Global Existence and Long-Time Behavior of Solutions to the Full Compressible Euler Equations with Damping and Heat Conduction in R3

School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou 550001, China School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing, Xiamen University, Xiamen, Fujian 361005, China South China Research Center for Applied Mathematics and Interdisciplinary Studies, School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China


Introduction
We study the following Cauchy problem of the full compressible Euler equations with damping and heat conduction: for ðx, tÞ ∈ ℝ 3 × ½0,∞Þ, Here, the unknown variables ρ = ρðx, tÞ,u = uðx, tÞ, θ = θðx, tÞ, P = Pðx, tÞ denote the density, the velocity, the absolute temperature, and the pressure, respectively. The total energy per unit mass E = ð1/2Þjuj 2 + 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][2][3]. Assume that the gas is perfect and polytropic, then where S is the entropy, R > 0 and A > 0 are the universal gas constants, γ > 1 is the adiabatic exponent, and c v > 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][7][8][9][10][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][9][10][11] for L ∞ entropy-weak solutions and to [13,[18][19][20] for small smooth solutions. In addition, there are some results on the 1D compressible nonisentropic Euler equations with damping (see [1][2][3][21][22][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][25][26][27][28][29][30][31][32][33][34][35] and the references cited therein). Recently, the free boundary problem of the Euler equations with damping was considered (cf. [36][37][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 ðρ, u, θÞ-system ρ t + div ρu ð Þ= 0, or the ðP, u, SÞ-system where ρ and θ are given by When κ > 0, Chen et al. [39] considered the ðρ, u, θÞ -system (3) and then used Fourier analysis methods together with energy methods to prove the global existence and timedecay 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 ðP, u, SÞ-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 ðP, uÞ and S are decoupled in the linearized ðP, u, SÞ-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 the ðρ, u, θÞ-system and prove the global existence and uniqueness of the smooth solution to the Cauchy problem (1) near a constant equilibrium state ð1, 0, 1Þ 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 . As a byproduct, we give the optimal L p -L q -type decay rates of solutions (see Corollary 3).
We employ the notation A ≲ B to mean that A ≤ CB for a generic positive constant C. We denote A~B if A ≲ B and B ≲ A. We use C 0 to denote a positive constant depending additionally on the initial data. For simplicity, we write kðA, BÞk X ≔ kAk X + kBk X and Ð f ≔ Ð ℝ 3 f dx. The notation C k ð0, T ; XÞðk ≥ 0Þ denotes the space of X-valued k-times continuously differentiable functions on ½0, T.
The main results in this paper can be stated as follows. or for some small constant δ 0 > 0. Then, the Cauchy problem (1) admits a unique global solution ðρ, u, θÞðtÞ such that for all t ≥ 0 and 3 ≤ ℓ ≤ N, By Lemma 9 and Lemmas 13 and 14, we easily obtain the following L p -L q -type decay rates.

Corollary 3. Under the assumptions of Theorem
We give some remarks for Theorems 1 and 2 and Corollary 3.
Remark 4. From Theorem 1, when N ≥ 5, we only require that the H 4 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 velocity u in Theorem 2 and Corollary 3 are optimal in the sense that they are consistent with those in the linearized case.
Remark 6. By Corollary 3, we prove the optimal L p -L q -type time-decay rates without the smallness assumption on the L p ð1 ≤ p < 2Þ norm of the initial data.
Remark 7. Compared with the decay results of the full compressible Navier-Stokes equations [44,45] 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 L 2 norm of the velocity is improved to ð1 + tÞ −ð5/4Þ (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 ðP, u, SÞ-system was adopted. For the case of κ > 0, we believe that it is more convenient to deal with the ðρ, u, θÞ-system, which is a forthcoming work.
The arrangement of this paper is as follows. In Section 2, we list some useful lemmas which will be frequently used. In Section 3, we establish some refined energy estimates (see  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.

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.
We give the commutator and product estimates.
The following lemma gives the convenient L p estimates for well-prepared functions.
As a byproduct of Lemma 11, we immediately have the following.
Corollary 12. Assume that kϱk L ∞ ≤ 1. Let gðϱÞ be a smooth function of ϱ with bounded derivatives of any order, then for any integer k ≥ 1 and 2 ≤ p ≤ ∞, Finally, we list some useful estimates or interpolation inequalities involving negative Sobolev or Besov spaces.

