# Global Well Posedness for the Thermally Radiative Magnetohydrodynamic Equations in 3D

Authors: Peng Jiang; Fei Yu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we study the thermally radiative magnetohydrodynamic equations in 3D, which describe the dynamical behaviors of magnetized fluids that have nonignorable energy and momentum exchange with radiation under the nonlocal thermal equilibrium case. By using exquisite energy estimate, global existence and uniqueness of classical solutions to Cauchy problem inℝ3 or T3 are established when initial data is a small perturbation of some given equilibrium. We can further prove that the rates of convergence of solution toward the equilibrium state are algebraic in ℝ3 and exponential in T3 under some additional conditions on initial data. The proof is based on the Fourier multiplier technique.


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## Body

## 1. Introduction

In the study of plasma physics, due to the high temperature and high pressure environment, the motion of charged particles flow is usually regarded as compressible fluids, and their dynamics is very often shaped and controlled by magnetic fields and high temperature radiation effects. Meanwhile, it is known that the radiation energy is carried by photons. When the distribution of photon is almost isotropic, based on the standard hydrodynamics, such dynamics can be described by the following 3D thermally radiative magnetohydrodynamic equations (cf. [1, 2]):
(1)ρt+divρu=0,ρut+u·∇u+∇P=μΔu+λ+μ∇divu+H·∇H−12∇H2,Ht−νΔH+u·∇H−H·∇u+Hdivu=0,divH=0,cνρθt+u·∇θ=κΔθ−Pdivu+λdivu2+2μD·D+ν∇×H2−θ4+n,nt−Δn=θ4−n,where ρ=ρx,t>0, u=ux,t∈ℝ3, θ=θx,t>0, H=Hx,t∈ℝ3, n=nx,t≥0 for x∈Ω,t≥0 denote the mass density, velocity field of the fluid, mass temperature, magnetic field, and radiation field, respectively, and P=Rρθ denotes the material pressure. The spatial domain Ω=ℝ3 or T3. The parameter R>0 is the perfect gas constant, cν>0 is the specific heat at constant volume, and κ is the heat conductivity coefficient; λ and μ are the viscosity coefficients of the flow satisfying μ>0 and 3λ+2μ≥0 and ν>0 is the magnetic diffusion coefficient. Throughout this paper, we assume that λ,μ,ν,κ are all positive constants. D=Du is the deformation tensor
(2)Dij≔12∂ui∂xj+∂uj∂xi,D·D≔∑i,j=13Dij2.When the magnetic is ignored (i.e.,H=0 in (1)), system (1) can be reduced to the nonequilibrium diffusion approximation model in radiation hydrodynamics. This model describes the energy flow due to radiative process in a semiquantitative sense and is particularly accurate if the specific intensity of radiation is almost isotropic (cf. [3–5]). There are some mathematical results on this model. For the global existence of smooth solution for one-dimensional case, see [6]; for the global well-posedness and large time behavior of classical solutions for multidimensional case, see [7]. For the inviscid case, in [8], the authors considered a 1D model and showed the existence of shock profiles for inviscid nonequilibrium gases provided that the initial strength is suitably small. For the local existence of smooth solutions for multidimensional system, see [9]. System ((1)) is the compressible MHD equations coupled with the radiative transport equation with nonlocal terms and is very difficult to solve both numerically and analytically. Ducomet and Feireisl consider the thermally radiative MHD system first and show the existence of global weak solutions for the multidimensional case in [10] (also see, for instance, Li and Guo [11]). For the one-dimensional case, this model has been studied by many authors under the various growth constraints on the heat conductivity κ [12–14].In this paper, we are focused on the asymptotic and global existence of classical solutions of system (1) with the initial data:
(3)ρ,u,H,θ,nt=0=ρ0,u0,H0,θ0,n0x,x∈Ω.It is easy to check thatρ,u,H,θ,n≡1,0,0,1,1 is an equilibrium state of (1). Therefore, it is natural to introduce the transforms
(4)ρ=1+ϱ,θ=1+Θ,n=1+η.Without loss of generality, we assume the positive constantsR=cν=κ=μ=λ=ν≡1. Then we can rewrite the system (1) as
(5)ϱt+1+ϱdivu+∇ϱ·u=0,(6)ut+u·∇u+1+Θ1+ϱ∇ρ+∇Θ=Δu1+ϱ+2∇divu1+ϱ+H·∇H−1/2∇H21+ϱ,(7)Ht+u·∇H−H·∇u+Hdivu=ΔH,divH=0,(8)Θt+u·∇Θ=ΔΘ1+ϱ−1+Θdivu+divu21+ϱ+2D·D1+ϱ−1+Θ41+ϱ+1+η1+ϱ+∇×H21+ϱ,(9)ηt−Δη=1+Θ4−1+η.with initial data
(10)ϱ,u,H,Θ,ηt=0=ϱ0,u0,H0,Θ0,η0x=ρ0−1,u0,H0,θ0−1,n0−1x.Then, the main results in this paper read as follows:Theorem 1.
LetΩ=ℝ3. Suppose that ∥ϱ,u0,H0,Θ0,η0∥H4 is small enough. Then, the Cauchy problem (5)–(10) admits a unique global classical solution ϱ,u,H,Θ,η satisfying
(11)ϱ,u,H,Θ,η∈C0,∞;H4,supt≥0∥ϱ,u,H,Θ,η∥H4≤C∥ϱ0,u0,H0,Θ0,η0∥H4.Theorem 2.
Under the conditions of Theorem1, if we further assume that ∥ρ0,u0,H0,Θ0,η0∥L1 is sufficiently small, then
(12)∥ϱ,u,H,Θ,η∥L2≤C1+t−3/4,(13)∥∇ϱ,u,H,Θ,η∥H3≤C1+t−5/4,for all t≥0.Theorem 3.
LetΩ=T3. Suppose that ∥ϱ0,u0,H0,Θ0,η0∥H4 is small enough, and
(14)∫T3ϱ0dx=0,∫T31+ϱ0u0dx=0,∫T3H0dx=0,∫T3121+ϱ0u02+12H02+ϱ0+Θ0+ρ0Θ0+η0dx=0.
Then, the problem (5)–(10) admits a unique global solution ϱ,u,H,Θ,η satisfying
(15)ϱ,u,H,Θ,η∈C0,∞;H4,supt≥0eγt∥ρ,u,H,Θ,η∥H4≤C∥ρ0,u0,H0,Θ0,η0∥H4,where γ>0 is a constant.The proof of the global existence of classical solution to (5)–(10) relies on the global a priori estimates together with the local existence of classical solutions and continuum argument. The main difficulty in establishing prior estimates in high-order Sobolev spaces is how to control the linear term in (5)–(9), such as 4Θ/1+ϱ, η/1+ϱ in (8) and 4Θ,η in (1.6). We develop the method in [15] and use the structure of system (5)–(9) itself to construct novel dissipation term 4Θ−η to overcome this difficulty. To prove Theorem 2, we first use a Fourier multiplier technique to establish the Lp−Lq time-decay property of linearized system (47)–(52). Then, the time decay rate can be given by combining the global a priori estimate obtained in Theorem 1 and the above Lp−Lq property and applying the energy estimate technique to the nonlinear problem (5)–(10), whose solutions can be represented by the solution-semigroup operator for the linearized system (47)–(52) by using the Duhamel principle. Here, some nonlinear terms of magnetic field H involved in (6) and (8) may lead to difficulties to gain the desired rate of convergence of solutions. Thus, we will construct some novel functionals such as (79) and (80) and adopt with modification some techniques motivated by [16–18] combined to some vector analysis formula to obtain expected decay rates.The remainder of this paper is organized as follows. In Section2, we derive the uniform-in-time a priori estimates and then establish the existence of a global classical solution. In Section 3, we investigate the decay rates of solutions. In Section 4, we adapt our proof to the periodic domain case. Throughout this paper C denotes a positive (generally large) constant and γ a positive (generally small) constant, where both C and γ may take different values in different places. The symbol A~B means CA≤B≤1/CA for a generic constant C>0. For simplicity, we shall use ∥·∥ to denote norm L2ℝ3.


