Convergence to the Barenblatt Solution for the Compressible Euler Equations with Damping and Vacuum

Transcription

1 Arch. Rational Mech. Anal. 176 (5 1 4 Digital Object Identifier (DOI 1.17/s y Convergence to the Barenblatt Solution for the Compressible Euler Equations with Damping Vacuum Feimin Huang, Pierangelo Marcati & Ronghua Pan Communicated by T.-P. Liu Abstract We study the asymptotic behavior of a compressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping. As t, the density is conjectured to obey the well-known porous medium equation the momentum is expected to be formulated by Darcy s law. In this paper, we give a definite answer to this conjecture without any assumption on smallness or regularity for the initial data. We prove that any L weak entropy solution to the Cauchy problem of damped Euler equations with finite initial mass converges, strongly in L p with decay rates, to matching Barenblatt s profile of the porous medium equation. The density function tends to the Barenblatt s solution of the porous medium equation while the momentum is described by Darcy s law. 1. Introduction We continue our study on the asymptotic behavior of compressible isentropic flow through a porous medium when vacuum occurs initially. The model system is the compressible Euler equation with frictional damping. As t, the density is conjectured to obey the well-known porous medium equation the momentum is expected to be formulated by Darcy s law. Although, many contributions have been made to the small smooth solutions or piecewise smooth Riemann solutions away from vacuum since the pioneer work of Nishida [9], some key problems in this topic remain open. Among them, the large-time asymptotic behavior for the solutions with vacuum has been a long-sting open problem. This work, together with the work of Huang & Pan [18 ], will give a complete answer to this problem. In fact, we showed that the L weak entropy solutions with vacuum selected by the physical entropy-flux pairs, converge strongly in L p with decay rates, to the similarity solutions of the porous medium equation, determined by the end-states of the initial data initial mass. New approaches are developed to deal This revised version was published in April 5. The volume number has now been inserted into the citation line.

2 Feimin Huang, Pierangelo Marcati & Ronghua Pan with the nonlinear convection, nonlinear coupling the singularity near vacuum based on the conservation of mass, the structure of the convection the existence of mechanical energy function. This approach seems remarkable since we do not need smallness assumptions on the solutions. We now formulate our results. Consider the compressible Euler equation with frictional damping, with the initial data ρ t + (ρu x =, (ρu t + (ρu + P(ρ x = αρu, (1.1 ρ(x, = ρ (x, m(x, = m (x. (1. Such a system occurs in the mathematical modeling of compressible flow though a porous medium. Here ρ, u P denote respectively the density, velocity, pressure; m = ρuis the momentum the constant α> models friction.assuming the flow is a polytropic perfect gas, then P(ρ = P ρ γ, 1 <γ <3, with P a positive constant, γ the adiabatic gas exponent. Without loss of generality, α P are normalized to be 1 throughout this paper. System (1.1 is hyperbolic with two characteristic speeds λ 1 = u P (ρ λ = u + P (ρ. Furthermore, (1.1 is strictly hyperbolic at the point away from vacuum where two characteristics coincide. Thus, this simple system involves three mechanisms: nonlinear convection, lower-order dissipation of damping, the resonance due to vacuum. The interaction of these mechanisms leads to the big difference in qualitative behaviors of solutions from those of strictly hyperbolic conservation laws. For instance, the long-time behavior of the solutions to the Cauchy problem for strictly hyperbolic conservation laws was known to be that of the corresponding Riemann solutions, while the nonlinear diffusive phenomena should be expected in the large-time behavior of solutions to (1.1, (1.. In experiments, Darcy s law was observed in the same process. Thus, we have another model: ρ t = P(ρ xx, m = P(ρ x, (1.3 where the second equation is the famous Darcy law the first equation is the well-known porous medium equation. So, it is natural to expect some relationship between system (1.1 system (1.3. Actually, we have the following conjecture; see []. Conjecture. As t, the system (1.1 is equivalent to the system (1.3. In the case away from vacuum, system (1.1 can be transferred to the p-system with damping by changing to the Lagrangian coordinates; see [35]. The conjecture has been justified by Hsiao & Liu [1, 13] for small smooth solutions away from vacuum, based on the energy estimates for derivatives. Since then, this problem has attracted considerable attention; see [11, 14, 15, 6, 8, 3 3, 36]. However, all