Energy Estimates
By a simple calculation, the Cauchy problem (1) becomes Without loss of generality, we assume R = c v = α = κ = 1 and choose the constant equilibrium state ð1, 0, 1Þ. Define the perturbations Then, problem (24) is reformulated as We will derive the a priori estimates for the problem (26) by assuming that for sufficiently small δ > 0 and some T > 0, where N = 3 or 4. By Sobolev's inequality, (27) implies First, we derive the energy estimates for ðϱ, u, ΘÞ up to order N − 1, which contain the dissipation estimates for u and Θ up to order N − 1 and N, respectively.
Proof. It is trivial for k = 0. Next, we will prove (29) (26) Now, we estimate the terms I 1 -I 4 . For the term I 1 , by integrating by parts; Hölder's, Sobolev's, and Cauchy's inequalities; and (17) of Lemma 10, we obtain For the term I 2 , by integrating by parts; Hölder's, Sobolev's, and Cauchy's inequalities; Lemma 10; and Corollary 12, we obtain In light of (32) and (33), we have As with the term I 1 , we obtain For the term I 4 , by integrating by parts; Hölder's, Sobolev's, and Young's inequalities; Lemmas 9 and 10; and Corollary 12, we obtain Plugging the estimates for I 1 -I 4 into (30), we deduce (29).
Next, we derive the N-th-order energy estimates for ðϱ, u, ΘÞ, which contain the dissipation estimates for u and Θ of order N and N + 1, respectively.

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Plugging the estimates for J 9 -J 10 into (53), we obtain d dt Adding (55) to (50), noting since δ is small, we deduce (37). Now, we prove (i). Note that all the estimates for J 2 -J 10 in the proof of (ii) also hold under the assumptions of N = 3 and sup 0≤t≤T kðϱ, u, ΘÞðtÞk H 3 < δ. Next, we only need to estimate the term J 1 for N = 3 under the condition of sup 0≤t≤T kðϱ, u, ΘÞðtÞk H 3 < δ. Note that from (26) 1,2 For N = 3, if sup 0≤t≤T kðϱ, u, ΘÞðtÞk H 3 < δ, then we can estimate where we have used the interpolation estimate Thus, combining the new estimate (58) with the estimates for J 2 -J 10 with N = 3, we can deduce from (42) that (37) holds for N = 3.
We shall derive the dissipation estimates for ϱ up to order N.
Proof. Rewrite (26) 2 as Applying ∇ k to (61) and multiplying the resulting identity by ∇∇ k ϱ and then integrating over ℝ 3 by parts and by Hölder's and Cauchy's inequalities, we have By (26) 1 , we integrate by parts to obtain where we have used the product estimates (17) of Lemma 10 to estimate By Lemmas 10 and 11, we have Plugging (63)-(66) into (62), we deduce (60).

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Finally, we collect all the dissipation estimates for ðϱ, u, ΘÞ in Lemmas 17-19 to derive the following energy inequality with the minimum derivative counts. Lemma 20. Let N ≥ 3 and T > 0. Then, there exists an energy functional E N l ðtÞ, which is equivalent to k∇ l ðϱ, u, ΘÞðtÞk 2 H N−l , such that for any t ∈ ½0, T and 0 ≤ l ≤ N − 1, under the assumption of Proof. Let N ≥ 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 Summing up (60) of Lemma 19 from k = l to k = N − 1, we obtain d dt Multiplying (71) by 2C 2 δ/C 3 and then adding it to (70), since δ is small, we deduce We define Note that 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 ∈ ½0, T, Hence, the proof of Lemma 20 is completed.

Global Solution
In this section, we will prove the existence and uniqueness of the global solution, namely, Theorem 1. We first record the local solution (cf. [52]).
where C 1 > 1 is some fixed constant. Here, Then, we construct the a priori estimates by using the energy estimates given in Lemma 20.
Proposition 22 (a priori estimates). Let N ≥ 3 and T > 0. Assume that for some sufficiently small δ > 0, or 9 Advances in Mathematical Physics Then, we have for any t ∈ ½0, T and 3 ≤ ℓ ≤ N, where C 2 > 1 is some fixed constant.
Finally, we perform a continuous argument to extend the local solution given in Proposition 21 to the global one. We only consider the case N ≥ 4. It is similar for N = 3.

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 ðϱ, u, ΘÞðtÞ can be controlled by the initial data.