# Optimal Convergence Rates for the Strong Solutions to the Compressible MHD Equations with Potential Force

Authors: Miao Ouyang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, the large-time behavior of solutions to the Cauchy problem for the 3D compressible MHD equations is considered with the effect of external force. We construct the global unique solution with the small initial data near the stationary profile. The optimalLp-L2(1≤p≤2) time decay rates of the solution to the system are built in multifrequency decompositions.


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## Body

## 1. Introduction

Consider the following Cauchy problem of the full compressible MHD equations affected by the external potential force inR3:(1)ρt+∇·ρu=0,ρut+u·∇u+∇Pρ,θ=μΔu+μ+λ∇∇·u+∇×b×b+ρF,bt-∇×u×b=-∇×∇×νb,ρcvθt+u·∇θ+θPθρ,θ∇·u=κΔθ+Ψu,∇·b=0where(2)ρ,u,b,θ0,x=ρ0,u0,b0,θ0x→ρ∞,0,0,θ∞,asx→∞. Here ρ≥0, u,b,θ∈R3, and p(ρ,θ) are density, velocity, magnetic field, temperature, and pressure, respectively. The constants μ, λ are the shear and bulk viscosity coefficients of the flow, satisfying the physical restrictions μ>0 and 2μ+3λ≥0. The resistivity coefficient ν>0 is inversely proportional to the electrical conductivity constant and acts as the magnetic diffusivity of magnetic fields. By the positive constant κ we denote the heat conduction coefficient and cv>0 is the specific heat at constant volume. In addition, F=F(x) is an external force and Ψ=Ψ[u] is the dissipation function:(3)Ψu=μ2∑i,j=13∂iuj+∂jui2+λ∑j=13∂juj2.It will be assumed thatP(ρ,θ) is smooth in a neighborhood of (ρ∞,θ∞) with Pρ(ρ∞,θ∞)>0 and Pθ(ρ∞,θ∞)>0 throughout this paper. Both ρ∞ and θ∞ are positive constants.We only consider the potential forceF=-∇Φ(x) in this work. Reference [1] has proofed the existence of the stationary solution to problem (1) and (2) under aforementioned assumptions. We give the solution (ρ∗,u∗,b∗,θ∗) in a neighborhood of (ρ∞,0,0,θ∞) as(4)∫ρ∞ρ∗xPρη,θ∞ηdx+Φx=0,u∗x=0,b∗x=0,θ∗x=θ∞, and satisfy(5)ρ∗-ρ∞HkR3≤CΦHkR3,0≤k≤4,(6)∑k=141+x∇kρ∗-ρ∞L2R3≤C∑k=141+x∇kΦL2R3.The global unique solution is constructed to (1) near the steady state (ρ∗,0,0,θ∞) when the initial perturbation belongs to the Sobolev space H2(R3). The main results are as follows.Theorem 1.
Suppose that(ρ0-ρ∞,u0,b0,θ0-θ∞)∈H2(R3), there exists constant ε>0 such that if(7)ρ0-ρ∞,u0,b0,θ0-θ∞H2R3+ΦH4R3+∑k=141+x∇kΦL2R3≤ε,then the initial value problem (1) and (2) admits a unique solution (ρ,u,b,θ) globally in time which satisfies(8)ρ-ρ∗∈C0,∞;H2R3∩C10,∞;H1R3,u,b,θ-θ∞∈C0,∞;H2R3∩C10,∞;L2R3.Moreover, if the initial data (ρ0-ρ∞,u0,b0,θ0-θ∞) is bounded in Lp(R3) for any given 1≤p≤2, the solution (ρ,u,b,θ) satisfies the following decay-in-time estimates:(9)∇ρ-ρ∗,u,b,θ-θ∞H1R3≤C1+t-3/21/p-1/2∀t≥0,(10)ρ-ρ∗,u,b,θ-θ∞LqR3≤C1+t-3/21/p-1/q∀t≥0,2≤q≤6,(11)∂tρ-ρ∗,u,b,θ-θ∞L2R3≤C1+t-3/21/p-1/2∀t≥0,for some positive constant C.Remark 2.
In Theorem1, using the Sobolev imbedding inequalities in Lemma 3, (7) together with (5) and (6) yields(12)ρ-ρ∗H4R3+∑k=131+x∇kρ∗-ρ∞L2R3∩L3R3≤Cε.Due to its importance in mathematics and physics, there are a lot of literatures devoted to the mathematical theory of the MHD fluid system, including the global existence, unique, and time decay rates of solutions with or without external forces; see [2–20] and references therein. For the initial value problem for three-dimensional isentropic MHD system ((1) with θ=const.), Li and Yu [13] obtained the global existence and the L2 decay rate of classical solutions under the small oscillations on small initial data. Chen and Tan [4] obtained the decay estimate of solutions with the initial data in Lp(p∈[1,6/5)). Noting the special construction of (1), the authors in [18] obtained decay estimate of solutions when the initial data belongs to negative Sobolev space. For the initial boundary value problem for three-dimensional isentropic MHD system (1), we refer to [11]. Pu and Guo [17] extended the global existence and decay rate of solutions in [4, 13] to three-dimensional incompressible nonisentropic MHD system (1). For the initial boundary value for three-dimensional nonisentropic MHD system, the global existence of weak solutions has been established in [9, 10].If the magnetic field disappear (b=0), system (1) deduces to Navier-Stokes equations. We only refer to time decay rates of solutions to the compressible Navier-Stokes equations. If there is no external force, Matsumura and Nishida [21] obtained the convergence rate for the compressible viscous and heat-conductive fluid in R3:(13)ρ-ρ∞,u,θ-θ∞t2≤C1+t-3/4,t≥0.For the same system, Ponce [22] gave the optimal Lp convergence rate(14)∇lρ-ρ∞,u,θ-θ∞tLp≤C1+t-n/21-1/p-l/2,l=0,1,2,t≥0,for the space dimension n=2 or 3. When there is an external potential force F=-∇Φ(x), the first work to give explicit estimates for the decay rates for solutions to problem (1) and (2) was represented by Deckelnick [23]. In [23], when the initial perturbation only belongs to the Sobolev space H3(Ω), the following decay estimates were established in an unbounded domain Ω⊂R3 (the half space R+3=x∈R3x3>0 or the exterior of a bounded domain with smooth boundary):(15)∇ρ-ρ∗,u,θ-θ∞L2Ω≤C1+t-1/4,∂tρ-ρ∗,u,θ-θ∞L2Ω≤C1+t-1/2.Duanet al. [24] obtained a similar result for the nonisentropic case(16)∇kρ-ρ∗,u,θ-θ∞tLp≤C1+t-3/21-1/p,2≤p≤6,ρ-ρ∗,u,θ-θ∞tL∞≤C1+t-5/4,∇ρ-ρ∞,u,θ-θ∞t2≤C1+t-5/4,ρt,ut,θtt≤C1+t-5/4.Recently, Wang [25] constructed the global unique solution near the stationary profile to the system for the small H2 initial data and the optimal Lp-L2(1≤p≤2) time decay rates of the solution:(17)∇ρ-ρ∗,u,b,θ-θ∞H1R3≤C1+t-3/21/p-1/2,ρ-ρ∗,u,b,θ-θ∞LqR3≤C1+t-3/21/p-1/q,2≤q≤6,∂tρ-ρ∗,u,b,θ-θ∞L2R3≤C1+t-3/21/p-1/2.In this paper, we will generalize the result in [25] to the three-dimensional magnetomicropolar fluid system (1). We will give the global existence and the optimal Lp-L2(1≤p≤2) time decay rates of the solution. Notice that the first three equations in (1) have been studied in [25]. The magnetic field b(t,x) satisfies the heat equation, which can be solved in Fourier space, b^=exp(-ξ2t)b^0, whose decay estimate has been proved in many papers, for instance, [26, 27]. However, compared with the work in [25], we have to deal with the difficulties in the present paper caused by the strong coupling between the velocity vector field and the magnetic field, for instance, the influence of the nonlinear term (∇×H)×H on the velocity vector field.Notations.Denote C as a generic positive constant. For multi-indices α=(α1,α2,α3) and β=(β1,β2,β3), Cαβ=α!/(α-β)!β! with β≤α which means βi≤αi for all 1≤i≤n. ∂xα=∂x1α1∂x2α2∂x3α3,∇=(∂1,∂2,∂3),∂i=∂xi,i=1,2,3, and α=Σi=13αi. Denote ∇m as a set composed of all mth partial derivatives with respect to the variable x. Hm(R3), m∈Z+, denotes the usual Sobolev space with its norm(18)fHmR3≜Σk=0m∇kfL2R3.In particular,·m means ·Hm(R3) and ·m means ·L2(R3). As usual, ·,· denotes the inner-product in L2(R3). The Fourier transform to the variable x∈R3 is applied by f^(ξ,t)=∫Rnf(x,t)e--1x·ξdx and the inverse Fourier transform to the variable ξ∈R3 by (F-1f^(x,t)=(2π)-n∫R3f^(ξ,t)e-1x·ξdξ.The outlet of paper is organized as follows. In Section2, we reformulate the Cauchy problem (1)-(2) into a more suitable form. In Section 3, we establish the global existence of solutions to the problem in H2-framework. Section 4 is devoted to the proof of the Lp-L2 time decay rates of the solution and we complete the proof of Theorem 1.