3 Convergence to the Barenblatt Solution for the Compressible Euler Equations 3 these results are away from vacuum /or require small smooth initial data. For more references on the p-system with damping, we refer to [6, 16, 17, 3, 7, 34, 38]. When a vacuum occurs in the solution, the difficulty of the problem is greatly increased. The main difficulties come from the interaction of nonlinear convection, lower-order dissipation of damping, the resonance due to vacuum. It is known that the nonlinearity is the reason for shock formation in a hyperbolic system. For hyperbolic conservation laws, the self-similarity is an important feature in constructing fundamental Riemann solutions in describing the large-time behaviors of solutions. The damping presents weak dissipation; it prevents the formation of singularity if the data is small smooth. However, it breaks the self-similarity of the system. This is crucial for the large solutions. Another difficulty is due to the resonance near vacuum which develops a new singularity. In fact, Liu & Yang [4, 5] observed that the local smooth solutions of (1.1 blow up in finite time before shock formation. This implies the moving of the interface between the vacuum the gas. Due to this new singularity, it is very difficult to obtain the solutions with any degree of regularity. This makes (1.1 difficult to underst analytically makes the construction of effective numerical methods for computing solutions a highly non-trivial problem. Indeed, the only global weak solutions are constructed in L space by using the method of compensated compactness; see Ding, Chen & Luo [8] for 1 <γ 5 3 Huang & Pan [18] for 1 γ<3. Thus, to study the large-time behavior of the solution of (1.1, (1. with vacuum, it is suitable to consider the L weak solution. Definition 1. We call (ρ, m(x, t L an entropy weak solution of (1.1 (1., if, for any non-negative test function φ D(R+, (ρφ t + mφ x dxdt+ ρ (xφ(x, dx=, t> R [mφ t + t> ρ + P (ρφ x mφ] dxdt + m (xφ(x, dx=, R (η e φ t + q e φ x ρu φ dxdt + η e (x, φ(x, dx. t> R (1.4 Here, the entropy-flux pair (η e,q e is associated with mechanical energy: η e = 1 1 ρu + (γ 1 ργ, q e = 1 ρu3 + γ γ 1 ργ u. (1.5 As the L weak solution does not have any degree of regularity, the methods for the case away from vacuum are not applicable here. Recently, some essential progress was made by Huang & Pan [18]; the authors followed the rescaling argument due to Serre & Hsiao [34] obtained the first justification for the conjecture for the vacuum case. They showed that the density in the L weak entropy solutions of (1.1, (1. converges to the similarity solution of the porous

4 4 Feimin Huang, Pierangelo Marcati & Ronghua Pan medium equation along the level curve of the diffusive similarity profiles provided that one of the initial end-states is nonzero. The long-time behavior of the momentum is not known however. This is far from satisfactory. In [19] [], Huang & Pan developed the new technique based on the conservation of mass entropy analysis to attack this conjecture. They showed that the L weak entropy solutions with vacuum converge, strongly in L p (R (p p for any p with decay rates, to the similarity solution of the porous medium equation determined uniquely by the end-states the mass distribution of the initial data provided that one of the end-states is away from vacuum. However, the case where the initial data are L 1 remains as an important open problem. This case has particular interest since the asymptotic behavior is expected to be the famous Barenblatt solution of the porous media equation. We give a definite answer to this expectation in the following theorem. Theorem 1.1. Suppose ρ (x L 1 (R, M = ρ (x dx. Let (ρ, m be an L entropy weak solution of the Cauchy problem (1.1, (1., satisfying the estimates ρ(x,t C, m(x, t Cρ(x, t, (1.6 let be the Barenblatt solution of (1.3 with mass M m = P( x. Then L = O(1(1 + t 1 γ L γ = γ +1, P( dx = O(1(1 + t γ 1 γ +1. (1.7 Define y = x (ρ (r, t dr.ify(x, L (R, then there exist positive γ constants k 1 = min{, γ 1 (γ +1 γ }, k γ = min{, 1 (γ +1 γ } C such that for any ε>, where (ρ (x, t L C(1 + t k 1+ε (ρ (x, t γ L γ C(1 + t k +ε if 1 <γ, if γ >, (1.8 k 1 > 1 γ + 1 k > γ 1 γ + 1 for any γ> 1 + 5, if γ<1 +. (1.9 Furthermore, for 1 <γ <1 +, P(ρdecays as fast as P( in the sense And for 1 <γ <, P(ρdx = O(1(1 + t γ 1 γ +1. (1.1 (ρ γ L γ C(1 + t γ 1 γ +1. (1.11

