# Global Existence and Long-Time Behavior of Solutions to the Full Compressible Euler Equations with Damping and Heat Conduction inℝ3

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

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## Abstract

We study the Cauchy problem of the three-dimensional full compressible Euler equations with damping and heat conduction. We prove the existence and uniqueness of the global smallHNN≥3 solution; in particular, we only require that the H4 norms of the initial data be small when N≥5. Moreover, we use a pure energy method to show that the global solution converges to the constant equilibrium state with an optimal algebraic decay rate as time goes to infinity.


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## Body

## 1. Introduction

We study the following Cauchy problem of the full compressible Euler equations with damping and heat conduction: forx,t∈ℝ3×0,∞,
(1)ρt+divρu=0,ρut+divρu⊗u+∇P=−αρu,ρEt+divρuE+uP=−αρu2+κΔθ,ρ,u,θx,tt=0=ρ0,u0,θ0x⟶1,0,1,x⟶∞.Here, the unknown variablesρ=ρx,t,u=ux,t,θ=θx,t, 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 constants α>0 and κ>0 are the friction damping coefficient and the thermal conductivity, respectively.The system (1) can be used to model a compressible gas flow through a porous medium [1–3]. Assume that the gas is perfect and polytropic, then
(2)P=Rρθ=AeS/cvργ,e=cvθ,cv=Rγ−1,where S is the entropy, R>0 and A>0 are the universal gas constants, γ>1 is the adiabatic exponent, and cv>0 is the specific heat at constant volume.We review the known results about the compressible Euler equations with damping. There are a lot of research works on the compressible isentropic Euler equations with damping in dimension one. For the Cauchy problems, readers can refer to [4, 5] for the existence of the global BV solutions, to [6–11] for the global L∞ entropy-weak solutions with vacuum, and to [12, 13] for small smooth solutions. For the initial-boundary value problems, readers can refer to [14, 15] for the existence of the global L∞ entropy-weak solutions and to [2, 16, 17] for small smooth solutions. For the asymptotic convergence of solutions, we refer to [8–11] for L∞ entropy-weak solutions and to [13, 18–20] for small smooth solutions. In addition, there are some results on the 1D compressible nonisentropic Euler equations with damping (see [1–3, 21–23]). The global existence and long-time behavior of solutions to the multidimensional compressible isentropic Euler equations with damping were studied by many researchers (cf. [15, 24–35] and the references cited therein). Recently, the free boundary problem of the Euler equations with damping was considered (cf. [36–38]).To the best of our knowledge, there are few results on the three-dimensional full compressible damped Euler equations (1). We first notice that the system (1) can be equivalently reduced to the ρ,u,θ-system
(3)ρt+divρu=0,ut+u·∇u+Rθ∇ρρ+R∇θ=−αu,cvθt+cvu·∇θ+Rθdivu=κΔθρ,or the P,u,S-system
(4)Pt+u·∇P+cv+RcvPdivu=RκcvΔθ,ut+u·∇u+∇Pρ=−αu,St+u·∇S=RκΔθP,where ρ and θ are given by
(5)ρ=A−cv/cv+RPcv/cv+Re−S/cv+R,θ=R−1Acv/cv+RPR/cv+ReS/cv+R.Whenκ>0, Chen et al. [39] considered the ρ,u,θ-system (3) and then used Fourier analysis methods together with energy methods to prove the global existence and time-decay rates of small smooth solutions. For the case of κ=0, the temperature equation in (3) has no dissipation, and thus, the method used in [39] is not applicable. To overcome the difficulties arising from the nondissipation of θ, the researchers in [40, 41] studied the P,u,S-system (4) with κ=0 and thus proved the similar results as the case of κ>0. An important observation is that the linear parts of P,u and S are decoupled in the linearized P,u,S-system, which helps to derive the desired estimates as done in [40, 41]. With regard to the corresponding initial-boundary value problem for κ=0 in a bounded domain, Zhang and Wu [42] and Wu [43] independently obtained the global existence and the exponential stability of small smooth solutions.In the present paper, we shall choose theρ,u,θ-system and prove the global existence and uniqueness of the smooth solution to the Cauchy problem (1) near a constant equilibrium state 1,0,1 for the initial data with various regularities. At the same time, we will use a pure energy method developed in [29, 44] to derive the optimal time-decay rates of solutions as well as their spatial derivatives of any order. Compared with the Fourier analysis method used in [39], the pure energy method can be used to obtain the optimal time-decay rates under the weak regularity assumptions, which can be seen from Lp⊂Ḣ−s1<p≤2,0≤s<3/2 or Lp⊂Ḃ2,∞−s1≤p<2,0<s≤3/2. As a byproduct, we give the optimal Lp–Lq-type decay rates of solutions (see Corollary 3).Notation.
Throughout this paper,∇k with an integer k≥0 represents the usual any 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 HNℝ3 for some integer N≥0 to denote the usual Sobolev spaces with the norm ·HN. 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 review the homogeneous Besov spaces. Letϕ∈C0∞ℝξ3 satisfy 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.We employ the notationA≲B to mean 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.The main results in this paper can be stated as follows.Theorem 1.
