# OptimalLp–Lq-Type Decay Rates of Solutions to the Three-Dimensional Nonisentropic Compressible Euler Equations with Relaxation

Authors: Rong Shen; Yong Wang
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this paper, we consider the three-dimensional Cauchy problem of the nonisentropic compressible Euler equations with relaxation. Following the method of Wu et al. (2021, Adv. Math. Phys. Art. ID 5512285, pp. 1–13), we show the existence and uniqueness of the global smallHkk⩾3 solution only under the condition of smallness of the H3 norm of the initial data. Moreover, we use a pure energy method with a time-weighted argument to prove the optimal Lp–Lq1⩽p⩽2,2⩽q⩽∞-type decay rates of the solution and its higher-order derivatives.


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## Body

## 1. Introduction

In this paper, we shall study the nonisentropic compressible Euler equations with relaxation (cf. [1]):
(1)ρt+divρu=0,ρut+divρu⊗u+∇P=−1τρu,ρEt+divρuE+uP=−1τρu2+ρτgρ−θ.Here,x,t∈ℝ3×0,∞, and the unknown variables ρ=ρx,t, u=ux,t, θ=θx,t, and P=Px,t denote the density, the velocity, the absolute temperature, and the pressure, respectively. The total energy per unit mass E=1/2u2+e, and e is the internal energy per unit mass. The constant τ>0 is the relaxation parameter. The function gρ is smooth with respect to ρ. The system (1) can be used to model a compressible gas flow through a porous medium [2–4]. Assuming that the gas is perfect and polytropic, then
(2)P=ρθ,e=1γ−1θ,where γ>1 is the adiabatic exponent. Using the constitutive relations (2), the system (1) is reduced to
(3)ρt+divρu=0,ut+u·∇u+θ∇ρρ+∇θ=−1τu,1γ−1θt+1γ−1u·∇θ+θdivu=1τgρ−θ.We supplement (3) with the initial condition
(4)ρ,u,θx,tt=0=ρ0,u0,θ0x⟶1,0,1,x⟶∞.Now, we review the known research results for the compressible Euler equations with relaxation. When considering the isentropic or isothermal case, the system (1) becomes
(5)ρt+divρu=0,ρut+divρu⊗u+∇P=−1τρu.There are a lot of research works on the system (5). For the one-dimensional Cauchy problem, one can refer to [5, 6] for the existence of the global BV solutions, to [7–12] for the global L∞ entropy-weak solutions with vacuum, and to [13, 14] for small smooth solutions. For the one-dimensional initial-boundary value problem, one can refer to [15, 16] for the existence of the global L∞ entropy-weak solutions and to [17–19] for small smooth solutions. For the asymptotics of solutions, we refer to [9–12] for L∞ entropy-weak solutions and to [14, 20–22] for small smooth solutions. In addition, there are some results on the one-dimensional nonisentropic compressible Euler equations with relaxation (cf. [1–4, 23, 24]). The global existence and large-time behavior of solutions to the multidimensional isentropic compressible Euler equations with relaxation were studied by many researchers (cf. [16, 25–36] and the references cited therein).To the best of our knowledge, there are few results on the three-dimensional nonisentropic compressible Euler equations with relaxation (1). In this paper, following the similar discussions in [37], we shall use a delicate energy method to obtain a refined global existence and uniqueness result, in which we only require the initial H3 norm to be small. Moreover, we will prove the optimal Lp–Lq1⩽p⩽2,2⩽q⩽∞-type decay rates of solutions as well as its higher-order derivatives by employing the negative Sobolev or Besov estimates as well as some interpolation and time-weighted estimates.


### 1.1. Notation

Throughout this paper,∇k with an integer k⩾0 represents the spatial derivatives of order k. When k<0 or k is not a positive integer, ∇k means Λk defined by Λkf≔F−1ξkFf, where F is the usual Fourier transform operator and F−1 is its inverse. We denote by Lpℝ31⩽p⩽∞ the usual Lebesgue spaces with the norm ·Lp. For simplicity, we write ·=·L2. We use Hkℝ3 for some integer k⩾0 to denote the usual Sobolev spaces with the norm ∥·∥Hk. We use Ḣsℝ3s∈ℝ to denote the homogeneous Sobolev spaces with the norm ·Ḣs defined by fḢs≔∇sf. It is clear for H0=Ḣ0=L2.We introduce the homogeneous Besov spaces. Letϕ∈C0∞ℝξ3 satisfy that ϕξ=1 if ξ⩽1 and ϕξ=0 if ξ⩾2. Define φξ≔ϕξ−ϕ2ξ and φjξ≔φ2−jξ for j∈ℤ. Then, ∑j∈ℤφjξ=1 if ξ≠0. Define Δ̇jf≔F−1φj∗f. For s∈ℝ and 1⩽p⩽∞, we denote by Ḃp,∞sℝ3 the homogeneous Besov spaces with the norm ·Ḃp,∞s defined by fḂp,∞s≔supj∈ℤ2sjΔ̇jfLp.The notationA≲B means that A⩽CB for a generic positive constant C. We denote A~B if A≲B and B≲A. We use C0 to denote a positive constant depending additionally on the initial data. For simplicity, we write A,BX≔AX+BX and ∫f≔∫ℝ3fdx. The notation Ck0,T;Xk⩾0 denotes the space of X-valued k-times continuously differentiable functions on 0,T.Our main results are stated in the following.Theorem 1.
Letk⩾3 be an integer. Assume that ρ0−1,u0,θ0−1∈Hk satisfying
(6)ρ0−1,u0,θ0−1H3<δ0,for some small constant δ0>0. Suppose that gρ is a smooth function of ρ satisfying g1=1 and g′1=0. Then, the Cauchy problem (3)–(4) admits a unique global solution ρ,u,θt such that for all t⩾0 and 3⩽ℓ⩽k,
(7)ρ−1,u,θ−1tHℓ+∫0t∇ρςHℓ−12+u,θ−1ςHℓ2dς1/2⩽Cρ0−1,u0,θ0−1Hℓ,where C>0 depends only on γ and τ.Theorem 2.
Under the assumptions of Theorem1 and g′′1=0, if further ρ0−1,u0,θ0−1∈Ḣ−s for some s∈0,3/2 or ρ0−1,u0,θ0−1∈Ḃ2,∞−s for some s∈0,3/2, then for all t⩾0,
(8)∇lρ−1t⩽C01+t−l+s/2,0⩽l⩽k,(9)∇lu,θ−1t⩽C01+t−l+1+s/2,0⩽l⩽k−1,∇ku,θ−1t⩽C01+t−k+s/2.By LemmasA.1, A.5, and A.6, we easily obtain the following Lp–Lq-type decay rates.Corollary 3.
Under the assumptions of Theorem2, if ρ0−1,u0,θ0−1∈Lp for some p∈1,2, then for 2⩽q⩽∞,
(10)∇lρ−1tLq⩽C01+t−3/21/p−1/q−l/2,0⩽l⩽k−2,∇lu,θ−1tLq⩽C01+t−3/21/p−1/q−l+1/2,0⩽l⩽k−3,∇k−2u,θ−1tLq⩽C01+t−3/21/p−1/q−2k−3/4−1/2q.Some remarks for Theorems1 and 2 and Corollary 3 are given in the following.Remark 4.
From Theorem1, when k⩾3, we only require that the H3 norms of the initial density, velocity, and temperature are small, while the higher-order Sobolev norms can be arbitrarily large.Remark 5.
We claim that the decay rates in Theorem2 and Corollary 3 are optimal in the sense that they are consistent with those in the linearized case.Remark 6.
By Corollary3, we prove the optimal Lp–Lq-type decay rates without the smallness assumption on the Lp norm of the initial data.Remark 7.
The additional restrictiong′′1=0 in Theorem 2 is necessary to obtain the negative Sobolev or Besov estimates of solutions since the density perturbation is degenerately dissipative.The rest of this paper is organized as follows. In Section2, we establish some refined energy estimates (see Lemmas 8–11) which help us to derive important energy estimates with the minimum derivative counts (see Lemma 12). Then, we prove the global solution (Theorem 1) and the decay rates (Theorem 2) in Sections 3 and 4, respectively. In Appendix A, we list some useful lemmas which will be frequently used in the previous sections. The detailed proof of Lemma 15 is given in Appendix B.