5 Convergence to the Barenblatt Solution for the Compressible Euler Equations 5 Remark. (1 Condition (1.6 is fulfilled if the solutions are in the physical region initially. The invariant region theory verifies (1.6; see [4]. ( The explicit form of the Barenblatt solution of (1.3 is given in Section. (3 Theorem 1.1 states that any L entropy weak solutions of (1.1 (1. satisfying the conditions in Theorem 1.1 must converge to the related Barenblatt solution of (1.3 with the same mass. Although there is not uniqueness for the solutions, our results indicate the unique asymptotic profile determined by the initial mass. Equation (1.1 indicates that ρ L γ decays to zero as fast as the L γ norm of Barenbaltt s solution. (4 Since Barenblatt s solution decays itself, it is interesting to compare the decay rate of with that of ρ. The inequality (1.9 shows that ρ L decays faster than L when 1+ 5 <γ <, ρ L γ decays faster than L γ if γ<1 +. The inequality (1.11 shows that ρ L γ decays at least as fast as L γ if 1 <γ <. This covers most interesting physical cases. However, the estimates in Section 4 strongly suggest that ρ L γ decays faster than the rates given in this Theorem faster than the decay rates of the L γ for any γ>1. This will be carried out in a future paper. Let us explain the basic ideas of this paper. Two main difficulties are the lack of regularity the singularity near vacuum. Our ideas are based on the nature of the system: the conservation of mass, the structure of the pressure law, the dissipation of damping the existence of a convex entropy (the mechanical energy. We want to explore these features to control the singularity nonlinearity. Our first observation is that the mechanical energy will give a uniform estimate for the solutions (ρ, m. However, this estimate is not useful in the proof of the long-time behavior. We thus construct the proper functions by exping the entropy around the Barenblatt profile (, m; this might give the estimate for the difference between our solutions Barenblatt s profiles (ρ,m m. In order to obtain the large time convergence, higher-order estimates are necessary. There are several ways to get higher-order estimates if the solutions are smooth. However, our solutions are rather rough. Luckily, when we observe the conservation of the mass, we find the mass difference between our solutions Barenblatt s profiles are zero. So, it is possible to introduce anti-derivative y(x, tfor (ρ (x, t. Thus, our entropy estimate becomes the derivatives estimate for y. Furthermore, the equation of y is wave equation with source term. Thus, the normal energy method will give some kind of estimate on y its derivatives. Coupling these two estimates in a clever way, the uniform estimates for both y its derivatives are possible. However, life is not so easy. The singularity near vacuum makes our goal much harder to reach. In order to control the singularity near vacuum, we explore the structure of the convection find some useful inequalities near vacuum; see Lemma 3.1 below. With the help of these inequalities, careful analysis on our two estimates gives the desired results. Then a weighted entropy estimate will give the decay rates. Our proof is somehow tricky technical, this is due to the difficulties of the problem. Our argument becomes neat simple when it is applied to the case away from vacuum. Since (1.1 is hyperbolic, the entropy estimate is much more natural than the parabolic-type energy method used in [1]. Such an entropy analysis goes

6 6 Feimin Huang, Pierangelo Marcati & Ronghua Pan back to Dafermos [5] DiPerna [9], see the books by Dafermos [7] Serre [33] for more references. Our proof may be compared with the proof by Liu & Hsiao [1] for smooth small solutions away from vacuum. In [1], the estimates were obtained by a normal parabolic-type energy method for wave equations. To weaken decouple the nonlinearity, smallness the third-order estimates are necessary in order to close the argument. We can check that such a method is not applicable for our case. The nonlinear terms in convection cannot be controlled without higher-order derivative estimates. Here, we succeed in closing our argument on first-order estimates for large rough solutions. This is one of the remarkable advantages of our approach. The arrangement of the present paper is as follows. In Section, some knowledge on the Barenblatt solutions are prepared carefully. The crucial uniform estimates are made in Section 3 the decay estimates are done in Section 4.. The Barenblatt solution We suspect that the large-time behavior of the solutions to (1.1 (1. could be described by the fundamental solutions of porous media equations, i.e., the Barenblatt solution. By the results of [1], the solution of t = ( γ xx, ( 1,x = Mδ(x, M >, (.1 should take the form (x,t = (t + 1 γ +1 1 {(A Bξ + } γ 1 1, (. with ξ = x(t + 1 γ +1 1, (f + = max{,f}, B = γ 1 γ(γ+1 A determined by A γ +1 (γ 1 B 1 π Here is a weak solution to (.1 such that (cos θ γ +1 γ 1 dθ = M. (.3 dx= M, (.4 =, if ξ A/B. (.5 Hence, for any finite time T >, has compact support. This is the property of finite speed of propagation for the porous medium equation. Furthermore, the derivatives of are not continuous across the interface between the gas vacuum. This is because the porous medium equation is parabolic away from vacuum is not at vacuum. For the definition of the weak solution to (.1, we refer to [1,, 1]. Kamin proved in [1] that (.1 admits at most one solution. Here, we addressed the initial data at t = 1 to avoid the singularity at t =. Thus, we have the following lemmas from (.1 (.5.

7 Convergence to the Barenblatt Solution for the Compressible Euler Equations 7 Lemma.1. If M is finite, then there is one only one solution (x,t to (.1. Furthermore, (1 (x,t is continuous on R, ( there is a number b = ( B A 1 1 >, such that (x,t > if x <bt γ +1 1 (x,t = if x bt γ +1, 1 (3 (x,t is smooth if x <bt γ +1. In terms of the explicit form of, it is easy to check the following estimates. Lemma.. For defined in (. t>, C(1 + t γ +1 1, ( γ 1 x C(1 + t γ γ +1, ( γ 1 t C(1 + t γ γ +1, ( γ x C(1 + t 1, ( γ γ +1 t C(1 + t γ +1, (.6 ( γ 1 x dx C(1 + t 1 γ +1, γ dx C(1 + t γ 1 γ +1, dx C(1 + t γ 1 γ +1, ( γ 1 4γ 1 t dx C(1 + t ( γ x γ +1, dx C(1 + t γ +1 γ +1, ( γ 4γ +1 t dx C(1 + t γ +1. (.7 3. Uniform Estimates In this section, we are going to establish the basic estimates for the difference between solutions of (1.1, (1. the related Barenblatt profile. Our approaches are based on the conservation of mass, the analysis of entropy inequality the control of the singularity near vacuum states. First of all, we give a generalized version of Lemma 4.1 of Huang & Pan [18]. These simple inequalities play an important role in controlling the singularity near vacuum. Lemma 3.1. Let a,b <. There are positive constants C 1 C such that (1 a b γ +1 (a b(p (a P(b,