LetN≥3 be an integer. Assume that ρ0−1,u0,θ0−1∈HN satisfying
(6)ρ0−1,u0,θ0−1H3<δ0,ifN=3,or
(7)ρ0−1,u0,θ0−1H4<δ0,ifN≥4,for some small constant δ0>0. Then, the Cauchy problem (1) admits a unique global solution ρ,u,θt such that for all t≥0 and 3≤ℓ≤N,
(8)ρ−1,u,θ−1tHℓ+∫0t∇ρτHℓ−12+u,∇θτHℓ2dτ1/2≤Cρ0−1,u0,θ0−1Hℓ.Theorem 2.
Under the assumptions of Theorem1, 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,
(9)∇lρ−1,θ−1t≤C01+t−l+s/2,0≤l≤N−1,∇Nρ−1,θ−1t≤C01+t−N−1+s/2,and
(10)∇lut≤C01+t−l+1+s/2,0≤l≤N−2,∇N−1u,∇Nut≤C01+t−N−1+s/2.
By Lemma9 and Lemmas 13 and 14, 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≤∞,
(11)∇lρ−1,θ−1tLq≤C01+t−3/21/p−1/q−l/2,0≤l≤N−3,∇lutLq≤C01+t−3/21/p−1/q−l+1/2,0≤l≤N−4,∇N−3utLq≤C01+t−3/21/p−1/q−2N−5/4−1/2q.
We give some remarks for Theorems1 and 2 and Corollary 3.Remark 4.
From Theorem1, when N≥5, we only require that the H4 norms of the initial density, velocity, and temperature be small, while the higher-order Sobolev norms can be arbitrarily large.Remark 5.
We claim that the decay rates except the velocityu in Theorem 2 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 time-decay rates without the smallness assumption on the Lp1≤p<2 norm of the initial data.Remark 7.
Compared with the decay results of the full compressible Navier-Stokes equations [44, 45]
(12)ρ−1,u,θ−1≤C01+t−3/4,the density and temperature of the full compressible damped Euler equations have the same decay rates (see (9) with l=0 and s=3/2); however, the decay of the L2 norm of the velocity is improved to 1+t−5/4 (see (10) with l=0 and s=3/2) due to the damping effect.Remark 8.
With regard to the initial-boundary value problem of the three-dimensional full compressible damped Euler equations (1), the case of κ=0 was solved in [42, 43], and the corresponding P,u,S-system was adopted. For the case of κ>0, we believe that it is more convenient to deal with the ρ,u,θ-system, which is a forthcoming work.The arrangement of this paper is as follows. In Section2, we list some useful lemmas which will be frequently used. In Section 3, we establish some refined energy estimates (see Lemmas 17–19) which help us to derive important energy estimates with the minimum derivatives counts (see Lemma 20). Then, we prove the global solution (Theorem 1) and the time-decay rates (Theorem 2) in Sections 4 and 5, respectively.


## 2. Preliminaries

In this section, we will give some lemmas which are often used in the later sections. We first recall the Gagliardo-Nirenberg-Sobolev inequality.Lemma 9.
Let0≤m,α≤l, and 2≤p≤∞. Then, we have
(13)∇αfLp≲∇mf1−ϑ∇lfϑ,where 0≤ϑ≤1 and α satisfy
(14)α+312−1p=m1−ϑ+lϑ.
Here, we require that0<ϑ<1, m≤α+1, and l≥α+2, when p=∞.Proof.
(see [46], Theorem, p.125).We give the commutator and product estimates.Lemma 10.
Letm be an integer. Define the commutator
(15)∇m,fg≔∇mfg−f∇mg.
Then, we have form≥1,
(16)∇m,fgLp≲∇fLp1∇m−1gLp2+∇mfLp3gLp4,and for m≥0,
(17)∇mfgLp≲fLp1∇mgLp2+∇mfLp3gLp4,where p,p1,p2,p3,p4∈1,∞ and 1/p=1/p1+1/p2=1/p3+1/p4.Proof.
(refer to [47], Lemma 3.1, or [48], Lemma A.4).The following lemma gives the convenientLp estimates for well-prepared functions.Lemma 11.
Assume thatϱL∞≤1 and ΘL∞≤1. Let gϱ,Θ be a smooth function of ϱ,Θ with bounded derivatives of any order, then for any integer k≥1 and 2≤p≤∞,
(18)∇kgϱ,ΘLp≲∇kϱLp+∇kΘLp.Proof.
(see [49], Lemma A.2).As a byproduct of Lemma11, we immediately have the following.Corollary 12.
Assume thatϱL∞≤1. Let gϱ be a smooth function of ϱ with bounded derivatives of any order, then for any integer k≥1 and 2≤p≤∞,
(19)∇kgϱLp≲∇kϱLp.
Finally, we list some useful estimates or interpolation inequalities involving negative Sobolev or Besov spaces.Lemma 13.