## 2. Global Existence

In what follows, our analysis is based on the Cauchy problem (5)–(10). To obtain the global existence, the most important point is to establish the uniform-in-time a priori estimates.


### 2.1. A Priori Estimates

Now, we begin to establish the global a priori estimates in the case of the whole spaceℝ3 under the assumption
(16)supt≥0∥ϱ,u,H,Θ,η∥H4≤δ,where 0<δ<1 is a generic constant small enough and ρ,u,H,Θ,η is the smooth solution to the Cauchy problem (5)–(10) on 0≤t<T for T>0. Firstly, we list two important lemmas in Sobolev space.Lemma 4 (see [16, 19]).
There exist a positiveconstant C, such that for any f,g∈H4ℝ3 and any multi-index α with 1≤∣α∣≤4,
(17)∥f∥L∞ℝ3≤C∥∇f∥L2ℝ31/2∥∇2f∥L2ℝ31/2,∥f∥Lpℝ3≤C∥∇f∥H1ℝ3,where2≤p≤6,∥fg∥H3ℝ3≤C∥f∥H3ℝ3∥∇g∥H3ℝ3,∥∂αfg∥L2ℝ3≤C∥∇f∥H3ℝ3∥∇g∥H3ℝ3.Lemma 5 (Moser-type calculus inequalities) (see [20]).
Lets≥1 be an integer. Suppose f∈Hsℝ3, ∇f∈L∞ℝ3 and g∈Hs−1ℝ3⋂L∞ℝ3. Then, for all multi-index α with ∣α∣≤s, we have
(18)∥∂αfg−f∂αg∥≤Cs∥∇f∥L∞∥Ds−1g∥+∥Dsf∥∥g∥L∞,∥Dsf∥=∑∣α∣=s∥∂αf∥.
Then, we begin to give the priori estimate ofϱ,u,H,Θ,η.Lemma 6.
Suppose thatϱ,u,H,Θ,η be a smooth solution to (5)–(10). Then, for all0≤t≤T with any fixed T>0, it holds
(19)12ddt∥2ρ,2u,H,2Θ,η∥2+γ∥∇u,H,Θ,η∥2+∥divu∥2+∥4Θ−η∥2≤C∥ϱ,u,H,Θ,η∥H2+∥ϱ,u,H,Θ,η∥H22∥∇ϱ,u,H,Θ,η∥2+∥divu∥2+∥4Θ−η∥2.Proof.
Multiplying (5)–(9) by 4ρ,4u,H,4Θ, and η and then taking integration and summation, we get
(20)12ddt∥2ϱ∥2+∥2u∥2+∥H∥2+∥2Θ∥2+∥η∥2+∫4∇u21+ρdx+∫8divu21+ϱdx+∫4∇Θ21+ϱdx+∥∇H∥2+∥∇η∥2+∥4Θ−η∥2=−∫2ρ2divudx+∫4ρ−Θ1+ϱ∇ρ·udx−∫4u·∇u·udx−∫4∇11+ϱ·∇u·udx−∫8∇11+ϱ·udivudx−∫4Θ2divudx−∫4Θ∇Θ·udx+∫4Θdivu21+ϱdx+∫8ΘD·D1+ϱdx−∫4Θ∇11+ϱ·∇Θdx+∫4Θ∇×H21+ϱdx+∫4ρΘ4Θ−η1+ϱdx−∫4Θ−η6Θ2+4Θ3+Θ41+ϱdx+∫ρη6Θ2+4Θ3+Θ41+ϱdx+∫H·∇H−12∇H24u1+ϱdx+∫−u·∇H+H·∇H−Hdivu·Hdx≡∑j=116Ij.
ForI1 to I12, using Hölder’s and Sobolev’s inequalities, we have
(21)I1≤C∥ϱ∥L3∥divu∥∥ϱ∥L6≤C∥ρ∥H1∥∇ϱ∥∥divu∥,I2≤C∥ϱ∥L3+∥Θ∥L3∥∇ϱ∥∥u∥L6≤C∥ϱ∥H1+∥Θ∥H1∥∇ϱ∥∥∇u∥,I3≤C∥u∥L3∥∇u∥∥u∥L6≤C∥u∥H1∥∇u∥2,I4+I5≤C∥u∥L6∥∇u∥L3+∥divu∥L3∇11+ρ≤C∥u∥H2∥∇ϱ∥∥∇u∥,I6≤C∥Θ∥L3∥divu∥∥Θ∥L6≤C∥Θ∥H1∥divu∥∥∇Θ∥,I7≤C∥u∥L3∥∇Θ∥∥Θ∥L6≤C∥u∥H1∥∇Θ∥2,I8≤C∥Θ∥L∞∥divu∥2≤C∥Θ∥H2∥divu∥2,I9=∑i,j=13∫2Θuxji+uxij21+ϱdx≤C∑i,j=13∥Θ∥L∞∥uxji∥∥uxij∥≤C∥Θ∥H2∥∇u∥2,I10≤C∇11+ϱ∥∇Θ∥L3∥Θ∥L6≤C∥Θ∥H2∥∇ϱ∥∥∇Θ∥,I11≤C∥Θ∥L∞∥∇×H∥2≤C∥Θ∥H2∥∇H∥2,I12≤C∥ϱ∥L3∥4Θ−η∥∥Θ∥L6≤C∥ϱ∥H1∥4Θ−η∥∥∇Θ∥.
ForI13, I14, under the assumption (16), one also has
(22)I13≤C∥4Θ−η∥∥Θ∥L3∥Θ∥L6+∥Θ2∥L3∥Θ∥L6+∥Θ∥L∞∥Θ2∥L3∥Θ∥L6≤C∥4Θ−η∥∥Θ∥H1∥∇Θ∥+∥∇Θ∥3+∥Θ∥H2∥∇Θ∥3≤C∥Θ∥H2+∥Θ∥H22∥4Θ−η∥2+∥∇Θ∥2,I14≤C∥ϱ∥H1∥η∥H1∥∇Θ∥2+∥∇Θ∥3+∥∇Θ∥4≤C∥ϱ∥H12+∥Θ∥H12+∥η∥H12∥∇Θ∥2.
At last, forI15 and I16, we have
(23)I15≤C∥u∥L3∥∇H∥∥H∥L6+C∥H∥L3∥divu∥∥H∥L6≤C∥u∥H1∥∇H∥2+C∥H∥H1∥∇H∥∥∇u∥,I16≤C∥u∥L3∥∇H∥∥H∥L6+C∥H∥L3∥∇u∥∥H∥L6+C∥H∥L3∥divu∥∥H∥L6≤C∥u∥H1∥∇H∥2+C∥H∥H1∥∇H∥∥∇u∥.
Plugging all the above estimates into (20), we obtain (19).Lemma 7.
Suppose thatϱ,u,H,Θ,η be a smooth solution to (5)–(10). Then, for all0≤t≤T with any fixed T>0, it holds
(24)12ddt∑1≤∣α∣≤4∥2∂αϱ∥2+∥2∂αu∥2+∥∂αH∥2+∥2∂αΘ∥2+∥∂αη∥2+γ∑1≤∣α∣≤4∥∇∂αu∥2+∥div∂αu∥2+∥∇∂αH∥2+∥∇∂αΘ∥2+∥∇∂αη∥2+∥∂α4Θ−η∥2≤C∥ϱ,u,H,Θ,η∥H4+∥ϱ,u,H,Θ,η∥H42∥∇ϱ,u,H,Θ,η∥H32+∥divu∥H32.Proof.
Applying∂α with 1≤∣α∣≤4 to (5)–(9) and multiplying by 4∂αϱ,4∂αu,∂αH,4∂αΘ, and ∂αη, respectively, then taking integration and summation, we have