8 8 Feimin Huang, Pierangelo Marcati & Ronghua Pan ( C 1 a b [P(a P(b P (b(a b] C a b γ if 1 <γ, (3 C 1 a b γ [P(a P(b P (b(a b] C a b if γ>. Proof. It suffices to prove [P(a P(b P (b(a b] { C a b γ if 1 <γ, C a b if γ>, because all the other inequalities are given in [18]. Without loss of generality, we assume that a b thus x = a b x.forγ>, P (x M, we define F(x = P(b+ x P(b P (bx Mx which satisfies F( =, F (x = P (b + x P (b Mx (P (b + θx Mx for θ [, 1] Mx. Hence, we have F(x for x [, ]. And thus [P(a P(b P (b(a b] M(a b. For 1 <γ, we define f(x = P(b+ x P(b P (bx γx γ which satisfies f( =. We observe that f(x = (1 γx γ ifb =. Assume b b 1 >. We compute f (x = γ(b+ x γ 1 γb γ 1 γ x γ 1 γ [(b + x γ 1 b γ 1 γx γ 1 ]. Hence f ( =. For x (, ], we compute f (x = γ(γ 1[(b + x γ γx γ ] = γ(γ 1x γ x [( b + x γ γ ] γ(γ 1 x γ <, lim x +f (x <. Thus, we have f (x for x. This implies f(x for x b. Therefore, [P(a P(b P (b(a b] γ a b γ. Suppose that (ρ, m is a weak entropy solution of (1.1 (1. satisfying the conditions in Theorem 1.1. Let be the Barenblatt solution of (.1 such that M = (x,t dx = Due to the conservation of mass, it is easy to see M = ρ(x,t dx. ρ (x dx.

9 Convergence to the Barenblatt Solution for the Compressible Euler Equations 9 Let m = P( x w = ρ, z = m m, (3. which satisfies Define We have z t + ρ x w t + z x = + (P (ρ P( x + z = m t, (3.3 w(x, t dx =. x y = w(r, tdr. (3.4 y x = w, z = y t. (3.5 Therefore the second equation of (3.3 turns into a wave equation with source term: y tt + + (P (ρ P( x + y t = m t. (3.6 ρ x Multiplying y with (3.6 integrating over [,t] (,, wehave (y t y + 1 y dx yt dxdτ + (P (ρ P((ρ dxdτ C (y, (m m(x, L + m ρ y x dxdτ + y x ( γ t dxdτ. (3.7 Due to Lemma 3.1 Lemma., we have the following inequalities: (P (ρ P((ρ ρ γ +1 = y x γ +1, (3.8 y x ( γ γ +1 t dxdτ (1 + τ γ +1 y x L 1 dτ C. (3.9

10 1 Feimin Huang, Pierangelo Marcati & Ronghua Pan Thus, (3.7 (3.9 imply the following lemma. Lemma 3.. Let y be the function defined in (3.4. Ify(x, L (R, then (y t y + 1 y dx yt dxdt + y x γ +1 dxdt C + m ρ y x dxdt. (3.1 In order to deal with the nonlinearity singularity near vacuum, we now use the entropy inequality, rather than the usual energy method. This, together with (3.1, will give one of our desired estimates in the following theorem. Theorem 3.3. Let y be the function defined in (3.4 such that y(x, L (R. Then (y + yt + y x dx + ( y x γ +1 + yt dxdτ C if 1 <γ, (y + y t + y x γ dx + ( y x γ +1 + y t dxdτ C if γ >. (3.11 In order to prove Theorem 3.3, we choose η e = m ρ + 1 γ 1 P(ρ to be the mechanical energy q e the related flux as in Definition 1. Then we define η = η e 1 γ 1 P ( (ρ 1 P(. (3.1 γ 1 It is easy to check that 1 γ 1 P( t = ( x m. (3.13 Thus, by the definition of weak entropy solution, the following entropy inequality holds in the sense of distribution: η t + 1 γ 1 [P ( (ρ ] t + q ex + ( x + ρ m. (3.14 By the theory of divergence-measure fields (see Chen & Frid [3], we have d η (x, t dx + d 1 dt dt γ 1 [P ( (ρ ] dx + ρ m dx. (3.15

