Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 41, pp. 1–29. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: https://doi.org/10.58997/ejde.2023.41 SPACE-TIME DECAY RATES OF A TWO-PHASE FLOW MODEL WITH MAGNETIC FIELD IN R3 QIN YE, YINGHUI ZHANG Abstract. We investigate the space-time decay rates of strong solution to a two-phase flow model with magnetic field in the whole space R3. Based on the temporal decay results by Xiao [24] we show that for any integer ` ≥ 3, the space-time decay rate of k(0 ≤ k ≤ `)-order spatial derivative of the strong solution in the weighted Lebesgue space L2 γ is t− 3 4 − k 2 +γ . Moreover, we prove that the space-time decay rate of k(0 ≤ k ≤ `−2)-order spatial derivative of the difference between two velocities of the fluid in the weighted Lebesgue space L2 γ is t− 5 4 − k 2 +γ , which is faster than ones of the two velocities themselves. 1. Introduction and main results In this article, we study the space-time decay rates of strong solutions to the compressible isothermal Euler equations coupled with compressible magnetohydro- dynamic (MHD) system through a drag forcing term in the whole space R3. The coupled system models the motions of particles immersed in the electrically con- ducting fluid with the effect of magnetic field. The system takes the following form ρt + div(ρu) = 0, (ρu)t + div(ρu⊗ u) +∇ρ = −ρ(u− v), nt + div(nv) = 0, (nv)t + div(nv ⊗ v) +∇P (n)− µ∆v − (µ+ λ)∇ div v = ρ(u− v) + (∇×B)×B, Bt −∇× (v ×B) = −ν∇× (∇×B), divB = 0, (1.1) where (x, t) ∈ R3 × R+ is the spatial coordinate and time. Let ρ(x, t) and n(x, t) denote the densities of fluid and let u(x, t) and v(x, t) be the corresponding velocities of ρ(x, t) and n(x, t) respectively. P = P (n) = Ana(A > 0, a ≥ 1) represents the pressure. µ and λ are the shear viscosity and the bulk viscosity coefficients of the 2020 Mathematics Subject Classification. 35Q31, 35K65, 76N10. Key words and phrases. Compressible Euler equations; Two-phase flow model; Space-time decay rate; Weighted Sobolev space. ©2023. This work is licensed under a CC BY 4.0 license. Submitted September 21, 2022. Published June 23, 2023. 1 2 Q. YE, Y. ZHANG EJDE-2023/41 fluid satisfying the following physical restrictions µ > 0, 2 3 µ+ λ ≥ 0. Here B represents magnetic field and ν denotes the coefficient of magnetic diffusivity acting as a magnetic diffusion. We consider the coupled system (1.1) with initial data (ρ, u, n, v,B)|t=0 = (ρ0(x), u0(x), n0(x), v0(x), B0(x)) , x ∈ R3, (1.2) satisfying (ρ0(x), u0(x), n0(x), v0(x), B0(x))→ (ρ̄, 0, n̄, 0, 0) as ‖x| → ∞, where the positive constants ρ̄ and n̄ are the reference densities. 1.1. History of the problem. When the magnetic field is not taken into account (B = 0) in (1.1), system (1.1) reduces to the two-phase fluid model ρt + div(ρu) = 0, (ρu)t + div(ρu⊗ u) +∇ρ = −ρ(u− v), nt + div(nv) = 0, (nv)t + div(nv ⊗ v) +∇P (n)− µ∆v − (µ+ λ)∇divv = ρ(u− v). (1.3) We notice that Choi [4] firstly addressed the formal derivation of the coupled hy- drodynamic system (1.3) from kinetic-fluid equations, which is a type of Vlasov- Fokker-Planck/compressible Navier-Stokes equations. When the magnetic field is taken into account (B 6= 0) as in (1.1), its derivation is slightly different from (1.3). Denote the distribution of particles at the position-velocity (x, ω) ∈ R3×R3 and at time t ∈ R+ by f(x, ω, t), the isentropic compressible fluid density and velocity by n(x, t) and v(x, t) respectively. We intend to study the following kinetic-fluid equa- tions with local alignment and noise forces for the particles to model the motions of particles immersed in the compressible and electrically conducting fluid with the effect of magnetic field: ft + ω · ∇xf +∇ω · ((v − ω)f) = −α∇ω · ((uf − ω)f) + σ∆ωf, nt +∇x · (nv) = 0, (nv)t +∇x · (nv ⊗ v) +∇xP (n)− µ∆xv − (µ+ λ)∇x∇x · v = (∇×B)×B + ∫ R3 (ω − v)fdω, Bt −∇× (v ×B) = −ν∇× (∇×B), (1.4) where uf is the averaged local velocity defined by uf (x, t) := ∫ R3 ωf(x, ω, t)dω∫ R3 f(x, ω, t)dω . (1.5) We consider a regime where the local alignment and noise forces are strong, i.e., α = σ = ε−1. Let (fε, nε, vε, Bε) be the solution to the system (1.4) with α = σ = ε−1. It follows from (1.4)1 that −∇ω · ((ufε − ω)fε) + ∆ωf ε = ε(fεt + ω · ∇xfε +∇ω · ((vε − ω)fε))→ 0 EJDE-2023/41 SPACE-TIME DECAY RATES 3 as ε→ 0. So fε(x, ω, t) is expected to converge to f(x, ω, t) = ρf (x, t) (2π)3/2 e− |uf (x,t)−ω|2 2 , (1.6) where ρf (x, t) = ∫ R3 f(x, ω, t)dω. (1.7) Integrating (1.4)1 with respect to ω over R3, if ρfε → ρf and ufε → uf as ε → 0 then we obtain the continuity equation (1.1)1. According to the (1.5)-(1.7), we have∫ R3 (ufε − ω)fεdω = 0. (1.8) Multiplying (1.4)1 by ω, integrating the resulting equation with respect to ω over R3 and combining (1.8), we have d dt ∫ R3 ωfεdω = ∫ R3 ω(−∇x · (ωfε)−∇ω · ((vε − ω)fε))dω − 1 ε ∫ R3 ω(∇ω · ((ufε − ω)fε)−∆ωf ε)dω = −∇x · ∫ R3 ω ⊗ ωfεdω + ∫ R3 (vε − ω)fεdω := Iε1 + Iε2 . (1.9) Substituting (1.5) and (1.7) into (1.9), one has Iε1 = −∇x · (∫ R3 (ω − ufε)⊗ (ω − ufε)fεdω − ∫ R3 ufε ⊗ ufεfεdω + ∫ R3 ω ⊗ ufεfεdω + ∫ R3 ufε ⊗ ωfεdω ) and Iε2 = ρfε(v ε − ufε). Using (1.5)-(1.7) and assuming the appropriate convergence of solutions, we can deduce that d dt ∫ R3 ωfεdω → d dt ∫ R3 ωfdω = d dt (ρfuf ), and Iε1 → I1 = −∇x · (∫ R3 (ω − uf )⊗ (ω − uf ) ρf (2π)3/2 e− |uf−ω| 2 2 dω − ∫ R3 uf ⊗ uf ρf (2π)3/2 e− |uf−ω| 2 2 dω + ∫ R3 ω ⊗ uffdω + ∫ R3 uf ⊗ ωfdω ) = −∇x · ( ρf I3 − ρfuf ⊗ uf + ρfuf ⊗ uf + ρfuf ⊗ uf ) = −∇xρf −∇x · (ρfuf ⊗ uf ), where I3 = ∫ R3 (ω − uf )⊗ (ω − uf ) 1 (2π)3/2 e− |uf−ω| 2 2 dω,∫ R3 1 (2π)3/2 e− |uf−ω| 2 2 dω = 1. 