Let1<p≤2 and 1/2+s/3=1/p. Then, 0≤s<3/2 and
(20)fḢ−s≲fLp.Proof.
It follows from the Hardy-Littlewood-Sobolev theorem (cf. [50], Theorem 1, p.119).Lemma 14.
Let1≤p<2 and 1/2+s/3=1/p. Then, 0<s≤3/2 and
(21)fḂ2,∞−s≲fLp.Proof.
(see [51], Lemma 4.1).Lemma 15.
Lets≥0 and l≥0. Then,
(22)∇lf≲∇l+1f1−ϑfḢ−sϑ,ϑ=1l+s+1.Proof.
(see [44], Lemma A.4).Lemma 16.
Lets>0 and l≥0. Then,
(23)∇lf≲∇l+1f1−ϑfḂ2,∞−sϑ,ϑ=1l+s+1.Proof.
We refer to [51], Lemma 4.2, by noting that Ḃ2,p−s⊂Ḃ2,q−s for p≤q.


## 3. Energy Estimates

By a simple calculation, the Cauchy problem (1) becomes
(24)ρt+divρu=0,ut+u·∇u+Rθ∇ρρ+R∇θ=−αu,cvθt+cvu·∇θ+Rθdivu=κΔθρ,ρ,u,θt=0=ρ0,u0,θ0.Without loss of generality, we assumeR=cv=α=κ=1 and choose the constant equilibrium state 1,0,1. Define the perturbations
(25)ϱ=ρ−1,u=u−0,Θ=θ−1.Then, problem (24) is reformulated as
(26)ϱt=−u·∇ϱ−1+ϱdivu,ut+u+∇Θ=−u·∇u−1+Θ1+ϱ∇ϱ,Θt−ΔΘ=−u·∇Θ−1+Θdivu−ϱ1+ϱΔΘ,ϱ,u,Θt=0=ϱ0,u0,Θ0≔ρ0−1,u0,θ0−1.We will derive the a priori estimates for the problem (26) by assuming that for sufficiently small δ>0 and some T>0,
(27)sup0≤t≤Tϱ,u,ΘtHN<δ,where N=3 or 4. By Sobolev’s inequality, (27) implies
(28)12≤1+ϱ≤32,12≤1+Θ≤32.First, we derive the energy estimates forϱ,u,Θ up to order N−1, which contain the dissipation estimates for u and Θ up to order N−1 and N, respectively.Lemma 17.
LetN≥3 and δ≪1. If sup0≤t≤Tϱ,u,ΘtH3<δ, then for 0≤k≤N−1,
(29)ddt∇kϱ,u,Θ2+∇ku2+∇k+1Θ2≲δ∇k+1ϱ2.Proof.
It is trivial fork=0. Next, we will prove (29) for 1≤k≤N−1. Applying ∇k to (26)1, (26)2, and (26)3 and multiplying the resulting identities by ∇kϱ, ∇ku, ∇kΘ, respectively; summing them up; and then integrating over ℝ3 by parts, we get
(30)12ddt∇kϱ,u,Θ2+∇ku2+∇k+1Θ2=−∫∇kdivϱu·∇kϱ−∫∇ku·∇u+Θ−ϱ1+ϱ∇ϱ·∇ku−∫∇kdivΘu·∇kΘ−∫∇kϱ1+ϱΔΘ·∇kΘ≔I1+I2+I3+I4.
Now, we estimate the termsI1–I4. For the term I1, by integrating by parts; Hölder’s, Sobolev’s, and Cauchy’s inequalities; and (17) of Lemma 10, we obtain
(31)I1=∫∇k−1divϱu·∇k−1Δϱ≲∇kϱu∇k+1ϱ≲∇kϱL6uL3+ϱL∞∇ku∇k+1ϱ≲δ∇k+1ϱ2+∇ku2.
For the termI2, by integrating by parts; Hölder’s, Sobolev’s, and Cauchy’s inequalities; Lemma 10; and Corollary 12, we obtain
(32)∫∇ku·∇u·∇ku=∫u·∇∇ku·∇ku+∇k,u·∇u·∇ku=−12∫divu∇ku2+∫∇k,u·∇u·∇ku≲divuL∞∇ku2+∇k,u·∇u∇ku≲uH3∇ku2+∇uL∞∇ku2≲δ∇ku2(33)∫∇kΘ−ϱ1+ϱ∇ϱ·∇ku≲∇kΘ−ϱ1+ϱ∇ϱ∇ku≲∇k∇ϱ1+ϱΘ+∇kϱ1+ϱ∇ϱ∇ku≲∇k∇ϱ1+ϱΘL∞+∇ϱ1+ϱL3∇kΘL6∇ku+∇kϱ1+ϱL6∇ϱL3+ϱ1+ϱL∞∇k+1ϱ∇ku≲δ∇k+1ϱ,Θ2+∇ku2.
In light of (32) and (33), we have
(34)I2≲δ∇k+1ϱ,Θ2+∇ku2.
As with the termI1, we obtain
(35)I3≲δ∇k+1Θ2+∇ku2.
