)+‖∇4ϱh(t)‖H12+‖∇4uh(t)‖L22+‖∇4Bh(t)‖L22≲(1+t)-112,

where we have used the fact that T₁ is large enough. On the other hand, it is easy to see that

(2.64) ‖∇4ϱh(t)‖H12+‖∇4uh(t)‖L22+‖∇4Bh(t)‖L22≥Cℰ(t).

Therefore, (1.8) follows from (2.63), (2.64), Lemma 2.3 and Gronwall's argument immediately.

3 Proof of Theorem 1.3

This section is concerned with the optimal convergence rate of the solution, which is stated on Theorem 1.3. First, we will derive optimal convergence rate on ‖∇⁵ ϱ^l‖_L².

Step 1. Low–frequency L² energy estimate. By virtue of the equation (2.1)₁, Theorem 1.2, Duhamel’s principle, Plancherel theorem, Hölder’s inequality and Hausdorff–Young’s inequality, we have

(3.1) ∥∇5ϱl(t)∥L2≲ (1+t)-134∥(ϱ,u,B)(0)∥L1+∫0t2(1+t-τ)-134∥G(τ)∥L1dτ+∫t2t(1+t-τ)-54∥|ξ|4G^l(τ)∥L∞dτ.

For ‖G(τ)‖_L¹, we can see the proof of (2.8). On the other hand, we can use (1.8) to bound the third term on the right-hand side of (3.1) as follows

(3.2) ‖|ξ|4G^l(τ)‖L∞≲‖∇4Gl(τ)‖L1≲‖∇3( div (ϱu))(τ)‖L1+‖∇3(u⋅∇u)(τ)‖L1 +‖∇2(ϱϱ+1[μΔu+(μ+λ)∇ div u])(τ)‖L1+‖∇3([P′(ϱ+1)ϱ+1-P′(1)]∇ϱ)(τ)‖L1 +‖∇2(ϑ24ϱϱ+1∇Δϱ)(τ)‖L1+‖∇3(ϑ24(|∇ϱ|2∇ϱ(1+ϱ)3-∇ϱΔϱ(1+ϱ)2-∇ϱ⋅∇2ϱ(1+ϱ)2))(τ)‖L1 +‖∇3(11+ϱ((∇×B)×B))(τ)‖L1+‖∇3(∇×(u×B)(τ))‖L1≲‖(ϱ,u)(τ)‖L2‖∇4(ϱ,u)(τ)‖L2+‖∇2u(τ)‖L22+‖ϱ(τ)‖L2‖∇5ϱ(τ)‖L2 +‖∇ϱ(τ)‖L2‖∇5ϱ(τ)‖L2+‖∇ϱ(τ)‖L2‖∇4ϱ(τ)‖L2+‖∇2ϱ(τ)‖L2‖∇3ϱ(τ)‖L2 +‖∇B(τ)‖L2‖∇3B(τ)‖L2+‖(u,B)(τ)‖L2‖∇4(u,B)(τ)‖L2≲(1+t)-72.

Substituting (3.2) into (3.1), it arrives at

(3.3) ‖∇5ϱl‖L2≲(1+t)-134.

Step 2. High–frequency L² energy estimate. In this part, we will make full use of (1.8) to achieve this goal. It is not difficult to find that the following calculations are consistent with the second section except replacing the L² decay rate (1+t)-94 by (1+t)-114 for the fourth order spatial derivative of solution. Similar to the proof of (2.48), we have

(3.4) ddtℰ(t)+‖∇4ϱh(t)‖H12+‖∇4uh(t)‖L22+‖∇4Bh(t)‖L22≲(1+t)-132.

Finally, we conclude that

(3.5) ‖∇4ϱh(t)‖H1+‖∇4uh(t)‖L2+‖∇4Bh(t)‖L2≲(1+t)-134,

which together with (3.3) implies (1.9) directly.

4 Analytic tools

Now, we state some auxiliary lemmas, which will be frequently used in this paper.

Lemma 4.1

(Gagliardo-Nirenberg’s inequality) Given 2 ≤ p ≤ + ∞ and 0 ≤ k, m ≤ ℓ, then for any f ∈ H^ℓ(ℝ³), we have

∥∇kf∥Lp≲∥∇mf∥L2α∥∇𝓁f∥L21-α,

where α ∈ [0, 1] satisfies

k3-1p=(m3-12)α+(𝓁3-12)(1-α).

Proof

This is the special case of [23].

Lemma 4.2

Let k ≥ 1 be an integer, then it holds that

‖∇k(fg)‖Lp≲‖f‖Lp1‖∇kg‖Lp2+‖g‖Lp3‖∇kf‖Lp4,

where p, p₂, p₃ ∈ (1, +∞) and

1p=1p1+1p2=1p3+1p4.

Proof

For p = p₂ = p₃ = 2, it can be proved by using Lemma 4.1. For the general case, one may refer to [12]

Lemma 4.3

If f ∈ L^r (ℝ³) for any 2 ≤ r ≤ ∞, then we have

∥fl∥Lr+∥fh∥Lr≲∥f∥Lr.

Proof

For 2 ≤ r ≤ ∞, by Young inequality’s for convolutions, for the low frequency, it holds

∥fl∥Lr≲∥𝔉-1ϕ∥L1∥f∥Lr≲∥f∥Lr,

and hence

∥fh∥Lr≲∥f∥Lr+∥fl∥Lr≲∥f∥Lr.

Lemma 4.4

Let r₁, r₂ > 0, then it holds that

∫0t(1+t-s)-r1(1+s)-r2ds≤C(1+t)-min{r1,r2,r1+r2-1-∫},

for any arbitrarily small ε > 0.

Acknowledgments

This work is partially supported by Guangxi Natural Science Foundation #2019JJG110003, #2019AC20214, and National Natural Science Foundation of China #11771150.

Conflict of interest: Authors state no conflict of interest.

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Received: 2021-09-24

Accepted: 2021-12-19

Published Online: 2022-02-09

This work is licensed under the Creative Commons Attribution 4.0 International License.

Optimal decay rate for higher–order derivatives of solution to the 3D compressible quantum magnetohydrodynamic model

Abstract

1 Introduction

1.1 History of the problem

1.2 Notation

1.3 Main results

Theorem 1.1

Theorem 1.2

Theorem 1.3

Remark 1.4

Remark 1.5

2 Proof of Theorem 1.2

Lemma 2.1

Lemma 2.2

Lemma 2.3

Proof

Proof of Theorem 1.2

3 Proof of Theorem 1.3

4 Analytic tools

Lemma 4.1

Proof

Lemma 4.2

Proof

Lemma 4.3

Proof

Lemma 4.4

Acknowledgments

References

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