Global Existence and Long-Time Behavior of Solutions to the Full Compressible Euler Equations with Damping and Heat Conduction in
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 small solution; in particular, we only require that the norms of the initial data be small when . 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.
We study the following Cauchy problem of the full compressible Euler equations with damping and heat conduction: for ,
Here, the unknown variables , denote the density, the velocity, the absolute temperature, and the pressure, respectively. The total energy per unit mass , and is the internal energy per unit mass. The constants and 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 where is the entropy, and are the universal gas constants, is the adiabatic exponent, and 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 solutions, to [6–11] for the global 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 entropy-weak solutions and to [2, 16, 17] for small smooth solutions. For the asymptotic convergence of solutions, we refer to [8–11] for entropy-weak solutions and to [13, 18–20] for small smooth solutions. In addition, there are some results on the 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 -system or the -system where and are given by
When , Chen et al.  considered the -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 , the temperature equation in (3) has no dissipation, and thus, the method used in  is not applicable. To overcome the difficulties arising from the nondissipation of , the researchers in [40, 41] studied the -system (4) with and thus proved the similar results as the case of . An important observation is that the linear parts of and are decoupled in the linearized -system, which helps to derive the desired estimates as done in [40, 41]. With regard to the corresponding initial-boundary value problem for in a bounded domain, Zhang and Wu  and Wu  independently obtained the global existence and the exponential stability of small smooth solutions.
In the present paper, we shall choose the -system and prove the global existence and uniqueness of the smooth solution to the Cauchy problem (1) near a constant equilibrium state 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 , the pure energy method can be used to obtain the optimal time-decay rates under the weak regularity assumptions, which can be seen from or . As a byproduct, we give the optimal –-type decay rates of solutions (see Corollary 3).
Notation. Throughout this paper, with an integer represents the usual any spatial derivatives of order . When or is not a positive integer, means defined by , where is the usual Fourier transform operator and is its inverse. We denote by the usual Lebesgue spaces with the norm . For simplicity, we write . We use for some integer to denote the usual Sobolev spaces with the norm . We use to denote the homogeneous Sobolev spaces with the norm defined by . It is clear for .
We review the homogeneous Besov spaces. Let satisfy if and if . Define and for . Then, if . Define . For and , we denote by the homogeneous Besov spaces with the norm defined by .
We employ the notation to mean that for a generic positive constant . We denote if and . We use to denote a positive constant depending additionally on the initial data. For simplicity, we write and . The notation denotes the space of -valued -times continuously differentiable functions on .
The main results in this paper can be stated as follows.
Theorem 1. Let be an integer. Assume that satisfying or for some small constant . Then, the Cauchy problem (1) admits a unique global solution such that for all and ,
Remark 4. From Theorem 1, when , we only require that the norms of the initial density, velocity, and temperature be small, while the higher-order Sobolev norms can be arbitrarily large.
Remark 6. By Corollary 3, we prove the optimal –-type time-decay rates without the smallness assumption on the 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 and ); however, the decay of the norm of the velocity is improved to (see (10) with and ) 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 was solved in [42, 43], and the corresponding -system was adopted. For the case of , we believe that it is more convenient to deal with the -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 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.
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. Let , and . Then, we have where and satisfy Here, we require that , , and when .
Proof. (see , Theorem, p.125).
We give the commutator and product estimates.
Lemma 10. Let be an integer. Define the commutator Then, we have for , and for , where and
The following lemma gives the convenient estimates for well-prepared functions.
Lemma 11. Assume that and . Let be a smooth function of with bounded derivatives of any order, then for any integer and ,
Proof. (see , Lemma A.2).
As a byproduct of Lemma 11, we immediately have the following.
Corollary 12. Assume that . Let be a smooth function of with bounded derivatives of any order, then for any integer and , Finally, we list some useful estimates or interpolation inequalities involving negative Sobolev or Besov spaces.
Lemma 13. Let and . Then, and
Proof. It follows from the Hardy-Littlewood-Sobolev theorem (cf. , Theorem 1, p.119).
Lemma 14. Let and . Then, and
Proof. (see , Lemma 4.1).
Lemma 15. Let and . Then,
Proof. (see , Lemma A.4).
Lemma 16. Let and . Then,
Proof. We refer to , Lemma 4.2, by noting that for .
3. Energy Estimates
By a simple calculation, the Cauchy problem (1) becomes
Without loss of generality, we assume and choose the constant equilibrium state . Define the perturbations
Then, problem (24) is reformulated as
First, we derive the energy estimates for up to order , which contain the dissipation estimates for and up to order and , respectively.
Lemma 17. Let and . If , then for ,
Proof. It is trivial for . Next, we will prove (29) for . Applying to (26)1, (26)2, and (26)3 and multiplying the resulting identities by , , , respectively; summing them up; and then integrating over by parts, we get Now, we estimate the terms –. For the term , by integrating by parts; Hölder’s, Sobolev’s, and Cauchy’s inequalities; and (17) of Lemma 10, we obtain For the term , 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 , we obtain For the term , 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 – into (30), we deduce (29).
Next, we derive the -th-order energy estimates for , which contain the dissipation estimates for and of order and , respectively.
Lemma 18. Let . Then, we have under the assumption of or Here,
Proof. We first prove (ii). For equations (26)1–(26)3, computing and integrating by parts, we have Now, we estimate the terms –. By (26)1, (26)3, (28), and Hölder’s and Sobolev’s inequalities, we have By the commutator notation (15), the commutator estimate (16), integrating by parts, and Lemma 11, we have By Hölder’s, Sobolev’s, and Cauchy’s inequalities and Lemma 11, we obtain By Lemma 10, integrating by parts, and Hölder’s, Sobolev’s, and Cauchy’s inequalities, we have Note that where the double dots mean that for two matrices and . By Hölder’s, Sobolev’s, and Cauchy’s inequalities and Lemma 10, we estimate By integrating by parts and by the product estimates (17) of Lemma 10 and Corollary 12, we have