For the termI4, by integrating by parts; Hölder’s, Sobolev’s, and Young’s inequalities; Lemmas 9 and 10; and Corollary 12, we obtain
(36)I4=∫∇k−1ϱ1+ϱΔΘ·∇k−1ΔΘ≲∇k−1ϱ1+ϱΔΘ∇k+1Θ≲∥∇k−1ϱ1+ϱ∥L6∥ΔΘ∥L3+∥ϱ1+ϱ∥L∞∇k+1Θ∇k+1Θ≲∇kϱ∇2ΘL3+ϱH2∇k+1Θ∇k+1Θ≲∇k+1ϱk/k+1ϱ1/k+1∇k+1ϱ1/k+1∇3k+1/2kΘk/k+1∇k+1Θ+δ∇k+1Θ2≲ϱ,ΘH3∇k+1ϱ+∇k+1Θ∇k+1Θ+δ∇k+1Θ2≲δ∇k+1ϱ,Θ2.
Plugging the estimates forI1–I4 into (30), we deduce (29).Next, we derive theN-th-order energy estimates for ϱ,u,Θ, which contain the dissipation estimates for u and Θ of order N and N+1, respectively.Lemma 18.
LetN≥3. Then, we have
(37)ddt∫ENt+∇Nu2+∇N+1Θ2≲δ∇Nϱ,Θ2,under the assumption of
(38)sup0≤t≤Tϱ,u,ΘtH3<δ≪1,ifN=3,or
(39)sup0≤t≤Tϱ,u,ΘtH4<δ≪1,ifN≥4.
Here,(40)ENt≔1+Θ1+ϱ∇Nϱ2+1+ϱ∇N−1divu2+∇N−1curlu2+1+ϱ1+Θ∇NΘ2.Proof.
We first prove (ii). For equations (26)1–(26)3, computing
(41)∫1+Θ1+ϱ∇Nϱ·∇N261+1+ϱ∇N−1divu·∇N−1div262+1+ϱ1+Θ∇NΘ·∇N263and integrating by parts, we have
(42)12ddt∫1+Θ1+ϱ∇Nϱ2+1+ϱ∇N−1divu2+1+ϱ1+Θ∇NΘ2+∫1+ϱ∇N−1divu2+1+ϱ1+Θ∇N+1Θ2=12∫∂t1+Θ1+ϱ∇Nϱ2+∂tϱ∇N−1divu2+∂t1+ϱ1+Θ∇NΘ2−∫1+Θ1+ϱ∇Nϱ·∇N1+ϱdivu+1+ϱ∇N−1divu·∇N−1div1+Θ1+ϱ∇ϱ−∫1+ϱ∇N−1divu·∇N−1ΔΘ+1+ϱ1+Θ∇NΘ·∇N1+Θdivu−∫∇1+ϱ1+Θ·∇NΘ·∇N+1Θ−∫1+Θ1+ϱ∇Nϱ·∇Nu·∇ϱ−∫1+ϱ∇N−1divu·∇N−1divu·∇u−∫1+ϱ1+Θ∇NΘ·∇Nu·∇Θ−∫1+ϱ1+Θ∇NΘ·∇Nϱ1+ϱΔΘ≔∑i=18Ji.
Now, we estimate the termsJ1–J8. By (26)1, (26)3, (28), and Hölder’s and Sobolev’s inequalities, we have
(43)J1≲ΘtL∞+ϱtL∞∇Nϱ,u,Θ2≲ϱ,uH3+ΘH4∇Nϱ,u,Θ2≲δ∇Nϱ,u,Θ2.
By the commutator notation (15), the commutator estimate (16), integrating by parts, and Lemma 11, we have
(44)J2=−∫1+Θ∇Nϱ·∇Ndivu+1+Θ∇N−1divu·∇N−1Δϱ−∫1+Θ1+ρ∇Nϱ·∇N,1+ϱdivu+1+ϱ∇N−1divu·∇N−1div,1+Θ1+ϱ∇ϱ≲∫∇Θ∇Nϱ∇Nu+∫∇Nϱ∇N,1+ϱ∇u+∫∇Nu∇N,1+Θ1+ϱ∇ϱ≲∇ΘL∞∇Nϱ∇Nu+∇Nϱ∇N,1+ϱ∇u+∇Nu∇N,1+Θ1+ϱ∇ϱ≲∇ΘL∞∇Nu∇Nϱ+∇Nϱ∇ϱL∞∇Nu+∇Nϱ∇uL∞+∇Nu∇1+Θ1+ϱL∞∇Nϱ+∇N1+Θ1+ϱ∇ϱL∞≲δ∇Nϱ,u,Θ2,J3=−∫1+ϱ∇N−1divu·∇N−1ΔΘ+1+ϱ∇NΘ·∇Ndivu−∫1+ϱ1+Θ∇NΘ·∇N,1+Θdivu≲∫∇ϱ∇NΘ∣∇N−1divu∣+∫∇NΘ∇N,1+Θ∇u≲∇ϱL∞∇NΘ∇Nu+∇NΘ∇ΘL∞∇Nu+∇NΘ∇uL∞≲δ∇Nu,Θ2.