4 Q. YE, Y. ZHANG EJDE-2023/41 By (1.5) and (1.6), we have Iε2 → I2 = ρf (v − uf ), which implies (1.1)2. Finally, as ε→ 0, we can obtain (1.1)3-(1.1)5. System (1.4) models the interactions between particles and a fluid. This type of the kinetic-fluid system has attracted a lot of attention because of applications in biotechnology, medicine, sedimentation phenomena, compressibility of droplets of a spray, diesel engines, etc. We can refer [2, 16] for more physical background. Kinetic theory in the mathematical study of nonlinear partial differential equa- tions has attracted considerable attention in the last few decades. There is much progress on the topics of the kinetic-fluid equations and related models. For system (1.4) without magnetic field and local alignment force (i.e., B = 0 and α = 0), the existence of weak global solution was established by Mellet and Vasseur [15]. Baranger and Desvillettes [1] studied the local-in-time existence of classical solution for the Vlasov/compressible Euler equations. For the existence of global classical solution near equilibrium to Vlasov-Fokker-Planck/Euler equation, Duan and Liu discussed it in [6]. We also refer the readers to [5, 10] for the study of Vlasov- Fokker-Planck equations with the local alignment force. For the three-dimensional compressible MHD system, if the initial density has a uniform positive lower bound, Vol’pert and Hudjaev investigated the local well- posedness of the Cauchy problem in [20]. The result was extended by Fan and Yu in [8], where the initial density does not need to be positive and may vanish in an open set. When the initial data are discontinuous and have large oscillations, Wu, Zhang and Zou [23] showed the optimal time-decay rates of the weak solutions in Lr(2 ≤ r ≤ ∞)-norm and the first-order derivative of the velocity and magnetic field in L2-norm. We also refer the interested readers to study the multidimensional case in [25], when the initial data are close to a stable equilibrium state in Besov spaces, the authors established the existence and uniqueness of a global strong solution. When the magnetic field is not involved (B = 0) in (1.1), system (1.1) is reduced to (1.3). When the initial data are small in H`-norm, Choi established the existence of global strong solutions for both the periodic domain T3 and the whole space R3 in [4]. The author also obtained the large-time behavior of strong solutions for the periodic domain case, but the strategy in [4] can not be applied to the whole space case, as the Poincaré’s inequality is not available for the whole space case. Recently, Wu, Zhang, and Zou [22] solved this problem in a perturbation framework. They proved that the perturbation and its k-order derivative decay in L2-norm are t−3/4 and t− 3 4− k 2 in the whole space R3 respectively. They also showed that the time decay rate is optimal, which coincides with that of the heat equation. We also refer [26] for the existence and large-time behavior of the system (1.3). For the two-fluid system with magnetic field (1.1), Xiao obtained the existence and large time behavior of global strong solution for the 3D Cauchy problem in [24]. His main results are as follows. Let (ρ− ρ̄, u, n− n̄, v, B) be the strong solution to equations (1.1)-(1.2) and assume that (ρ0 − ρ̄, u0, n0 − n̄, v0, B0) ∈ H` ( R3 ) for an integer ` ≥ 3. Then there exists a constant δ0 > 0 such that if ‖ ( ρ0 − ρ̄, u0, n0 − n̄, v0, B0 ) ‖H` ≤ δ0, (1.10) EJDE-2023/41 SPACE-TIME DECAY RATES 5 then the Cauchy problem (1.1)-(1.2) admits a unique globally classical solution (ρ, u, n, v,B) such that ‖(ρ− ρ̄, u, n− n̄, v, B)(t)‖2H` + ∫ t 0 ( ‖∇(ρ− ρ̄, u, n− n̄, B)(τ)‖2H`−1 + ‖(u− v,∇v)(τ)‖2H` ) dτ ≤ C‖ (ρ0 − ρ̄, u0, n0 − n̄, v0, B0) ‖2H` , t ≥ 0. Upper bounds: If additionally, (1.10) holds for a small constant δ0 > 0 and ‖ (ρ0 − ρ̄, u0, n0 − n̄, v0, B0) ‖L1 <∞, then ‖∇k(ρ− ρ̄, u, n− n̄, v, B)(t)‖L2 ≤ C(1 + t)− 3 4− k 2 , (1.11) ‖(ρ− ρ̄, u, n− n̄, v, B)(t)‖Lp ≤ C(1 + t)− 3 2 (1− 1 p ), (1.12) for t is large enough, 0 ≤ k ≤ ` and 2 ≤ p ≤ ∞, where C is a positive constant independent of t. The author also proved the lower bound optimal decay rates. The space-time decay rate of the strong solution has attracted more and more attention. In the following, we will state the progress on the topic about the space- time decay in the weighted Sobolev space H` γ . Takahashi first established the space- time decay of strong solutions to the Navier-Stokes equations in [18]. In [11, 13], Kukavica et al. used the parabolic interpolation inequality to obtain the sharp decay rates of the higher-order derivatives for the solutions in the weighted Lebesgue space L2 γ . In [12, 14], Kukavica et al. also established the strong solution’s space-time decay rate in Lpγ(2 ≤ p ≤ ∞) and extended the result to n(n ≥ 2) dimensions. Using the Fourier splitting method, Gao, Lyu and Yao obtained space-time decay rate for the compressible Hall-MHD equations in [9]. However, to the best of our knowledge, up to now, there is no result on the space- time decay rate of the two-fluid system with the magnetic field (1.1). The main motivation of this paper is to give a definite answer to this issue. More precisely, we establish the space-time decay rates of the k(0 ≤ k ≤ `)-order derivative of strong solution to the Cauchy problem (1.1)-(1.2) in the weighted Lebesgue space L2 γ and the sharp space-time decay rate of k(0 ≤ ` − 2)-order derivative of the difference between two velocities of the fluid in the weighted Lebesgue space L2 γ . 1.2. Notation. We use Lp and H` to denote the usual Lebesgue space Lp ( R3 ) and Sobolev spaces H` ( R3 ) = W `,2 ( R3 ) with norms ‖·‖Lp and ‖·‖H` respectively. We denote ‖(f, g)‖X := ‖f‖X + ‖g‖X for simplicity. The notation f . g means that f ≤ Cg for a generic positive constant C > 0 that only depends on the parameters coming from the problem. We often drop x-dependence of differential operators, that is ∇f = ∇xf = (∂x1 f, ∂x2 f, ∂x3 f) and ∇k denotes any partial derivative ∂α with multi-index α, |α| = k. For any γ ∈ R, denote the weighted Lebesgue space by Lpγ ( R3 ) (2 ≤ p < +∞) with respect to the spatial variables: Lpγ(R3) := { f(x) : R3 → R, ‖f‖p Lpγ(R3) := ∫ R3 |x|pγ |f(x)|p dx < +∞ } . Then, we can define the weighted Sobolev space: Hs γ ( R3 ) , { f ∈ L2 γ ( R3 ) | ‖f‖2Hsγ(R3) := ∑ k≤s ‖∇ku‖2L2 γ(R3) < +∞ } . 6 Q. YE, Y. ZHANG EJDE-2023/41 Let Λs be the pseudo differential operator defined by Λsf = F−1 ( |ξ|sf̂ ) , for s ∈ R, where f̂ and F(f) are the Fourier transform of f . The homogenous Sobolev space Ḣs ( R3 ) with norm given by ‖f‖Ḣs , ‖Λ sf‖L2 . The function f is in Schwartz class S, if it is infinitely differentiable and if all of its derivatives decrease rapidly at infinity. That is, sup x |xαDβf(x)| <∞, for all α, β ∈ N3. 