Acta Mathematicae Applicatae Sinica, English Series

Vol. 29, No. 3 (2013) 509–516

DOI: 10.1007/s10255-011-0097-3http://www.ApplMath.com.cn & www.SpringerLink.com

Acta Mathema�cae Applicatae Sinica,English Series© The Editorial Office of AMAS & Springer-Verlag Berlin Heidelberg 2013

Large Time Behavior of Entropy Solutions to SomeHyperbolic System with Dissipative StructureHui-min YU

Department of Mathematics, Shandong Normal University, Jinan 250014, China (E-mail: hmyu@amss.ac.cn)

Abstract In this paper the large time behavior of the global L∞ entropy solutions to the hyperbolic system

with dissipative structure is investigated. It is proved that as t → ∞ the entropy solutions tend to a constant

equilibrium state in L2 norm with exponential decay even when the initial values are arbitrarily large. As an

illustration, a class of 2 × 2 system is studied.

Keywords large time behavior, fully dissipative system, entropy solutions

2000 MR Subject Classification 35D05; 35L65; 35M10

1 Introduction

In this paper we consider the hyperbolic system of balance laws:

∂u(x, t)∂t

+∂f(u(x, t))

∂x+ g(u(x, t)) = 0, (1.1)

with initial datau(x, 0) = u0(x) ∈ L∞(R), −∞ < x < ∞, (1.2)

where u = (u1, u2, · · · , uN)T is an N -vector field. The flux function f and the source term gare smooth mappings from RN to RN, and the initial data u0 satisfy

limx→±∞u0(x) = u±. (1.3)

Throughout this paper, we use the notation u ∈ L∞(·) to denote ui ∈ L∞(·) for i = 1, 2, · · · , N .We also use the notation u ∈ L2(·) to denote ui ∈ L2(·) for i = 1, 2, · · · , N .

We assume that the system is strictly hyperbolic, i.e. ∇f(u) has N real distinct eigenvaluesλ1(u) < λ2(u) < · · · < λN (u), and thereby N linearly independent right eigenvectors ri(u), i =1, · · · , N . We also assume that the system is endowed with a C2 strictly convex entropy-entropyflux pair (η, q).

When g(u) ≡ 0, the System (1.1) becomes a pure hyperbolic system of conservation laws.It is well known that the solutions to such system develop shocks or singularities in finitetime no matter how smooth the initial data is. Therefore, it is important to study weaksolutions in the sense of distribution. However, in the class of weak solutions, one has toimpose admissibility condition to exclude the non-physical ones. This condition is commonlycalled entropy condition. In the past decades, several useful approaches like Glimm scheme (see[11, 18–20]), front tracking method[1,4], vanishing viscosity approach[2] and the compensated

Manuscript received December 9, 2008. Revised October 17, 2010.This project is supported by the National Natural Science Foundation of China (Grant No. 10901095) and thePromotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (Grant No.BS2010SF025).

510 H.M. YU

compactness (see [7–10, 13, 16, 17]) are developed to study weak entropy solutions in BV orL∞ space.

When g �= 0, System (1.1) is a hyperbolic balance law. In many physical examples whichhave the form of (1.1) the source term g has some dissipation mechanism. And the dissipationmechanism may prevent the shock formation for smooth and small initial data[12,19]. Yet, ifthe initial data is large (even it is smooth), the dissipation is not strong enough to preventthe formation of singularities. Therefore, it is significant to study the weak entropy solutionsto the hyperbolic system with dissipative structure. When g satisfies a suitable dissipativecondition, Dafermos-Hsiao[6] and Christoforou[5] established the global existence of small BVsolutions respectively by Glimm scheme and vanishing viscosity approach with small initialdata. However, the authors did not investigate the large time behavior of weak solutions in [5,6]. For the large time behavior of small smooth solutions we refer to [3] and the referencestherein.

