GLOBAL EXISTENCE AND CONVERGENCE RATES OF SOLUTIONS FOR THE COMPRESSIBLE EULER EQUATIONS WITH DAMPING

. The Cauchy problem for the 3D compressible Euler equations with damping is considered. Existence of global-in-time smooth solutions is estab- lished under the condition that the initial data is small perturbations of some given constant state in the framework of Sobolev space H 3 ( R 3 ) only, but we don’t need the bound of L 1 norm. Moreover, the optimal L 2 - L 2 convergence rates are also obtained for the solution. Our proof is based on the beneﬁt of the low frequency and high frequency decomposition, here, we just need spectral analysis of the low frequency part of the Green function to the linearized system, so that we succeed to avoid some complicate analysis.


1.
Introduction. The compressible Euler equations with damping are used to model and simulate the compressible flow through a porous medium. In the present paper, we consider the compressible Euler equations with damping    ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) + aρu = 0, (ρ, u)(x, t)| t=0 = (ρ 0 , u 0 )(x), x ∈ R 3 , where ρ = ρ(t, x), u = u(t, x) represent the density and velocity functions respectively, x ∈ R 3 is the space variable, t > 0 is the time variable. Furthermore, the pressure p is suitably smooth function of ρ. Assuming the flow is a polytropic perfect gas, then p(ρ) = Aρ r with constants A > 0 and r > 1 the adiabatic exponent; the constant a > 0 models friction and 1/a may be regarded as the relaxation time for some physical flows.
There are many important progresses on the investigation of the global existence and large time behavior of solutions to (1). For one dimension, the global existence of a smooth solution with small data was proved by Nishida [13,14], and the behavior of the smooth solution was studied in many papers; see the excellent survey paper by Dafermos [3], the book by Hsiao [5], the papers [6,7,9,10,15,21,25,26], and their references.
We note that the multi-dimensional compressible Euler equations with damping describe more realistic phenomena and carry some unique features, such as the effect of vorticity, which make the problem more mathematically challenging. Due to its physical importance and significant mathematical challenge, a deep research on the multi-dimensional model (1) is of great importance. In three dimensions, Wang and Yang [22] proved the global existence of the small smooth solutions and obtained the pointwise estimates of the solutions, then [17] used a different energy method to obtain the similar results. The authors in [17] proved that the L 2 -norm of the solution decays at the rate (1 + t) −3/4 provided that the initial data is small in L 1 , while the authors in [20] proved that the L 2 -norm of the solution decays at the rate (1 + t) −3/4−s/2 provided that the initial data is small in B −s 1,∞ . [18] studied the global existence and time-asymptotic behavior of small smooth solutions to the system (1) by a refined pure energy method. [2] studied in more details the effect of the damping on the decay rate of the solutions, along with this end, they removed the smallness of those low frequency assumption of the initial data and showed the optimal decay rates of the higher-order spatial derivatives. For initial boundary value problem, refer for instance to [8,16] and the references therein.
In this paper, we prove the global existence and optimal convergence rates of the solution when H 3 (R 3 )-norm of the initial perturbation around a constant state is sufficiently small only. There are two main differences between the analysis of this paper and the known results for Navier-Stokes(-Poisson) equations [4,19]. For Navier-Stokes equations, there are strong dissipative terms (i.e.viscous of fluid) in the momentum equations, which is helpful for energy estimates. However, there is only a weak dissipative term (i.e. damping term) in the momentum equations of Euler equations, which causes troubles for energy estimates. Another difference is that we introduce the low frequency and high frequency decomposition u = u l + u h to consider the decay rates, where the two terminologies(i.e.the low frequency part u l and the high frequency part u h ) have been used in [23]. In particular, we do not require that the L p norm of initial data is sufficiently small.
We reformulate the Cauchy problem of the compressible Euler system (1) as in [17]. The main point is to obtain a symmetric system. Introduce the sound speed µ(ρ) = p (ρ), and setμ = µρ corresponding to the sound speed at a background densityρ > 0. Define Then the Euler equations (1) are transformed into the following system: where F 1 : = −u · ∇n − νndivu, The main result of the present paper is stated in the following theorem: is sufficiently small. Then the initial value problem (2) admits a unique global solution (n, u) satisfying that for all t ≥ 0, Moreover, there is a constant C such that for any t ≥ 0, the solution (n, u) enjoys the decay properties Notations. Throughout this paper, C denotes a generic positive constant which may vary in different estimates. The norms in the usual L p and Sobolev Space H m on R 3 are denoted by · L p and · H m respectively. Moreover, we use · to denote the inner product in L 2 (R 3 ). ∂ j stands for ∂ xj , ∇ l with an integer l ≥ 0 stands for the usual any spatial derivatives of order l, when l < 0 or l is not a positive integer, ∇ l stands for Λ l defined by Λ s u = F −1 (|ξ| sû (ξ)), whereû is the Fourier transform of u and F −1 its inverse. We will employ the notation A B to mean that A ≤ CB for a universal constant C > 0 that only depends on the parameters coming from the problem. For the sake of conciseness, we write (A, B) X := A X + B X .