## 2. Preliminary

We are going to reformulate the Cauchy problem of the compressible MHD system (1) with the initial condition (2) and give some lemmas in this section.Define(19)ρ~x,t=ρx,t-ρ∗x,u~x,t=ux,t,b~x,t=bx,t,θ~x,t=θx,t-θ∞,ρ¯x=ρ∗x-ρ∞.Then, by using (4), the MHD equations (1) are transformed as the following:(20)ρt~+ρ∞∇·u~=S~1,ut~-μρ∞Δu~-μ+λρ∞∇∇·u~+Pρρ∞,θ∞ρ∞∇ρ~+Pθρ∞,θ∞ρ∞∇θ~=S~2,b~t-νΔb~=S~3,θ~t-κcvρ∞Δθ~+θ∞Pθρ∞,θ∞cvρ∞∇·u~=S~4,and the initial condition (2) turns into(21)ρ~,u~,b~,θ~x,tt=0=ρ0-ρ∗,u0,b0,θ0x→0,0,0,0,asx→∞, where S~1, S~2, S~3, and S~4 are the source terms with(22)S~1=-∇·ρ~u~-∇·ρ¯u~,(23)S~2=-u~·∇u~+μρ~+ρ∗-μρ∞Δu~+μ+λρ~+ρ∗-μ+λρ∞∇∇·u~-Pρρ~+ρ∗,θ~+θ∞ρ~+ρ∗-Pρρ∞,θ∞ρ∞∇ρ~-Pθρ~+ρ∗,θ~+θ∞ρ~+ρ∗-Pθρ∞,θ∞ρ∞∇θ~-Pρρ~+ρ∗,θ~+θ∞ρ~+ρ∗-Pρρ∗,θ∞ρ∗∇ρ¯+1ρ~+ρ∗-1ρ∞b·∇b-12∇b2+1ρ∞b·∇b-12∇b2,(24)S~3=b~·∇u~-b~∇·u~-u~·∇b~,(25)S~4=-u~·∇θ~-θ~+θ∞Pθρ~+ρ∗,θ~+θ∞cvρ~+ρ∗-θ∞P∞ρ∞,θ∞cvρ∞∇·u~+κcvρ~+ρ∗-κcvρ∞Δθ~+1cvρ~+ρ∗-1cvρ∞Ψu~+1cvρ∞Ψu~.For obtaining a symmetric system, we denote(26)σx,t=ρ~x,t,ωx,t=P1ρ∞u~x,t,mx,t=P1ρ∞b~x,t,zx,t=P2ρ∞P1P3θ~x,t,μ1=μρ∞,μ2=μ+λρ∞,γ1=P1ρ∞,γ2=P2P3,κ1=κcvP2P1P3ρ∞,with(27)P1=Pρρ∞,θ∞ρ∞,P2=Pθρ∞,θ∞ρ∞,P3=θ∞Pθρ∞,θ∞cvρ∞.Then (20) and (21) can be reformulated as(28)σt+γ1∇·ω=S1,ωt-μ1Δω-μ2∇∇·ω+γ1∇σ+γ2∇z=S2,mt-νΔm=S3,zt-κ1Δz+γ2∇·ω=S4,σ,ω,m,zx,tt=0=σ0,ω0,m0,z0x,where S1=S~1, S2=P1/ρ∞S~2, S3=P1/ρ∞S~3, S4=P2ρ∞/P1P3S~4, and(29)σ0,ω0,m0,z0x=ρ0-ρ∗,ρ∞P1u0,ρ∞P1b0,P2ρ∞P1P3θ0-θ∞x→0,0,0,0,asx→∞.Then, we will focus on considering the global existence and time decay rates of the solution(ρ,u,b,θ) to the steady state (ρ∗,0,0,θ∞), that is, the existence and decay rates of the perturbed solution (σ,ω,m,z) to problem (28). First of all, some inequalities are listed as follows for late use.Lemma 3 (see [28]).
Letf∈H2(R3). Then(30)ifL∞≤C∇f1/2∇fH11/2≤C∇fH1;iifL6≤C∇f;iiifLq≤CfH1,2≤q≤6.Lemma 4 (see [24]).
Letr1,r2>0, then it holds that(31)∫0t1+t-τ-r11+τ-r2dτ≤Cr1,r21+t-min⁡r1,r2,r1+r2-1-η,for an arbitrarily small η>0.