## 1.1. Notation

Throughout this paper,∇k with an integer k⩾0 represents the spatial derivatives of order k. When k<0 or k is not a positive integer, ∇k means Λk defined by Λkf≔F−1ξkFf, where F is the usual Fourier transform operator and F−1 is its inverse. We denote by Lpℝ31⩽p⩽∞ the usual Lebesgue spaces with the norm ·Lp. For simplicity, we write ·=·L2. We use Hkℝ3 for some integer k⩾0 to denote the usual Sobolev spaces with the norm ∥·∥Hk. We use Ḣsℝ3s∈ℝ to denote the homogeneous Sobolev spaces with the norm ·Ḣs defined by fḢs≔∇sf. It is clear for H0=Ḣ0=L2.We introduce the homogeneous Besov spaces. Letϕ∈C0∞ℝξ3 satisfy that ϕξ=1 if ξ⩽1 and ϕξ=0 if ξ⩾2. Define φξ≔ϕξ−ϕ2ξ and φjξ≔φ2−jξ for j∈ℤ. Then, ∑j∈ℤφjξ=1 if ξ≠0. Define Δ̇jf≔F−1φj∗f. For s∈ℝ and 1⩽p⩽∞, we denote by Ḃp,∞sℝ3 the homogeneous Besov spaces with the norm ·Ḃp,∞s defined by fḂp,∞s≔supj∈ℤ2sjΔ̇jfLp.The notationA≲B means that A⩽CB for a generic positive constant C. We denote A~B if A≲B and B≲A. We use C0 to denote a positive constant depending additionally on the initial data. For simplicity, we write A,BX≔AX+BX and ∫f≔∫ℝ3fdx. The notation Ck0,T;Xk⩾0 denotes the space of X-valued k-times continuously differentiable functions on 0,T.Our main results are stated in the following.Theorem 1.
Letk⩾3 be an integer. Assume that ρ0−1,u0,θ0−1∈Hk satisfying
(6)ρ0−1,u0,θ0−1H3<δ0,for some small constant δ0>0. Suppose that gρ is a smooth function of ρ satisfying g1=1 and g′1=0. Then, the Cauchy problem (3)–(4) admits a unique global solution ρ,u,θt such that for all t⩾0 and 3⩽ℓ⩽k,
(7)ρ−1,u,θ−1tHℓ+∫0t∇ρςHℓ−12+u,θ−1ςHℓ2dς1/2⩽Cρ0−1,u0,θ0−1Hℓ,where C>0 depends only on γ and τ.Theorem 2.
Under the assumptions of Theorem1 and g′′1=0, if further ρ0−1,u0,θ0−1∈Ḣ−s for some s∈0,3/2 or ρ0−1,u0,θ0−1∈Ḃ2,∞−s for some s∈0,3/2, then for all t⩾0,
(8)∇lρ−1t⩽C01+t−l+s/2,0⩽l⩽k,(9)∇lu,θ−1t⩽C01+t−l+1+s/2,0⩽l⩽k−1,∇ku,θ−1t⩽C01+t−k+s/2.By LemmasA.1, A.5, and A.6, we easily obtain the following Lp–Lq-type decay rates.Corollary 3.
Under the assumptions of Theorem2, if ρ0−1,u0,θ0−1∈Lp for some p∈1,2, then for 2⩽q⩽∞,
(10)∇lρ−1tLq⩽C01+t−3/21/p−1/q−l/2,0⩽l⩽k−2,∇lu,θ−1tLq⩽C01+t−3/21/p−1/q−l+1/2,0⩽l⩽k−3,∇k−2u,θ−1tLq⩽C01+t−3/21/p−1/q−2k−3/4−1/2q.Some remarks for Theorems1 and 2 and Corollary 3 are given in the following.Remark 4.
From Theorem1, when k⩾3, we only require that the H3 norms of the initial density, velocity, and temperature are small, while the higher-order Sobolev norms can be arbitrarily large.Remark 5.
We claim that the decay rates in Theorem2 and Corollary 3 are optimal in the sense that they are consistent with those in the linearized case.Remark 6.
By Corollary3, we prove the optimal Lp–Lq-type decay rates without the smallness assumption on the Lp norm of the initial data.Remark 7.
The additional restrictiong′′1=0 in Theorem 2 is necessary to obtain the negative Sobolev or Besov estimates of solutions since the density perturbation is degenerately dissipative.The rest of this paper is organized as follows. In Section2, we establish some refined energy estimates (see Lemmas 8–11) which help us to derive important energy estimates with the minimum derivative counts (see Lemma 12). Then, we prove the global solution (Theorem 1) and the decay rates (Theorem 2) in Sections 3 and 4, respectively. In Appendix A, we list some useful lemmas which will be frequently used in the previous sections. The detailed proof of Lemma 15 is given in Appendix B.