(25)12ddt∥2∂αρ∥2+∥2∂αu∥2+∥∂αH∥2+∥2∂αΘ∥2+∥∂αη∥2+∫4∇∂αu21+ϱdx+∫8div∂αu21+ϱdx+∫4∇∂αΘ21+ϱdx+∥∇∂αH∥2+∥∇∂αη∥2+∥∂α4Θ−η∥2=∫4−∂α,ϱdivu∂αϱdx+∫4−∂α,u·∇ϱ∂αρdx+∫2∂αϱ2divudx−∫4ϱ∂αρdiv∂αudx+∫4−∂α,u·∇u∂αudx+∫4−∂α,1+Θ1+ϱ∇ϱ∂αudx+∫2∂αu2divudx+∫4Θ−ρ1+ϱ∂αϱdiv∂αudx+∫4∂αϱ∇1+Θ1+ϱ·∂αudx−∫4∇11+ϱ·∇∂αu·∂αudx+∑0≤β<αCα,β∫4∂α−β11+ϱ∂βΔu·∂αudx−∫8∇11+ϱ·∂αudiv∂αudx+∑0≤β<αCα,β∫8∂α−β11+ϱ∂β∇divu·∂αudx−∫∂αu·∇H−H·∇u+Hdivu∂αHdx+∫4∂α11+ϱH·∇H−12∇∣H∣2∂αudx+∫4−∂α,u·∇Θ∂αΘdx+∫2∂αΘ2divudx+∫4−∂α,Θdivu∂αΘdx−∫4∇11+ϱ·∇∂αΘ∂αΘdx−∫4Θdiv∂αu∂αΘdx+∑0≤β<αCα,β∫4∂α−β11+ϱ∂βΔΘ∂αΘdx+∫16ρ∂αΘ21+ϱdx−∫4ρ∂αη∂αΘ1+ϱdx+∑0≤β≤αCα,β∫4∂α−β11+ϱ∂βdivu2+2D·D+∇×H2∂αΘdx−∫4∂α6Θ2+4Θ3+Θ41+ϱ∂αΘdx+∑0≤β<αCα,β∫4∂α−β11+ϱ∂βη∂αΘdx−∑0≤β<αCα,β∫16∂α−β11+ϱ∂βΘ∂αΘdx+∫∂α6Θ2+4Θ3+Θ4∂αηdx≡∑j=128Ij,where A,B denotes the commutator AB−BA for two operators A and B; Cα,β is constant depending only on α and β. We now bound each term on the right-hand side of (25). Utilizing Lemma 5, we get
(26)I1≤C∥ϱ∥H4∥∇ϱ∥H3∥divu∥H3,I2≤C∥ϱ∥H4∥∇ϱ∥H3∥∇u∥H3,I5≤C∥u∥H4∥∇u∥H32,I6≤C∥u∥H4∥∇ϱ∥H3∥∇Θ∥H3+∥∇ϱ∥H32+C∥ϱ∥H4∥u∥H4∥∇ϱ∥H3∥∇Θ∥H3,I16≤C∥Θ∥H4∥∇u∥H3∥∇Θ∥H3,I18≤C∥Θ∥H4∥divu∥H3∥∇Θ∥H3.
ForI11, we have
(27)I11≤C∥u∥H4∥∇ϱ∥H3∥∇∂αu∥≤ε∥∇∂αu∥2+Cε∥u∥H42∥∇ϱ∥H32,with ε>0 a small constant, where the first inequality follows that for β<α,
(28)∫∂α−β11+ρ∂βΔu∂αu≤∥∂α11+ϱ∥∥Δu∥L∞∥∂αu∥β=0,∥∂α−β11+ϱ∥L3∥∂βΔu∥L6∥∂αu∥β=1,∥∂α−β11+ϱ∥L∞∥∂βΔu∥∥∂αu∥β≥2,and Sobolev’s and Young’s inequalities were further used. Similarly, we have
(29)I13≤ε∥∇∂αu∥2+Cε∥u∥H42∥∇ϱ∥H32,I21≤ε∥∇∂αΘ∥2+Cε∥Θ∥H42∥∇ϱ∥H32,I24≤∥u∥H4∥Θ∥H4∥∇ϱ∥H3∥∇u∥H3+∥divu∥H3+∥∇H∥H3+ε∥∇∂αu∥2+∥div∂αu∥2+∥∇∂αH∥2+Cε∥Θ∥H42∥∇u∥H22+∥divu∥H22+∥∇H∥H22,I26+I27≤C∥Θ∥H4∥∇ρ∥H3∥∇η∥H2+∥∇Θ∥H2.
ForI15, we have
(30)I15=∫41+ϱ∂αH·∇H∂αudx+∑0≤β<αCα,β∫4∂α−β11+ϱ∂βH·∇H∂αudx−∫21+ϱ∂α∇H2∂αudx−∑0≤β<αCα,β∫2∂α−β11+ϱ∂β∇H2∂αudx≡∑j=14I15j.
By Lemma4, we obtain
(31)I151≤C∥∇H∥H3∥∇∂αH∥∥∂αu∥≤ε∥∇∂αH∥2+Cε∥u∥H42∥∇H∥H32,I152≤C∥ϱ∥H4∥u∥H4∥∇H∥H32,since
(32)∫∂α−β11+ϱ∂βH·∇H∂αu≤∥∂α11+ϱ∥∥H·∇H∥L∞∥∂αu∥β=0,∥∂α−β11+ϱ∥L3∥∂βH·∇H∥L6∥∂αu∥β=1,∥∂α−β11+ϱ∥L∞∥∂βH·∇H∥∥∂αu∥∣β∣≥2.
Similarly, we can deduce that(33)I153≤ε∥∇∂αH∥2+Cε∥u∥H42∥∇H∥H32,I154≤C∥ρ∥H4∥u∥H4∥∇H∥H32.
Therefore,(34)I15≤ε∥∇∂αH∥2+C1+∥ϱ∥H4∥u∥H4∥∇H∥H32.
Using Hölder’s, Sobolev’s, and Young’s inequalities and Lemma4, we can get following bounds:
(35)I3+I7+I17≤C∥divu∥L∞∥∂αϱ∥2+∥∂αu∥2+∥∂αΘ∥2≤C∥u∥H3∥∇ϱ∥H32+∥∇u∥H32+∥∇Θ∥H32,I4≤ε∥div∂αu∥2+Cε∥ϱ∥H42∥∇ρ∥H12,I8≤ε∥div∂αu∥2+Cε∥ϱ∥H42∥∇ϱ∥H12+∥∇Θ∥H12,I9≤C∥ϱ∥H4∥u∥H4∥∇ϱ∥H2∥∇Θ∥H1+C∥ϱ∥H4∥∇u∥H3∥∇ϱ∥H2+∥∇u∥H2,I10≤ε∥∇∂αu∥2+Cε∥u∥H42∥∇ϱ∥H22,I12≤ε∥div∂αu∥2+Cε∥u∥H42∥∇ϱ∥H22,I14≤ε∥∇∂αu∥2+∥div∂αu∥2+∥∇∂αH∥2+Cε∥H∥H42∥∇u∥H32+∥∇H∥H32,I19≤ε∥∇∂αΘ∥2+Cε∥Θ∥H42∥∇ϱ∥H22,I20≤ε∥div∂αu∥2+Cε∥Θ∥H42∥∇Θ∥H12,I22+I23≤C∥ϱ∥H2∥∇Θ∥H3∥∇Θ∥H3+∥∇η∥H3.For the remaining terms, under the assumption (16), one also has
(36)I25≤C∥Θ∥H4∥∇ϱ∥H3∥∇Θ∥H3+∥∇Θ∥H32+∥∇Θ∥H32≤C∥Θ∥H4∥∇ϱ∥H3∥∇Θ∥H3,I28≤C∥η∥H4∥Θ∥H32+∥Θ∥H33+∥Θ∥H34≤C∥η∥H4∥∇Θ∥H32.Putting all the above estimates into (25) and taking the sum over 1≤∣α∣≤4, then (24) follows, and thus, Lemma 7 is proven.Next, we will give the dissipation rate ofρ.Lemma 8.
Suppose thatϱ,u,H,Θ,η be a smooth solution to (5)–(10). Then, for all 0≤t≤T with any fixed T>0, it holds
(37)ddt∑∣α∣≤3∫∇∂αϱ·∂αudx+γ∥∇ϱ∥H32≤C∥∇u∥H42+∥divu∥H32+∥∇Θ∥H32+C∥ϱ,u,H,Θ∥H4+∥ϱ,u,H,Θ∥H42∥∇ϱ,u,H,Θ∥H32.Proof.
Taking differentiation∂α∣α∣≤3 to (8) and multiplying by ∇∂αϱ, then taking integration, one can get
(38)∫∇∂αϱ2dx=−∫∇∂αϱ·∂αutdx−∫∇∂αϱ∂αu·∇udx−∫∇∂αϱ·∂αΘ−ϱ1+ϱ∇ϱdx−∫∇∂αϱ·∇∂αΘdx+∫∇∂αϱ·∂αΔu1+ϱdx+∫2∇∂αϱ·∂α∇divu1+ϱdx+∫∇∂αϱ·∂α11+ϱH·∇H−12∇H2dx≡∑j=17Ij.
ForI1, applying (5), we have
(39)I1=−ddt∫∇∂αϱ·∂αudx+∫∂αdivu∂α1+ϱdivu+∇ϱ·udx≤−ddt∫∇∂αϱ·∂αudx+C∥divu∥H32+C∥ϱ∥H3∥∇u∥H32.
Using Hölder’s, Sobolev’s, and Young’s inequalities, we obtain(40)I2≤C∥ϱ∥H4∥∇u∥H32,I3≤C∥ϱ∥H42∥∇ϱ∥H3∥∇ϱ∥H3+∥∇Θ∥H3,I4≤14∥∇∂αϱ∥2+C∥∇Θ∥H32,I5+I6≤C∥ϱ∥H4∥∇ϱ∥H3∥∇u∥H3+C∥∇∂αϱ∥∥∇u∥H4≤14∥∇∂αϱ∥2+C∥ϱ∥H4∥∇ϱ∥H3∥∇u∥H3+C∥∇u∥H42,I7≤C∥ϱ∥H4∥∇H∥H32+C∥ϱ∥H42∥∇H∥H32.Putting these estimates into (38) and taking the sum over ∣α∣≤3 gives (37), and Lemma 8 is proven.