## 3. Global Existence

We will use the energy method to establish the global existence of solutions to the problem (28) in H2-framework in this section. Let us define the solution space and the solution norm of the initial value problem (25) by(32)X0,T=σ,ω,m,z∣σ∈C00,T;H2R3∩C10,T;H1R3,ω,m,z∈C00,T;H2R3∩C10,T;L2R3,∇σ∈L20,T;H1R3,∇ω,m,z∈L20,T;H2R3,N0,T2=sup0≤t≤T⁡σ,ω,m,zt22+∫0T∇σt12+∇ω,m,zt22dt,where T∈[0,+∞]. By the standard continuity argument, the global existence of solutions to (25) will be obtained from the combination of the local existence result with some a priori estimates.Proposition 5 (local existence).
Suppose that the initial data satisfy(σ0,ω0,m0,z0)∈H2(R3) and (7). Then there exists a positive constant T1>0 depending on σ0,ω0,m0,z02, such that the initial value problem (25) has a unique solution (σ,ω,m,z)∈X(0,T1) which satisfies N(0,T1)≤2N(0,0).Proof.
The proof can be stated by using the standard iteration arguments. Refer to [2], for instance.Proposition 6 (a priori estimates).
Suppose that the initial value problem (28) has a solution (σ,ω,m,z)∈X(0,T), where T is a positive constant. Under the assumptions of Theorem 1, there exist a positive constant C0 and a small positive constant δ which are independent of T, such that if(33)sup0≤t≤T⁡σ,ω,m,zt2≤δ, where δ>max⁡{ε,C0ε} and ε is given by (7), then it holds that, for any t∈[0,T],(34)σ,ω,m,zt22+∫0t∇στ12+∇ω,m,zτ22dτ≤C0σ0,ω0,m0,z022<δ2,where C0 is independent of δ.Remark 7.
The global existence and uniqueness of the solutions stated in Theorem1 follow from Propositions 5 and 6. By (12), (33), and the Sobolev inequality, we have(35)12ρ∞≤σ+ρ∗≤2ρ∞.This will be used in following statement. Before proving Proposition 6, we supply Lemmas 8 and 9.Lemma 8.
For0≤k≤2, it holds that(36)12ddt∇kσ,ω,m,zt2+μ12∇k∇ωt2+μ22∇k∇·ωt2+ν2∇k∇mt2+k12∇k+1zt2≤Cδ∇σt12+∇ω,m,zt22, for any 0≤t≤T.Proof.
For each multi-indexα with α=k, operating ∂xα to the first 4 equations of (28), then multiplying the resulting equations by ∂xασ, ∂xαω, ∂xαm, and ∂xαz, respectively, summing up, and then integrating the result over R3, we obtain(37)12ddt∂xασ,ω,m,zt2+μ1∇∂xαωt2+μ2∇·∂xαωt2+ν∇∂xαmt2+κ1∇∂xαzt2=∂xασt,∂xαS1t+∂xαωt,∂xαS2t+∂xαmt,∂xαS3t+∂xαzt,∂xαS4t≜I1kt+I2kt+I3kt+I4kt, where the nonlinear terms in (22)-(25) have the following equivalence properties:(38)S1~∂iσωi+σ∂iωi+∂iρ¯ωi+ρ¯∂iωi,S2~ωi∂iωj+σ∂i∂iωj+σ∂j∂iωi+σ∂jσ+z∂jσ+σ∂jz+z∂jz+ρ¯∂i∂iωj+ρ¯∂j∂iωi+ρ¯∂jσ+σ∂jρ¯+z∂jρ¯+ρ¯∂jz+σ+ρ¯+1mi∂imj+mj∂imj,S3~ωi∂imj+mi∂iωj+mj∂iωi,S4~ωi∂iz+σ∂i∂iz+σ∂iωi+z∂iωi+σΨω+Ψω+ρ¯∂i∂iz+ρ¯∂iωi+ρ¯Ψω.