## 2. Energy Estimates

We choose the constant equilibrium state1,0,1. Define the perturbations
(11)ϱ=ρ−1,u=u−0,Θ=θ−1.Then, the Cauchy problem (3)–(4) is equivalently written as
(12)ϱt=−u·∇ϱ−1+ϱdivu,(13)ut+1τu+∇Θ=−u·∇u−1+Θ1+ϱ∇ϱ,(14)1γ−1Θt+1τΘ=−1γ−1u·∇Θ−1+Θdivu+1τgϱ+1−1,with
(15)ϱ,u,Θt=0=ϱ0,u0,Θ0≔ρ0−1,u0,θ0−1.Next, we will derive the a priori estimates for equations (12)–(14) by assuming that for sufficiently small δ>0 and some T>0,
(16)sup0⩽t⩽Tϱ,u,ΘtH3<δ.By Sobolev’s inequality, (16) implies
(17)12⩽1+ϱ⩽32,12⩽1+Θ⩽32.By Taylor’s expansion, we have(18)gϱ+1=g1+g′1ϱ+Oϱ2=1+Oϱ2,where we have used the assumptions that g1=1 and g′1=0. Thus, we have
(19)gϱ+1−1=Oϱ2.First, we derive the zero-order energy estimate forϱ,u,Θ.Lemma 8.
Letδ≪1. If sup0⩽t⩽Tϱ,u,ΘtH3<δ, then
(20)ddtϱ,u,1γ−1Θ2+Cu,Θ2≲δ∇ϱ,Θ2.Proof.
Multiplying equations (12)–(14) by ϱ, u, and Θ, respectively, summing the resulting identities up, and then integrating over ℝ3 by parts, by (19), we get
(21)12ddtϱ,u,1γ−1Θ2+1τu2+1τΘ2=−∫divϱuϱ−∫u·∇u+Θ−ϱ1+ϱ∇ϱ·u−∫Θdivu+1γ−1u·∇Θ−1τOϱ2Θ.
Now, we estimate (21) term by term. For the term −∫divϱuϱ, by integrating by parts, Hölder’s, Sobolev’s, and Cauchy’s inequalities, we obtain
(22)−∫divϱuϱ=∫ϱu·∇ϱ⩽ϱL6uL3∇ϱ≲δ∇ϱ2.
By Hölder’s, Sobolev’s, and Cauchy’s inequalities and (17), we obtain
(23)−∫u·∇u·u⩽∇uL∞u2≲δu2,−∫Θ−ϱ1+ϱ∇ϱ·u⩽Θ−ϱ1+ϱL∞∇ϱu≲δ∇ϱ2+u2.
Similarly, we have(24)−∫Θdivu+1γ−1u·∇ΘΘ≲δu2+∇Θ2,(25)∫1τOϱ2Θ≲ϱL3ϱL6Θ≲δ∇ϱ2+Θ2.
Plugging the estimates (22)–(25) into (21), since δ≪1, we deduce (20).Next, we construct the higher-order energy estimates forϱ,u,Θ, which include the dissipation estimates for u and Θ of order k.Lemma 9.
Letk⩾3 and δ≪1. If sup0⩽t⩽Tϱ,u,ΘtH3<δ, then for 3⩽ℓ⩽k,
(26)ddt∫Eℓt+C∇ℓu,Θ2≲δ∇ℓϱ2,where
(27)Eℓt≔1+Θ1+ϱ∇ℓϱ2+1+ϱ∇ℓ−1divu2+∇ℓ−1curlu2+1γ−11+ϱ1+Θ∇ℓΘ2.Proof.
For equations (12)–(14), computing
(28)∫1+Θ1+ϱ∇ℓϱ·∇ℓequation12+1+ϱ∇ℓ−1divu·∇ℓ−1divequation13+1+ϱ1+Θ∇ℓΘ·∇ℓequation14,by (19), we have
(29)12ddt∫1+Θ1+ϱ∇ℓϱ2+1+ϱ∇ℓ−1divu2+1γ−11+ϱ1+Θ∇ℓΘ2+1τ∫1+ϱ∇ℓ−1divu2+1+ϱ1+Θ∇ℓΘ2=12∫∂t1+Θ1+ϱ∇ℓϱ2+∂tϱ∇ℓ−1divu2+1γ−1∂t1+ϱ1+Θ∇ℓΘ2−∫1+Θ1+ϱ∇ℓϱ·∇ℓ1+ϱdivu+1+ϱ∇ℓ−1divu·∇ℓ−1div1+Θ1+ϱ∇ϱ−∫1+ϱ∇ℓ−1divu·∇ℓ−1ΔΘ+1+ϱ1+Θ∇ℓΘ·∇ℓ1+Θdivu−∫1+Θ1+ϱ∇ℓϱ·∇ℓu·∇ϱ−∫1+ϱ∇ℓ−1divu·∇ℓ−1divu·∇u−1γ−1∫1+ϱ1+Θ∇ℓΘ·∇ℓu·∇Θ+1τ∫1+ϱ1+Θ∇ℓΘ·∇ℓOϱ2≔∑i=17Ii.
Now, we estimate the termsI1–I7. By (12), (14), and (17) and Hölder’s and Sobolev’s inequalities, we have
(30)I1≲ΘtL∞+ϱtL∞∇ℓϱ,u,Θ2≲ϱ,u,ΘH3∇ℓϱ,u,Θ2≲δ∇ℓϱ,u,Θ2.
By the commutator notation (A.3), the commutator estimates (A.4), (17), integrating by parts, and Lemma A.3, we have