In 2003, Huang and Pan[14] proposed a framework to study the large time behavior of largeweak solution for compressible Euler equations with damping. Later, Huang, Pan and Yu[15]

generalized the framework of [14] to Euler-Poisson system for semiconductor devices. In thispaper, motivated by [14, 15], we establish a framework to study the large time behavior oflarge L∞ weak entropy solutions for general hyperbolic balance laws (1.1)–(1.3) with somedissipative structures. For this, we need to assume the L∞ entropy solutions exist and beuniformly bounded with respect to time t, i.e.

‖u(x, t)‖L∞(R2+) ≤ M, (1.4)

for some constant M , where R2+ = {(x, t)|x ∈ R, 0 ≤ t < +∞}.

Before stating the main result, we first give some definitions which will be used later.

Definition 1. The bounded measurable function u(x, t) is called an entropy solution of (1.1)–(1.3) if it satisfies the Problem (1.1)–(1.3) and the following entropy inequality

ηt(u) + qx(u) + ∇η · g(u) ≤ 0

in the distributional sense, where (η, q) is any strictly convex entropy-entropy flux pair.

Definition 2. The System (1.1) is called fully dissipative if the matrix (∇g + ∇gT )(u) ispositive definite for any u ∈ L∞(R).

Definition 3. The constant equilibrium state of System (1.1) is the constant vector u ∈ RN

such that g(u) = 0.In Definition 2, we give a global dissipative condition which is different from that in [5] and

[6], where the dissipation is defined near the constant equilibrium state. Because our purpose isto study the large solution, the dissipation near the constant equilibrium state is not sufficientto study the existence and large time behavior of L∞ weak solutions. It is noted that for fullydissipative system, the constant equilibrium state is uniquely defined by the System (1.1) itself.

In this paper, for the fully dissipative system, we prove that any L∞ weak entropy solutionssatisfying (1.4) tend to the unique constant equilibrium state with the exponential decay rate.Our main result is:

Theorem 1. Assume that for the fully dissipative System (1.1) there exists a C2 strictlyconvex entropy η satisfying

∇2η(u)∇g(v) = ∇g(v)∇2η(u), for any u, v. (1.5)

Let the constant vector u is the equilibrium state of (1.1), u is any L∞ weak entropy solutionof (1.1)–(1.3) satisfying (1.4). Suppose

u0 − u ∈ L2(R),

Large Time Behavior of Entropy Solutions to Some Hyperbolic System with Dissipative Structure 511

then there exist two constants C and ˜C, which are independent of time t, such that∫ ∞

−∞|u − u|2dx ≤ Ce−C̃t

∫ ∞

−∞|u0 − u|2dx. (1.6)

The paper is organized as follows: Section 2 is devoted to proving the main result. InSection 3 an example is given to illustrate Theorem 1.

2 The Proof of the Main Result

Before proving Theorem 1, we prove some lemmas.

Lemma 1. Suppose the N ×N matrices A and B are symmetric, positive definite and satisfyAB = BA, then there exists a positive constant θ such that for any X ∈ RN

XT ABX ≥ θ|X |2. (2.1)

Proof. Since A, B are symmetric and satisfy AB = BA, we have AB = BA = BT AT =(AB)T . So the matrix AB is symmetric. In order to prove (2.1) we only need to prove thesymmetric matrix AB is positive definite, i.e. all the eigenvalues of AB are positive.

In fact, noticing A and B are positive definite, we can find two non-singular matrices C andD such that

A = CT C, B = DT D. (2.2)

Let λ be any eigenvalue of matrix AB and X be the corresponding eigenvector, then we have

CT CDT DX = ABX = λX, (2.3)

that is(DCT CDT )DX = λDX. (2.4)

Obviously, CDT is non-singular and DCT CDT is positive definite. Then λ > 0. Lemma 1 isproved. �

Lemma 2. Let the N ×N symmetric matrices A and B +BT are positive definite and satisfyAB = BA, then there exists a θ > 0 such that for any X ∈ RN

XT ABX ≥ θ|X |2.Proof. Since the matrix A is symmetric and AB = BA, then we have

A(B + BT ) = (B + BT )A.

Using Lemma 1, we know that there exists a positive constant θ1 such that for any X ∈ RN

XT A(B + BT )X ≥ θ1|X |2. (2.5)

Again from AB = BA, we have

(AB − ABT )T = BT A − BA = −(AB − ABT ). (2.6)

Suppose AB − ABT = (Cij), then from (2.6) we have Cii = 0. Moreover, for any X =(x1, x2, · · · , xN ) ∈ RN

XT (AB − ABT )X =N

∑

i=1

Ciix2i = 0. (2.7)

512 H.M. YU

ThenXT ABX = XT ABT X for any X ∈ RN. (2.8)

Then our result follows from (2.5) and (2.8). �

Proof of Theorem 1. From the assumption, the System (1.1) has a strictly C2 convex entropy-entropy flux (η, q). In terms of Definition 1, the weak entropy solution u satisfies

ηt + qx + ∇η · g ≤ 0 (2.9)

in the sense of distribution. Let

η∗(u) = η(u) − η(u) −∇η(u) · (u − u),q∗(u) = q(u) − q(u) −∇η(u) · (f(u) − f(u)),

(2.10)

then η∗ ∼ |u − u|2, that is there exist two positive constants C1, C2 such that

C1|u − u|2 ≤ η∗(u) ≤ C2|u − u|2. (2.11)

From the entropy inequality (2.9) we have

η∗t(u) + q∗x(u) ≤−∇η(u) · g(u) −∇η(u)(ut + f(u)x)= − (∇η(u) −∇η(u)) · (g(u) − g(u)) in D′. (2.12)

We compute

(∇η(u) −∇η(u)) · (g(u) − g(u))

=(u − u)T

∫ 1

0

∇2η(u + s(u − u))ds

∫ 1

0

∇g(u + s(u − u))ds(u − u). (2.13)

Since the symmetric matrix ∇2η(u) is positive definite for any u ∈ RN, then it is easy to seethat the matrix

∫ 1

0

∇2η(u + s(u − u))ds =(

∫ 1

0

(∇2η(u + s(u − u)))ijds)

(2.14)

is symmetric and positive definite. Similarly, by Definition 2, we know that the symmetricmatrix

∫ 1

0

(∇g + (∇g)T )(u + s(u − u))ds (2.15)

is also positive definite. In view of (1.5), we obtain

∫ 1

0

∇2η(u + s(u − u))ds

∫ 1

0

∇g(u + s(u − u))ds

=∫ 1

0

∇g(u + s(u − u))ds

∫ 1

0

∇2η(u + s(u − u))ds. (2.16)

From (2.14)–(2.16) and Lemma 2, we know there exists a positive constant θ such that

(∇η(u) −∇η(u)) · (g(u) − g(u)) ≥ θ|u − u|2. (2.17)

Then (2.12) becomesη∗t(u) + q∗x(u) + θ|u − u|2 ≤ 0 in D′. (2.18)

Large Time Behavior of Entropy Solutions to Some Hyperbolic System with Dissipative Structure 513

Integrate (2.18) from −∞ to +∞, we have

(

∫ ∞

−∞η∗(u)dx

)

t+ θ

∫ ∞

−∞|u − u|2dx ≤ 0. (2.19)

Then from (2.11) we get the inequality (1.6), and Theorem 1 is proved. �

3 An Example

Consider a class of fluid induced by fading memory{

wt(x, t) − vx(x, t) + w(x, t) = 0,

vt(x, t) − σx(w(x, t)) + v(x, t) = 0, σ′ > 0,(3.1)

(w, v)(x, 0) = (w0, v0) ∈ L∞, limx→±∞(w0, v0) = (w±, v±), (3.2)

where σ and w satisfysign(wσ′′) > 0 if w �= 0.