1.3. Main results. Inspired by the work of [21], we investigate the space-time decay rates of strong solution in the weighted Lebesgue space L2 γ as follows. Theorem 1.1. Let (ρ− ρ̄, u, n− n̄, v, B) be the strong solution to the equations (1.1)-(1.2) with initial data (ρ0 − ρ̄, u0, n0 − n̄, v0, B0) belonging to the Schwartz class S. In addition, for any integer ` ≥ 3, assume that (ρ0 − ρ̄, u0, n0 − n̄, v0, B0) ∈ H` ( R3 ) ∩H` γ ( R3 ) ∩ L1 ( R3 ) . If there exists a small constant δ0 > 0 such that ‖ (ρ0 − ρ̄, u0, n0 − n̄, v0, B0) ‖H` ≤ δ0, then there exists a large enough T such that ‖∇k(ρ− ρ̄, u, n− n̄, v, B)(t)‖L2 γ ≤ Ct− 3 4− k 2+γ , (1.13) for all t > T , 0 ≤ k ≤ ` and γ ≥ 0, where C is a positive constant independent of t. Remark 1.2. Applying Gagliardo-Nirenberg-Sobolev inequality, we can obtain the space-time decay rates of smooth solution in the weighted Lebesgue space Lpγ as follows. For any f ∈ L2 ( R3 ) ∩ Ḣ2 ( R3 ) , we have ‖f‖L∞(R3) ≤ ‖f‖ 1/4 L2(R3)‖f‖ 3 4 H2(R3). So we can obtain the estimate ‖|x|γ∇k(ρ − ρ̄, u, n − n̄, v, B)(t)‖L∞(k ∈ [0, ` − 2]) from the estimates ‖|x|γ∇k(ρ − ρ̄, u, n − n̄, v, B)(t)‖L2 and ‖|x|γ∇k(ρ − ρ̄, u, n − n̄, v, B)(t)‖Ḣ2 . Using the interpolation inequality, we can show that there exists a large enough T such that ‖∇k(ρ− ρ̄, u, n− n̄, v, B)(t)‖Lpγ ≤ Ct − 3 2 (1− 1 p )− k2+γ , for t > T , 2 ≤ p ≤ ∞ and 0 ≤ k ≤ `−2, where C is a positive constant independent of t. Remark 1.3. Under the assumptions of Theorem 1.1, applying the similar method as [26], we can easily show that there exists a large enough T such that ‖∇k(u− v)(t)‖L2 ≤ C(1 + t)− 5 4− k 2 , (1.14) for all t > T and 0 ≤ k ≤ `− 2, where C is a positive constant independent of t. Theorem 1.4. Let (ρ− ρ̄, u, n− n̄, v, B) be the strong solution to the system (1.1)- (1.2) with initial data (ρ0 − ρ̄, u0, n0 − n̄, v0, B0) belonging to the Schwartz class S. Under the assumptions in Theorem 1.1, then there exists a large enough T such that ‖∇k(u− v)(t)‖L2 γ ≤ C(1 + t)− 5 4− k 2+γ , (1.15) EJDE-2023/41 SPACE-TIME DECAY RATES 7 for all t > T , 0 ≤ k ≤ `− 2 and γ ≥ 0, where C is a positive constant independent of t. Now, let us outline the strategies for proving Theorem 1.1 and 1.4, and explain the main difficulties in the process. Proof of Theorem 1.1. We employ delicate weighted energy estimates, the strategy of induction and interpolation trick. Firstly, using of several lemmas in Section 2 and (1.11), we obtain d dt E(t) ≤ C0t −5/4E(t) + C1t − 3 4γ E(t) 2γ−1 2γ + C2t − 3 2γ E(t) γ−1 γ , (1.16) for t is large enough and γ > 3 2 , where E(t) := ‖(m,u, σ, v,B)‖2L2 γ and C0, C1, C2 are positive constants independent of t. Applying Lemma 2.5 for (1.16) and the interpolation trick, we show that the Theorem 1.1 holds for case k = 0. Secondly, Using the similar method as k = 0 and Minkowski’s inequality, we can show that the Theorem 1.1 holds for k = 1. Finally, according to the strategy of induction, we prove that Theorem 1.1 holds for 0 ≤ k ≤ `. The main difficulties come from those terms like α2n̄ ∫ R3 |x|2γ∇k(u · ∇m) · ∇km dx and n̄ ∫ R3 |x|2γ∇k ( (α1 − P ′(n) n )∇σ ) · ∇kv dx, which contain three difficulties. The first difficulty is the absence of the dissipation ‖∇k+1m‖2L2 γ . We use integration by parts to overcome this difficulty (see (3.38)). The second difficulty is that we can only obtain the decay rate of ‖∇`−1u‖H1 rather than the decay rate of ‖∇`u‖H1 , we make different estimates for∑k−2 j=2 ‖|x|2γ∇ju∇k−j+1m∇km‖L1 and ‖|x|2γ∇k−1u∇2m∇km‖L1 (see (3.40) and (3.41)). The last difficulty is that the Lemma 2.2 does not hold in the weighted Lebesgue space L2 γ , so we have to estimate ∇j ( (α1 − P ′(n) n )∇σ ) in Lp-norm (see (3.49)-(3.52)). Overcoming all of these difficulties, applying some lemmas in Section 2 and (1.11), we have d dt E(t) ≤ C0t −5/4E(t) + C1t −( 3 4+ k 2 ) 1 γ E(t) 2γ−1 2γ + C2t −( 3 4+ k 2 ) 2 γ E(t) γ−1 γ + C3t − 5 2−k+2γ , (1.17) where E(t) := ‖∇k(m,u, σ, v,B)‖2L2 γ and C0, C1, C2, C3 are positive constants in- dependent of t. Using the Gronwall-type Lemma 2.5 for (1.17), we prove that Theorem 1.1 is true for γ > 3+2k 2 . Applying the interpolation trick, we complete the proof of Theorem 1.1. � Proof of Theorem 1.4. We make full use of energy estimates, the result of Theorem 1.1 and (1.11) to obtain d dt ‖∇k(u− v)‖2L2 γ + C ′‖∇k(u− v)‖2L2 γ . (1 + t)− 5 2−k+2γ , (1.18) for t is large enough, where C ′ is a positive constant independent of t. We note that there exist terms like −µ̄〈∇k∆v, |x|2γ∇k(u − v)〉 in (4.2) so 0 ≤ k ≤ ` − 2 in Theorem 1.4. Using the Gronwall’s inequality of differential form to (1.18), we complete the proof. � The paper will be organized as follows. In section 2, we rewrite the Cauchy problem (1.1)-(1.2) and present some lemmas, which are used frequently throughout 8 Q. YE, Y. ZHANG EJDE-2023/41 this paper. In section 3, using the strategy of induction, we prove the Theorem 1.1. In section 4, applying the energy estimate, we prove the Theorem 1.4. 2. Reformulation and preliminaries 2.1. Reformulation. In this section, we reformulate the Cauchy problem (1.1)- (1.2). We denote m = ln ρ− ln ρ̄, σ = n− n̄, α1 = P ′(n̄) n̄ , α2 = ρ̄ n̄ , µ̄ = µ n̄ , λ̄ = λ n̄ . Then system (1.1)-(1.2) can be rewritten as mt + div u = F1, ut +∇m+ (u− v) = F2, σt + n̄div v = F3, vt + α1∇σ − µ̄∆v − (µ̄+ λ̄)∇div v − α2(u− v) = F4, Bt − ν∆B = F5, divB = 0, (2.1) and (m,u, σ, v,B) ∣∣ t=0 = ( m0(x), u0(x), σ0(x), v0(x), B0(x) ) , (2.2) where F1 = −u · ∇m, F2 = −u · ∇u, F3 = −v · ∇σ − σ div v, F4 = −v · ∇v + ( α1 − P ′(n) n ) ∇σ + (µ n − µ̄ ) ∆v + (µ+ λ n − (µ̄+ λ̄) ) ∇ div v + ( ρ n − α2 ) (u− v) + (divB)B +B · ∇B − 1 2 ∇|B|2, F5 = −(div v)B +B · ∇v − v · ∇B. (2.3) It is noted that in the homogeneous case (i.e. Fi = 0 for i = 1, 2, . . . , 5, (2.1)5 is decoupled from (2.1)1-(2.1)4. This key observation enables us to employ the sharp linear estimates for (m,u, σ, v) obtained in [26] and