(31)I2=−∫1+Θ∇ℓϱ·∇ℓdivu+1+Θ∇ℓ−1divu·∇ℓ−1Δϱ−∫1+Θ1+ϱ∇ℓϱ·∇ℓ,1+ϱdivu+1+ϱ∇ℓ−1divu·∇ℓ−1div,1+Θ1+ϱ∇ϱ≲∫∇Θ∇ℓϱ∇ℓu+∫∇ℓϱ∇ℓ,1+ϱ∇u+∫∇ℓu∇ℓ,1+Θ1+ϱ∇ϱ≲∇ΘL∞∇ℓϱ∇ℓu+∇ℓϱ∇ℓ,1+ϱ∇u+∇ℓu∇ℓ,1+Θ1+ϱ∇ϱ≲∇ΘL∞∇ℓu∇ℓϱ+∇ℓϱ∇ϱL∞∇ℓu+∇ℓϱ∇uL∞+∇ℓu∇1+Θ1+ϱL∞∇ℓϱ+∇ℓ1+Θ1+ϱ∇ϱL∞≲δ∇ℓϱ,u,Θ2,I3=−∫1+ϱ∇ℓ−1divu·∇ℓ−1ΔΘ+1+ϱ∇ℓΘ·∇ℓdivu−∫1+ϱ1+Θ∇ℓΘ·∇ℓ,1+Θdivu≲∫∇ϱ∇ℓΘ∇ℓ−1divu+∫∇ℓΘ∇ℓ,1+Θ∇u≲∇ϱL∞∇ℓΘ∇ℓu+∇ℓΘ∇ΘL∞∇ℓu+∇ℓΘ∇uL∞≲δ∇ℓu,Θ2.
By (17), Lemma A.2, integrating by parts, and Hölder’s, Sobolev’s, and Cauchy’s inequalities, we have
(32)I4=−∫1+Θ1+ϱ∇ℓϱ·∇ℓu·∇ϱ=−12∫1+Θ1+ϱu·∇∇ℓϱ2−∫1+Θ1+ϱ∇ℓϱ·∇ℓ,u·∇ϱ=12∫div1+Θ1+ϱu∇ℓϱ2−∫1+Θ1+ϱ∇ℓϱ·∇ℓ,u·∇ϱ≲δ∇ℓϱ2+∇ℓϱ∇ℓ,u·∇ϱ≲δ∇ℓϱ2+∇ℓϱ∇uL∞∇ℓϱ+∇ℓu∇ϱL∞≲δ∇ℓϱ,u2.
Note that the vector formula(33)divu·∇u=∇u:∇uT+u·∇divu,where the double dots : means that A:B=∑i,j=13aijbij for two 3×3 matrices A=aij and B=bij. By integrating by parts, (17) and (33), Hölder’s, Sobolev’s, and Cauchy’s inequalities, and Lemma A.2, we estimate
(34)I5=−∫1+ϱ∇ℓ−1divu·∇ℓ−1divu·∇u=−∫1+ϱ∇ℓ−1divu·∇ℓ−1∇u:∇uT+u·∇divu=−∫1+ϱ∇ℓ−1divu·∇ℓ−1∇u:∇uT−∫1+ϱ∇ℓ−1divu·∇ℓ−1u·∇divu=−∫1+ϱ∇ℓ−1divu·∇ℓ−1∇u:∇uT−12∫1+ϱu·∇∇ℓ−1divu2−∫1+ϱ∇ℓ−1divu·∇ℓ−1,u·∇divu=−∫1+ϱ∇ℓ−1divu·∇ℓ−1∇u:∇uT+12∫div1+ϱu∇ℓ−1divu2−∫1+ϱ∇ℓ−1divu·∇ℓ−1,u·∇divu≲∇ℓ−1divu∇ℓ−1∇u:∇uT+div1+ϱuL∞∇ℓ−1divu2+∇ℓ−1divu∇ℓ−1,u·∇divu≲∇ℓu∇uL∞∇ℓu+δ∇ℓu2+∇ℓu∇ℓ−1uL6∇divuL3+∇uL∞∇ℓ−1divu≲δ∇ℓu2,I6=−1γ−1∫1+ϱ1+Θ∇ℓΘ·∇ℓu·∇Θ=−1γ−1∫1+ϱ1+Θ∇ℓΘ·u·∇∇ℓΘ−1γ−1∫1+ϱ1+Θ∇ℓΘ·∇ℓ,u·∇Θ=121γ−1∫div1+ϱ1+Θu∇ℓΘ2−1γ−1∫1+ϱ1+Θ∇ℓΘ·∇ℓ,u·∇Θ≲div1+ϱ1+ΘuL∞∇ℓΘ2+1+ϱ1+ΘL∞∇ℓΘ∇ℓ,u·∇Θ≲δ∇ℓΘ2+∇ℓΘ∇ℓu∇ΘL∞+∇uL∞∇ℓ−1∇Θ≲δ∇ℓu,Θ2.
By (17), the product estimates (A.5) of Lemma A.2, and Corollary A.4, we have
(35)I7=1τ∫1+ϱ1+Θ∇ℓΘ·∇ℓOϱ2≲∫∇ℓΘ∇ℓϱOϱ≲δ∇ℓϱ,Θ2.
Plugging the estimates forI1–I7 into (29), by (17), since δ is small, we deduce
(36)ddt∫1+Θ1+ϱ∇ℓϱ2+1+ϱ∇ℓ−1divu2+1+ϱ1+Θ∇ℓΘ2+C∇ℓ−1divu2+∇ℓΘ2≲δ∇ℓϱ,u2.
Rewrite equation (13) as
(37)ut+1τu=−∇ϱ−∇Θ−u·∇u+ϱ−Θ1+ϱ∇ϱ.
Applying curl to (37), we obtain
(38)curlut+1τcurlu=−curlu·∇u+∇ϱ−Θ1+ϱ×∇ϱ,where × represents the cross product of vectors. Applying ∇ℓ−1 to (38), multiplying the resulting identity by ∇ℓ−1curlu, and integrating over ℝ3, we obtain
(39)12ddt∫∇ℓ−1curlu2+1τ∇ℓ−1curlu2=−∫∇ℓ−1curlu·∇u·∇ℓ−1curlu+∫∇ℓ−1∇ϱ−Θ1+ϱ×∇ϱ·∇ℓ−1curlu≔I8+I9.
By (17), integrating by parts, Hölder’s, Sobolev’s, and Cauchy’s inequalities, and Lemmas A.2 and A.3, we estimate
(40)I8=−∫∇ℓ−1curlu·∇u·∇ℓ−1curlu=−12∫u·∇∇ℓ−1curlu2−∫∇ℓ−1curl,u·∇u·∇ℓ−1curlu=12∫divu∇ℓ−1curlu2−∫∇ℓ−1curl,u·∇u·∇ℓ−1curlu≲δ∇ℓ−1curlu2+∫∇ℓ,u·∇u∇ℓu≲δ∇ℓ−1curlu2+∇ℓ,u·∇u∇ℓu≲δ∇ℓ−1curlu2+∇uL∞∇ℓu2≲δ∇ℓu2,I9=∫∇ℓ−1∇ϱ−Θ1+ϱ×∇ϱ·∇ℓ−1curlu∇ℓ−1∇ϱ−Θ1+ϱ×∇ϱ∇ℓ−1curlu≲∇ℓϱ−Θ1+ϱ∇ϱL∞+∇ϱ−Θ1+ϱL∞∇ℓϱ∇ℓ−1curlu≲∇ℓϱ,ΘϱH3+ϱ,ΘH3∇ℓϱ∇ℓu≲δ∇ℓϱ,u,Θ2.
Plugging the estimates forI8 and I9 into (39), we obtain
(41)ddt∫∇ℓ−1curlu2+C∇ℓ−1curlu2≲δ∇ℓϱ,u,Θ2.
Adding (41) to (36), noting
(42)∇ℓu2=∇ℓ−1divu2+∇ℓ−1curlu2,since δ≪1, we deduce (26).Corollary 10.
Letk⩾3 and δ≪1. If sup0⩽t⩽Tϱ,u,ΘtH3<δ, then for 1⩽ℓ⩽k−1,