Theorem 2. If (w0, v0) ∈ L2(R) ∩ L∞(R), then Problem (3.1)–(3.2) has a weak entropysolution (w, v) satisfying

‖w‖L∞(R2+) + ‖v‖L∞(R2

+) ≤ M,

where M depends only on the initial data. Moreover, (w, v) converges to zero strongly in L2

norm with exponential time decay rate, i.e.∫ +∞

−∞(|w(x, t)|2 + |v(x, t)|2)dx ≤ Ce−C̃t

∫ +∞

−∞(|w0(x)|2 + |v0(x)|2)dx.

We first use the compensated compactness theory of Diperna[9] to prove the existence ofthe �L∞ weak entropy solutions and then apply Theorem 1 to get the large time behavior of theweak solutions. We now list two lemmas, which will be used in the following.

Murat Lemma[21]. Ω ∈ R2 is a bounded open set. If a sequence {gε} satisfy gε ∈ ((compactset of H−1(Ω))+(bounded set of M)), and if gε ∈ bounded set of W−1,∞(Ω), then gε ∈ compactset of H−1

loc (Ω).

Diperna Lemma[9]. Suppose (wε, vε) be a sequence of functions which lie in a bounded setof L∞. Assume η(wε, vε)t + q(wε, vε)x lie in a compact subset of H−1

loc as ε varies, whereη = 1

2v2 + p(w), q = −σ(w)v are the mechanical entropy-entropy flux (η, q) of System (3.1)and p′(w) = σ(w). Then there exist a subsequence of (wε, vε) (still denoted by (wε, vε)) suchthat (wε, vε) pointwisely converges to some functions (w, v) ∈ L∞(R2

+) as ε → 0.First we consider the homogenous system

{

wt(x, t) − vx(x, t) = 0,

vt(x, t) − σx(w(x, t)) = 0,(3.3)

whose eigenvalues and Riemann invariants are

λ1 = −√

σ′(w), λ2 =√

σ′(w), (3.4)

andG1 = v −

∫ w

0

√

σ′(ξ)dξ, G2 = v +∫ w

0

√

σ′(ξ)dξ. (3.5)

514 H.M. YU

Adding viscosity terms to the right-hand side of (3.1) yields the following parabolic system:{

wεt − vε

x + wε = εwεxx,

vεt − σx(wε) + vε = εvε

xx,(3.1ε)

with the initial data given by

(wε, vε)(x, 0) = (w0 ∗ jε, v0 ∗ jε), (3.2ε)

where jε is a standard mollifier.Since the existence and uniqueness of local smooth solution to Problem (3.1ε) − (3.2ε) are

classical, we omit here. The global existence and uniqueness are based on the local existenceand a priori estimate of (wε, vε). We shall use the invariant region theory[20] to show the apriori estimate.

First the System (3.1ε) can be written in a standard form

uεt = M(uε)uε

x + εuεxx + h(uε), (3.6)

where uε = (wε, vε)T , M(uε) =(

0 1

σ′(wε) 0

)

and h(uε) = (−wε,−vε)T . Obviously, ∇G1 and∇G2 are left eigenvectors of M . Define a region in the plane of (w, v)

∑

={

(w, v) : |G1(w, v)| ≤ C0, |G2(w, v)| ≤ C0

}

, (3.7)

where C0 = max{|G1(w0, v0)|, |G2(w0, v0)|} (see Figure 1 below).

Figure 1.

It is straightforward to check that∑

is a convex region by

sign(wσ′′) > 0 if w �= 0.

To show∑

is an invariant region of the system (3.1ε), we only need to show that for any point(w, v) ∈ ∂

∑

, (∇Gi · h)(w, v) < 0, i = 1, 2 when v ≥ 0 and (∇Gi · h)(w, v) > 0, i = 1, 2 whenv ≤ 0, see Theorem 14.7 of [20]. It is easy to see the above statement is true. For example,if v ≥ 0 and w ≥ 0, then G2(w, v) = C0 which implies that w and v are not zero at the sametime. We have

(∇G2 · h)(w, v) = −w√

σ′(w) − v < 0. (3.8)

Therefore the solution (wε, vε) of (3.1ε) always lies in the region∑

for any x and t.