Whenk=0, by using the Hölder inequality, Lemma 3, (12), the a priori assumption (33), and the Young inequality, we obtain(39)I10t≤Cσt,∂iσωit+σt,σ∂iωit+σt,∂iρ¯ωit+σt,ρ¯∂iωit≤CσL6∂iσωiL3+σL6σL3∂iω+σ1+x1+x∂iρ¯L3ωiL6+σL6ρ¯L3∂iωi≤C∇σ2ω1+∇σσ1∇ω+∇σ1+x∂iρ¯1∇ω+∇σρ¯1∇ω≤Cδ∇σ2+∇ω2,where we have used the Hardy inequality as follows:(40)σ1+x≤∇σ.In a similar way, we get(41)I20t≤Cδ∇σ2+∇ω12+∇m2+∇z2,I30t≤Cδ∇ω2+∇m2,I40t≤Cδ∇z12+∇ω2.Putting (39), (41) into (37) and taking δ sufficiently small, we have(42)12ddtσ,ω,m,zt2+μ12∇ωt2+μ22∇·ωt2+ν2∇mt2+κ12∇zt2≤Cδ∇σt2+∇2ωt2+∇2zt2.
When1≤k≤2, the terms on the RHS of (37) can be estimated as follows. For I1k(t), from (38), we get(43)I1kt≤C∂xασt,∂xα∂iσωit+∂xασt,∂xασ∂iωit+∂xασt,∂xα∂iρ¯ωit+∂xασt,∂xαρ¯∂iωit.
For the first term on the RHS of (43), by using integration by parts, the Hölder inequality, the Young inequality, Lemma 3, and the a priori assumption (33), we have(44)∂xασt,∂xα∂iσωit≤C∂xασ,∂xα∂iσωi+C∑β≤α,β≤α-1Cαβ∂xασ,∂xβ∂iσ∂xα-βωi=C2∂xασ2,∂iωi+C∑β=0+∑1≤β≤α-1∂xασ,∂xβ∂iσ∂xα-βωi≤C∂iωiL∞∂xασ2+C∂xασ∂iσL3∂xαωiL6+δ∂xασ2+Cδ∑1≤β≤α-1∇∂xα-βωi2∂xβ∂iσ2≤Cδ∂xασ∇ω2+Cδ∂xασ∇∂xαω+δ∂xασ2+Cδ∇ω12≤Cδ∂xασ2+∇ω22, where the terms including the sum of β with 1≤β≤α-1 will be vanished if α=1. Similarly, we have(45)∂xασt,∂xασ∂iωit=∑β=0+∑1≤β≤α-1+∑β=α⁡Cαβ∂xασ,∂xα-βσ∂xβ∂iωi≤C∂iωiL∞∂xασ2+C∑1≤β≤α∂xασ∂xα-βσL6∂xβ∂iωiL3+C∂xασσL∞∂xα∂iω≤Cδ∂xασ∂iωi2+δ∂xασ2+Cδ∑1≤β≤α-1∂xα-β∇σ2∂xβ∂iωi12+Cδ∂xασ∂xα∂iωi≤Cδ∂xασ2+∇ω22. By the Hölder inequality, the Young inequality, Lemma 3, and (12), the third term on the RHS of (43) can be estimated as follows:(46)∂xασt,∂xα∂iρ¯ωit≤∑β≤α∂xασ,∂xβ∂iρ¯∂xα-βωi≤C∑β≤α∂xασ∂xβ∂iρ¯L3∂xα-βωiL6≤δ∂xασ2+Cδ∑β≤α∂xβ∂iρ¯12∂xα-β∇ωi2≤Cδ∂xασ2+∇ω22. By a similar argument, the fourth term on the RHS of (43) has following estimations:(47)∂xασt,∂xαρ¯∂iωit=∑β=0+∑1≤β≤α∂xασ,∂xβρ¯∂xα-β∂iωi≤C∂xασρ¯L∞∂xα∂iωi+C∑1≤β≤α∂xασ∂xβρ¯L3∂xα-β∂iωiL6≤Cδ∂xασ2+∇ω22.Therefore, putting (44)-(47) into (43) yields(48)I1kt≤Cδ∂xασ2+∇ω22.
ForI2k(t), from (38), it is obvious that(49)I2kt≤∂xαω,∂xαωi∂iωj+∂xαω,∂xασ∂i∂iωj+∂xαω,∂xασ∂j∂iωi+∂xαω,∂xασ∂jσ+∂xαω,∂xαz∂jσ+∂xαω,∂xασ∂jz+∂xαω,∂xαz∂jz+∂xαω,∂xαρ¯∂i∂iωj+∂xαω,∂xαρ¯∂j∂iωi+∂xαω,∂xαρ¯∂jσ+∂xαω,∂xασ∂jρ¯+∂xαω,∂xαz∂jρ¯+∂xαω,∂xαρ¯∂jz+∂xαω,∂xασ+ρ¯+1mj∂imj+mi∂imi.
For the second term on the RHS of (49), based on integration by parts, the Hölder inequality, Lemma 3, and (33), we have(50)∂xαωt,∂xασ∂i∂iωj=∑β=0+∑1≤β≤α∂xαω,∂xβσ∂xα-β∂i∂iωj≤∂xα∂iω2,σ+∂xαω,∂iσ∂xα∂iωj+∑1≤β≤α-1∂xαω,∂xβσ∂xα-β∂i∂iωj+∂xαω,∂xασ∂i∂iωj≤CσL∞∂xα∂iω2+∂xαωL6∂iσL3∂xα∂iωj+∑1≤β≤α-1∂xαωL6∂xβσL3∂xα-β∂i∂iωj+∂xαωL6∂xασ∂i∂iωjL3≤Cδ∂xα∇ω2+∇2ω12.