estimates for B obtained in [27]. We can refer [24] for more details. Here are several useful tools, which will be frequently used in the whole article. Lemma 2.1 (Gagliardo-Nirenberg inequality). Let 0 ≤ i, j ≤ k, then ‖∇if‖Lp . ‖∇jf‖1−aLq ‖∇ kf‖aLr , where a satisfies i 3 − 1 p = ( j 3 − 1 q ) (1− a) + (k 3 − 1 r ) a. Especially, when p = 3, q = r = 2, i = j = 0, k = 1, combining Cauchy’s inequality, we have ‖f‖L3 . ‖f‖1/2L2 ‖∇f‖1/2L2 . ‖f‖H1 , (2.4) when p =∞, q = r = 2, i = 0, j = 1, k = 2, combining Cauchy’s inequality, we have ‖f‖L∞ . ‖∇f‖1/2L2 ‖∇2f‖1/2L2 . ‖∇f‖H1 , (2.5) while i = j = 0, k = 1, a = 1, p = q = r = 2 and using Minkowski’s inequality, we have ‖f‖L6 γ . ‖∇ (|x|γf) ‖L2 . ( ‖∇f‖L2 γ + ‖f‖L2 γ−1 ) . (2.6) EJDE-2023/41 SPACE-TIME DECAY RATES 9 Proof. This is a special case of [17] and some inequalities based on our needs. � Lemma 2.2. Assume that the function f(%) satisfies f(%) ∼ % and ‖f (k)(%)‖ ≤ Ck for all k ≥ 1, then for any integer k ≥ 0 and p ≥ 2, we have ‖∇kf(%)‖Lp ≤ Ck‖∇k%‖Lp , where Ck is a constant independent of t. Especially, in this paper, α1 − P ′(n) n ∼ σ, µ n − µ̄ ∼ σ, µ+ λ n − (µ̄+ λ̄) ∼ σ, ρ n − α2 ∼ (m,σ). For a proof of the above lemma, we refer to [3, Lemma A.4] for p = 2, and to [19, Lemma 2.2] for p ≥ 2. Lemma 2.3. For each vector function f ∈ C∞0 ( R3 ) , and bounded scalar function g, we have ∣∣ ∫ R3 ( ∇|x|2γ ) · fg dx ∣∣ . ‖g‖L2 γ ‖f‖L2 γ−1 . Proof. The left side of the above inequality can be rewritten as∣∣2γ ∫ R3 |x|2γ−2xj∂ixjgfi dx ∣∣. Then using Hölder’s inequality, we have the desired inequality � Lemma 2.4 (Interpolation inequality with weights). If p, r > 1, s + n/r, α + n/p, β + n/q > 0, and 0 6 θ 6 1 then ‖f‖Lrs ≤ ‖f‖ θ Lpα ‖f‖1−θ Lqβ , for f ∈ C∞0 (Rn) provided that 1 r = θ p + 1− θ q , and s = θα + (1 − θ)β. In particular when s = p = q = 2, θ = γ−1 γ , s = γ − 1, α = γ, β = 0, we have ‖f‖L2 γ−1 ≤ ‖f‖ γ−1 γ L2 γ ‖f‖1/γL2 . (2.7) Proof. By computations, we have∫ U |x|sr|f |r dx = ∫ U |x|αθr|f |θr|x|β(1−θ)r|f |(1−θ)r dx ≤ (∫ U ( |x|αθr|f |θr ) p θr dx )θr/p(∫ U ( |x|β(1−θ)r|f |(1−θ)r ) q (1−θ)r dx ) (1−θ)r q . This completes the proof. � 10 Q. YE, Y. ZHANG EJDE-2023/41 Lemma 2.5 (Gronwall-type Lemma). Let α0 > 1, α1 < 1, α2 < 1, β1 < 1, and β2 < 1. Assume that a continuously differential function F : [1,∞) → [0,∞) satisfies d dt F (t) ≤ C0t −α0F (t) + C1t −α1F (t)β1 + C2t −α2F (t)β2 + C3t γ1−1, t ≥ 1 F (1) ≤ K0, where C0, C1, C2, C3,K0 ≥ 0 and γi = 1−αi 1−βi > 0 for i = 1, 2. Assume that γ1 ≥ γ2, then there exists a constant C∗ depending on α0, α1, β1, α2, β2, K0, Ci, i = 1, 2, 3, such that F (t) ≤ C∗tγ1 , for all t ≥ 1. Fr a proof of the above lemma, see [21, Lemma 2.1]. Lemma 2.6 (Gronwall’s inequality of differential form). Let η(·) be a nonnegative, absolutely continuous function on [0, T ], which satisfies for a.e. t the differential inequality η′(t) ≤ φ(t)η(t) + ψ(t), where φ(t) and ψ(t) are nonnegative and summable functions on [0, T ]. Then η(t) ≤ e ∫ t 0 φ(s)ds [ η(0) + ∫ t 0 ψ(s)ds ] , for 0 ≤ t ≤ T . For a proof of the above lemma, see [7, Appendix B.2.j]. 3. The proof of Theorem 1.1 Inspired by the work in [21], we will address the space-time decay rate of the strong solution of the coupled system (1.1)-(1.2). Under the assumptions of Theo- rem 1.1, it is clear that there exists a large enough T , such that ‖∇k(m,u, σ, v,B)(t)‖L2 ≤ Ct− 3 4− k 2 , (3.1) for all t > T and 0 ≤ k ≤ `, where C is a positive constant independent of t. Lemma 3.1. Under the assumptions of Theorem 1.1, there exists a large enough T such that the solution (m,u, σ, v,B) of system (2.1)-(2.2) has the estimate ‖ (m,u, σ, v,B) (t)‖L2 γ ≤ Ct− 3 4+γ , (3.2) for all t > T and γ ≥ 0, where C is a positive constant independent of t. Proof. Multiplying (2.1)1–(2.1)5 by α2n̄|x|2γm, α2n̄|x|2γu, α1|x|2γσ, n̄|x|2γv and |x|2γB respectively, and the summing them, integrating over R3, and then using integration by parts, one has 1 2 d dt ( α2n̄‖m‖2L2 γ + α2n̄‖u‖2L2 γ + α1‖σ‖+L2 γ n̄‖v‖2L2 γ + ‖B‖2L2 γ ) + α2n̄‖u− v‖2L2 γ + n̄µ̄‖∇v‖2L2 γ + n̄(λ̄+ µ̄)‖div v‖2L2 γ + ν‖∇B‖2L2 γ = α2n̄ ∫ R3 ∇ ( |x|2γ ) ·mudx+ α1n̄ ∫ R3 ∇ ( |x|2γ ) · σv dx EJDE-2023/41 SPACE-TIME DECAY RATES 11 − n̄µ̄ ∫ R3 ∇ ( |x|2γ ) · v∇v dx− n̄(λ̄+ µ̄) ∫ R3 ∇ ( |x|2γ ) · v div v dx + ∫ R3 ( α2n̄|x|2γm · F1 ) dx+ ∫ R3 ( α2n̄|x|2γu · F2 ) dx + ∫ R3 ( α1|x|2γσ · F3 ) dx+ ∫ R3 ( n̄|x|2γv · F4 ) dx + ∫ R3 ( |x|2γB · F5 ) dx = α2n̄ ∫ R3 ∇ ( |x|2γ ) ·mudx+ α1n̄ ∫ R3 ∇ ( |x|2γ ) · σv dx − n̄µ̄ ∫ R3 ∇ ( |x|2γ ) · v∇v dx− n̄(λ̄+ µ̄) ∫ R3 ∇ ( |x|2γ ) · v div vdx − α2n̄ ∫ R3 |x|2γ(u · ∇m)mdx− α2n̄ ∫ R3 |x|2γ(u · ∇u)udx − α1 ∫ R3 |x|2γ(v · ∇σ)σdx− α1 ∫ R3 |x|2γ(σ div v)σdx − n̄ ∫ R3 |x|2γ(v · ∇v)v dx+ n̄ ∫ R3 |x|2γ ( (α1 − P ′(n) n )∇σ ) · v dx + n̄ ∫ R3 |x|2γ ( ( µ n − µ̄)∆v ) · v dx + n̄ ∫ R3 |x|2γ ( ( µ+ λ n − (µ̄+ λ̄))∇div v ) · v dx + n̄ ∫ R3 |x|2γ ( ( ρ n − α2)(u− v) ) · v dx + n̄ ∫ R3 |x|2γ((divB)B) · v dx+ n̄ ∫ R3 |x|2γ(B · ∇B) · v dx − 1 2 n̄ ∫ R3 |x|2γ ( ∇|B|2 ) · v dx− ∫ R3 |x|2γ((div v)B) ·B dx + ∫ R3 |x|2γ(B · ∇v) ·B dx− ∫ R3 |x|2γ(v · ∇B) ·B dx := 19∑ j=1 I1,j . (3.3) Applying Lemma 2.3, we have |I1,1|+ |I1,2| . ‖∇(|x|2γ)mu‖L1 + ‖∇(|x|2γ)σv‖L1 . ‖(m,σ)‖L2 γ ‖(u, v)‖L2 γ−1 . (3.4) Using Lemma 2.3 and Cauchy’s inequality, we have |I1,3| . ‖∇(|x|2γ)v∇v‖L1 . ‖∇v‖L2 γ ‖v‖L2 γ−1 ≤ n̄µ̄ 4 C‖∇v‖2L2 γ + C(n̄µ̄)‖v‖2L2 γ−1 . (3.5) 12 Q. YE, Y. ZHANG EJDE-2023/41 Applying a method similar to the one for (3.5), one has |I1,4| ≤ n̄(λ̄+ µ̄) 4 C‖ div v‖2L2 γ + C(n̄λ̄µ̄)‖v‖2L2 γ−1 . (3.6) Using Hölder’s inequality, Lemma 2.1 (Gagliardo-Nirenberg inequality), Cauchy’s inequality and (3.1), we have |I1,5| . ‖∇m‖L∞‖u‖L2 γ ‖m‖L2 γ . ‖∇2m‖H1‖u‖L2 γ ‖m‖L2 γ . t− 7 4 ‖(m,u)‖2L2 γ . (3.7) Applying a method similar to the one for (3.7), one has 9∑ j=6 |I1,j |+ 19∑ j=14 |I1,j | . t− 7 4 ‖(u, v, σ,B)‖2L2 γ . (3.8) Applying integration by parts, Minkowski’s inequality, Hölder’s inequality, Lemma 2.3, Lemma 2.2, Lemma 2.1 (Gagliardo-Nirenberg inequality), Cauchy’s inequality and (3.1), we have |I1,10| . ‖∇ ( |x|2γ ( α1 − P ′(n) n ) v ) σ‖L1 . ‖ ( α1 − P ′(n) n ) ‖L∞‖v‖L2 γ ‖σ‖L2 γ−1 + ‖∇ ( α1 − P ′(n) n ) ‖L∞‖v‖L2 γ ‖σ‖L2 γ + ‖ ( α1 − P ′(n) n ) ‖L∞‖∇v‖L2 γ ‖σ‖L2 γ . ‖∇σ‖H1‖∇v‖2Lγ + ‖(∇σ,∇2σ)‖H1‖(σ, v)‖2Lγ + ‖∇σ‖H1‖σ‖2L2 γ−1 . t−5/4‖∇v‖2Lγ + t−5/4‖(σ, v)‖2Lγ + ‖σ‖2L2 γ−1 . (3.9) Applying a method similar to the one for (3.9), one has 12∑ j=11 |I1,j | . t−5/4‖(∇v,div v)‖2Lγ + t−5/4‖v‖2Lγ + ‖(σ, v)‖2L2 γ−1 . (3.10) Using Hölder’s inequality, Lemma 2.2, Lemma 2.1 (Gagliardo-Nirenberg inequality), Triangle inequality, Cauchy’s inequality and (3.1), we obtain |I1,13| . ‖( ρ n − α2)‖L∞‖(u− v)‖L2 γ ‖v‖L2 γ . ‖∇(m,σ)‖H1‖(u, v)‖2L2 γ . t−5/4‖(u, v)‖2L2 γ . (3.11) EJDE-2023/41 SPACE-TIME DECAY RATES 13 Substituting (3.4)-(3.11) into (3.3), we have 1 2 d dt ( α2n̄‖m‖2L2 γ + α2n̄‖u‖2L2 γ + α1‖σ‖2L2 γ + n̄‖v‖2L2 γ + ‖B‖2L2 γ ) + α2n̄‖u− v‖2L2 γ + n̄µ̄‖∇v‖2L2 γ + n̄(λ̄+ µ̄)‖ div v‖2L2 γ + ν‖∇B‖2L2 γ ≤ n̄µ̄ 4 ‖∇v‖2L2 γ + n̄(λ̄+ µ̄) 4 ‖ div v‖2L2 γ + Ct−5/4‖(∇v,div v)‖2Lγ + Ct−5/4‖(m,u, σ, v,B)‖2L2 γ + C‖(m,σ)‖L2 γ ‖(u, v)‖L2 γ−1 + C‖(σ, v)‖2L2 γ−1 , (3.12) For t is large enough, we have t−5/4 ≤ min { n̄µ̄ 4 , n̄(λ̄+ µ̄) 4 } . (3.13) Substituting (3.13) into (3.12), there exists a large enough T such that d dt ( α2n̄‖m‖2L2 γ + α2n̄‖u‖2L2 γ + α1‖σ‖2L2 γ + n̄‖v‖2L2 γ + ‖B‖2L2 γ ) + α2n̄‖u− v‖2L2 γ + n̄µ̄‖∇v‖2L2 γ + n̄(λ̄+ µ̄)‖ div v‖2L2 γ + ν‖∇B‖2L2 γ . t−5/4‖(m,u, σ, v,B)‖2L2 γ + ‖(m,σ)‖L2 γ ‖(u, v)‖L2 γ−1 + ‖(σ, v)‖2L2 γ−1 , (3.14) for all t > T . Substituting (2.7) and (3.1) into (3.14), we have d dt ( α2n̄‖m‖2L2 γ + α2n̄‖u‖2L2 γ + α1‖σ‖2L2 γ + n̄‖v‖2L2 γ + ‖B‖2L2 γ ) + α2n̄‖u− v‖2L2 γ + n̄µ̄‖∇v‖2L2 γ + n̄(λ̄+ µ̄)‖ div v‖2L2 γ + ν‖∇B‖2L2 γ . t−5/4‖(m,u, σ, v,B)‖2L2 γ + ‖(m,σ)‖L2 γ ‖(u, v)‖ γ−1 γ L2 γ ‖(u, v)‖1/γL2 + ‖(σ, v)| 2(γ−1) γ L2 γ ‖(σ, v)‖2/γL2 . t−5/4‖(m,u, σ, v,B)‖2L2 γ + t− 3 4γ ‖(m,σ)‖ 2γ−1 γ L2 γ + t− 3 2γ ‖(σ, v)‖ 2(γ−1) γ L2 γ . Denoting E(t) := ‖(m,u, σ, v,B)‖2L2 γ , we obtain d dt E(t) ≤ C0t −5/4E(t) + C1t − 3 4γ E(t) 2γ−1 2γ + C2t − 3 2γ E(t) γ−1 γ , where C0, C1, C2 are positive constants independent of t. If γ > 3 2 , then we can apply Lemma 2.5 with α0 = 5 4 > 1, α1 = 3 4γ < 1, β1 = 2γ−1 2γ < 1, α2 = 3 2γ < 1, β2 = γ−1 γ < 1,γ1 = 1−α1 1−β1 = − 3 2 + 2γ > γ2 = 1−α2 1−β2 = − 3 2 + γ to obtain E(t) ≤ Ct− 3 2+2γ , (3.15) for all t > T . Lemma 3.1 is proved for all γ > 3 2 and the conclusion for the case of [0, 3/2] is proved by Lemma 2.4(interpolation inequality with weights). More precisely, combining (3.1) and (3.15), we have ‖(m,u, σ, v,B)(t)‖L2 γ0 ≤ C‖(m,u, σ, v,B)(t)‖1− γ0 γ L2 ‖(m,u, σ, v,B)(t)‖ γ0 γ L2 γ ≤ Ct− 3 4+γ0 , for all t > T and γ0 ∈ [0, γ](γ > 3 2 ). Thus, the proof is complete. � 14 Q. YE, Y. ZHANG EJDE-2023/41 Lemma 3.2. Under the assumptions of Theorem 1.1 and (3.1), there exists a large enough T such that the solution (m,u, σ, v,B) of coupled system (2.1)-(2.2) satisfies ‖∇ (m,u, σ, v,B) (t)‖L2 γ ≤ Ct− 5 4+γ , (3.16) for all t > T and γ ≥ 0, where C is a positive constant independent of t. Proof. Applying ∇ to each equation of (2.1)1–(2.1)5, then multiplying equations (2.1)1–(2.1)5 by α2n̄|x|2γ∇m, α2n̄|x|2γ∇u, α1|x|2γ∇σ, n̄|x|2γ∇v, and |x|2γ∇B re- spectively, summing them and then integrating over R3, and using integration by parts, we have 1 2 d dt ( α2n̄‖∇m‖2L2 γ + α2n̄‖∇u‖2L2 γ + α1‖∇σ‖2L2 γ + n̄‖∇v‖2L2 γ + ‖∇B‖2L2 γ ) + α2n̄‖∇(u− v)‖2L2 γ + n̄µ̄‖∇2v‖2L2 γ + n̄(λ̄+ µ̄)‖∇ div v‖2L2 γ + ν‖∇2B‖2L2 γ = α2n̄ ∫ R3 ∇(|x|2γ) · ∇m∇udx+ α1n̄ ∫ R3 ∇(|x|2γ) · ∇σ∇v dx − n̄µ̄ ∫ R3 ∇(|x|2γ) · ∇v∇2v dx− n̄(λ̄+ µ̄) ∫ R3 ∇(|x|2γ) · ∇v∇div v dx + ∫ R3 ( α2n̄|x|2γm · ∇F1 ) dx+ ∫ R3 ( α2n̄|x|2γu · ∇F2 ) dx + ∫ R3 ( α1|x|2γσ · ∇F3 ) dx+ ∫ R3 ( n̄|x|2γv · ∇F4 ) dx+ ∫ R3 ( |x|2γB · ∇F5 ) dx = α2n̄ ∫ R3 ∇(|x|2γ) · ∇m∇udx+ α1n̄ ∫ R3 ∇(|x|2γ) · ∇σ∇v dx − n̄µ̄ ∫ R3 ∇(|x|2γ) · ∇v∇2v dx− n̄(λ̄+ µ̄) ∫ R3 ∇(|x|2γ) · ∇v∇div v dx − α2n̄ ∫ R3 |x|2γ∇(u · ∇m)∇m dx− α2n̄ ∫ R3 |x|2γ∇(u · ∇u)∇udx − α1 ∫ R3 |x|2γ∇(v · ∇σ) · ∇σdx− α1 ∫ R3 |x|2γ∇(σ div v) · ∇σdx − n̄ ∫ R3 |x|2γ∇(v · ∇v) · ∇v dx+ n̄ ∫ R3 |x|2γ∇ ( (α1 − P ′(n) n )∇σ ) · ∇v dx + n̄ ∫ R3 |x|2γ∇ ( ( µ n − µ̄)∆v ) · ∇v dx + n̄ ∫ R3 |x|2γ∇ ( ( µ+ λ n − (µ̄+ λ̄))∇divv ) · ∇v dx + n̄ ∫ R3 |x|2γ∇ ( ( ρ n − α2)(u− v) ) · ∇v dx+ n̄ ∫ R3 |x|2γ∇ ((divB)B) · ∇v dx + n̄ ∫ R3 |x|2γ∇ (B · ∇B) · ∇v dx− 1 2 n̄ ∫ R3 |x|2γ∇ ( ∇|B|2 ) · ∇v dx − ∫ R3 |x|2γ∇ ((div v)B) · ∇B dx+ ∫ R3 |x|2γ∇ (B · ∇v) · ∇B dx − ∫ R3 |x|2γ∇ (v · ∇B) · ∇B dx EJDE-2023/41 SPACE-TIME DECAY RATES 15 := 19∑ j=1 I2,j . (3.17) Applying Lemma 2.3, we have 2∑ i=1 |I2,j | . ‖∇(m,σ)‖L2 γ ‖∇(u, v)‖L2 γ−1 . (3.18) Using Lemma 2.3 and Cauchy’s inequality, we have |I2,3| . ‖∇(|x|2γ)∇v∇2v‖L1 . ‖∇2v‖L2 γ ‖∇v‖L2 γ−1 ≤ n̄µ̄ 4 C‖∇2v‖2L2 γ + (n̄µ̄)C‖∇v‖2L2 γ−1 . (3.19) Applying a method similar to the one for (3.19), one has |I2,4| ≤ n̄(λ̄+ µ̄) 4 C‖∇ div v‖2L2 γ + C(n̄λ̄µ̄)‖∇v‖2L2 γ−1 . (3.20) Using Minkowski’s inequality, integration by parts, Hölder’s inequality, Lemma 2.3, Lemma 2.1 (Gagliardo-Nirenberg inequality), Cauchy’s inequality and (3.1), we have |I2,5| . ‖|x|2γ∇u|∇m|2‖L1 + ‖|x|2γu∇2m∇m‖L1 . ‖|x|2γ∇u|∇m|2‖L1 + ‖∇(|x|2γu)|∇m|2‖L1 . ‖|x|2γ∇u|∇m|2‖L1 + ‖∇(|x|2γ)u|∇m|2‖L1 . ‖∇u‖L∞‖∇m‖2L2 γ + ‖u‖L∞‖∇m‖L2 γ ‖∇m‖L2 γ−1 . ‖∇2‖H1‖∇m‖2L2 γ + ‖∇u‖H1 [ ‖∇m‖2L2 γ + ‖∇m‖2L2 γ−1 ] . t−5/4‖∇m‖2L2 γ + ‖∇m‖2L2 γ−1 . (3.21) Applying a method similar to the one for (3.21), we have 7∑ j=6 |I2,j |+ |I2,9| . t−5/4‖∇(m,u, σ, v)‖2L2 γ + ‖∇(m,u, σ, v)‖2L2 γ−1 . (3.22) Applying Minkowski’s inequality, Hölder’s inequality, Lemma 2.1, Cauchy’s inequal- ity and (3.1), we have |I2,8| . ‖|x|2γ div v|∇σ|2‖L1 + ‖|x|2γ∇ div vσ∇σ‖L1 . ‖∇v‖L∞‖∇σ‖2L2 γ + ‖σ‖L∞‖∇ div v‖L2 γ ‖∇σ‖L2 γ . ‖∇σ‖H1‖∇ div v‖2L2 γ + ‖(∇σ,∇2v)‖H1‖∇σ‖2L2 γ . t−5/4‖∇ div v‖2L2 γ + t−5/4‖∇σ‖2L2 γ . (3.23) Using Minkowski’s inequality, integration by parts, Hölder’s inequality, Lemma 2.3, Lemma 2.1 (Gagliardo-Nirenberg inequality), Lemma 2.2, Cauchy’s inequality and (3.1), we have |I2,10| . ‖|x|2γ ( α1 − P ′(n) n ) ∇2σ∇v‖L1 + ‖|x|2γ∇ ( α1 − P ′(n) n ) ∇σ∇v‖L1 . ‖∇(|x|2γ ( α1 − P ′(n) n ) ∇v)∇σ‖L1 + ‖|x|2γ∇ ( α1 − P ′(n) n ) ∇σ∇v‖L1 16 Q. YE, Y. ZHANG EJDE-2023/41 . ‖ ( α1 − P ′(n) n ) ‖L∞‖∇v‖L2 γ ‖∇σ‖L2 γ−1 + ‖∇ ( α1 − P ′(n) n ) ‖L∞‖∇v‖L2 γ ‖∇σ‖L2 γ + ‖ ( α1 − P ′(n) n ) ‖L∞‖∇2v‖L2 γ ‖∇σ‖L2 γ . ‖∇σ‖H1‖∇2v‖2L2 γ + ‖(∇σ,∇2σ)‖H1‖∇(σ, v)‖2L2 γ + ‖∇σ‖H1‖∇v‖2L2 γ−1 . t−5/4‖∇2v‖2L2 γ + t−5/4‖∇(σ, v)‖2L2 γ + ‖∇v‖2L2 γ−1 . (3.24) Applying method similar to the one for (3.24), we have 12∑ j=11 |I2,j | . t−5/4‖(∇2v,∇ div v)‖2Lγ + t−5/4‖∇(σ, v)‖2Lγ + ‖∇v‖2L2 γ−1 . (3.25) Using Hölder’s inequality, Lemma 2.1 (Gagliardo-Nirenberg inequality), Lemma 2.2, Triangle inequality, Cauchy’s inequality and (3.1), we have |I2,13| . ‖|x|2γ( ρ n − α2)∇(u− v)∇v‖L1 + ‖|x|2γ∇( ρ n − α2)(u− v)∇v‖L1 . ‖( ρ n − α2)‖L∞‖∇(u− v)‖L2 γ ‖∇v‖L2 γ + ‖∇( ρ n − α2)‖L∞‖(u− v)‖L2 γ ‖∇v‖L2 γ . ∇‖(m,σ)‖H1 [ ‖∇u‖L2 γ ‖∇v‖L2 γ + ‖∇v‖2L2 γ ] + ‖∇2(m,σ)‖H1‖(u, v)‖L2 γ ‖∇v‖L2 γ . t−5/4‖∇(u, v)‖L2 γ + t− 7 4 × t− 3 4+γ‖∇(u, v)‖L2 γ . t−5/4‖∇(u, v)‖2L2 γ + t− 7 2+2γ . (3.26) Using the same method as in (3.23), we have 19∑ j=14 |I2,j | . t−5/4‖∇2(v,B)‖2L2 γ + t−5/4‖∇B‖2L2 γ . (3.27) Substituting estimates (3.18)-(3.27) into (3.17), we have 1 2 d dt ( α2n̄‖∇m‖2L2 γ + α2n̄‖∇u‖2L2 γ + α1‖∇σ‖2L2 γ + n̄‖∇v‖2L2 γ + ‖∇B‖2L2 γ ) + α2n̄‖∇(u− v)‖2L2 γ + n̄µ̄‖∇2v‖2L2 γ + n̄(λ̄+ µ̄)‖∇ div v‖2L2 γ + ν‖∇2B‖2L2 γ ≤ Ct−5/4‖(∇2v,∇div v,∇2B)‖2L2 γ + n̄µ̄ 4 ‖∇2v‖2L2 γ + n̄(λ̄+ µ̄) 4 ‖ div v‖2L2 γ + Ct−5/4‖∇(m,u, σ, v,B)‖2L2 γ + C‖∇(m,σ)‖L2 γ ‖∇(u, v)‖L2 γ−1 + ‖∇(m,u, σ, v)‖2L2 γ−1 + Ct− 7 2+2γ , (3.28) where C is a positive constant independent of t. For t is large enough, we have t−5/4 ≤ 1 2 min { n̄µ̄ 2 , n̄(λ̄+ µ̄) 2 , ν } . (3.29) EJDE-2023/41 SPACE-TIME DECAY RATES 17 Substituting (3.29) into (3.28), there exists a large enough T such that d dt ( α2n̄‖∇m‖2L2 γ + α2n̄‖∇u‖2L2 γ + α1‖∇σ‖2L2 γ + n̄‖∇v‖2L2 γ + ‖∇B‖2L2 γ ) + α2n̄‖∇(u− v)‖2L2 γ + n̄µ̄‖∇2v‖2L2 γ + n̄(λ̄+ µ̄)‖∇ div v‖2L2 γ + ν‖∇2B‖2L2 γ . t−5/4‖∇(m,u, σ, v,B)‖2L2 γ + ‖∇(m,σ)‖L2 γ ‖∇(u, v)‖L2 γ−1 + ‖∇(m,u, σ, v)‖2L2 γ−1 + t− 7 2+2γ , (3.30) for all t > T . Substituting (2.7) and (3.1) into (3.30), we have d dt ( α2n̄‖∇m‖2L2 γ + α2n̄‖∇u‖2L2 γ + α1‖∇σ‖2L2 γ + n̄‖∇v‖2L2 γ + ‖∇B‖2L2 γ ) + α2n̄‖∇(u− v)‖2L2 γ + n̄µ̄‖∇2v‖2L2 γ + n̄(λ̄+ µ̄)‖∇ div v‖2L2 γ + ν‖∇2B‖2L2 γ . t−5/4‖∇(m,u, σ, v,B)‖2L2 γ + ‖∇(m,σ)‖L2 γ ‖∇(u, v)‖ γ−1 γ L2 γ ‖∇(u, v)‖1/γL2 + ‖∇(m,u, σ, v)‖ 2(γ−1) γ L2 γ ‖∇(m,u, σ, v)‖2/γL2 + t− 7 2+2γ . t−5/4‖∇(m,u, σ, v,B)‖2L2 γ + t− 5 4γ ‖∇(m,u, σ, v)‖ 2γ−1 γ L2 γ + t− 5 2γ ‖∇(m,u, σ, v)‖ 2(γ−1) γ L2 γ + t− 7 2+2γ . Denoting E(t) := ‖∇(m,u, σ, v,B)‖2L2 γ , we obtain d dt E(t) ≤ C0t −5/4E(t) + C1t − 5 4γ E(t) 2γ−1 2γ + C2t − 5 2γ E(t) γ−1 γ + C3t − 7 2+2γ , for all t > T , where C0, C1, C2, C3 are positive constants independent of t. If γ > 5 2 , then we can apply Lemma 2.5 with α0 = 5 4 > 1, α1 = 5 4γ < 1, β1 = 2γ−1 2γ < 1, α2 = 5 2γ < 1, β2 = γ−1 γ < 1, γ1 = 1−α1 1−β1 = − 5 2 +2γ > γ2 = 1−α2 1−β2 = − 5 2 +γ to obtain E(t) ≤ Ct− 5 2+2γ , (3.31) for all t > T . The Lemma 3.2 is proved for all γ > 5/2 and the conclusion for the case of [0, 52 ] is proved by Lemma 2.4(interpolation inequality with weights). Thus, the proof is complete. � Lemma 3.3. Under the assumptions of Theorem 1.1 and (3.1), there exists a large enough T such that the solution (m,u, σ, v,B) of the coupled system (2.1)-(2.2) satisfies ‖∇k (m,u, σ, v,B) (t)‖L2 γ ≤ Ct− 3 4− k 2+γ , (3.32) for all t > T , 0 ≤ k ≤ ` and γ ≥ 0, where C is a positive constant independent of t. Proof. We use induction to prove the for estimate (3.32). In fact, inequalities (3.2) and (3.16) imply (3.32) when k = 0 and k = 1. By the general step of induction, assume that the estimate (3.32) holds for 0 ≤ j ≤ k − 1(2 ≤ k ≤ `), i.e., ‖∇j (m,u, σ, v,B) (t)‖L2 γ ≤ Ct− 3 4− j 2+γ , (3.33) for 0 ≤ j ≤ k − 1. Then, we need to verify that (3.33) holds for j = k. Applying ∇k to each equation of (2.1)1–(2.1)5, multiplying the equations (2.1)1–(2.1)5 by α2n̄|x|2γ∇km, α2n̄|x|2γ∇ku, α1|x|2γ∇kσ, n̄|x|2γ∇kv, and |x|2γ∇kB respectively, 18 Q. YE, Y. ZHANG EJDE-2023/41 summing them and then integrating over R3, using integration by parts the terms of the left side in (2.1), we have 1 2 d dt ( α2n̄‖∇km‖2L2 γ + α2n̄‖∇ku‖2L2 γ + α1‖∇kσ‖2L2 γ + n̄‖∇kv‖2L2 γ + ‖∇kB‖2L2 γ ) + α2n̄‖∇k(u− v)‖2L2 γ + n̄µ̄‖∇k+1v‖2L2 γ + n̄(λ̄+ µ̄)‖∇k div v‖2L2 γ + ν‖∇k+1B‖2L2 γ = α2n̄ ∫ R3 ∇(|x|2γ) · ∇km∇kudx+ α1n̄ ∫ R3 ∇(|x|2γ) · ∇kσ∇kv dx − n̄µ̄ ∫ R3 ∇(|x|2γ) · ∇kv∇k+1v dx− n̄(λ̄+ µ̄) ∫ R3 ∇(|x|2γ) · ∇kv∇k div v dx + ∫ R3 ( α2n̄|x|2γm · ∇kF1 ) dx+ ∫ R3 ( α2n̄|x|2γu · ∇kF2 ) dx + ∫ R3 ( α1|x|2γσ · ∇kF3 ) dx+ ∫ R3 ( n̄|x|2γv · ∇kF4 ) dx+ ∫ R3 ( |x|2γB · ∇kF5 ) dx = α2n̄ ∫ R3 ∇(|x|2γ) · ∇km∇kudx+ α1n̄ ∫ R3 ∇(|x|2γ) · ∇kσ∇kv dx − n̄µ̄ ∫ R3 ∇(|x|2γ) · ∇kv∇k+1v dx− n̄(λ̄+ µ̄) ∫ R3 ∇(|x|2γ) · ∇kv∇k div v dx − α2n̄ ∫ R3 |x|2γ∇k(u · ∇m) · ∇km dx− α2n̄ ∫ R3 |x|2γ∇k(u · ∇u) · ∇kudx − α1 ∫ R3 |x|2γ∇k(v · ∇σ) · ∇kσdx− α1 ∫ R3 |x|2γ∇k(σ div v) · ∇kσdx − n̄ ∫ R3 |x|2γ∇k(v · ∇v) · ∇kv dx+ n̄ ∫ R3 |x|2γ∇k ( (α1 − P ′(n) n )∇σ ) · ∇kv dx + n̄ ∫ R3 |x|2γ∇k ( ( µ n − µ̄)∆v ) · ∇kv dx + n̄ ∫ R3 |x|2γ∇k ( ( µ+ λ n − (µ̄+ λ̄))∇divv ) · ∇kv dx + n̄ ∫ R3 |x|2γ∇k ( ( ρ n − α2)(u− v) ) · ∇kv dx + n̄ ∫ R3 |x|2γ∇k ((divB)B) · ∇kv dx+ n̄ ∫ R3 |x|2γ∇k (B · ∇B) · ∇kv dx − 1 2 n̄ ∫ R3 |x|2γ∇k ( ∇|B|2 ) · ∇kv dx− ∫ R3 |x|2γ∇k ((div v)B) · ∇kB dx + ∫ R3 |x|2γ∇k (B · ∇v) · ∇kB dx− ∫ R3 |x|2γ∇k (v · ∇B) · ∇kB dx := 19∑ j=1 I3,j . (3.34) Applying Lemma 2.3, we have 2∑ i=1 |I3,j | . ‖∇k(m,σ)‖L2 γ ‖∇k(u, v)‖L2 γ−1 . (3.35) EJDE-2023/41 SPACE-TIME DECAY RATES 19 Using Lemma 2.3 and Cauchy’s inequality, we have |I3,3| . ‖∇ ( |x|2γ ) ∇kv∇k+1v‖L1 . ‖∇k+1v‖L2 γ ‖∇kv‖L2 γ−1 ≤ n̄µ̄ 4 C‖∇k+1v‖2L2 γ + C(n̄µ̄)‖∇kv‖2L2 γ−1 . (3.36) Applying a similar method to the one for (3.36), we have |I3,4| ≤ n̄(λ̄+ µ̄) 4 C‖∇k div v‖2L2 γ + C(n̄λ̄µ̄)‖∇kv‖2L2 γ−1 . (3.37) Using integration by parts, we have − 1 α2n̄ I3,5 = ∫ R3 |x|2γu∇k+1m · ∇km dx+ k∑ j=1 Cjk ∫ R3 |x|2γ∇ju∇k−j+1m∇kmdx = −1 2 ∫ R3 ∇ ( |x|2γ ) u|∇km|2 dx+ (k − 1 2 ) ∫ R3 |x|2γ∇u|∇km|2 dx + k−2∑ j=2 Cjk ∫ R3 |x|2γ∇ju∇k−j+1m∇km dx + ∫ R3 |x|2γ∇ku∇m∇kmdx+ k ∫ R3 |x|2γ∇k−1u∇2m∇km dx := 5∑ j=1 I3,5,j . (3.38) Applying Hölder’s inequality, Lemma 2.3, Lemma 2.1 (Gagliardo-Nirenberg in- equality), (3.33), Cauchy’s inequality, and (3.1), we have |I3,5,1| . ‖u‖L∞‖∇km‖L2 γ ‖∇km‖L2 γ−1 . ‖∇u‖H1‖∇km‖L2 γ ‖∇km‖L2 γ−1 . t−5/4‖∇km‖2L2 γ + ‖∇km‖2L2 γ−1 . (3.39) j=4∑ j=2 |I3,5,j | . ‖∇u‖L∞‖∇km‖2L2 γ + k−2∑ j=2 ‖∇ju‖L∞‖∇k−j+1m‖L2 γ ‖∇km‖L2 γ + ‖∇ku‖L2 γ ‖∇m‖L∞‖∇km‖L2 γ . ‖∇2u‖H1‖∇km‖2L2 γ + k−2∑ j=2 ‖∇j+1u‖H1‖∇k−j+1m‖L2 γ ‖∇km‖L2 γ + ‖∇2m‖H1‖∇km‖L2 γ ‖∇ku‖L2 γ . t− 7 4 ‖∇k(m,u)‖2L2 γ + t− 5 4− k 2+γ × t−5/4‖∇k(m,u)‖L2 γ . t−5/4‖∇km‖2L2 γ + t− 5 2−k+2γ . (3.40) 20 Q. YE, Y. ZHANG EJDE-2023/41 |I3,5,5| . ‖∇k−1u‖L3‖∇2m‖L6 γ ‖∇km‖L2 γ . ‖∇k−1u‖H1(‖∇3m‖L2 γ + ‖∇2m‖L2 γ−1 )‖∇km‖L2 γ . t− 5 4− k 2+γ × t− 9 4 ‖∇km‖L2 γ . t−5/4‖∇km‖2L2 γ + t− 5 2−k+2γ . (3.41) Combining (3.38)-(3.41), we have |I3,5| . t−5/4‖∇k(m,u)‖2L2 γ + ‖∇km‖2L2 γ−1 + t− 5 2−k+2γ . (3.42) Applying a method similar to the one in I3,5, we have 7∑ j=6 |I3,j |+ |I3,9| . t−5/4‖∇k(u, σ, v)‖2L2 γ + ‖∇k(u, v, σ)‖2L2 γ−1 + t− 5 2−k+2γ . (3.43) According to (3.34), we have − 1 α1 I3,8 = ∫ R3 |x|2γσ∇k div v · ∇kσ dx+ k∑ j=1 Cjk ∫ R3 |x|2γ∇jσ∇k−j div v∇kσ dx := I3,5,1 + I3,5,2. (3.44) Using Minkowski’s inequality, Hölder’s inequality, Lemma 2.1 (Gagliardo-Nirenberg inequality), Cauchy’s inequality and (3.1), we have |I3,8,1| . ‖|x|2γσ∇k div v∇kσ‖L1 . ‖σ‖L∞‖∇k div v‖L2 γ ‖∇kσ‖L2 γ . ‖∇σ‖H1‖∇k div v‖L2 γ ‖∇kσ‖L2 γ . t−5/4‖∇k div v‖2L2 γ + t−5/4‖∇kσ‖2L2 γ . (3.45) Applying the similar method to (3.40)-(3.41), one has |I3,8,2| . t−5/4‖∇k(σ, v)‖2L2 γ + t− 5 2−k+2γ . (3.46) Combining (3.45) and (3.46), we have |I3,8| . t−5/4‖∇k div v‖2L2 γ + t−5/4‖∇k(σ, v)‖2L2 γ + t− 5 2−k+2γ . (3.47) Applying integration by parts, one has 1 n̄ I3,10 = ∫ R3 |x|2γ ( α1 − P ′(n) n ) ∇k+1σ · ∇kv dx + k∑ j=1 Cjk ∫ R3 |x|2γ∇j ( α1 − P ′(n) n ) ∇k−j+1σ∇kv dx = − ∫ R3 ∇ ( |x|2γ ) ( α1 − P ′(n) n ) ∇kv∇kσ dx − ∫ R3 |x|2γ ( α1 − P ′(n) n ) ∇k+1v∇kσ dx + (k − 1) ∫ R3 |x|2γ∇ ( α1 − P ′(n) n ) ∇kv∇kσ dx EJDE-2023/41 SPACE-TIME DECAY RATES 21 + k−1∑ j=2 Cjk ∫ R3 |x|2γ∇j ( α1 − P ′(n) n ) ∇k−j+1σ∇kv dx + ∫ R3 |x|2γ∇k ( α1 − P ′(n) n ) ∇σ∇kv dx := 5∑ j=1 I3,10,j . (3.48) Using Hölder’s inequality, Lemma 2.3, Lemma 2.1 (Gagliardo–Nirenberg inequal- ity), Lemma 2.2, (3.1), and Cauchy inequality, we have |I3,10,1| . ‖∇ ( |x|2γ ) ( α1 − P ′(n) n ) ∇kv∇kσ‖L1 . ‖ ( α1 − P ′(n) n ) ‖L∞‖∇kv‖L2 γ ‖∇kσ‖L2 γ−1 . ‖∇σ‖H1‖∇kv‖L2 γ ‖∇kσ‖L2 γ−1 . t−5/4‖∇kv‖2L2 γ + ‖∇kσ‖2L2 γ−1 . (3.49) Applying a method similar to the on in (3.49), we have |I3,10,2|+ |I3,10,3| . t−5/4‖∇k+1v‖2L2 γ + t−5/4 ‖∇k(σ, v)‖2L2 γ . (3.50) We note that Lemma 2.2 can not be applied in the weighted space. For I3,10,4 and I3,10,5, applying Hölder’s inequality, Lemma 2.1 (Gagliardo–Nirenberg inequality) and (3.33) skillfully, we have |I3,10,4| . k−1∑ j=2 ‖|x|2γ∇j ( α1 − P ′(n) n ) ∇k−j+1σ∇kv‖L1 . k−1∑ j=2 ‖∇j ( α1 − P ′(n) n ) ‖L3‖∇k−j+1σ‖L2 γ ‖∇kv‖L6 γ . k−1∑ j=2 ‖∇jσ‖H1‖∇k−j+1σ‖L2 γ ( ‖∇k+1v‖L2 γ + ‖∇kv‖L2 γ−1 ) . t− 5 4− k 2+γ × t−3/4 ( ‖∇k+1v‖L2 γ + ‖∇kv‖L2 γ−1 ) . t−5/4‖∇k+1v‖2L2 γ + ‖∇kv‖2L2 γ−1 + t− 5 2−k+2γ . (3.51) 22 Q. YE, Y. ZHANG EJDE-2023/41 |I3,10,5| . ‖|x|2γ∇k ( α1 − P ′(n) n ) ∇σ∇kv‖L1 . ‖∇k ( α1 − P ′(n) n ) ‖L2‖|x|γ∇σ‖L∞‖∇kv‖L2 γ . ‖∇kσ‖L2 ( ‖∇ (|x|γ∇σ) ‖1/2L2 ‖∇2 (|x|γ∇σ) ‖1/2L2 ) ‖∇kv‖L2 γ . ‖∇kσ‖L2 ( ‖∇2 (|x|γ∇σ) ‖L2 + ‖∇ (|x|γ∇σ) ‖L2 ) ‖∇kv‖L2 γ . ‖∇kσ‖L2 ( ‖∇3σ‖L2 γ + ‖∇2σ‖L2 γ−1 + ‖∇σ‖L2 γ−2 + ‖∇2σ‖L2 γ + ‖∇σ‖L2 γ−1 ) ‖∇kv‖L2 γ . t− 5 4− k 2+γ × t−5/4‖∇kv‖L2 γ . t−5/4‖∇kv‖2L2 γ + t− 5 2−k+2γ . (3.52) Combining (3.48)-(3.52), we have |I3,10| . t−5/4‖∇k+1v‖2L2 γ + t−5/4‖∇k(σ, v)‖2L2 γ + ‖∇k(σ, v)‖2L2 γ−1 + t− 5 2−k+2γ . (3.53) Using the same method as in |I3,10|, one has |I3,11|+ |I3,12| . t−5/4‖∇k+1v‖2L2 γ + t−5/4‖∇k div v‖2L2 γ + t−5/4‖∇kv‖2L2 γ + ‖∇kv‖2L2 γ−1 + t− 5 2−k+2γ . (3.54) According to (3.3), we have 1 n̄ I3,13 = ∫ R3 |x|2γ( ρ n − α2)∇k(u− v)∇kv dx + k−1∑ j=1 Cjk ∫ R3 |x|2γ∇j( ρ n − α2)∇k−j(u− v)∇kv dx + ∫ R3 |x|2γ∇k( ρ n − α2)(u− v)∇kv dx := 3∑ j=1 I3,13,j . (3.55) Applying Minkowski’s inequality, Hölder’s inequality, Lemma 2.1, Lemma 2.2, Tri- angle inequality, (3.1), and (3.33), we have |I3,13,1| . ‖( ρ n − α2)‖L∞‖∇k(u− v)‖L2 γ ‖∇kv‖L2 γ . ‖∇(m,σ)‖H1‖∇k(u, v)‖L2 γ ‖∇kv‖L2 γ . t−5/4‖∇k(u, v)‖2L2 γ . (3.56) EJDE-2023/41 SPACE-TIME DECAY RATES 23 |I3,13,2| . k−1∑ j=1 ‖∇j( ρ n − α2)‖L3‖∇k−j(u− v)‖L2 γ ‖∇kv‖L6 γ . k−1∑ j=2 ‖∇j(m,σ)‖H1‖∇k−j(u, v)‖L2 γ ( ‖∇k+1v‖L2 γ + ‖∇kv‖L2 γ−1 ) . t− 5 4− k 2+γ × t− 1 4 ( ‖∇k+1v‖L2 γ + ‖∇kv‖L2 γ−1 ) . t−1/2‖∇k+1v‖2L2 γ + ‖∇kv‖2L2 γ−1 + t− 5 2−k+2γ . (3.57) |I3,13,3| . ‖∇k( ρ n − α2)‖L2‖|x|2γ(u− v)‖L∞‖∇kv‖L2 γ . ‖∇k(m,σ)‖L2 ( ‖∇ (|x|γ(u+ v)) ‖1/2L2 ‖∇2 (|x|γ(u+ v)) ‖1/2L2 ) ‖∇kv‖L2 γ . ‖∇k(m,σ)‖L2 ( ‖∇2 (|x|γ(u+ v)) ‖L2 + ‖∇ (|x|γ(u+ v)) ‖L2 ) ‖∇kv‖L2 γ . ‖∇k(m,σ)‖L2 ( ‖∇2(u, v)‖L2 γ + ‖∇(u, v)‖L2 γ−1 + ‖(u, v)‖L2 γ−2 + ‖∇(u, v)‖L2 γ + ‖(u, v)‖L2 γ−1 ) ‖∇kv‖L2 γ . t− 5 4− k 2+γ × t−3/4‖∇kv‖L2 γ . t−5/4‖∇kv‖2L2 γ + t− 5 2−k+2γ . (3.58) Combining (3.55)–(3.58), one has |I3,13| . t−1/2‖∇k+1v‖2L2 γ + t−5/4‖∇k(u, v)‖2L2 γ + ‖∇kv‖2L2 γ−1 + t− 5 2−k+2γ . (3.59) According to (3.3), we have 1 n̄ I3,14 = ∫ R3 |x|2γB∇k divB∇kv dx+ k ∫ R3 |x|2γ∇B∇k−1 divB∇kv dx + k−1∑ j=2 Cjk ∫ R3 |x|2γ∇jB∇k−j divB∇kv dx + ∫ R3 |x|2γ∇kB divB∇kv dx := 4∑ j=1 I3,14,j . (3.60) Using Minkowski’s inequality, Hölder’s inequality, Lemma 2.1 (Gagliardo–Nirenberg inequality) (3.1), (3.33), and Cauchy’s inequality, we have |I3,14,1| . ‖B‖L∞‖∇k+1B‖L2 γ ‖∇kv‖L2 γ . ‖∇B‖H1‖∇k+1B‖L2 γ ‖∇kv‖L2 γ . t−5/4‖∇k+1B‖2L2 γ + t−5/4‖∇kv‖2L2 γ . (3.61) Applying a method similar to the one for (3.61), we have |I3,14,2|+ |I3,14,4| . t−5/4‖∇k(v,B)‖2L2 γ . (3.62) 24 Q. YE, Y. ZHANG EJDE-2023/41 For I3,14,3, we have |I3,14,3| . k−1∑ j=2 ‖∇jB‖L3‖∇k−j+1B‖L2 γ ‖∇kv‖L6 γ . k−1∑ j=2 ‖∇jB‖H1‖∇k−j+1B‖L2 γ ( ‖∇k+1v‖L2 γ + ‖∇kv‖L2 γ−1 ) . t− 5 4− k 2+γ × t−3/4 ( ‖∇k+1v‖L2 γ + ‖∇kv‖L2 γ−1 ) . t−5/4‖∇k+1v‖2L2 γ + ‖∇kv‖2L2 γ−1 + t− 5 2−k+2γ . (3.63) Combining (3.60)–(3.63), one has |I3,14| . t−5/4‖∇k+1(v,B)‖2L2 γ + t−5/4‖∇k(v,B)‖2L2 γ + ‖∇kv‖2L2 γ−1 + t− 5 2−k+2γ . (3.64) Using the same method as I3,14, we have 19∑ j=15 |I3,j | . t−5/4‖∇k+1(v,B)‖2L2 γ + t−5/4‖∇k(v,B)‖2L2 γ + ‖∇k(v,B)‖2L2 γ−1 + t− 5 2−k+2γ . (3.65) Substituting the estimates (3.35)–(3.65) into (3.34), we have 1 2 d dt ( α2n̄‖∇km‖2L2 γ + α2n̄‖∇ku‖2L2 γ + α1‖∇kσ‖2L2 γ + n̄‖∇kv‖2L2 γ + ‖∇kB‖2L2 γ ) + α2n̄‖∇k(u− v)‖2L2 γ + n̄µ̄‖∇k+1v‖2L2 γ + n̄(λ̄+ µ̄)‖∇k div v‖2L2 γ + ν‖∇k+1B‖2L2 γ ≤ n̄µ̄ 4 ‖∇k+1v‖2L2 γ + n̄(λ̄+ µ̄) 4 ‖∇k div v‖2L2 γ + Ct−5/4‖(∇k div v,∇k+1B)‖2L2 γ + Ct−1/2‖∇k+1v‖2L2 γ + Ct−5/4‖∇k(m,u, σ, v,B)‖2L2 γ + C‖∇k(m,u, σ, v,B)‖2L2 γ−1 + C‖∇k(m,σ)‖L2 