### 2.2. Proof of Global Existence

In this section, we will show there exists a unique global in time solution to the problem (5)–(10). Firstly, combining estimates obtained in Lemmas 6–8, one can finish the proof of uniform-in-time a priori estimates as follows. Define a total temporal energy functional Et and the corresponding dissipation rate functional Dt by
(41)Et=∥2ϱ∥2+∥2u∥2+∥H∥2+∥2Θ∥2+∥η∥2+∑1≤∣α∣≤4∥2∂αϱ∥2+∥2∂αu∥2+∥2∂αΘ∥2+∥∂αH∥2+∥∂αη∥2+τ1∑∣α∣≤3∫∇∂αϱ·∂αudx,(42)Dt=∥∇ρ∥H32+∥∇u,H,Θ,η∥H42+∥divu∥H42+∥4Θ−η∥H42,where 0<τ1≪1 is a small constant. Under the assumption (16), then
(43)Et~∥ϱ,u,H,Θ,ηt∥H42,holds true uniformly for all 0≤t<T. Furthermore, summing ((19)) and ((24)) and τ1× ((37)) and noticing thatτ1is sufficiently small, we have
(44)ddtEt+γDt≤CE1/2t+EtDt,for all 0≤t<T. With the help of (16), one has E1/2t+Et≤Cδ+δ2 with 0<δ<1 being small enough. Then, the time integration of (44) yields
(45)Et+γ∫0tDsds≤E0,for all 0≤t<T. Besides, (16) can be justified by choosing
(46)E0~∥ϱ0,u0,H0,Θ0,η0∥H42,which is sufficiently small. For brevity, the proof for local existence of smooth solutions is omitted. Then, the global existence and uniqueness of solutions follow from (45) together with the local existence as well as application of the continuity argument.