11 Convergence to the Barenblatt Solution for the Compressible Euler Equations 11 Since we integrate (3.15 over [,t] obtain P ( (ρ dx CM, (3.16 Lemma 3.4. For any t>, η (x, t dx + ρ m dx C. (3.17 At this moment, no conclusion can be drawn from (3.17 since m ρ in sign. One role in the proof of Theorem 3.3 is the study of the term m ρ the Taylor expansion of m ρ around ( m,,wehave m varies m.by where m ρ = m m + z (ρ + Q, (3.18 m Q = m ρ due to the convexity of m ρ. Then we have Lemma 3.5. For any t>, ( m m m + m ρ = ρ m ρ, (3.19 η (x, t dx + Proof. By Lemma 3.4 (3.18, it is clear that Q(x, τ dx C. (3. η (x, t dx + Q(x, τ dx C + m zdxdτ t + m (ρ dxdτ. (3.1 However, m (ρ dxdτ C m (ρ L 1 dτ L CM (1 + τ γ γ +1 dτ C, (3.

12 1 Feimin Huang, Pierangelo Marcati & Ronghua Pan = C m zdxdτ m y t dxdτ ( γ 1 t y x t dxdτ + C C + C (1 + τ γ γ +1 dτ ( γ 1 t y x dxdτ C, due to y x L 1 C. Therefore, (3. follows from (3.1 (3.3. Since Q, it is obvious that Corollary 3.6. For any t>, η (x, t dx C, Q(x, τ dxdτ C, ρ m dxdτ C. Now we are able to improve the estimate in Lemma 3. as follows. (3.3 Lemma 3.7. For any t>, (y t y + 1 y dx 3 yt dxdt + y x γ +1 dxdt C. Proof. We are going to bound the last term in (3.1. From (3.18 we have m ρ y m x = (Q + z + m y x y x + m y x, thus by the estimates in (3. (3.3, we have m ρ y x dxdτ ] m = [(Q + z + m y x y x + m y x dxdτ C + C (Q y x + m zy x + m y x dxdτ ( 1 C + z + C m y x dxdτ C + 1 y t dxdτ. (3.4

13 Convergence to the Barenblatt Solution for the Compressible Euler Equations 13 Then, (3.4 (3.1 imply Lemma 3.7. We now proceed with further analysis on ρ 1 ={(x, t : ρ(x,t < }, 1t ={x : ρ(x,t < }, m. Let ={(x, t : ρ(x,t }, t ={x : ρ(x,t }, F( 1 = 1 ρ m dxdτ, F( = In 1 (t, wehave y x m ρ m = m ρ m dxdτ. ρ y x + 1 (m m z m = + z + y x m (Q + z + m y x + m. (3.5 Hence, we have z dxdτ 1 [ F( 1 + z m 1 + y ] x m (Q + z + m y x + m dxdτ C + F( 1 + y m 1 x zdxdτ C + F( z m dxdτ + C 1 y x 1 y x dxdτ C + F( dxdτ, (3.6 which implies that 1 z z dxdτ C + F( 1. (3.7 1 Here, the integral in (3.6 could be bounded in the same manner as in the case for the domain [,t] (,. On the other h, in (t,wehave y x ρ m ρ m = z ρ + m ρ z + m ρ y x = z ρ + m z + m ρ zy x + m ρ y x. (3.8

14 14 Feimin Huang, Pierangelo Marcati & Ronghua Pan Thus, we have z ρ dxdτ ( C + F( + m z + m ρ zy x + m ρ y x dxdτ C + F( + C m ρ zy x dxdτ + C m dxdτ C + F( + 1 z ρ dxdτ + C m y x ρ y x dxdτ C + F( + 1 z ρ dxdτ, which implies that z ρ dxdτ C + F(. (3.9 We thus proved the following result. Lemma 3.8. Let y be defined as in (3.4. Then, y t dxdτ C. (3.3 We now turn to explore the estimates on η. In Corollary 3.6, we have η dx C. By Lemma 3.1 y x C, we thus have ρ + y x dx C if 1 <γ, ρ + y x γ dx C if γ>, (3.31 hence m dx C. (3.3 ρ In 1t,wehave y x (3.5. Therefore, ( z dx m 1t 1t ρ m m ρ y x m z dx C + m 1t zdx C + C m 1t dx C,

15 Convergence to the Barenblatt Solution for the Compressible Euler Equations 15 which implies 1t z dx C yt dx C. (3.33 1t On the other h, in t,wehave y x ρ (3.8. Therefore, z t ( ρ dx m t ρ m m z m z y x ρ m y x dx ρ C, which implies Thus, (3.31 (3.34 gives Lemma 3.9. For any t, t z ρ dx C t y t dx C. (3.34 ( y x + yt dx C if 1 <γ, ( y x γ + yt dx C if γ. Lemmas 3.7, imply Theorem Decay rates We now prove the decay rate for the difference between the solutions of damped Euler equations the Barenblatt solutions for porous media equations. Our main technique is the weighted entropy estimates. In fact, we have Theorem 4.1. For any t, there are constants k 1 = min{ min{ γ, 1 (γ +1 γ } C> such that for any ε>, (1 + t k 1 ε (1 + t k ε γ, γ 1 (γ +1 (η + yt (x, t dx C if 1 <γ, γ }, k = (η + yt (x, t dx C if γ. (4.1