By Hölder’s, Sobolev’s, and Cauchy’s inequalities and Lemma11, we obtain
(45)J4≲∇1+ϱ1+ΘL3∇NΘL6∇N+1Θ≲δ∇N+1Θ2.
By Lemma10, integrating by parts, and Hölder’s, Sobolev’s, and Cauchy’s inequalities, we have
(46)J5=−∫1+Θ1+ϱ∇Nϱ·∇Nu·∇ϱ=−12∫1+Θ1+ϱu·∇∇Nϱ2−∫1+Θ1+ϱ∇Nϱ·∇N,u·∇ϱ=12∫div1+Θ1+ϱu∇Nϱ2−∫1+Θ1+ϱ∇Nϱ·∇N,u·∇ϱ≲δ∇Nϱ2+∇Nϱ∇N,u·∇ϱ≲δ∇Nϱ2+∇Nϱ∇uL∞∇Nϱ+∇Nu∇ϱL∞≲δ∇Nϱ,u2.
Note that(47)divu·∇u=∇u:∇uT+u·∇divu,where the double dots : mean that A:B=∑i,j=13aijbij for two 3×3 matrices A=aij and B=bij. By Hölder’s, Sobolev’s, and Cauchy’s inequalities and Lemma 10, we estimate
(48)J6=−∫1+ϱ∇N−1divu·∇N−1divu·∇u=−∫1+ϱ∇N−1divu·∇N−1∇u:∇uT+u·∇divu=−∫1+ϱ∇N−1divu·∇N−1∇u:∇uT−∫1+ϱ∇N−1divu·∇N−1u·∇divu=−∫1+ϱ∇N−1divu·∇N−1∇u:∇uT−12∫1+ϱu·∇∇N−1divu2−∫1+ϱ∇N−1divu·∇N−1,u·∇divu=−∫1+ϱ∇N−1divu·∇N−1∇u:∇uT+12∫div1+ϱu∇N−1divu2−∫1+ϱ∇N−1divu·∇N−1,u·∇divu≲∇N−1divu∇N−1∇u:∇uT+div1+ϱuL∞∇N−1divu2+∇N−1divu∇N−1,u·∇divu≲∇Nu∇uL∞∇Nu+δ∇Nu2+∇Nu∇N−1uL6∇divuL3+∇uL∞∇N−1divu≲δ∇Nu2,J7=−∫1+ϱ1+Θ∇NΘ·∇Nu·∇Θ≲∇NΘ∇Nu·∇Θ≲∇NΘuL∞∇N+1Θ+∇Nu∇ΘL∞≲δ∇Nu,Θ2+∇N+1Θ2.
By integrating by parts and by the product estimates (17) of Lemma 10 and Corollary 12, we have
(49)J8=−∫1+ϱ1+Θ∇NΘ·∇Nϱ1+ϱΔΘ=∫div1+ϱ1+Θ∇NΘ·∇N−1ϱ1+ϱΔΘ=∫∇1+ϱ1+Θ∇NΘ+1+ϱ1+Θ∇N+1Θ·∇N−1ϱ1+ϱΔΘ≲∇1+ϱ1+ΘL3∇NΘL6+∇N+1Θ∇N−1ϱ1+ϱΔΘ≲∇1+ϱ1+ΘL3∇NΘL6+∇N+1Θ∇N−1ϱ1+ϱL6ΔΘL3+ϱL∞∇N+1Θ≲ϱ,ΘH3∇N+1Θ+∇N+1Θ∇NϱΘH3+ϱH3∇N+1Θ≲δ∇Nϱ2+∇N+1Θ2.
Plugging the estimates forJ1–J8 into (42), by (28), since δ is small, we deduce
(50)ddt∫1+Θ1+ϱ∇Nϱ2+1+ϱ∇N−1divu2+1+ϱ1+Θ∇NΘ2+∇N−1divu2+∇N+1Θ2≲δ∇Nϱ,u,Θ2.
Rewrite (26)2 as
(51)ut+u=−∇ϱ−∇Θ−u·∇u+ϱ−Θ1+ϱ∇ϱ.
Applyingcurl to (51), we obtain
(52)curlut+curlu=−curlu·∇u+∇ϱ−Θ1+ϱ×∇ϱ,where × represents the cross product of vectors. Applying ∇N−1 to (52), multiplying the resulting identity by ∇N−1curlu and integrating over ℝ3, we obtain
(53)12ddt∫∇N−1curlu2+∇N−1curlu2=−∫∇N−1curlu·∇u·∇N−1curlu+∫∇N−1∇ϱ−Θ1+ϱ×∇ϱ·∇N−1curlu≔J9+J10.