## 2.1. A Priori Estimates

Now, we begin to establish the global a priori estimates in the case of the whole spaceℝ3 under the assumption
(16)supt≥0∥ϱ,u,H,Θ,η∥H4≤δ,where 0<δ<1 is a generic constant small enough and ρ,u,H,Θ,η is the smooth solution to the Cauchy problem (5)–(10) on 0≤t<T for T>0. Firstly, we list two important lemmas in Sobolev space.Lemma 4 (see [16, 19]).
There exist a positiveconstant C, such that for any f,g∈H4ℝ3 and any multi-index α with 1≤∣α∣≤4,
(17)∥f∥L∞ℝ3≤C∥∇f∥L2ℝ31/2∥∇2f∥L2ℝ31/2,∥f∥Lpℝ3≤C∥∇f∥H1ℝ3,where2≤p≤6,∥fg∥H3ℝ3≤C∥f∥H3ℝ3∥∇g∥H3ℝ3,∥∂αfg∥L2ℝ3≤C∥∇f∥H3ℝ3∥∇g∥H3ℝ3.Lemma 5 (Moser-type calculus inequalities) (see [20]).
Lets≥1 be an integer. Suppose f∈Hsℝ3, ∇f∈L∞ℝ3 and g∈Hs−1ℝ3⋂L∞ℝ3. Then, for all multi-index α with ∣α∣≤s, we have
(18)∥∂αfg−f∂αg∥≤Cs∥∇f∥L∞∥Ds−1g∥+∥Dsf∥∥g∥L∞,∥Dsf∥=∑∣α∣=s∥∂αf∥.
Then, we begin to give the priori estimate ofϱ,u,H,Θ,η.Lemma 6.
Suppose thatϱ,u,H,Θ,η be a smooth solution to (5)–(10). Then, for all0≤t≤T with any fixed T>0, it holds
(19)12ddt∥2ρ,2u,H,2Θ,η∥2+γ∥∇u,H,Θ,η∥2+∥divu∥2+∥4Θ−η∥2≤C∥ϱ,u,H,Θ,η∥H2+∥ϱ,u,H,Θ,η∥H22∥∇ϱ,u,H,Θ,η∥2+∥divu∥2+∥4Θ−η∥2.Proof.
Multiplying (5)–(9) by 4ρ,4u,H,4Θ, and η and then taking integration and summation, we get
(20)12ddt∥2ϱ∥2+∥2u∥2+∥H∥2+∥2Θ∥2+∥η∥2+∫4∇u21+ρdx+∫8divu21+ϱdx+∫4∇Θ21+ϱdx+∥∇H∥2+∥∇η∥2+∥4Θ−η∥2=−∫2ρ2divudx+∫4ρ−Θ1+ϱ∇ρ·udx−∫4u·∇u·udx−∫4∇11+ϱ·∇u·udx−∫8∇11+ϱ·udivudx−∫4Θ2divudx−∫4Θ∇Θ·udx+∫4Θdivu21+ϱdx+∫8ΘD·D1+ϱdx−∫4Θ∇11+ϱ·∇Θdx+∫4Θ∇×H21+ϱdx+∫4ρΘ4Θ−η1+ϱdx−∫4Θ−η6Θ2+4Θ3+Θ41+ϱdx+∫ρη6Θ2+4Θ3+Θ41+ϱdx+∫H·∇H−12∇H24u1+ϱdx+∫−u·∇H+H·∇H−Hdivu·Hdx≡∑j=116Ij.
ForI1 to I12, using Hölder’s and Sobolev’s inequalities, we have
(21)I1≤C∥ϱ∥L3∥divu∥∥ϱ∥L6≤C∥ρ∥H1∥∇ϱ∥∥divu∥,I2≤C∥ϱ∥L3+∥Θ∥L3∥∇ϱ∥∥u∥L6≤C∥ϱ∥H1+∥Θ∥H1∥∇ϱ∥∥∇u∥,I3≤C∥u∥L3∥∇u∥∥u∥L6≤C∥u∥H1∥∇u∥2,I4+I5≤C∥u∥L6∥∇u∥L3+∥divu∥L3∇11+ρ≤C∥u∥H2∥∇ϱ∥∥∇u∥,I6≤C∥Θ∥L3∥divu∥∥Θ∥L6≤C∥Θ∥H1∥divu∥∥∇Θ∥,I7≤C∥u∥L3∥∇Θ∥∥Θ∥L6≤C∥u∥H1∥∇Θ∥2,I8≤C∥Θ∥L∞∥divu∥2≤C∥Θ∥H2∥divu∥2,I9=∑i,j=13∫2Θuxji+uxij21+ϱdx≤C∑i,j=13∥Θ∥L∞∥uxji∥∥uxij∥≤C∥Θ∥H2∥∇u∥2,I10≤C∇11+ϱ∥∇Θ∥L3∥Θ∥L6≤C∥Θ∥H2∥∇ϱ∥∥∇Θ∥,I11≤C∥Θ∥L∞∥∇×H∥2≤C∥Θ∥H2∥∇H∥2,I12≤C∥ϱ∥L3∥4Θ−η∥∥Θ∥L6≤C∥ϱ∥H1∥4Θ−η∥∥∇Θ∥.
ForI13, I14, under the assumption (16), one also has
(22)I13≤C∥4Θ−η∥∥Θ∥L3∥Θ∥L6+∥Θ2∥L3∥Θ∥L6+∥Θ∥L∞∥Θ2∥L3∥Θ∥L6≤C∥4Θ−η∥∥Θ∥H1∥∇Θ∥+∥∇Θ∥3+∥Θ∥H2∥∇Θ∥3≤C∥Θ∥H2+∥Θ∥H22∥4Θ−η∥2+∥∇Θ∥2,I14≤C∥ϱ∥H1∥η∥H1∥∇Θ∥2+∥∇Θ∥3+∥∇Θ∥4≤C∥ϱ∥H12+∥Θ∥H12+∥η∥H12∥∇Θ∥2.
At last, forI15 and I16, we have
(23)I15≤C∥u∥L3∥∇H∥∥H∥L6+C∥H∥L3∥divu∥∥H∥L6≤C∥u∥H1∥∇H∥2+C∥H∥H1∥∇H∥∥∇u∥,I16≤C∥u∥L3∥∇H∥∥H∥L6+C∥H∥L3∥∇u∥∥H∥L6+C∥H∥L3∥divu∥∥H∥L6≤C∥u∥H1∥∇H∥2+C∥H∥H1∥∇H∥∥∇u∥.
Plugging all the above estimates into (20), we obtain (19).Lemma 7.
Suppose thatϱ,u,H,Θ,η be a smooth solution to (5)–(10). Then, for all0≤t≤T with any fixed T>0, it holds
(24)12ddt∑1≤∣α∣≤4∥2∂αϱ∥2+∥2∂αu∥2+∥∂αH∥2+∥2∂αΘ∥2+∥∂αη∥2+γ∑1≤∣α∣≤4∥∇∂αu∥2+∥div∂αu∥2+∥∇∂αH∥2+∥∇∂αΘ∥2+∥∇∂αη∥2+∥∂α4Θ−η∥2≤C∥ϱ,u,H,Θ,η∥H4+∥ϱ,u,H,Θ,η∥H42∥∇ϱ,u,H,Θ,η∥H32+∥divu∥H32.Proof.
Applying∂α with 1≤∣α∣≤4 to (5)–(9) and multiplying by 4∂αϱ,4∂αu,∂αH,4∂αΘ, and ∂αη, respectively, then taking integration and summation, we have