(43)ddt∇ℓϱ,u,1γ−1Θ2+C∇ℓu,Θ2≲δ∇ℓ+1ϱ,Θ2.
Combining with Lemma8, we have for 0⩽ℓ⩽k−1,
(44)ddt∇ℓϱ,u,1γ−1Θ2+C∇ℓu,Θ2≲δ∇ℓ+1ϱ,Θ2.Proof.
Applying∇ℓ to equations (12)–(14), then multiplying the resulting identities by ∇ℓϱ, ∇ℓu, and ∇ℓΘ, respectively, summing them up, and then integrating over ℝ3 by parts, we obtain
(45)12ddt∇ℓϱ,u,1γ−1Θ2+1τ∇ℓu,Θ2=−∫∇ℓdivϱu·∇ℓϱ−∫∇ℓu·∇u·∇ℓu+∇ℓΘ−ϱ1+ϱ∇ϱ·∇ℓu−∫1γ−1∇ℓu·∇Θ·∇ℓΘ+∇ℓΘdivu·∇ℓΘ−1τ∇ℓOϱ2·∇ℓΘ.
Then, as in the proof of Lemma9, we easily deduce (43) from (45).We shall derive the dissipation estimates forϱ up to order k.Lemma 11.
Letk⩾3 and δ≪1. If sup0⩽t⩽Tϱ,u,ΘtH3<δ, then for 0⩽ℓ⩽k−1,
(46)ddt∫∇ℓu·∇∇ℓϱ+∇ℓ+1ϱ2≲∇ℓu2+∇ℓ+1u,Θ2.Proof.
Rewrite equation (13) as
(47)∇ϱ=−ut−1τu−∇Θ−u·∇u+ϱ−Θ1+ϱ∇ϱ.
Applying∇ℓ to (47), multiplying the resulting identity by ∇∇ℓϱ, and then integrating over ℝ3, by Hölder’s and Cauchy’s inequalities, we have
(48)∇ℓ+1ϱ2≲−∫∂t∇ℓu·∇∇ℓϱ+∇ℓu2+∇ℓ+1Θ2+∇ℓu·∇u2+∇ℓϱ−Θ1+ϱ∇ϱ2.
By (12), we integrate by parts to obtain
(49)−∫∂t∇ℓu·∇∇ℓϱ=−ddt∫∇ℓu·∇∇ℓϱ+∫∇ℓu·∇∇ℓϱt=−ddt∫∇ℓu·∇∇ℓϱ+∫∇ℓdivu·∇ℓdivu+∇ℓdivϱu≲−ddt∫∇ℓu·∇∇ℓϱ+∥∇ℓ+1u∥2+∥∇ℓ+1ϱu∥2≲−ddt∫∇ℓu·∇∇ℓϱ+∇ℓ+1u2+δ∇ℓ+1ϱ,u2,where we have used the product estimates (A.5) of Lemma A.2 to estimate
(50)∇ℓ+1ϱu≲ϱL∞∇ℓ+1u+∇ℓ+1ϱuL∞≲δ∇ℓ+1ϱ,u.
By LemmasA.2 and A.3, we have
(51)∇ℓu·∇u≲uL∞∇ℓ+1u+∇ℓuL6∇uL3≲δ∇ℓ+1u,(52)∇ℓϱ−Θ1+ϱ∇ϱ≲ϱ,ΘL∞∇ℓ+1ϱ+∇ℓϱ−Θ1+ϱL6∇ϱL3≲δ∇ℓ+1ϱ,Θ.
Plugging (49)–(52) into (48), we deduce (46).Finally, we collect all the dissipation estimates forϱ,u,Θ in Lemma 9, Corollary 10, and Lemma 11 to derive the following energy inequality with the minimum derivative counts.Lemma 12.
Letk⩾3 and T>0. If sup0⩽t⩽Tϱ,u,ΘtH3<δ≪1, then there exists an energy functional Elkt, which is equivalent to ∇lϱ,u,ΘtHk−l2, such that for any t∈0,T and 0⩽l⩽k−1,
(53)ddtElkt+∇l+1ϱtHk−l−12+∇lu,ΘtHk−l2⩽0.Proof.
Letk⩾3 and 0⩽l⩽k−1. Summing up (44) of Corollary 10 from ℓ=l to ℓ=k−1 and adding the resulting identity to (26) of Lemma 9 which ℓ=k, since δ is small, we obtain
(54)ddt∇lϱ,u,1γ−1ΘtHk−l−12+∫Ekt+C1∇lu,ΘtHk−l2⩽C2δ∇l+1ϱtHk−l−12.
Summing up (46) of Lemma 11 from ℓ=l to ℓ=k−1, we obtain
(55)ddt∫∑ℓ=lk−1∇ℓu·∇∇ℓϱ+C3∇l+1ϱtHk−l−12⩽C4∇lutHk−l2+∇l+1ΘtHk−l−12.
Multiplying (55) by 2C2δ/C3 and then adding it to (54), since δ≪1, we deduce
(56)ddt∇lϱ,u,1γ−1ΘtHk−l−12+∫Ekt+2C2δC3∫∑ℓ=lk−1∇ℓu·∇∇ℓϱ+C5∇l+1ϱtHk−l−12+∇lu,ΘtHk−l2⩽0.
We define(57)Elkt≔1C5∇lϱ,u,1γ−1ΘtHk−l−12+∫Ekt+2C2δC3∫∑ℓ=lk−1∇ℓu·∇∇ℓϱ.
Note that(58)Ekt=1+Θ1+ϱ∇kϱ2+1+ϱ∇k−1divu2+∇k−1curlu2+1γ−11+ϱ1+Θ∇kΘ2.
By (17) and (42), since δ≪1, we can deduce from (57) and (58) that there exists a positive constant c such that for any t∈0,T,
(59)1c∇lϱ,u,ΘtHk−l2⩽Elkt⩽c∇lϱ,u,ΘtHk−l2.
Hence, the proof of Lemma12 is completed.