For the 11th term on the RHS of (49), by using the Hölder inequality, the Hardy inequality, Lemma 3, and (12), we obtain(51)∂xαωt,∂xασ∂jρ¯=∑β=0+∑1≤β≤α⁡∂xαωt,∂xβσ∂xα-β∂jρ¯≤C∂xαωL61+x∂xα∂jρ¯L3σ1+x+∑1≤β≤α∂xαωL6∂xβσ∂xα-β∂jρ¯L3≤Cδ∂xα∇ω∇σ+Cδ∑1≤β≤α∂xα∇ω∂xβσ≤Cδ∂xα∇ω2+∇σ12,∂xαωt,∂xασmj∂imj=∑β=0+∑1≤β≤α⁡∂xαωt,∂xβσ∂xα-βσmj∂imj≤CσL∞∂xαmj∂imj∂xαωt+∑1≤β≤α∂xαωL6∂xβσ∂xα-βmj∂imjL3≤CσL∞mL∞∂xαω∂xα∇m+∑1≤β≤α∂xα∇ωmL6∂xβσ∂xα-β∇mL6≤Cδ∂xα∇ω2+∂xα∇m2,where we used (33). Similarly,(52)∂xαωt,∂xασmi∂imj≤Cδ∂xα∇ω2+∂xα∇m2,∂xαω,∂xαmj∂imj+mi∂imi≤Cδ∂xα∇ω2+∂xα∇m2.The other terms on the RHS of (49) can be estimated in some similar way.(53)I2kt≤Cδ∇σ12+∂xαω2+∂xα∇ω2+∂xα∇m2+∇ω22+∇z12.
Now, let us estimateI3k(t),I4k(t). Similar to the estimate before, I3k(t) and I3k(t) hold that(54)I3kt≤Cδ∂xαm2+∂xα∇m2,(55)I4kt≤Cδ∂xαz2+∂xα∇z2+∇ω22+∇z22.
Thus (37) together with (48), (53), (54), and (55) yields(56)12ddt∂xασ,ω,m,zt2+μ12∇∂xαωt2+μ22∇·∂xαωt2+ν2∇∂xαmt2+κ12∇∂xαzt2≤Cδ∂xασ,ω,m,z2+∂xα∇ω,m,z2+∇σ12+∇ω,m,z22,where we have taken δ>0 sufficiently small. Therefore, (42) and (56) give (36).Now, the following lemma will give the dissipative estimates for∇σ.Lemma 9.
For0≤k≤1, it holds that(57)ddt∇kωt,∇k∇σt+γ12∇k∇σt2≤C∇k∇ωt12+∇k∇zt2+Cδ∇σt12+∇ω,m,zt22, for any 0≤t≤T.Proof.
From(28)2, we get(58)γ1∇σ=-ωt+μ1△ω+μ2∇∇·ω-γ2∇z+S2.By applying ∂xα(α=k) to (58), multiplying it by ∇∂xασ, and then integrating over R3, we have(59)γ1∇∂xασ2=-∂xαωt,∇∂xασ+μ1∂xα△ω,∇∂xασ+μ2∂xα∇∇·ω,∇∂xασ-γ2∂xα∇z,∇∂xασ+∂xαS2,∇∂xασ.
Using integration by parts and(28)1, we obtain(60)-∂xαωt,∇∂xασ=-ddt∂xαω,∇∂xασ+∂xαω,∇∂xασt=-ddt∂xαω,∇∂xασ-γ1∂xαω,∇∂xα∇·ω+∂xαω,∇∂xαS1. Adding (59) and (60) gives(61)γ1∇∂xασ2+ddt∂xαω,∇∂xασ=μ1∂xα△ω,∇∂xασ+μ2∂xα∇∇·ω,∇∂xασ-γ2∂xα∇z,∇∂xασ+∂xαS2,∇∂xασ-γ1∂xαω,∇∂xα∇·ω+∂xαω,∇∂xαS1.Then by using the Young inequality and integration by parts, it follows from the above equality that(62)3γ14∇∂xασ2+ddt∂xαω,∇∂xασ≤C∂xα∇2ω2+∂xα∇·ω2+∂xα∇z2+C∇∂xαω,∂xαS1+C∇∂xασ,∂xαS2.Similar to the estimates of I1k(t) and I2k(t) in the proof of Lemma 8, we have(63)∇∂xαω,∂xαS1≤C∇∂xαω2+Cδ∇σ2+∇ω22,(64)∇∂xασ,∂xαS2≤Cδ∇∂xασ2+∇σ12+∇ω22+∇m22+∇z12.Putting (63) and (64) into (62) and taking δ>0 sufficiently small yield(65)γ12∇∂xασ2+ddt∂xαω,∇∂xασ≤C∂xα∇ω12+∂xα∇z2+Cδ∇σ12+∇ω22+∇m22+∇z12.Thus, the proof of Lemma 9 is finished.Let0≤l≤1. By summing up (36) from k=l to 2, since δ is small, there exists a constant C1>0 such that(66)ddt∑l≤k≤2∇kσ,ω,m,zt2+∑l≤k≤2∇k+1ω,m,zt2≤C1δ∇σt12.Summing up (57) for from k=l to 1, since δ is small, we have(67)ddt∑l≤k≤1∇kωt,∇k∇σt+C2∑l≤k≤1∇k∇σt2≤C∑l≤k≤2∇k+1ωt2+∑l≤k≤1∇k+1zt2+δ∇σ,ω,m,zt2, where C2>0 is a constant. Multiplying (67) by 2C1δ/C2<1/2, adding it with (66), taking l=0, and using the smallness of δ, we have(68)ddt∑0≤k≤2∇kσ,ω,m,zt2+2C1δC2∑0≤k≤1∇kωt,∇k∇σt+C∑0≤k≤1∇k+1σt2+∑0≤k≤2∇k+1ω,m,zt2≤0.Integrating (68) with respect to t and using the smallness of δ, we get (34). The proof of Proposition 6 is complete.