(25)12ddt∥2∂αρ∥2+∥2∂αu∥2+∥∂αH∥2+∥2∂αΘ∥2+∥∂αη∥2+∫4∇∂αu21+ϱdx+∫8div∂αu21+ϱdx+∫4∇∂αΘ21+ϱdx+∥∇∂αH∥2+∥∇∂αη∥2+∥∂α4Θ−η∥2=∫4−∂α,ϱdivu∂αϱdx+∫4−∂α,u·∇ϱ∂αρdx+∫2∂αϱ2divudx−∫4ϱ∂αρdiv∂αudx+∫4−∂α,u·∇u∂αudx+∫4−∂α,1+Θ1+ϱ∇ϱ∂αudx+∫2∂αu2divudx+∫4Θ−ρ1+ϱ∂αϱdiv∂αudx+∫4∂αϱ∇1+Θ1+ϱ·∂αudx−∫4∇11+ϱ·∇∂αu·∂αudx+∑0≤β<αCα,β∫4∂α−β11+ϱ∂βΔu·∂αudx−∫8∇11+ϱ·∂αudiv∂αudx+∑0≤β<αCα,β∫8∂α−β11+ϱ∂β∇divu·∂αudx−∫∂αu·∇H−H·∇u+Hdivu∂αHdx+∫4∂α11+ϱH·∇H−12∇∣H∣2∂αudx+∫4−∂α,u·∇Θ∂αΘdx+∫2∂αΘ2divudx+∫4−∂α,Θdivu∂αΘdx−∫4∇11+ϱ·∇∂αΘ∂αΘdx−∫4Θdiv∂αu∂αΘdx+∑0≤β<αCα,β∫4∂α−β11+ϱ∂βΔΘ∂αΘdx+∫16ρ∂αΘ21+ϱdx−∫4ρ∂αη∂αΘ1+ϱdx+∑0≤β≤αCα,β∫4∂α−β11+ϱ∂βdivu2+2D·D+∇×H2∂αΘdx−∫4∂α6Θ2+4Θ3+Θ41+ϱ∂αΘdx+∑0≤β<αCα,β∫4∂α−β11+ϱ∂βη∂αΘdx−∑0≤β<αCα,β∫16∂α−β11+ϱ∂βΘ∂αΘdx+∫∂α6Θ2+4Θ3+Θ4∂αηdx≡∑j=128Ij,where A,B denotes the commutator AB−BA for two operators A and B; Cα,β is constant depending only on α and β. We now bound each term on the right-hand side of (25). Utilizing Lemma 5, we get
(26)I1≤C∥ϱ∥H4∥∇ϱ∥H3∥divu∥H3,I2≤C∥ϱ∥H4∥∇ϱ∥H3∥∇u∥H3,I5≤C∥u∥H4∥∇u∥H32,I6≤C∥u∥H4∥∇ϱ∥H3∥∇Θ∥H3+∥∇ϱ∥H32+C∥ϱ∥H4∥u∥H4∥∇ϱ∥H3∥∇Θ∥H3,I16≤C∥Θ∥H4∥∇u∥H3∥∇Θ∥H3,I18≤C∥Θ∥H4∥divu∥H3∥∇Θ∥H3.
ForI11, we have
(27)I11≤C∥u∥H4∥∇ϱ∥H3∥∇∂αu∥≤ε∥∇∂αu∥2+Cε∥u∥H42∥∇ϱ∥H32,with ε>0 a small constant, where the first inequality follows that for β<α,
(28)∫∂α−β11+ρ∂βΔu∂αu≤∥∂α11+ϱ∥∥Δu∥L∞∥∂αu∥β=0,∥∂α−β11+ϱ∥L3∥∂βΔu∥L6∥∂αu∥β=1,∥∂α−β11+ϱ∥L∞∥∂βΔu∥∥∂αu∥β≥2,and Sobolev’s and Young’s inequalities were further used. Similarly, we have
(29)I13≤ε∥∇∂αu∥2+Cε∥u∥H42∥∇ϱ∥H32,I21≤ε∥∇∂αΘ∥2+Cε∥Θ∥H42∥∇ϱ∥H32,I24≤∥u∥H4∥Θ∥H4∥∇ϱ∥H3∥∇u∥H3+∥divu∥H3+∥∇H∥H3+ε∥∇∂αu∥2+∥div∂αu∥2+∥∇∂αH∥2+Cε∥Θ∥H42∥∇u∥H22+∥divu∥H22+∥∇H∥H22,I26+I27≤C∥Θ∥H4∥∇ρ∥H3∥∇η∥H2+∥∇Θ∥H2.
ForI15, we have
(30)I15=∫41+ϱ∂αH·∇H∂αudx+∑0≤β<αCα,β∫4∂α−β11+ϱ∂βH·∇H∂αudx−∫21+ϱ∂α∇H2∂αudx−∑0≤β<αCα,β∫2∂α−β11+ϱ∂β∇H2∂αudx≡∑j=14I15j.
By Lemma4, we obtain
(31)I151≤C∥∇H∥H3∥∇∂αH∥∥∂αu∥≤ε∥∇∂αH∥2+Cε∥u∥H42∥∇H∥H32,I152≤C∥ϱ∥H4∥u∥H4∥∇H∥H32,since
(32)∫∂α−β11+ϱ∂βH·∇H∂αu≤∥∂α11+ϱ∥∥H·∇H∥L∞∥∂αu∥β=0,∥∂α−β11+ϱ∥L3∥∂βH·∇H∥L6∥∂αu∥β=1,∥∂α−β11+ϱ∥L∞∥∂βH·∇H∥∥∂αu∥∣β∣≥2.
Similarly, we can deduce that(33)I153≤ε∥∇∂αH∥2+Cε∥u∥H42∥∇H∥H32,I154≤C∥ρ∥H4∥u∥H4∥∇H∥H32.
Therefore,(34)I15≤ε∥∇∂αH∥2+C1+∥ϱ∥H4∥u∥H4∥∇H∥H32.
Using Hölder’s, Sobolev’s, and Young’s inequalities and Lemma4, we can get following bounds:
(35)I3+I7+I17≤C∥divu∥L∞∥∂αϱ∥2+∥∂αu∥2+∥∂αΘ∥2≤C∥u∥H3∥∇ϱ∥H32+∥∇u∥H32+∥∇Θ∥H32,I4≤ε∥div∂αu∥2+Cε∥ϱ∥H42∥∇ρ∥H12,I8≤ε∥div∂αu∥2+Cε∥ϱ∥H42∥∇ϱ∥H12+∥∇Θ∥H12,I9≤C∥ϱ∥H4∥u∥H4∥∇ϱ∥H2∥∇Θ∥H1+C∥ϱ∥H4∥∇u∥H3∥∇ϱ∥H2+∥∇u∥H2,I10≤ε∥∇∂αu∥2+Cε∥u∥H42∥∇ϱ∥H22,I12≤ε∥div∂αu∥2+Cε∥u∥H42∥∇ϱ∥H22,I14≤ε∥∇∂αu∥2+∥div∂αu∥2+∥∇∂αH∥2+Cε∥H∥H42∥∇u∥H32+∥∇H∥H32,I19≤ε∥∇∂αΘ∥2+Cε∥Θ∥H42∥∇ϱ∥H22,I20≤ε∥div∂αu∥2+Cε∥Θ∥H42∥∇Θ∥H12,I22+I23≤C∥ϱ∥H2∥∇Θ∥H3∥∇Θ∥H3+∥∇η∥H3.For the remaining terms, under the assumption (16), one also has
(36)I25≤C∥Θ∥H4∥∇ϱ∥H3∥∇Θ∥H3+∥∇Θ∥H32+∥∇Θ∥H32≤C∥Θ∥H4∥∇ϱ∥H3∥∇Θ∥H3,I28≤C∥η∥H4∥Θ∥H32+∥Θ∥H33+∥Θ∥H34≤C∥η∥H4∥∇Θ∥H32.Putting all the above estimates into (25) and taking the sum over 1≤∣α∣≤4, then (24) follows, and thus, Lemma 7 is proven.Next, we will give the dissipation rate ofρ.Lemma 8.
Suppose thatϱ,u,H,Θ,η be a smooth solution to (5)–(10). Then, for all 0≤t≤T with any fixed T>0, it holds
(37)ddt∑∣α∣≤3∫∇∂αϱ·∂αudx+γ∥∇ϱ∥H32≤C∥∇u∥H42+∥divu∥H32+∥∇Θ∥H32+C∥ϱ,u,H,Θ∥H4+∥ϱ,u,H,Θ∥H42∥∇ϱ,u,H,Θ∥H32.Proof.
Taking differentiation∂α∣α∣≤3 to (8) and multiplying by ∇∂αϱ, then taking integration, one can get
(38)∫∇∂αϱ2dx=−∫∇∂αϱ·∂αutdx−∫∇∂αϱ∂αu·∇udx−∫∇∂αϱ·∂αΘ−ϱ1+ϱ∇ϱdx−∫∇∂αϱ·∇∂αΘdx+∫∇∂αϱ·∂αΔu1+ϱdx+∫2∇∂αϱ·∂α∇divu1+ϱdx+∫∇∂αϱ·∂α11+ϱH·∇H−12∇H2dx≡∑j=17Ij.
ForI1, applying (5), we have
(39)I1=−ddt∫∇∂αϱ·∂αudx+∫∂αdivu∂α1+ϱdivu+∇ϱ·udx≤−ddt∫∇∂αϱ·∂αudx+C∥divu∥H32+C∥ϱ∥H3∥∇u∥H32.
Using Hölder’s, Sobolev’s, and Young’s inequalities, we obtain(40)I2≤C∥ϱ∥H4∥∇u∥H32,I3≤C∥ϱ∥H42∥∇ϱ∥H3∥∇ϱ∥H3+∥∇Θ∥H3,I4≤14∥∇∂αϱ∥2+C∥∇Θ∥H32,I5+I6≤C∥ϱ∥H4∥∇ϱ∥H3∥∇u∥H3+C∥∇∂αϱ∥∥∇u∥H4≤14∥∇∂αϱ∥2+C∥ϱ∥H4∥∇ϱ∥H3∥∇u∥H3+C∥∇u∥H42,I7≤C∥ϱ∥H4∥∇H∥H32+C∥ϱ∥H42∥∇H∥H32.Putting these estimates into (38) and taking the sum over ∣α∣≤3 gives (37), and Lemma 8 is proven.