16 16 Feimin Huang, Pierangelo Marcati & Ronghua Pan Namely, ( ρ + (m m (x, t dx C(1 + t k 1+ε ( ρ γ + (m m (x, t dx C(1 + t k +ε if 1 <γ, if γ. (4. Proof. We multiply the equation (3.15 with (1 + t k to obtain (1 + t k d (η (x, t + 1 dt γ 1 [P ( (ρ ] dx +(1 + t k ρ m dx. (4.3 Thus, we have d + t dt (1 k η (x, tdx + d (1 + t k 1 dt γ 1 [P ( (ρ ] dx +(1 + t k ρ m dx k(1 + t k 1 η (x, t + 1 γ 1 [P ( (ρ ] dx. Integrating the above over [,t],wehave (1 + t k (η (x, t + 1 γ 1 [P ( (ρ ] dx + (1 + τ k ρ m dxdτ C + k(1 + τ k 1 η (x, τdxdτ +k (1 + τ k 1 1 γ 1 [P ( (ρ ] dxdτ. (4.4 First of all, we bound the terms on the right-h side of (4.4. We observe for some positive δ that k (1 + τ k 1 1 γ 1 [P ( (ρ ] dxdτ C +C C(δ + C (1 + τ (k +(1+δ ( γ 1 x dxdτ (1 + τ 1 δ y(,τ L dτ γ 1 k 1+δ (1 + τ γ +1 dτ C (4.5

17 Convergence to the Barenblatt Solution for the Compressible Euler Equations 17 if δ 1 4 Due to Lemma 3.1, we have k < γ 1 γ + 1 [P(ρ P( P ( (ρ ] ( γ 1 γ + 1 k. { C ρ γ if 1 <γ <, C ρ if γ. (4.6 where Hence, for γ, we have k k(1 + τ k 1 η (x, τdxdτ k +C + k +C k 1 m (1 + τ C 1 k 1 m (1 + τ ρ dxdτ (1 + τ k 1 y x dxdτ (1 + τ k 1 k 1 m (1 + τ ρ m dxdτ dxdτ (1 + τ k 1 yx dxdτ, (4.7 dxdτ C (1 + τ k 1 y x dxdτ +C C +C ((1 + τ k 1 y x γ 1 γ ( y x 1+γ γ γ k 1 (1 + τ γ +1 dτ C, (4.8 γ dxdτ γ γ 1 dxdτ (1 + τ (k 1( γ γ 1 y x dxdτ if (k 1( γ γ 1 < 1, i.e., k< 1 γ. y x γ +1 dxdτ C (4.9

18 18 Feimin Huang, Pierangelo Marcati & Ronghua Pan For 1 γ<, we have However, k(1 + τ k 1 η (x, τdxdτ k + k +C C + k +C (1 + τ k 1 C 3 k 1 m (1 + τ ρ m dxdτ (1 + τ k 1 y x γ dxdτ (1 + τ k 1 dxdτ ρ m dxdτ (1 + τ k 1 y x γ dxdτ. (4.1 (1 + τ k 1 y x γ dxdτ +C 4 C +C ((1 + τ k 1 y x 1 γ γ dxdτ ( y x (γ γ 1 γ γ 1 dxdτ (1 + τ γ(k 1 y x dxdτ y x γ +1 dxdτ C, (4.11 if γ(k 1 < 1, i.e., k<1 γ 1. It is observed that { γ 1 (γ +1 1 γ 1 if 1 <γ, γ 1 (γ +1 γ 1 if γ. Thus, we conclude from (4.4 (4.11 that for any ε>, (1 + t γ 1 γ ε (η (x, t + 1 γ 1 [P ( (ρ dx ( + (1 + τ γ 1 γ ε m ρ m dxdτ C if 1 <γ, (1 + t γ 1 ε (η (x, t + 1 γ 1 [P ( (ρ dx ( + (1 + τ γ 1 ε m ρ m dxdτ C if γ.

19 Convergence to the Barenblatt Solution for the Compressible Euler Equations 19 Since if k< (1 + t k 1 γ 1 [P ( (ρ ] dx C (1 + t k ( γ 1 x ydx (4.1 y dx + C γ 1 k C + C(1 + t γ +1 C γ 1 (γ +1, we arrive at (1 + t k η (x, tdx + (1 + t k ( γ 1 x dx ( k 1 (1 + τ k (1 + τ ρ m dxdτ C, (4.13 where { γ 1 k<q(γ= γ if 1 <γ, 1 γ if γ. From now on, we choose k<q(γ. By (3.18, we deduce that (1 + t k η (x, tdx + C + However, ( 1 ( 1 C C + C k (1 + τ k Q(x, τ dxdτ (1 + τ ( 1 k (1 + τ [ ] mz (1 + τ k + m y x k (1 + τ k m (1 + τ y x dxdτ ] [(1 γ +1 + τ k m γ dxdτ + C dxdτ. (4.14 y x γ +1 dxdτ (1 + τ k γ +1 γ + 1 γ +1 dτ C (4.15 if k< γ (γ +1. Here, we used the following estimate: ( m which is from Lemma.. γ +1 γ dx C(1 + t + 1 γ +1,