2.
Reformulations. In this subsection, we will consider the global existence and time decay rates of the solutions to the reformulated system (2). By the standard continuity argument, the global existence of solutions to (2) will be obtained by combining the local existence result with global a priori estimates. The following local existence result can be established using the arguments in [11,12]. Proposition 1. Suppose that the initial data satisfy (n 0 (x), u 0 (x)) ∈ H 3 (R 3 ), then there exists a unique local solution (n(x, t), u(x, t)) of the Cauchy problems (2) ) for some finite T > 0.
To prove global existence of a smooth solution with small initial data, we also need to establish the following a priori energy estimates. Proposition 2. Let (n 0 , u 0 ) ∈ H 3 (R 3 ), suppose that the initial value problem (2) has a solution (n, u)(x, t) on R 3 × [0, T ] for some T > 0. Then there exist a small constant δ > 0 and a constant C, which are independent of T , such that if then for any t ∈ [0, T ], it holds that 3. Some a priori estimates. In this section, we will establish some a priori estimates of the solution (n, u). We first make the a priori assumption that for sufficiently small δ > 0.
In terms of the semigroup theory for evolutional equation, the solution (n, u) can be expressed for U = (n, u) t as which gives rise to Applying the Fourier transform to system, we have and A(ξ) is defined as The eigenvalues of the matrix A are computed from the determinant The semigroup e tA is expressed as where the project operators P i can be computed as Denote then we have the following decomposition for (n, u) = G * U 0 as We are able to obtain that it holds for |ξ| 1, We are also able to obtain that it holds for |ξ| 1, Then we obtain that We can now derive the decay rates for the solution of the linear system (12).
where C > 0 is a positive constant independent of time.
Proof. With the help of the formula for Green's function in Fourier space and the asymptotical analysis on its elements, we have Therefore, we have the L 2 -decay rate on the derivatives of n as where 1 r = 1 q − 1 p , η are some positive constants. As for u, in the similar fashion, we have then we also obtain that The proof of the Lemma is completed.
Noticing the definition of f h (x) and using the Plancherel theorem, it is obvious that there exist two positive constants C 3 and C 4 such that if f (x) ∈ H 3 (R 3 ), 3.3. Estimates of the low-frequency part. To begin with, we first rewrite the solution of (2) 1,2 as Taking χ 0 (D x ) on both sides of (35), we have where S l (t) = χ 0 (D x )S(t) and U l (t) = (n l (t), u l (t)). Making use of Lemma 3.1, we have the L p − L q type of the time decay estimates of S l (t) as follows.
Lemma 3.2. Let k ≥ 0 be integers and p, q ∈ [1, ∞) and p ≥ q, then for any t > 0, it holds that where C > 0 is a positive constant independent of time.
In the following, we show the estimates on U l (t) = (n l (t), u l (t)). By the way, one can get the same estimates for U l (t), where U l (t) = (n l (t), u l (t)).

Lemma 3.3.
Under the assumptions of Proposition 2, it holds that for 2 ≤ p ≤ +∞ and any integer k ≥ 0 Proof. From (36) and Lemma 3.2, we obtain We shall estimate the second and third terms on the right hand side of (40), by Hölder's inequality and the a priori assumption (11), we have Putting (41)-(42) into (40), we get (38). Doing the similar argument to (39), we thus complete the proof of Lemma 3.3.