## 2.2. Proof of Global Existence

In this section, we will show there exists a unique global in time solution to the problem (5)–(10). Firstly, combining estimates obtained in Lemmas 6–8, one can finish the proof of uniform-in-time a priori estimates as follows. Define a total temporal energy functional Et and the corresponding dissipation rate functional Dt by
(41)Et=∥2ϱ∥2+∥2u∥2+∥H∥2+∥2Θ∥2+∥η∥2+∑1≤∣α∣≤4∥2∂αϱ∥2+∥2∂αu∥2+∥2∂αΘ∥2+∥∂αH∥2+∥∂αη∥2+τ1∑∣α∣≤3∫∇∂αϱ·∂αudx,(42)Dt=∥∇ρ∥H32+∥∇u,H,Θ,η∥H42+∥divu∥H42+∥4Θ−η∥H42,where 0<τ1≪1 is a small constant. Under the assumption (16), then
(43)Et~∥ϱ,u,H,Θ,ηt∥H42,holds true uniformly for all 0≤t<T. Furthermore, summing ((19)) and ((24)) and τ1× ((37)) and noticing thatτ1is sufficiently small, we have
(44)ddtEt+γDt≤CE1/2t+EtDt,for all 0≤t<T. With the help of (16), one has E1/2t+Et≤Cδ+δ2 with 0<δ<1 being small enough. Then, the time integration of (44) yields
(45)Et+γ∫0tDsds≤E0,for all 0≤t<T. Besides, (16) can be justified by choosing
(46)E0~∥ϱ0,u0,H0,Θ0,η0∥H42,which is sufficiently small. For brevity, the proof for local existence of smooth solutions is omitted. Then, the global existence and uniqueness of solutions follow from (45) together with the local existence as well as application of the continuity argument.