## 4. Decay Rates

In this section, we get decay rates of solutions to problem (25). Let A be the following 8×8 matrix of the differential operators of the form(69)A=0γ1div00γ1∇-μ1Δ-μ2∇div0γ2∇00-νΔ00γ2div0-κ1Δ.If we set(70)U~=σ~,ω~,m~,z~t,U0=σ0,ω0,m0,z0t,the linearized problem of (25) is written as(71)U~t+AU~=0,fort>0,U~t=0=U0.By taking the Fourier transform of (71) with respect to the x-variable and solving the ordinary differential equation with respect to t, we have(72)U~t=EtU0,t≥0, where E(t)=e-tA(t≥0) is the semigroup generated by the linear operator -A. We define E(t)f=F-1(e-tA^ξf^(ξ,t)) for function f(x,t) with(73)A^ξ=0-1γ1ξt00-1γ1ξμ1ξ2+μ2ξξt0-1γ2ξ00νξ200-1γ2ξt0κ¯ξ2,forξ=ξ1ξ2ξ3ξ4.In what following, we denote the solutions to the nonlinear problem (25) by(74)Ut=σt,ωt,mt,ztt. Then (28) can be rewritten as follows:(75)Ut+AU=SU,fort>0,Ut=0=U0,where S(U)≜(S1,S2,S3,S4)t. By using the Duhamel principle, we have(76)Ut=EtU0+∫0tEt-τSUτdτ,t≥0. Let φ0(ξ) be a function in C0∞(R3) such that(77)φ0ξ=1,forξ≤R2,0,forξ≥R, where R is some fixed positive constant. Based on the Fourier transform and (77), we can define the low frequency and the high frequency decomposition (fL(x),fH(x)) for f(x) as follows:(78)fL=F-1φ0ξf^ξ,fH=f-fL.By noticing definitions (77) and (78), using the Plancherel theorem, we can obtain the following estimates:(79)∇f≤C∇fL+∇fH,(80)C∇fH≤∇kfH,C∇kfH≤∇kf,fork≥1.Denote UL(t)≜F-1(φ0(ξ)U^(ξ,t)) and EL(t)f≜F-1(φ0(ξ)e-tA^(ξ)f^(ξ,t)) as the definition (78).Then from (76), we have(81)ULt=ELtU0+∫0tELt-τSUτdτ.To estimateUL(t), we need the Lp-Lq type of the time decay estimates on the low-frequency part of the semigroup E(t). Notice that the first, the second, and the fourth equations in (71) have been studied in [29]. The magnetic field H(t,x) satisfies the heat equation, which can be solved in Fourier space, H^=exp(-ξ2t)H^0, whose decay estimate has been proved in many papers, for instance, [26, 27]. Then the semigroup E(t) has the following properties on the decay in time.Lemma 10 (see [29]).
Letk≥0 be integers and 1≤p≤2≤q≤∞, then for any t>0, it holds that(82)∇kELtfLq≤Ck,p,q1+t-3/21/p-1/q-k/2fLp.


### 4.1. Decay Estimates of the Low-Frequency Part

In this subsection, we have the decay estimates ofUL(t) with the Lp-Lq type estimate (82) as follows.Lemma 11.
Fork≥0, it holds that(83)∇kULt≤C1+t-3/21/p-1/2-k/2K0+Cδ+ε∫0t1+t-τ-3/4-k/2∇Uτ1dτ, where K0=U0Lp is finite.Proof.
From (81) and Lemma 10, we have(84)∇kULt≤C1+t-3/21/p-1/2-k/2U0Lp+C∫0t1+t-τ3/4-k/2SUτL1dτ. For the terms including ρ¯ on the RHS of (84), by using the Hölder inequality, (12), and the Hardy inequality, we get(85)∇ρ¯·ωL1≤1+x∇ρ¯ω1+x≤Cε∇ω,ρ¯∇·ωL1≤ρ¯∇·ω≤Cε∇ω.Thus, it holds that(86)∇·ρ¯ωL1≤Cε∇ω.Similarly, the above inequalities give that(87)ρ¯m·∇m-12∇m2L1≤ε∇m1,∇ρ¯σL1≤Cε∇σ1,∇ρ¯zL1≤Cε∇z1,ρ¯∇∇·ωL1≤Cε∇2ω,ρ¯ΔzL1≤Cε∇2z.The other terms can be estimated easily. Then, from (38) and the a priori assumption (33), we have(88)SUtL1≤Cσ,ω,m1∇σ,ω,m,z1+∇·ρ¯ωL1+∇ρ¯σL1+ρ¯m·∇m-12∇m2L1+∇ρ¯zL1+ρ¯∇2ω,zL1≤Cδ+ε∇σ,ω,m,z1.Thus, putting (88) into (84), we get (83).