3.4.
Estimates of the high-frequency part. In this section, we will carry out the energy estimates on the high frequency component of the solution. First of all, taking the operator χ 1 (D x ) on both sides of (2) 1,2 , we have a problem of (p h (t), u h (t)) as follows In what follows, we show the energy estimate for (n h , u h ).

Lemma 3.4. For k=2, 3, it holds that
Proof. First we take ∇ k to (43), and multiply the equations by ∇ k n h and ∇ k u h respectively. After adding them together, we integrate the resulting over R 3 by parts We shall estimate each term on the right hand side of (45). The second and fourth terms on the right hand side of (45) can be rewritten as follows Using Lemma 6.1 and (32), the first two terms on the right hand side of (46) can be estimated as follows The last two terms on the right hand side of (46) can be estimated as follows For the first term on the right-hand side of (45), using the Paserval theorem and the decomposition (32), we obtain In the following, we will estimate the term I 1 . Using Lemma 6.1, it is obvious that I 1 can be rewritten as For I 1 1 , using integration by parts, Lemmas 6.1-6.2, the assumption (11) and the Plancherel theorem, we obtain where we have used the fact that n h (x, t) = χ 1 (D x )n h (x, t). By a similar argument, we have With the above two inequalities at hands, we obtain For I 2 , by Lemma 6.1, the assumption (11) and the Young inequality, we get

Then, it follows from integration by parts and the Young inequality that
Similar to the estimate on the terms on the right hand side of (45) in the proof of Lemma 3.3, we have Thus, putting (62) and (63) into (61) and noticing the smallness of δ, we get (57).
Lemma 3.6. There exists a suitably large constant D 2 > 0 such that for any 0 ≤ t ≤ T .
Proof. Multiplying (2) 1 , (2) 2 by n, u respectively, adding them together and then integrating the resulting over R 3 by parts, we obtain 1 2 The two terms on the right hand side of (67) can be estimated as follows. Firstly, for the first term, by Hölder's inequality, the Lemma 6.1, the Young inequality and the a priori assumption (11), we have n, F 1 n, ndivu + n, u · ∇n (68) For the second term, it follows from Lemma 6.1, the Hölder's inequality and (11) that u, F 2 u, u · ∇u) + u, n∇n Then, combining (67) with (68) and (69) yields From (10) 2 , we haveμ Multiplying it by ∇n, and then integrating it over R 3 , we have that Using integration by parts and (2) 1 , the first term on the right hand side of (72) can be rewritten as Adding (72) and (73) and using the Young inequality, we obtain In a similar way, the fourth term on the right hand side of (68) can be estimated as follows Since δ > 0 is sufficiently small, combining (74) and (75) with (76) yields Finally, multiplying (70) by D 2 suitably large and adding it to (77), one has (66) since δ > 0 is sufficiently small.
Our next goal is to deal with the higher order estimate of (n, u).
for any 0 ≤ t ≤ T .
4. Global existence. In this section, we will prove Proposition 2. We are aiming to close the a priori assumption about (n, u). Up to now, we only can get a priori decay-in-time estimates of the low frequency part (n l , u l ). Under a priori decay-intime assumptions blow, we continue to obtain a decay-in-time estimates on (n, u). Precisely, we have the following lemma.
Proof. Assume that the classical solution to the problem exists for t ∈ [0, T ] and denote that Define the temporal energy functional Then, there exist two positive constants C 5 and C 6 satisfying From (65), we have Then, using (87), it follows from (86) that Noticing the smallness of δ and the definition of F(t), (88) gives Using Lemma 3.3 with p = +∞, (32) and Lemma 6.2, for k ≥ 0 we have and for k ≥ 1, which implies that ∇n l W 2,∞ + ∇u l In a similar way, we get ∇n l W 3,∞ + ∇u l Noticing the definitions of M 1 (t) and using the smallness of δ, we get M 1 (t) 2 F(0) + (n 0 , u 0 ) 2 L 2 + δ 2 M 0 (t) 2 .