## 3. Global Solution

In this section, we will prove the existence and uniqueness of the global solution, namely, Theorem1. We first record the local solution (cf. [38]).Proposition 13 (local-in-time solution).
Assume thatϱ0,u0,Θ0∈H3 and infx∈ℝ3ϱ0x+1>0. Then, there exists a constant T>0 such that the Cauchy problem (12)–(15) admits a unique solution ϱ,u,Θt∈E0,T;H3 satisfying
(60)infx∈ℝ3,0⩽t⩽Tϱx,t+1>0,sup0⩽t⩽Tϱ,u,ΘtH3⩽C1ϱ0,u0,Θ0H3,where C1>1 is some fixed constant. Here,
(61)E0,T;H3≔ϱ,u,Θx,t:ϱ,u,Θx,t∈C00,T;H3∩C10,T;H2.Then, we construct the a priori estimates by using the energy estimates given in Lemma12.Proposition 14 (a priori estimates).
Letk⩾3 and T>0. Assume that for some sufficiently small δ>0,
(62)sup0⩽t⩽Tϱ,u,ΘtH3<δ.
Then, we have for anyt∈0,T and 3⩽ℓ⩽k,
(63)ϱ,u,ΘtHℓ+∫0t∇ϱςHℓ−12+u,ΘςHℓ2dς1/2⩽C2ϱ0,u0,Θ0Hℓ,where C2>1 is some fixed constant.Proof.
Letk⩾3 and 3⩽ℓ⩽k. Letting l=0 and k=ℓ in (53) of Lemma 12, we obtain
(64)ddtE0ℓt+∇ϱtHℓ−12+u,ΘtHℓ2⩽0.
Integrating (64) in time, we obtain
(65)E0ℓt+∫0t∇ϱςHℓ−12+u,ΘςHℓ2dς⩽E0ℓ0.
Lettingl=0 and k=ℓ in (59), we obtain for any t∈0,T and some c>0,
(66)1cϱ,u,ΘtHℓ2⩽E0ℓt⩽cϱ,u,ΘtHℓ2.
We immediately deduce (63) from (65) and (66).Finally, we perform a continuous argument to extend the local solution given in Proposition13 to the global one.Letk⩾3. Assume ϱ0,u0,Θ0∈Hk satisfying
(67)ϱ0,u0,Θ0H3<δC1C2,where C1>1, C2>1, and δ>0 are given by Propositions 13 and 14. Since
(68)ϱ0,u0,Θ0H3<δC1,by Proposition 13, there exists a constant T1>0 such that the Cauchy problem (12)–(15) has a unique local solution
(69)ϱ,u,Θt∈E0,T1;H3,which satisfies
(70)ϱ,u,ΘtH3⩽C1ϱ0,u0,Θ0H3<δ,∀t∈0,T1.By (70) and Proposition 14, we obtain for any t∈0,T1 and 3⩽ℓ⩽k,
(71)ϱ,u,ΘtHℓ⩽C2ϱ0,u0,Θ0Hℓ,which, together with (67), implies
(72)ϱ,u,ΘT1∈H3,ϱ,u,ΘT1H3<δC1.Then, choosingT1>0 as the new initial time instant, by Proposition 13 again, we obtain that the Cauchy problem (12)–(15) has a unique local solution
(73)ϱ,u,Θt∈ET1,2T1;H3,such that
(74)ϱ,u,ΘtH3⩽C1ϱ,u,ΘT1H3<δ,∀t∈T1,2T1.From the above, we have proved that the Cauchy problem (12)–(15) has a unique local solution
(75)ϱ,u,Θt∈E0,2T1;H3,such that
(76)ϱ,u,ΘtH3<δ,∀t∈0,2T1.By (76) and Proposition 14, we obtain for any t∈0,2T1 and 3⩽ℓ⩽k,
(77)ϱ,u,ΘtHℓ⩽C2ϱ0,u0,Θ0Hℓ,which, together with (67) again, implies
(78)ϱ,u,Θ2T1∈H3,ϱ,u,Θ2T1H3<δC1.By repeating the above procedures, we can extend the local solution to the global one only ifϱ0,u0,Θ0∈Hkk⩾3 satisfying that ϱ0,u0,Θ0H3 is suitably small, as (67). So, we can choose δ0=δ/C1C2 in Theorem 1. Hence, the proof of Theorem 1 is completed.


## 4. Decay Rates

In this section, we shall derive the decay rates (8) and (9) in Theorem 2. We will divide the proof into four parts.


### 4.1. Part 1: Decay Rates of the Solution Itself and Its Derivatives up to thek−1-Order

First, we show that the negative Sobolev or Besov norms of the solutionϱ,u,Θt can be bounded by the initial data.Lemma 15.
Under the assumptions of Theorem1 and g′′1=0, if further ϱ0,u0,Θ0∈Ḣ−s for some s∈0,3/2 or ϱ0,u0,Θ0∈Ḃ2,∞−s for some s∈0,3/2, then for all t⩾0,
(79)ϱ,u,ΘtḢ−s⩽C0,or
(80)ϱ,u,ΘtḂ2,∞−s⩽C0.Proof.
The detailed proof can be found in AppendixB.Then, we prove the following differential inequality with respect to time.Lemma 16.
Letk⩾3. Under the assumptions of Theorem 1, it holds that for all t⩾0 and 0⩽l⩽k−1,
(81)ddtElkt+∇l+1ϱtHk−l−12+∇lu,ΘtHk−l2⩽0,where
(82)Elkt~∇lϱ,u,ΘtHk−l2.Proof.
It follows from Lemma12 and Theorem 1.Next, we can use Lemmas15 and 16 to prove the decay rates of the solution itself and its derivatives up to the k−1-order. For 0⩽l⩽k−1, by Lemmas A.7 and A.8, we have
(83)∇l+1f≳fḢ−s−1/l+s∇lf1+1/l+s,(84)∇l+1f≳fḂ2,∞−s−1/l+s∇lf1+1/l+s.Combining (79) and (80) with (83) and (84), we have
(85)∇l+1ϱ≳C0∇lϱ1+1/l+s.This together with (7) infers for 0⩽l⩽k−1,
(86)∇l+1ϱtHk−l−12+∇lu,ΘtHk−l2⩾C0∇lϱ,u,ΘHk−l21+1/l+s.From the differential inequality (81) of Lemma 16, we obtain for 0⩽l⩽k−1,
(87)ddtElkt+C0Elkt1+1/l+s⩽0.Solving the above inequality directly, we get for0⩽l⩽k−1,
(88)Elkt⩽C01+t−l+s.By (82), we have for 0⩽l⩽k−1,
(89)∇lϱ,u,ΘtHk−l2⩽C01+t−l+s.Note that the decay rate of thek-order derivatives is the same as one of the k−1-order.