### 4.2. Decay Rates for the Nonlinear Problem

In this subsection, from the a priori estimates and the low frequency and high frequency decomposition (78), we derive a new Lyapunov-type inequality, firstly. From (66) and (67), by taking l=1, similar to the estimate (68), we get(89)dHtdt+∇2σ,ω,m,zt2+∇2ω,m,zt12≤Cδ∇σ,ω,m,zt2,where H(t)∑1≤k≤2∇kσ,ω,m,zt2+〈∇ω(t),∇2σ(t)〉 is an energy functional which is equivalent to ∇(σ,ω,z)(t)12; that means, there exists a positive constant C3 such that(90)1C3∇σ,ω,m,zt12≤Ht≤C3∇σ,ω,m,zt12.By using (79) and (80), from (89), there exists a constant C4>0 such that(91)dHtdt+C4∇σH,ωH,mH,zHt2+12∇2σ,ω,m,zt2+12∇2ω,m,zt12≤Cδ∇σL,ωL,mL,zLt2+∇σH,ωH,mH,zHt2. It follows from (91) and the smallness of δ that(92)dHtdt+C5∇σH,ωH,mH,zHt2+12∇2σ,ω,m,zt2+12∇2ω,m,zt12≤Cδ∇σL,ωL,mL,zLt2, where C5>0 is a constant. Adding C5∇σL,ωL,mL,zLt2 to the both sides of (92), with the definition of H(t), there exists a constant D>0 such that(93)dHtdt+DHt≤C∇ULt2.To prove the decay estimates stated in Theorem1, we first define(94)Mt=sup0≤τ≤t⁡1+τ31/p-1/2+1Hτ.M(t) is obviously nondecreasing and(95)∇Uτ1≤CHτ≤C1+τ-3/21/p-1/2-1/2Mt,0≤τ≤t. Then, from (83), by using Lemma 4 and (95), we obtain(96)∇ULt≤CK01+t-3/21/p-1/2-1/2+Cδ+ε∫0t1+t-τ-5/41+τ-3/21/p-1/2-1/2dτMt≤C1+t-3/21/p-1/2-1/2K0+δ+εMt.By Gronwall’s inequality and (96), from (93), hence(97)Ht≤H0e-Dt+C2∫0te-Dt-τ∇ULτ2dτ≤H0e-Dt+C∫0te-Dt-τ1+τ-3/21/p-1/2-1dτK02+δ+ε2Mt≤1+t-3/21/p-1/2-1H0+K02+δ+ε2Mt, where H(0) is equivalent to ∇(σ0,ω0,m0,z0)12.In terms of the definition ofM(t), it follows from (97) that(98)Mt≤CH0+K02+Cδ+ε2Mt,which implies that if δ and ε are sufficiently small then(99)Mt≤CH0+K02.By applying the definition of H(t) and M(t), we have(100)∇σ,ω,m,zt1≤C1+t-3/21/p-1/2-1/2.Then, we have (9).Then, we define the temporal energy functional(101)H1t=σ,ω,m,zt22+2C1δC2∑0≤k≤2∇kσt,∇k∇ωt,where H1(t) is equivalent to (σ,ω,m,z)(t)22, for the positive constants δ can be sufficiently small. Then, from (68), we have(102)dH1tdt+∇σ,ω,m,zt2+∇ω,m,zt12≤0.Similar to the estimate (93), from (102), there exists a positive constant D1 such that(103)dH1tdt+D1H1t≤CULt2.Takingk=0 in (83) and using (100), for 1≤p≤2, we obtain(104)ULt≤C1+t-3/21/p-1/2K0+Cδ+ε∫0t1+t-τ-3/41+τ-3/21/p-1/2-1/2dτ≤C1+t-3/21/p-1/2K0+C, where let r1=3/4,r2=(3/2)(1/p-1/2)+1/2 and η=1/4 in Lemma 4.From (103), by using Gronwall’s inequality and (104), we get(105)H1t≤H10e-D1t+C∫0te-D1t-τULτ2dτ≤H0e-D1t+C+CK02∫0te-D1t-τ1+τ-31/p-1/2dτ≤C1+t-31/p-1/2H10+C+K02, where H1(0) is equivalent to (σ0,ω0,m0,z0)22. Then, it follows from the definition of H1(t) that(106)σ,ω,m,zt≤C1+t-3/21/p-1/2. Finally, by the interpolation, for any 2≤q≤6, plugging (106) into (100) yields(107)UtLq≤UtL6η0Ut1-η0≤C∇Utη0Ut1-η0≤C1+t-3/21/p-1/q,whereη0=3(q-2)/2q. Then, we get (10).On the other hand, by using (25), Lemma 3, (9), and (12), we have(108)∂tσ,ω,m,ztL2≤C∇·ωtL2+S1tL2+C∇σtL2+C∇ztL2+CΔωtL2+C∇∇·ωtL2+S2tL2+CΔmtL2+S3tL2+CΔztL2+S4tL2≤C∇σtL2+∇ωtH1+∇mtH1+∇ztH1≤C1+t-3/21/p-1/2-1/2,for any 0≤t≤T.This completes the proof of Theorem1.


## 4.1. Decay Estimates of the Low-Frequency Part

In this subsection, we have the decay estimates ofUL(t) with the Lp-Lq type estimate (82) as follows.Lemma 11.
Fork≥0, it holds that(83)∇kULt≤C1+t-3/21/p-1/2-k/2K0+Cδ+ε∫0t1+t-τ-3/4-k/2∇Uτ1dτ, where K0=U0Lp is finite.Proof.
From (81) and Lemma 10, we have(84)∇kULt≤C1+t-3/21/p-1/2-k/2U0Lp+C∫0t1+t-τ3/4-k/2SUτL1dτ. For the terms including ρ¯ on the RHS of (84), by using the Hölder inequality, (12), and the Hardy inequality, we get(85)∇ρ¯·ωL1≤1+x∇ρ¯ω1+x≤Cε∇ω,ρ¯∇·ωL1≤ρ¯∇·ω≤Cε∇ω.Thus, it holds that(86)∇·ρ¯ωL1≤Cε∇ω.Similarly, the above inequalities give that(87)ρ¯m·∇m-12∇m2L1≤ε∇m1,∇ρ¯σL1≤Cε∇σ1,∇ρ¯zL1≤Cε∇z1,ρ¯∇∇·ωL1≤Cε∇2ω,ρ¯ΔzL1≤Cε∇2z.The other terms can be estimated easily. Then, from (38) and the a priori assumption (33), we have(88)SUtL1≤Cσ,ω,m1∇σ,ω,m,z1+∇·ρ¯ωL1+∇ρ¯σL1+ρ¯m·∇m-12∇m2L1+∇ρ¯zL1+ρ¯∇2ω,zL1≤Cδ+ε∇σ,ω,m,z1.Thus, putting (88) into (84), we get (83).