Large Time Behavior of Entropy Solutions to Some Hyperbolic System with Dissipative Structure 515

It is not difficult to show that a priori pointwise control on the amplitude leads to pointwisecontrol on the higher derivatives for the parabolic system (3.1ε), see [10] for details. Thereforewe have

Lemma 3. For any fixed ε > 0 the Cauchy problems (3.1ε)–(3.2ε) has a unique global solution(wε, vε) satisfying ‖wε‖L∞ ≤ C, ‖vε‖L∞ ≤ C for some certain constant C which is independentof ε.

To use Diperna Lemma, we must prove the H−1loc compactness of the entropy dissipation

measures ηt(wε, vε) + qx(wε, vε) with the mechanical entropy-entropy flux pair (η, q) and theviscosity solution (wε, vε).

Multiply (3.1ε) by ∇ηε = (σ(wε), vε)T , we get

ηεt + qε

x = εηεxx − ε(σ′(wε)(wε

x)2 + (vεx)2) −∇ηε · h(uε). (3.9)

Choose a function ϕL(x, t) ∈ C∞0 (R2

+) satisfying ϕL = 1 on [−L, L] × [0, T ], where L can beany positive constant. Multiply (3.9) by ϕL and integrate the result in R2

+, we have

∫

R

(ηεϕL)(x, T )dx −∫

R

(ηεϕL)(x, 0)dx −∫ T

0

∫

R

(ηεϕLt + qεϕLx)dxdt

= − ε

∫ T

0

∫

R

(σ′(wε)(wεx)2 + (vε

x)2)ϕLdxdt + ε

∫ T

0

∫

R

ηεϕLxxdxdt

−∫ T

0

∫

R

∇ηε · h(uε)ϕLdxdt. (3.10)

From Lemma 3 and (3.9), we have

ε

∫ T

0

∫ L

−L

(σ′(wε)(wεx)2 + (vε

x)2)dxdt ≤ C,

where the constant C depends on the initial data, L, T and ϕL, but independent of ε. Thenwe have √

εwεx and

√εvε

x are bounded in L2loc(R

2+) ⊂ Mloc. (3.11)

Therefore, for any bounded set Ω ⊂ R2+, we have

‖εηεxx‖w−1,2(Ω) = sup

ϕ∈H10 ;‖ϕ‖≤1

{

∫

Ω

εηεxxϕ(x, t)dxdt

}

=√

ε‖∇η(uε)‖L∞(Ω)‖√

εuεx‖L2(Ω) sup

ϕ‖ϕx‖L2 → 0, ε → 0, (3.12)

∇ηε · h(uε) ∈ L∞(Ω) ⊂ L1(Ω) ⊂ Mloc. (3.13)

Noticingη(uε)t + q(uε)x are bounded in W−1,∞

loc (Ω),

from Murat Lemma and (3.9)–(3.13), we have

η(uε)t + q(uε)x are compact in H−1loc (Ω).

From Diperna Lemma, we know there exists (w, v) ∈ L∞ such that (wε, vε) contains a subse-quence converging pointwise a.e. to (w, v) and (w, v) is the weak solution of (3.1) and (3.2).That is,

Lemma 4. The Cauchy problems (3.1)–(3.2) admits at least one L∞ weak entropy solution(w, v) satisfying ‖w‖L∞ ≤ C, ‖v‖L∞ ≤ C.

516 H.M. YU

Finally, in this example ∇g = I commutes with any 2×2 matrix, Theorem 2 can be directlyderived from Theorem 1 and Lemma 4.

Acknowledgements. The author would like to thank her advisor Professor Huang Feiminfor helpful suggestions.

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