### 4.2. Part 2: Higher Decay ofu and Θ

We can further improve the decay rates ofu,Θ as soon as we have the whole decay rate of ϱ,u,Θ by using the following processes.By (13) and (14), we have
(90)ut+1τu=−∇Θ−∇ϱ−u·∇u−Θ−ϱ1+ϱ∇ϱ,(91)1γ−1Θt+1τΘ=−divu−1γ−1u·∇Θ−Θdivu+1τOϱ3,where we have used g1=1 and g′1=g′′1=0 to obtain
(92)gϱ+1−1=Oϱ3.Letk⩾3 and 0⩽l⩽k−2. Applying ∇l to (90) and (91), multiplying the resulting identities by ∇lu and ∇lΘ, respectively, and integrating over ℝ3, we obtain
(93)12ddt∇lu2+1τ∇lu2=−∫∇l∇Θ+∇ϱ+u·∇u+Θ−ϱ1+ϱ∇ϱ·∇lu,1γ−112ddt∇lΘ2+1τ∇lΘ2=−∫∇ldivu+1γ−1u·∇Θ+Θdivu−1τOϱ3·∇lΘ.By Hölder’s, Sobolev’s, and Cauchy’s inequalities, LemmasA.2 and A.3, and (89), we obtain
(94)ddt∇lu2+C∇lu2≲∇l+1Θ,∇l+1ϱ,∇lu·∇u,∇lΘ−ϱ1+ϱ∇ϱ2≲∇l+1ϱ,Θ2+∇luL62∇uL32+uL∞2∇l+1u2+∇lΘ−ϱL62∇ϱL32+Θ−ϱL∞2∇l+1ϱ2≲∇l+1ϱ,Θ2+ϱ,u,ΘH22∇l+1ϱ,u,Θ2⩽C01+t−l+1+s,(95)ddt∇lΘ2+C∇lΘ2≲∇l+1u,∇lu·∇Θ,∇lΘdivu,∇lOϱ32≲∇l+1u2+∇luL62∇ΘL32+uL∞2∇l+1Θ2+∇lΘL62∇uL32+ΘL∞2∇l+1u2+∇lϱL62ϱL64≲∇l+1u2+ϱ,u,ΘH22∇l+1ϱ,u,Θ2≲C01+t−l+1+s.Applying Gronwall’s inequality to (94) and (95), we obtain for 0⩽l⩽k−2,
(96)∇lu2⩽∇lu02e−Ct+C0∫0te−Ct−ς1+ς−l+1+sdς⩽C01+t−l+1+s,(97)∇lΘ2⩽∇lΘ02e−Ct+C0∫0te−Ct−ς1+ς−l+1+sdς⩽C01+t−l+1+s.


### 4.3. Part 3: Decay Rates of thek-Order Derivatives of the Solution

We continue to derive the decay rate of thek-order derivatives by using a time-weighted argument recently developed in [39].Lemma 17.
Letk⩾3 and s be given in Lemma 15. It holds that for 0<ε0<1,
(98)1+tk+s∇kϱ,u,Θ2+C1+t−ε0∫0t1+τk+s+ε0∇ku,Θ2dτ⩽C0.Proof.
Letk⩾3. As with the proof of Lemma 9, by using the known decay rates (89), (96), and (97), we easily obtain
(99)ddt∫Ekt+C∇ku,Θ2⩽C1+t−1∇kϱ2,where
(100)Ekt=1+Θ1+ϱ∇kϱ2+1+ϱ∇k−1divu2+∇k−1curlu2+1γ−11+ϱ1+Θ∇kΘ2~∇kϱ,u,Θ2.
Let0<ε0<1. Multiplying (99) by 1+tk+s+ε0 and integrating over 0,t in time, by (100), we have
(101)1+tk+s+ε0∇kϱ,u,Θ2+C∫0t1+τk+s+ε0∇ku,Θ2dτ⩽∇kϱ0,u0,Θ02+C∫0t1+τk+s+ε0−1∇kϱ,u,Θ2dτ.
Next, we estimate the integral term on the right-hand side of (101). Letting l=k−1 in (81), we have
(102)ddtEk−1kt+∇kϱt2+∇k−1u,ΘtH12⩽0,where Ek−1kt~∇k−1ϱ,u,ΘtH12. Multiplying (102) by 1+tk+s+ε0−1, by (88), we have
(103)ddt1+tk+s+ε0−1Ek−1kt+1+tk+s+ε0−1∇kϱt2+∇k−1u,ΘtH12⩽1+tk+s+ε0−2Ek−1kt⩽C01+t−1+ε0.
Integrating (103) over 0,t in time, we have
(104)1+tk+s+ε0−1Ek−1kt+∫0t1+τk+s+ε0−1∇kϱ2+∇k−1u,ΘH12dτ⩽Ek−1k0+C0∫0t1+τ−1+ε0dτ⩽C01+tε0,which implies
(105)∫0t1+τk+s+ε0−1∇kϱ,u,Θ2dτ⩽C01+tε0.
Plugging (105) into (101), we have
(106)1+tk+s+ε0∇kϱ,u,Θ2+C∫0t1+τk+s+ε0∇ku,Θ2dτ⩽C01+tε0,which infers (98).By Lemma17, we have
(107)∇kϱ,u,Θt2⩽C01+t−k+s.


### 4.4. Part 4: Conclusion

Repeating Part 2 (Section4.2) with (107), we can obtain for 0⩽l⩽k−1,
(108)∇lu,Θt2⩽C01+t−l+1+s.Thus, the decay rates (8) and (9) of Theorem 2 hold from (89), (107), and (108). Hence, we complete the proof of Theorem 2.


## 4.1. Part 1: Decay Rates of the Solution Itself and Its Derivatives up to thek−1-Order

First, we show that the negative Sobolev or Besov norms of the solutionϱ,u,Θt can be bounded by the initial data.Lemma 15.
Under the assumptions of Theorem1 and g′′1=0, if further ϱ0,u0,Θ0∈Ḣ−s for some s∈0,3/2 or ϱ0,u0,Θ0∈Ḃ2,∞−s for some s∈0,3/2, then for all t⩾0,
(79)ϱ,u,ΘtḢ−s⩽C0,or
(80)ϱ,u,ΘtḂ2,∞−s⩽C0.Proof.
The detailed proof can be found in AppendixB.Then, we prove the following differential inequality with respect to time.Lemma 16.
Letk⩾3. Under the assumptions of Theorem 1, it holds that for all t⩾0 and 0⩽l⩽k−1,
(81)ddtElkt+∇l+1ϱtHk−l−12+∇lu,ΘtHk−l2⩽0,where
(82)Elkt~∇lϱ,u,ΘtHk−l2.Proof.
It follows from Lemma12 and Theorem 1.Next, we can use Lemmas15 and 16 to prove the decay rates of the solution itself and its derivatives up to the k−1-order. For 0⩽l⩽k−1, by Lemmas A.7 and A.8, we have
(83)∇l+1f≳fḢ−s−1/l+s∇lf1+1/l+s,(84)∇l+1f≳fḂ2,∞−s−1/l+s∇lf1+1/l+s.Combining (79) and (80) with (83) and (84), we have
(85)∇l+1ϱ≳C0∇lϱ1+1/l+s.This together with (7) infers for 0⩽l⩽k−1,
(86)∇l+1ϱtHk−l−12+∇lu,ΘtHk−l2⩾C0∇lϱ,u,ΘHk−l21+1/l+s.From the differential inequality (81) of Lemma 16, we obtain for 0⩽l⩽k−1,
(87)ddtElkt+C0Elkt1+1/l+s⩽0.Solving the above inequality directly, we get for0⩽l⩽k−1,
(88)Elkt⩽C01+t−l+s.By (82), we have for 0⩽l⩽k−1,
(89)∇lϱ,u,ΘtHk−l2⩽C01+t−l+s.Note that the decay rate of thek-order derivatives is the same as one of the k−1-order.