## 3. Convergence Rates

In this section, we consider the convergence rate of solutions obtained in Theorem1. In order to obtain the desired decay rate estimates in Theorem 2, we firstly consider the linearized system corresponding to (5)–(9):
(47)ϱt+divu=0,(48)ut+∇ρ+∇Θ−Δu−2∇divu=0,(49)Ht−ΔH=0,divH=0,(50)Θt−ΔΘ+divu+4Θ−η=0,(51)ηt−Δη+η−4Θ=0,with initial data
(52)ϱ,u,H,Θ,ηt=0=ϱ0,u0,H0,Θ0,η0x,x∈ℝ3.Denote byUt=ϱ,u,H,Θ,ηt to be the solution of the Cauchy problem (47)–(52), then Ut can be presented as
(53)Ut=AtU0,where At is named as the solution operator of (47)–(52) and U0=Ut=0. Then, we utilize the energy method to Cauchy problem (47)–(52) in the Fourier space to present that there is a time-frequency Lyapunov functional which is equivalent to U∧t,k2. This estimate can help us to establish the Lp−Lq time decay property of Ut as follows.Theorem 9.
Let1≤q≤2. For any α,α′ with α′≤α and m=∣α−α′∣,
(54)∥∂αAtU0∥L2≤C1+t−3/21/q−1/2−m/2∥∂α′U0∥Lq+∥∂αU0∥L2,hold for all t≥0.Proof.
By taking Fourier transforming inx for (47)–(51), one has
(55)ϱ^t+ik·u^=0,(56)u^t+ikρ^+ikΘ^+k2u^+2kk·u^=0,(57)H^t+k2H^=0,(58)Θ^t+k2Θ^+ik·u^+4Θ^−η^=0,(59)η^t+k2η^+η^−4Θ^=0,where k∈ℝ3, i=−1∈ℂ is the imaginary unit.
Multiplying (55)–(59) by 4ρ^¯, 4u^¯,H^¯,4Θ^¯,η^¯, respectively, its real part gives
(60)∂t2ϱ∧,2u∧,H∧,2Θ∧,η∧2+4k2u∧2+k2H∧2+8k·u∧2+4k2Θ∧2+k2η∧2+4Θ∧−η∧2=0.
Multiplying (56) by ikρ^¯, utilizing integration by parts in t, and replacing ∂tρ^ by (55), one has
(61)∂tu^∣ikϱ^+k2ϱ∧2=k·u∧2+3k2ik·u^ϱ^¯−k2Θ^ϱ^¯,here ·∣· means the complex inner product. For the real part of (61) and with the help of Cauchy-Schwarz inequality, one has
(62)∂tReu^∣ikϱ^+k2ϱ∧2≤k·u∧2+εk2ϱ∧2+Cεk2k·u∧2+εk2ϱ∧2+Cεk2Θ∧2,with ε>0 being a small constant. Multiplying it by 1/1+k2, we conclude that there exists γ>0 such that
(63)∂tReu^∣ikϱ^1+k2+γk2ϱ∧21+k2≤Cu∧2+Ck2Θ∧21+k2.
Now, we define the time-frequency Lyapunov functional as(64)EU^t,k=2ϱ∧,2u∧,H∧,2Θ∧,η∧2+τ2Reu^∣ikϱ^1+k2,where 0<τ2≪1 is sufficiently small. It also holds that EU^~U∧2. Moreover, by suitably choosing constants τ1, the sum of equations (19), (24), τ1 × (37) gives the linear combination (60) + τ1 × (63) which gives (60)
(65)∂tEU^t,k+γk21+k2EU^t,k≤0.As in [15, 21], the desired time decay estimates (54) and directly follows from the above estimate, and the detailed proof is omitted for brevity.Now, we prove the rate of convergence (12). We quote a technical lemma in [19] for later proofs.Lemma 10.
Given any0<β1≠1 and β2>1,
(66)∫0t1+t−s−β11+s−β2ds≤C1+t−minβ1,β2for all t≥0.By the Duhamel principle, the solution of nonlinear Cauchy problem (5)–(10) can be formally written as
(67)Ut=AtU0+∫0tAt−sY1,Y2,Y3,Y4,Y5ds,with
(68)Y1=−ϱdivu−∇ϱ·u,Y2=−u·∇u−Θ−ϱ1+ϱ∇ϱ−ϱ1+ϱΔu−2ϱ1+ϱ∇divu+11+ϱH·∇H−12∇H2,Y3=u·∇H−H·∇u+Hdivu,Y4=−u·∇Θ−ϱ1+ϱΔΘ−Θdivu+divu21+ϱ+2D·D1+ϱ−ϱη1+ϱ−4ρΘ1+ϱ−6Θ2+4Θ3+Θ41+ϱ+∇×H21+ϱ,Y5=6Θ2+4Θ3+Θ4.By the definitions ofEt and Dt in (41) and (42), respectively, we have
(69)Et≤CDt+∥U∥2.From (44), we have
(70)ddtEt+γCEt−∥U∥2≤ddtEt+γDt≤0,which implies
(71)ddtEt+γEt≤C∥U∥2.Gronwall’s inequality gives(72)Et≤e−γtE0+C∫0te−λt−s∥U∥2ds.Next, we give estimate of∥Ut∥. Firstly, we further rewrite (67) as
(73)Ut=AtU0+∫0tAt−sY1,Y2,Y3,0,0ds+∫0tAt−s0,0,0,Y4,Y5ds≡∑i=13Jit.Define(74)E∞t=sup0≤s≤t1+s3/2Es.One has that from (54) and Lemma 10,
(75)∥J1t∥≤C1+t−3/4∥U0∥L1∩L2,∥J2t∥≤C∫0t1+t−s−3/4∥Y1,Y2,Y3∥L1∩L2ds≤C∫0t1+t−s−3/4Esds≤C∫0t1+t−s−3/41+s−3/2dsE∞t≤C1+t−3/4E∞t,∥J3t∥≤C∫0t1+t−s−3/4∥Y4,Y5∥L1∩L2dsds≤C∫0t1+t−s−3/4Es+E2sds≤C1+t−3/4E∞t+E∞2t.Therefore, it follows that(76)∥U∥2≤C1+t−3/2∥U0∥L1∩L22+E∞2t+E∞4t.Substituting (76) into (72), we get
(77)E∞t≤C∥U0∥H4⋂L12+E∞2t+E∞4t,which implies that E∞t≤C∥U0∥H4⋂L12 for all t≥0, provided that ∥U0∥H4⋂L1 is sufficiently small. Thus,
(78)Et≤C1+t−3/2∥U0∥H4⋂L12.This can deduce (12).We continue to prove the rate of convergence (13). Firstly, we define the new energy functional and dissipation rate functional by
(79)Mt=∑1≤∣α∣≤4∥2∂αϱ∥2+∥2∂αu∥2+∥∂αH∥2+∥2∂αΘ∥2+∥∂αη∥2+τ1∑∣α∣≤3∫∇∂αϱ·∂αudx,(80)Nt=∑1≤∣α∣≤3∥∂α∇ϱ∥2+∥∂α∇4Θ−η∥2+∑1≤∣α∣≤4∥∂α∇u,H,Θ,η∥2+∥∂αdivu∥2.By using Lemma4 and similar arguments to those in the proof of Lemmas 6–8, we obtained
(81)ddtMt+γNt≤CM1/2t+MtNt.Adding the term∥∇U∥2 to both sides of (81) gives
(82)ddtMt+γMt≤C∥∇U∥2,ifMt is small enough. Being similar to the proof of ∥U∥ and defining M∞t=sup0≤s≤t1+s5/2Hs and by (73), we can deduce that
(83)∥∇U∥2≤C1+t−5/2∥U0∥L12+∥∇U0∥2+M∞2t+M∞4t.From (81), (83), and Gronwall’s inequality, we have
(84)Mt≤e−γtM0+C1+t−5/2∥∇U0∥2+∥U0∥L12+M∞2t+M∞4t,and hence
(85)M∞t≤C∥∇U0∥H32+∥U0∥L12+M∞2t+M∞4t.Thus, since∥∇U0∥H32+∥U0∥L12 can be small enough, this implies
(86)M∞t≤C∥∇U0∥H32+∥U0∥L12,for all t≥0, that is,
(87)Mt≤C1+t−5/2∥∇U0∥H32+∥U0∥L12,which means
(88)∥∇ρ,u,H,Θ,η∥H3≤C1+t−5/4,for all t≥0, this completes the proof of Theorem 2.


## 4. The Periodic Case

In this section, we deal with the spatial domainΩ=T3. For smooth solution of the system (1), it is not hard to get the following conservation laws in the case of torus,
(89)ddt∫T3ρdx=0,ddt∫T3ρudx=0,ddt∫T3Hdx=0,ddt∫T312ρu2+12H2+ρθ+ndx=0,and by the assumption (14), it follows that
(90)∫T3ρdx=0,∫T31+ρudx=0,∫T3Hdx=0,∫T3121+ρu2+12H2+ρ+Θ+ρΘ+ηdx=0,for all t≥0.Proof of Theorem 11.
We only give the proof of the global a priori estimates. Firstly, let the temporal energy functionalEt and the corresponding dissipation rate functional Dt be defined in the same way as in (41) and (42), respectively, for the case of the whole space Ω=ℝ3. Similarly, we conclude that
(91)ddtEt+γDt≤C∥ρ,u,H,Θ,η∥H4+∥ρ,u,H,Θ,η∥H42∥∇ρ,u,H,Θ,η∥H32+∥divu∥H32.
Thanks to Poincaré’s inequality and the conservation laws (90), we have
(92)∥ρ∥≤C∥∇ρ∥,∥H∥≤C∥∇H∥,(93)∥u∥≤∥u+ρu∥+∥ρu∥≤C∥∇u∥+C∥∇ρu∥+∥ρ∥∥u∥L∞≤C∥∇u∥+C∥u∥H2∥∇ρ∥+C∥ρ∥H2∥∇u∥,(94)Θ+η=121+ρu2+12H2+ρ+Θ+ρΘ+η+121+ρu2+12H2+ρ+ρΘ≤C∥u∥H2∥∇u∥+C∥ρ∥H2∥u∥H2∥∇u∥+C∥u∥H22∥∇ρ∥+C∥H∥H2∥∇H∥+C∥Θ∥H2∥∇ρ∥+C∥ρ∥H2∥∇Θ∥+C∥∇ρ∥+∥∇Θ∥+∥∇η∥.
Then, define(95)DTt=Dt+τ3∥ρ∥2+∥u∥2+∥H∥2+∥Θ+η∥2,where 0<τ3≪1 is sufficiently small. Notice
(96)DTt~∥ρ∥H42+∥u,H,Θ,η∥H52,uniformly for all t≥0. Combining (91)–(94) together, we have
(97)ddtEt+γDTt≤CE1/2t+Et+E2tDTt.
Using the fact thatEt is small enough and uniform in time, and Et≤CDTt, we then obtain
(98)ddtEt+γEt≤0,for all t≥0. Applying Gronwall’s inequality to (98), one has
(99)Et≤e−γtE0.This gives the desired exponential decay ofEt~∥ρ,u,H,Θ,η∥H42, and hence completes the proof of Theorem 11.



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*Source: 4748101-2020-07-10.xml*