By Hölder’s, Sobolev’s, and Cauchy’s inequalities and Lemmas10 and 11, we estimate
(54)J9=−∫∇N−1curlu·∇u·∇N−1curlu=−12∫u·∇∇N−1curlu2−∫∇N−1curl,u·∇u·∇N−1curlu=12∫divu∇N−1curlu2−∫∇N−1curl,u·∇u·∇N−1curlu≲δ∇N−1curlu2+∫∇N,u·∇u∇Nu≲δ∇N−1curlu2+∇N,u·∇u∇Nu≲δ∇N−1curlu2+∇uL∞∇Nu2≲δ∇Nu2,J10=∫∇N−1∇ϱ−Θ1+ϱ×∇ϱ·∇N−1curlu≤∇N−1∇ϱ−Θ1+ϱ×∇ϱ∇N−1curlu≲∇Nϱ−Θ1+ϱ∇ϱL∞+∇ϱ−Θ1+ϱL∞∇Nϱ∇N−1curlu≲∇Nϱ,ΘϱH3+ϱ,ΘH3∇Nϱ∇Nu≲δ∇Nϱ,u,Θ2.
Plugging the estimates forJ9–J10 into (53), we obtain
(55)ddt∫∇N−1curlu2+∇N−1curlu2≲δ∇Nϱ,u,Θ2.
Adding (55) to (50), noting
(56)∇Nu2=∇N−1divu2+∇N−1curlu2,since δ is small, we deduce (37).
Now, we prove (i). Note that all the estimates forJ2–J10 in the proof of (ii) also hold under the assumptions of N=3 and sup0≤t≤Tϱ,u,ΘtH3<δ. Next, we only need to estimate the term J1 for N=3 under the condition of sup0≤t≤Tϱ,u,ΘtH3<δ. Note that from (26)1,2(57)ϱt=−ϱdivu−divu−u·∇ϱ,Θt=−u·∇Θ−divu−Θdivu+11+ϱΔΘ.
ForN=3, if sup0≤t≤Tϱ,u,ΘtH3<δ, then we can estimate
(58)J1=12∫∂t1+Θ1+ϱ∇3ϱ2+∂tϱ∇2divu2+∂t1+ϱ1+Θ∇3Θ2≲ΘtL∞+ϱtL∞∇3ϱ,u,Θ2≲ϱL∞∇uL∞+∇uL∞+uL∞∇ϱL∞∇3ϱ,u,Θ2+uL∞∇ΘL∞+∇uL∞+ΘL∞∇uL∞∇3ϱ,u,Θ2+∇2ΘL∞∇3ϱ,u,Θ2≲δ∇3ϱ,u,Θ2+∇2ΘL∞∇3ϱ,u,Θ2≲δ∇3ϱ,u,Θ2+∇2ΘL∞2≲δ∇3ϱ,u,Θ2+∇4Θ2,where we have used the interpolation estimate
(59)∇2ΘL∞≲∇3Θ1/2∇4Θ1/2.
Thus, combining the new estimate (58) with the estimates for J2–J10 with N=3, we can deduce from (42) that (37) holds for N=3.We shall derive the dissipation estimates forϱ up to order N.Lemma 19.
LetN≥3 and δ≪1. If sup0≤t≤Tϱ,u,ΘtH3<δ, then for 0≤k≤N−1,
(60)ddt∫∇ku·∇∇kϱ+∇k+1ϱ2≲∇ku2+∇k+1u,Θ2.Proof.
Rewrite (26)2 as
(61)∇ϱ=−ut−u−∇Θ−u·∇u+ϱ−Θ1+ϱ∇ϱ.
Applying∇k to (61) and multiplying the resulting identity by ∇∇kϱ and then integrating over ℝ3 by parts and by Hölder’s and Cauchy’s inequalities, we have
(62)∇k+1ϱ2≲−∫∂t∇ku·∇∇kϱ+∇ku2+∇k+1Θ2+∇ku·∇u2+∇kϱ−Θ1+ϱ∇ϱ2.
By (26)1, we integrate by parts to obtain
(63)−∫∂t∇ku·∇∇kϱ=−ddt∫∇ku·∇∇kϱ+∫∇ku·∇∇kϱt=−ddt∫∇ku·∇∇kϱ+∫∇kdivu·∇kdivu+∇kdivϱu≲−ddt∫∇ku·∇∇kϱ+∇k+1u2+∇k+1ϱu2≲−ddt∫∇ku·∇∇kϱ+∇k+1u2+δ∇k+1ϱ,u2,where we have used the product estimates (17) of Lemma 10 to estimate
(64)∇k+1ϱu≲ϱL∞∇k+1u+∇k+1ϱuL∞≲δ∇k+1ϱ,u.
By Lemmas10 and 11, we have
(65)∇ku·∇u≲uL∞∇k+1u+∇kuL6∇uL3≲δ∇k+1u,(66)∇kϱ−Θ1+ϱ∇ϱ≲ϱ,ΘL∞∇k+1ϱ+∇kϱ−Θ1+ϱL6∇ϱL3≲δ∇k+1ϱ,Θ.
Plugging (63)–(66) into (62), we deduce (60).Finally, we collect all the dissipation estimates forϱ,u,Θ in Lemmas 17–19 to derive the following energy inequality with the minimum derivative counts.Lemma 20.