20 Feimin Huang, Pierangelo Marcati & Ronghua Pan We now choose k<min{q(γ, γ (γ +1 }. By (4.5, we have (1 + τ k m zdxdτ [(1 + τ k m ] y t dxdτ + t + (1 + τ k y x ( γ 1 t dxdτ C + C (1 + τ k y x ( γ 1 t dxdτ C + C C, y x γ +1 dxdτ + C k(1 + τ k 1 m ydxdτ [(1 + τ k ( γ 1 t ] γ +1 γ dxdτ (4.16 (1 + τ k 1 m zdxdτ C C. (1 + τ (k 1 ( x dxdτ + Thus, we conclude from (4.14 (4.17 that y t dxdτ (4.17 (1 + t k η (x, tdx + (1 + τ k Q(x, τ dxdτ C (4.18 (1 + t k η (x, tdx C (4.19 { γ for positive k such that k<min q(γ,. As the consequences of (4.19 Lemma 3.1, we have, for any ε>, (1 + t k 1 ε (1 + t k ε (γ +1 } ( y x + m dx C if 1 <γ, ρ ( y x γ + m dx C if γ, (4. ρ { } { } where k 1 = min γ, γ 1 (γ +1 γ, k = min γ, 1 (γ +1 γ. We now establish the decay estimates for y t based on (4..

21 Convergence to the Barenblatt Solution for the Compressible Euler Equations 1 In 1t,wehave y x (3.5. Therefore, (1 + t k z dx 1t [ C + m (1 + t k 1t ρ m m ρ y x m ] z dx C + k (1 + t zdx 1t m C, which implies (1 + t k 1t y t dx C. (4.1 On the other h, in t,wehave y x ρ (3.8. Therefore, (1 + t k z t ρ dx [ C + m (1 + t k t ρ m m z m z y ] x ρ m y x dx ρ C + k (1 + t zdx t m C. (4. The inequality (4. implies that (1 + t k t y t dx C. (4.3 We thus conclude from (4.1 (4.3 that (1 + t k yt dx C. (4.4 The inequalities (4.19 (4.4 give the results in Theorem 4.1. With the help of Theorem 4.1, a more careful analysis of η leads to the proof of Theorem 1.1. Proof of Theorem 1.1. First of all, it is easy to check that while γ (γ + 1 > γ 1 γ + 1, γ 1 > γ 1 γ γ + 1, (4.5 1 γ > γ 1 γ + 1 if 1 <γ <1 +, (4.6

22 Feimin Huang, Pierangelo Marcati & Ronghua Pan k 1 > 1 γ + 1 if <γ. (4.7 Hence, (1.8 (1.9 follow from Theorem 4.1. For (1.1, we assume 1 < γ < 1 + choose γ 1 γ +1 < k < { } γ min q(γ,, which is possible by (4.5 (4.7. We also have (4.19: (γ +1 (1 + t k η (x, tdx C. This implies the following estimates: (1 + t k F(ρ, (x, t dx C, (4.8 where F(ρ, = P(ρ P( P ( (ρ. We observe that F(ρ, for any ρ. Hence, P(ρdx= F(ρ, dx+ By (4.1 (4.19, we have F(ρ, dx C(1 + t k, P( dx+ P ( (ρ dx.(4.9 P ( (ρ dx C(1 + t k. Since P( dx decays at a rate (1 + γ 1 t γ +1, we conclude from (4.6 that P(ρdecays at the same rate as P(, P(ρdx = ρ γ dx = O(1(1 + t γ 1 γ +1. (4.3 This is for (1.1. We now prove (1.11. Assuming 1 <γ <, we observe that P(ρ P( F(ρ, + P ( (ρ, ρ γ P(ρ P(, which is obtained by dividing the first inequality of Lemma 3.1 with a b. Thus, we have (1 + t γ 1 γ +1 ρ γ dx (1 + t γ 1 γ +1 F(ρ, dx + (1 + t γ 1 γ +1 P ( ρ dx