## 4.2. Decay Rates for the Nonlinear Problem

In this subsection, from the a priori estimates and the low frequency and high frequency decomposition (78), we derive a new Lyapunov-type inequality, firstly. From (66) and (67), by taking l=1, similar to the estimate (68), we get(89)dHtdt+∇2σ,ω,m,zt2+∇2ω,m,zt12≤Cδ∇σ,ω,m,zt2,where H(t)∑1≤k≤2∇kσ,ω,m,zt2+〈∇ω(t),∇2σ(t)〉 is an energy functional which is equivalent to ∇(σ,ω,z)(t)12; that means, there exists a positive constant C3 such that(90)1C3∇σ,ω,m,zt12≤Ht≤C3∇σ,ω,m,zt12.By using (79) and (80), from (89), there exists a constant C4>0 such that(91)dHtdt+C4∇σH,ωH,mH,zHt2+12∇2σ,ω,m,zt2+12∇2ω,m,zt12≤Cδ∇σL,ωL,mL,zLt2+∇σH,ωH,mH,zHt2. It follows from (91) and the smallness of δ that(92)dHtdt+C5∇σH,ωH,mH,zHt2+12∇2σ,ω,m,zt2+12∇2ω,m,zt12≤Cδ∇σL,ωL,mL,zLt2, where C5>0 is a constant. Adding C5∇σL,ωL,mL,zLt2 to the both sides of (92), with the definition of H(t), there exists a constant D>0 such that(93)dHtdt+DHt≤C∇ULt2.To prove the decay estimates stated in Theorem1, we first define(94)Mt=sup0≤τ≤t⁡1+τ31/p-1/2+1Hτ.M(t) is obviously nondecreasing and(95)∇Uτ1≤CHτ≤C1+τ-3/21/p-1/2-1/2Mt,0≤τ≤t. Then, from (83), by using Lemma 4 and (95), we obtain(96)∇ULt≤CK01+t-3/21/p-1/2-1/2+Cδ+ε∫0t1+t-τ-5/41+τ-3/21/p-1/2-1/2dτMt≤C1+t-3/21/p-1/2-1/2K0+δ+εMt.By Gronwall’s inequality and (96), from (93), hence(97)Ht≤H0e-Dt+C2∫0te-Dt-τ∇ULτ2dτ≤H0e-Dt+C∫0te-Dt-τ1+τ-3/21/p-1/2-1dτK02+δ+ε2Mt≤1+t-3/21/p-1/2-1H0+K02+δ+ε2Mt, where H(0) is equivalent to ∇(σ0,ω0,m0,z0)12.In terms of the definition ofM(t), it follows from (97) that(98)Mt≤CH0+K02+Cδ+ε2Mt,which implies that if δ and ε are sufficiently small then(99)Mt≤CH0+K02.By applying the definition of H(t) and M(t), we have(100)∇σ,ω,m,zt1≤C1+t-3/21/p-1/2-1/2.Then, we have (9).Then, we define the temporal energy functional(101)H1t=σ,ω,m,zt22+2C1δC2∑0≤k≤2∇kσt,∇k∇ωt,where H1(t) is equivalent to (σ,ω,m,z)(t)22, for the positive constants δ can be sufficiently small. Then, from (68), we have(102)dH1tdt+∇σ,ω,m,zt2+∇ω,m,zt12≤0.Similar to the estimate (93), from (102), there exists a positive constant D1 such that(103)dH1tdt+D1H1t≤CULt2.Takingk=0 in (83) and using (100), for 1≤p≤2, we obtain(104)ULt≤C1+t-3/21/p-1/2K0+Cδ+ε∫0t1+t-τ-3/41+τ-3/21/p-1/2-1/2dτ≤C1+t-3/21/p-1/2K0+C, where let r1=3/4,r2=(3/2)(1/p-1/2)+1/2 and η=1/4 in Lemma 4.From (103), by using Gronwall’s inequality and (104), we get(105)H1t≤H10e-D1t+C∫0te-D1t-τULτ2dτ≤H0e-D1t+C+CK02∫0te-D1t-τ1+τ-31/p-1/2dτ≤C1+t-31/p-1/2H10+C+K02, where H1(0) is equivalent to (σ0,ω0,m0,z0)22. Then, it follows from the definition of H1(t) that(106)σ,ω,m,zt≤C1+t-3/21/p-1/2. Finally, by the interpolation, for any 2≤q≤6, plugging (106) into (100) yields(107)UtLq≤UtL6η0Ut1-η0≤C∇Utη0Ut1-η0≤C1+t-3/21/p-1/q,whereη0=3(q-2)/2q. Then, we get (10).On the other hand, by using (25), Lemma 3, (9), and (12), we have(108)∂tσ,ω,m,ztL2≤C∇·ωtL2+S1tL2+C∇σtL2+C∇ztL2+CΔωtL2+C∇∇·ωtL2+S2tL2+CΔmtL2+S3tL2+CΔztL2+S4tL2≤C∇σtL2+∇ωtH1+∇mtH1+∇ztH1≤C1+t-3/21/p-1/2-1/2,for any 0≤t≤T.This completes the proof of Theorem1.



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*Source: 6710508-2019-04-21.xml*