## 4.2. Part 2: Higher Decay ofu and Θ

We can further improve the decay rates ofu,Θ as soon as we have the whole decay rate of ϱ,u,Θ by using the following processes.By (13) and (14), we have
(90)ut+1τu=−∇Θ−∇ϱ−u·∇u−Θ−ϱ1+ϱ∇ϱ,(91)1γ−1Θt+1τΘ=−divu−1γ−1u·∇Θ−Θdivu+1τOϱ3,where we have used g1=1 and g′1=g′′1=0 to obtain
(92)gϱ+1−1=Oϱ3.Letk⩾3 and 0⩽l⩽k−2. Applying ∇l to (90) and (91), multiplying the resulting identities by ∇lu and ∇lΘ, respectively, and integrating over ℝ3, we obtain
(93)12ddt∇lu2+1τ∇lu2=−∫∇l∇Θ+∇ϱ+u·∇u+Θ−ϱ1+ϱ∇ϱ·∇lu,1γ−112ddt∇lΘ2+1τ∇lΘ2=−∫∇ldivu+1γ−1u·∇Θ+Θdivu−1τOϱ3·∇lΘ.By Hölder’s, Sobolev’s, and Cauchy’s inequalities, LemmasA.2 and A.3, and (89), we obtain
(94)ddt∇lu2+C∇lu2≲∇l+1Θ,∇l+1ϱ,∇lu·∇u,∇lΘ−ϱ1+ϱ∇ϱ2≲∇l+1ϱ,Θ2+∇luL62∇uL32+uL∞2∇l+1u2+∇lΘ−ϱL62∇ϱL32+Θ−ϱL∞2∇l+1ϱ2≲∇l+1ϱ,Θ2+ϱ,u,ΘH22∇l+1ϱ,u,Θ2⩽C01+t−l+1+s,(95)ddt∇lΘ2+C∇lΘ2≲∇l+1u,∇lu·∇Θ,∇lΘdivu,∇lOϱ32≲∇l+1u2+∇luL62∇ΘL32+uL∞2∇l+1Θ2+∇lΘL62∇uL32+ΘL∞2∇l+1u2+∇lϱL62ϱL64≲∇l+1u2+ϱ,u,ΘH22∇l+1ϱ,u,Θ2≲C01+t−l+1+s.Applying Gronwall’s inequality to (94) and (95), we obtain for 0⩽l⩽k−2,
(96)∇lu2⩽∇lu02e−Ct+C0∫0te−Ct−ς1+ς−l+1+sdς⩽C01+t−l+1+s,(97)∇lΘ2⩽∇lΘ02e−Ct+C0∫0te−Ct−ς1+ς−l+1+sdς⩽C01+t−l+1+s.


## 4.3. Part 3: Decay Rates of thek-Order Derivatives of the Solution

We continue to derive the decay rate of thek-order derivatives by using a time-weighted argument recently developed in [39].Lemma 17.
Letk⩾3 and s be given in Lemma 15. It holds that for 0<ε0<1,
(98)1+tk+s∇kϱ,u,Θ2+C1+t−ε0∫0t1+τk+s+ε0∇ku,Θ2dτ⩽C0.Proof.
Letk⩾3. As with the proof of Lemma 9, by using the known decay rates (89), (96), and (97), we easily obtain
(99)ddt∫Ekt+C∇ku,Θ2⩽C1+t−1∇kϱ2,where
(100)Ekt=1+Θ1+ϱ∇kϱ2+1+ϱ∇k−1divu2+∇k−1curlu2+1γ−11+ϱ1+Θ∇kΘ2~∇kϱ,u,Θ2.
Let0<ε0<1. Multiplying (99) by 1+tk+s+ε0 and integrating over 0,t in time, by (100), we have
(101)1+tk+s+ε0∇kϱ,u,Θ2+C∫0t1+τk+s+ε0∇ku,Θ2dτ⩽∇kϱ0,u0,Θ02+C∫0t1+τk+s+ε0−1∇kϱ,u,Θ2dτ.
Next, we estimate the integral term on the right-hand side of (101). Letting l=k−1 in (81), we have
(102)ddtEk−1kt+∇kϱt2+∇k−1u,ΘtH12⩽0,where Ek−1kt~∇k−1ϱ,u,ΘtH12. Multiplying (102) by 1+tk+s+ε0−1, by (88), we have
(103)ddt1+tk+s+ε0−1Ek−1kt+1+tk+s+ε0−1∇kϱt2+∇k−1u,ΘtH12⩽1+tk+s+ε0−2Ek−1kt⩽C01+t−1+ε0.
Integrating (103) over 0,t in time, we have
(104)1+tk+s+ε0−1Ek−1kt+∫0t1+τk+s+ε0−1∇kϱ2+∇k−1u,ΘH12dτ⩽Ek−1k0+C0∫0t1+τ−1+ε0dτ⩽C01+tε0,which implies
(105)∫0t1+τk+s+ε0−1∇kϱ,u,Θ2dτ⩽C01+tε0.
Plugging (105) into (101), we have
(106)1+tk+s+ε0∇kϱ,u,Θ2+C∫0t1+τk+s+ε0∇ku,Θ2dτ⩽C01+tε0,which infers (98).By Lemma17, we have
(107)∇kϱ,u,Θt2⩽C01+t−k+s.


## 4.4. Part 4: Conclusion

Repeating Part 2 (Section4.2) with (107), we can obtain for 0⩽l⩽k−1,
(108)∇lu,Θt2⩽C01+t−l+1+s.Thus, the decay rates (8) and (9) of Theorem 2 hold from (89), (107), and (108). Hence, we complete the proof of Theorem 2.



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