LetN≥3 and T>0. Then, there exists an energy functional ElNt, which is equivalent to ∇lϱ,u,ΘtHN−l2, such that for any t∈0,T and 0≤l≤N−1,
(67)ddtElNt+∇l+1ϱtHN−l−12+∇lutHN−l2+∇l+1ΘtHN−l2≤0,under the assumption of
(68)sup0≤t≤Tϱ,u,ΘtH3<δ≪1,ifN=3,or
(69)sup0≤t≤Tϱ,u,ΘtH4<δ≪1,ifN≥4.Proof.
LetN≥3 and 0≤l≤N−1. Summing up (29) of Lemma 17 from k=l to k=N−1 and adding the resulting identity to (37) of Lemma 18, since δ is small, we obtain
(70)ddt∇lϱ,u,ΘtHN−l−12+∫ENt+C1∇lutHN−l2+∇l+1ΘtHN−l2≤C2δ∇l+1ϱtHN−l−12.
Summing up (60) of Lemma 19 from k=l to k=N−1, we obtain
(71)ddt∫∑k=lN−1∇ku·∇∇kϱ+C3∇l+1ϱtHN−l−12≤C4∇lutHN−l2+∇l+1ΘtHN−l−12.
Multiplying (71) by 2C2δ/C3 and then adding it to (70), since δ is small, we deduce
(72)ddt∇lϱ,u,ΘtHN−l−12+∫ENt+2C2δC3∫∑k=lN−1∇ku·∇∇kϱ+C5∇l+1ϱtHN−l−12+∇lutHN−l2+∇l+1ΘtHN−l2≤0.
We define(73)ElNt≔1C5∇lϱ,u,ΘtHN−l−12+∫ENt+2C2δC3∫∑k=lN−1∇ku·∇∇kϱ.
Note that(74)ENt=1+Θ1+ϱ∇Nϱ2+1+ϱ∇N−1divu2+∇N−1curlu2+1+ϱ1+Θ∇NΘ2.
By (28) and (56), since δ is small, we can deduce from (73) to (74) that there exists a positive constant c such that for any t∈0,T,
(75)1c∇lϱ,u,ΘtHN−l2≤ElNt≤c∇lϱ,u,ΘtHN−l2.
Hence, the proof of Lemma20 is completed.


## 4. 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. [52]).Proposition 21 (local-in-time solution).
Letl=3 or 4. Assume that ϱ0,u0,Θ0∈Hl and infx∈ℝ3ϱ0x+1>0. Then, there exists a constant T>0 such that the Cauchy problem (26) admits a unique solution ϱ,u,Θt∈E0,T;Hl satisfying
(76)infx∈ℝ3,0≤t≤Tϱx,t+1>0,sup0≤t≤Tϱ,u,ΘtHl≤C1ϱ0,u0,Θ0Hl,where C1>1 is some fixed constant. Here,
(77)E0,T;Hl≔ϱ,u,Θx,t:ϱ,ux,t∈C00,T;Hl∩C10,T;Hl−1,Θx,t∈C00,T;Hl∩C10,T;Hl−2.
Then, we construct the a priori estimates by using the energy estimates given in Lemma20.Proposition 22 (a priori estimates).
LetN≥3 and T>0. Assume that for some sufficiently small δ>0,
(78)sup0≤t≤Tϱ,u,ΘtH3<δ,ifN=3,or
(79)sup0≤t≤Tϱ,u,ΘtH4<δ,ifN≥4.
Then, we have for anyt∈0,T and 3≤ℓ≤N,
(80)ϱ,u,ΘtHℓ+∫0t∇ϱτHℓ−12+u,∇ΘτHℓ2dτ1/2≤C2ϱ0,u0,Θ0Hℓ,where C2>1 is some fixed constant.Proof.
LetN≥3 and 3≤ℓ≤N. Letting l=0 and N=ℓ in (67) of Lemma 20, we obtain
(81)ddtE0ℓt+∇ϱtHℓ−12+u,∇ΘtHℓ2≤0.
Integrating (81) in time, we obtain
(82)E0ℓt+∫0t∇ϱτHℓ−12+u,∇ΘτHℓ2dτ≤E0ℓ0.
Lettingl=0 and N=ℓ in (75), we obtain for any t∈0,T and some c>0,
(83)1cϱ,u,ΘtHℓ2≤E0ℓt≤cϱ,u,ΘtHℓ2.