γ ‖∇k(u, v)‖L2 γ−1 + Ct− 5 2−k+2γ . (3.66) For t large enough, we have t−1/2 ≤ 1 2 min{ n̄µ̄ 2 , n̄(λ̄+ µ̄) 2 , ν}. (3.67) Substituting (3.67) into (3.66), there exists a large enough T such that d dt ( α2n̄‖∇km‖2L2 γ + α2n̄‖∇ku‖2L2 γ + α1‖∇kσ‖2L2 γ + n̄‖∇kv‖2L2 γ + ‖∇kB‖2L2 γ ) + α2n̄‖∇k(u− v)‖2L2 γ + n̄µ̄‖∇k+1v‖2L2 γ + n̄(λ̄+ µ̄)‖∇k div v‖2L2 γ + ν‖∇k+1B‖2L2 γ . t−5/4‖∇k(m,u, σ, v,B)‖2L2 γ + ‖∇k(m,σ)‖L2 γ ‖∇k(u, v)‖L2 γ−1 + ‖∇k(m,u, σ, v,B)‖2L2 γ−1 + t− 5 2−k+2γ , (3.68) EJDE-2023/41 SPACE-TIME DECAY RATES 25 for all t > T . Substituting (2.7) and (3.33) into (3.68), we have d dt ( α2n̄‖∇km‖2L2 γ + α2n̄‖∇ku‖2L2 γ + α1‖∇kσ‖2L2 γ + n̄‖∇kv‖2L2 γ + ‖∇kB‖2L2 γ ) + α2n̄‖∇k(u− v)‖2L2 γ + n̄µ̄‖∇k+1v‖2L2 γ + n̄(λ̄+ µ̄)‖∇k div v‖2L2 γ + ν‖∇k+1B‖2L2 γ . t−5/4‖∇k(m,u, σ, v,B)‖2L2 γ + ‖∇k(m,σ)‖L2 γ ‖∇k(u, v)‖ γ−1 γ L2 γ ‖∇k(u, v)‖1/γL2 + ‖∇k(m,u, σ, v,B)‖ 2(γ−1) γ L2 γ ‖∇k(m,u, σ, v,B)‖2/γL2 + t− 5 2−k+2γ . t−5/4‖∇k(m,u, σ, v,B)‖2L2 γ + t(− 3 4− k 2 ) 1 γ ‖∇k(m,u, σ, v)‖ 2γ−1 γ L2 γ + t(− 3 4− k 2 ) 2 γ ‖∇k(m,u, σ, v,B)‖ 2(γ−1) γ L2 γ + t− 5 2−k+2γ . Denoting E(t) := ‖∇k(m,u, σ, v,B)‖2L2 γ , we obtain d dt E(t) ≤ C0t −5/4E(t)+C1t (− 3 4− k 2 ) 1 γ E(t) 2γ−1 2γ +C2t (− 3 4− k 2 ) 2 γ E(t) γ−1 γ +C3t − 5 2−k+2γ , for all t > T , where C0, C1, C2, C3 are positive constants independent of t. If γ > 3+2k 2 , then we can apply Lemma 2.5 with α0 = 5 4 > 1, α1 = ( 3 4 + k 2 ) 1 γ < 1, β1 = 2γ−1 2γ < 1, α2 = ( 3 4 + k 2 ) 2 γ < 1, β2 = γ−1 γ < 1, γ1 = 1−α1 1−β1 = − 3 2 − k + 2γ > γ2 = 1−α2 1−β2 = − 3 2 − k + γ to obtain E(t) ≤ Ct− 3 2−k+2γ , (3.69) for all t > T and γ > 3+2k 2 . Lemma 3.3 is proved for all γ > 3+2k 2 and the conclusion for the case of [0, 3+2k 2 ] is proved by Lemma 2.4. Thus, the proof of Theorem 1.1 is complete. � 4. Proof of Theorem 1.4 Combining (1.11) and (1.13) and using direct energy estimate, we prove Theorem 1.4 as follows. Proof. Taking (2.1)2–(2.1)4, we have (u− v)t + (1 + α2)(u− v) = G, (4.1) where G is defined by G = F2 − F4 −∇m+ α1∇σ − µ̄∆v − (µ̄+ λ̄)∇div v. 26 Q. YE, Y. ZHANG EJDE-2023/41 For 0 ≤ k ≤ `−2, applying∇k to (4.1) and then multiplying (4.1) by |x|2γ∇k(u−v), integrating them over R3, we have 1 2 d dt ‖∇k(u− v)‖2L2 γ + (1 + α2)‖∇k(u− v)‖2L2 γ = −〈∇k(u · ∇u), |x|2γ∇k(u− v)〉+ 〈∇k(v · ∇v), |x|2γ∇k(u− v)〉 + 〈∇k[( µ n − µ̄)∆v], |x|2γ∇k(u− v)〉 − 〈∇k [( α1 − P ′(n) n ) ∇σ ] , |x|2γ∇k(u− v)〉 − 〈∇k [(µ+ λ n − (µ̄+ λ̄)∇div v )] , |x|2γ∇k(u− v)〉 − 〈∇k [( ρ n − α2 ) (u− v) ] , |x|2γ∇k(u− v)〉 − 〈∇k[(divB)B], |x|2γ∇k(u− v)〉 − 〈∇k(B.∇B), |x|2γ∇k(u− v)〉 + 〈∇k (1 2 ∇|B|2 ) , |x|2γ∇k(u− v)〉 − 〈∇k(∇m), |x|2γ∇k(u− v)〉+ α1〈∇k(∇σ), |x|2γ∇k(u− v)〉 − µ̄〈∇k∆v, |x|2γ∇k(u− v)〉 − (µ̄+ λ̄)〈∇k∇ div v, |x|2γ∇k(u− v)〉 = 13∑ j=1 I4,j . (4.2) Applying Minkowski’s inequality, Hölder’s inequality, (1.11), (1.13) and Cauchy’s inequality, we have |I4,1| . k∑ j=0 ‖∇ju∇k−j+1u∇k(u− v)‖L1 . k∑ j=0 ‖∇ju‖L∞‖∇k−j+1u‖L2 γ ‖∇k(u− v)‖L2 γ . k∑ j=0 ‖∇j+1u‖H1‖∇k−j+1u‖L2 γ ‖∇k(u− v)‖L2 γ . (1 + t)− 5 4− k 2+γ− 5 4 ‖∇k(u− v)‖L2 γ . (1 + t)− 5 2−k+2γ + (1 + t)− 5 2 ‖∇k(u− v)‖2L2 γ . (4.3) Applying a method similar to the one for (4.3), we have |I4,2| . (1 + t)− 5 2−k+2γ + (1 + t)− 5 2 ‖∇k(u− v)‖2L2 γ . (4.4) EJDE-2023/41 SPACE-TIME DECAY RATES 27 Using Minkowski’s inequality, Hölder’s inequality, Lemma 2.2, (1.11), (1.13), and Cauchy’s inequality, we have |I4,3| . k∑ j=0 ‖∇j(µ n − µ̄)∇k−j+2u∇k(u− v)‖L1 . k∑ j=0 ‖∇j(µ n − µ̄)‖L∞‖∇k−j+2u‖L2 γ ‖∇k(u− v)‖L2 γ . k∑ j=0 ‖∇jσ‖L∞‖∇k−j+2u‖L2 γ ‖∇k(u− v)‖L2 γ . k∑ j=0 ‖∇j+1σ‖H1‖∇k−j+2u‖L2 γ ‖∇k(u− v)‖L2 γ . (1 + t)− 5 4− k 2+γ− 7 4 ‖∇k(u− v)‖L2 γ . (1 + t)− 5 2−k+2γ + (1 + t)− 7 2 ‖∇k(u− v)‖2L2 γ . (4.5) Using the same method as (4.5), we obtain |I4,4| . (1 + t)− 5 2−k+2γ + (1 + t)− 5 2 ‖∇k(u− v)‖2L2 γ , (4.6) |I4,5| . (1 + t)− 5 2−k+2γ + (1 + t)− 7 2 ‖∇k(u− v)‖2L2 γ . (4.7) Applying Minkowski’s inequality, Hölder’s inequality, Lemma 2.2, Triangle inequal- ity, (1.11), (1.13), and Cauchy’s inequality, we have |I4,6| . k∑ j=0 ‖∇j( ρ n − α2)∇k−j(u− v)∇k(u− v)‖L1 . k∑ j=0 ‖∇j( ρ n − α2)‖L∞‖∇k−j(u− v)‖L2 γ ‖∇k(u− v)‖L2 γ . k∑ j=0 ‖∇j(m,σ)‖L∞‖∇k−j(u, v)‖L2 γ ‖∇k(u− v)‖L2 γ . k∑ j=0 ‖∇j+1(m,σ)‖H1‖∇k−j(u, v)‖L2 γ ‖∇k(u− v)‖L2 γ . (1 + t)− 5 4− k 2+γ− 3 4 ‖∇k(u− v)‖L2 γ . (1 + t)− 5 2−k+2γ + (1 + t)− 3 2 ‖∇k(u− v)‖2L2 γ . Applying a method similar to the one for (4.3), we have 9∑ j=7 |I4,j | . (1 + t)− 5 2−k+2γ + (1 + t)− 5 2 ‖∇k(u− v)‖2L2 γ . (4.8) 28 Q. YE, Y. ZHANG EJDE-2023/41 Using Hölder’s inequality, (1.13) and Cauchy’s inequality, we have |I4.10| . ‖∇k+1m‖L2 γ ‖∇k(u− v)‖L2 γ . (1 + t)− 5 4− k 2+γ‖∇k(u− v)‖L2 γ ≤ C(ε)(1 + t)− 5 2−k+2γ + ε‖∇k(u− v)‖2L2 γ . (4.9) Applying a method similar to one for (4.9), one has |I4.11| ≤ C(ε)(1 + t)− 5 2−k+2γ + ε‖∇k(u− v)‖2L2 γ . (4.10) Using Hölder’s inequality, (1.11) and (1.13), we have |I4,12|+ |I4,13| . ‖∇k+2v‖L2 γ ‖∇k(u− v)‖L2 γ . t− 7 4− k 2+γ‖∇k(u− v)‖L2 γ . t− 5 2−k+2γ + t−1‖∇k(u− v)‖2L2 γ . (4.11) Substituting (4.3)–(4.11) into (4.2), and noting that ε is small enough, then there exists large enough T such that d dt ‖∇k(u− v)‖2L2 γ + C ′‖∇k(u− v)‖2L2 γ . (1 + t)− 5 2−k+2γ , for all t > T , where C ′ is a positive constant independent of t. Using Lemma 2.6 (Gronwall’s inequality of differential form), we have ‖∇k(u− v)‖L2 γ . (1 + t)− 5 4− k 2+γ , for all t > T , 0 ≤ k ≤ `− 2 and γ ≥ 0. Thus, the proof is complete. � Acknowledgments. 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Ma; Global existence and time decay rates of the two-phase fluid system in R3. Z. Angew. Math. Phys., 72 (2021) (5): Paper No. 180, 28. [27] E. Zuazua; Large time asymptotics for heat and dissipative wave equations. Manuscript available at http://www. uam. es/enrique. zuazua, 2003. Qin Ye School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi 541004, China Email address: yeqin811@163.com Yinghui Zhang School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi 541004, China Email address: yinghuizhang@mailbox.gxnu.edu.cn 1. Introduction and main results 1.1. History of the problem 1.2. Notation 1.3. Main results 2. Reformulation and preliminaries 2.1. Reformulation 3. The proof of Theorem ?? 4. Proof of Theorem ?? Acknowledgments References