23 Convergence to the Barenblatt Solution for the Compressible Euler Equations 3 C + C(1 + t γ 1 γ +1 P ( L C. (4.31 This ends the proof of Theorem 1.1. Acknowledgements. We wish to thank S. Kawashima for kindly reminding us of the relations between the decay rate of Barenblatt s solution that of ρ. The research of Feimin Huang was partially supported by NSFC Grant. No References 1. D.G. Aronson: The porous media equations. In: Nonlinear Diffusion Problem. Lecture Notes in Math., Vol. 14(A. Fasano, M. Primicerio, Eds Springer-Verlag, Berlin, H. Brezis, M. Crall:Uniqueness of solutions of the initial-value problem for u t φ(u =. J. Math. pures et. appl. 58, ( G. Chen, H. Frid: Divergence-measure fields hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147, ( K. Chueh, C. Conley, J. Smoller: Positively invariant regions for systems of nonlinear diffusion equations. Indiana U. Math. J. 6, ( C.M. Dafermos: The second law of thermodynamics stability. Arch. Ration. Mech. Anal. 7, ( C.M. Dafermos: A system of hyperbolic conservation laws with frictional damping. Z. Angew. Math. Phys. 46 Special Issue, ( C.M. Dafermos: Hyperbolic conservation laws in continuum physics. Springer-Verlag, Berlin, 8. X. Ding, G. Chen, P. Luo: Convergence of the fractional step Lax-Friedrichs Godunov scheme for isentropic system of gas dynamics. Commun. Math. Phys 11, ( R. DiPerna: Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J. 8, ( C.J. van Duyn, L.A. Peletier: A class of similary solutions of the nonlinear diffusion equations. Nonlinear Analysis, TMA 1, 3 33 ( L. Hsiao: Quasilinear hyperbolic systems dissipative mechanisms. World Scientific, L. Hsiao, T. Liu: Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Comm. Math. Phys. 143, ( L. Hsiao, T. Liu: Nonlinear diffusive phenomena of nonlinear hyperbolic systems. Chin. Ann. of Math. Ser. B 14, ( L. Hsiao, T. Luo: Nonlinear diffusive phenomena of entropy weak solutions for a system of quasilinear hyperbolic conservation laws with damping. Q. Appl. Math. 56, ( L. Hsiao, R. Pan: The damped p-system with boundary effects. Contemporary Mathematics 55, ( 16. L. Hsiao, S. Tang: Construction qualitative behavior of solutions for a system of nonlinear hyperbolic conservation laws with damping. Q. Appl. Math. 53, ( L. Hsiao, S. Tang: Construction qualitative behavior of solutions of perturbated Riemann problem for the system of one-dimensional isentropic flow with damping. J. Differential Equations 13, ( F. Huang, R. Pan: Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum. Preprint ( 19. F. Huang, R. Pan: Convergence rate for compressible Euler equations with damping vacuum. Arch. Ration. Mech. Anal. 166, (3

24 4 Feimin Huang, Pierangelo Marcati & Ronghua Pan. F. Huang, R.H. Pan: Nonlinear diffusive phenomena in the solutions of compressible Euler equations with damping vacuum. Preprint (1 1. S. Kamin: Source-type solutions for equations of nonstationary filtration. J. Math. Anal. Appl. 64, (1978. T. Liu: Compressible flow with damping vacuum. Japan J. Appl. Math 13, 5 3 ( M. Luskin, B. Temple: The existence of a global weak solution to the nonlinear waterhammar problem. Comm. Pure Appl. Math ( T. Liu, T. Yang: Compressible Euler equations with vacuum. J. Differential Equations 14, 3 37 ( T. Liu, T. Yang: Compressible flow with vacuum physical singularity. Preprint ( T. Luo, T. Yang: Interaction of elementary waves for compressible Euler equations with frictional damping. J. Differential Equations 161, 4 86 ( 7. P. Marcati, A. Milani: The one dimensional Darcy s law as the limit of a compressible Euler flow. J. Differential Equations 84, ( P. Marcati, B. Rubino: Hyperbolic to Parabolic Relaxation Theory for Quasilinear First Order Systems. J. Differential Equations 16, ( 9. T. Nishida: Nonlinear hyperbolic equations related topics in fluid dynamics. Publ. Math. D Orsay ( K. Nishihara: Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping. J. Differential Equations 131, ( K. Nishihara, W. Wang, T. Yang: L p -convergence rate to nonlinear diffusion waves for p-system with damping. J. Differential Equations 161, ( 3. K. Nishihara, T. Yang: Boundary effect on asymptotic behavior of solutions to the p-system with damping. J. Differential Equations 156, ( D. Serre: Systems of hyperbolic conservation laws I, II. Cambridge University Press, Cambridge, 34. D. Serre, L. Xiao: Asymptotic behavior of large weak entropy solutions of the damped p-system. J. P. Diff. Equa. 1, ( J.A. Smoller: Shock waves reaction-diffusion equations. Springer-Verlag, H. Zhao: Convergence to strong nonlinear diffusion waves for solutions of p-system with damping. J. Differential Equations 174, 36 (1 37. Y. Zheng: Global smooth solutions to the adiabatic gas dynamics system with dissipation terms. Chinese Ann. of Math. Ser. A 17, ( C.J. Zhu: Convergence Rates to Nonlinear Diffusion Waves for Weak Entropy Solutions to p-system with Damping. Preprint ( Institute of Applied Mathematics,AMSS, Academia Sinica, Beijing18, China fhuang@amt.ac.cn Dipartmento di Matematica Pura ed Applicata Universita degli Studi di L Aquila 671, L Aquila, Italy marcati@univaq.it School of Mathematics, Georgia Institute of Technology, Atlanta, GA panrh@math.gatech.edu (Accepted August 4, 4 Published online 6 February, 5 Springer-Verlag (5