We immediately deduce (80) from (82) to (83).Finally, we perform a continuous argument to extend the local solution given in Proposition21 to the global one. We only consider the case N≥4. It is similar for N=3.LetN≥4. Assume ϱ0,u0,Θ0∈HN satisfying
(84)ϱ0,u0,Θ0H4<δC1C2,where C1>1,C2>1,δ>0 are given by Proposition 21 and Proposition 22. Since
(85)ϱ0,u0,Θ0H4<δC1,by Proposition 21, there exists a constant T1>0 such that the Cauchy problem (26) has a unique local solution
(86)ϱ,u,Θt∈E0,T1;H4,which satisfies
(87)ϱ,u,ΘtH4≤C1ϱ0,u0,Θ0H4<δ,∀t∈0,T1.By (87) and Proposition 22, we obtain for any t∈0,T1 and 3≤ℓ≤N,
(88)ϱ,u,ΘtHℓ≤C2ϱ0,u0,Θ0Hℓ,which, together with (84), implies
(89)ϱ,u,ΘT1∈H4,ϱ,u,ΘT1H4<δC1.Then, choosingT1>0 as the new initial time instant, by Proposition 21 again, we obtain that the Cauchy problem (26) has a unique local solution
(90)ϱ,u,Θt∈ET1,2T1;H4,such that
(91)ϱ,u,ΘtH4≤C1ϱ,u,ΘT1H4<δ,∀t∈T1,2T1.From the above, we have proven that the Cauchy problem (26) has a unique local solution
(92)ϱ,u,Θt∈E0,2T1;H4,such that
(93)ϱ,u,ΘtH4<δ,∀t∈0,2T1.By (93) and Proposition 22, we obtain for any t∈0,2T1 and 3≤ℓ≤N,
(94)ϱ,u,ΘtHℓ≤C2ϱ0,u0,Θ0Hℓ,which, together with (84) again, implies
(95)ϱ,u,Θ2T1∈H4,ϱ,u,Θ2T1H4<δC1.By repeating the above procedures, we can extend the local solution to the global one only ifϱ0,u0,Θ0∈HNN≥4 satisfying that ϱ0,u0,Θ0H4 is suitably small, as (84). So, we can choose δ0=δ/C1C2 in Theorem 1. Hence, the proof of Theorem 1 is completed.


## 5. Time-Decay Rates

In this section, we shall derive the time-decay rates (9) in Theorem 2. We first show that the negative Sobolev or Besov norms of the solution ϱ,u,Θt can be controlled by the initial data.Lemma 23.
Under the assumptions of Theorem1, 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,
(96)ϱ,u,ΘtḢ−s≤C0,or
(97)ϱ,u,ΘtḂ2,∞−s≤C0.Proof.
As with [29], Theorem 1.2, (1.7)–(1.8), we omit the details.To derive the time-decay rates of the solution and its higher-order derivatives, we prove the following differential inequality with respect to time.Lemma 24.
LetN≥3. Under the assumptions of Theorem 1, it holds that for all t≥0 and 0≤l≤N−1,
(98)ddtElNt+∇l+1ϱtHN−l−12+∇lutHN−l2+∇l+1ΘtHN−l2≤0,where
(99)ElNt~∇lϱ,u,ΘtHN−l2.Proof.
It follows from Lemma20 and Theorem 1.Now, we can use Lemmas23 and 24 to prove the time-decay rates (9) and (10) in Theorem 2.For0≤l≤N−1, by Lemmas 15 and 16, we have
(100)∇l+1f≳fḢ−s−1/l+s∇lf1+1/l+s,(101)∇l+1f≳fḂ2,∞−s−1/l+s∇lf1+1/l+s.Combining Lemma23 with (100) and (101), we have
(102)∇l+1ϱ,Θ≥C0∇lϱ,Θ1+1/l+s.This together with (8) infers for 0≤l≤N−1,
(103)∇l+1ϱtHN−l−12+∇lutHN−l2+∇l+1ΘtHN−l2≥C0∇lϱ,u,ΘHN−l21+1/l+s.From the differential inequality (98) of Lemma 24, we obtain for 0≤l≤N−1,
(104)ddtElNt+C0ElNt1+1/l+s≤0.Solving the above inequality directly, we get for0≤l≤N−1,
(105)ElNt≤C01+t−l+s.By (99), we have for 0≤l≤N−1,
(106)∇lϱ,u,ΘtHN−l2≤C01+t−l+s.Higher decay of the velocityu. Referring to (51), we know that the velocity u satisfies
(107)ut+u=−∇ϱ−∇Θ−u·∇u+ϱ−Θ1+ϱ∇ϱ.LetN≥3 and 0≤l≤N−2. Applying ∇l to (107), multiplying the resulting identity by ∇lu and integrating over ℝ3, by Hölder’s inequality, we obtain
(108)12ddt∫∇lu2+∇lu2≲∇l∇ϱ,∇Θ,u·∇u,ϱ−Θ1+ϱ∇ϱ∇lu,which infers
(109)ddt∇lu2+∇lu2≲∇l∇ϱ,∇Θ,u·∇u,ϱ−Θ1+ϱ∇ϱ2≲∇l+1ϱ,Θ2+ϱ,u,ΘH22∇l+1ϱ,u,Θ2≤C01+t−l+1+sby Cauchy’s inequality, Lemmas 10 and 11, and (106). Applying Gronwall’s inequality to (109), we obtain for 0≤l≤N−2,
(110)∇lut2≤∇lu02e−t+C0∫0te−t−τ1+τ−l+1+sdτ≤C01+t−l+1+s.Thus, we can deduce the decay rates (9) and (10) from (106) and (110). Hence, we complete the proof of Theorem 2.



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