Compactness Methods and Nonlinear Hyperbolic Conservation Laws

Compactness Methodsand

Nonlinear Hyperbolic Conservation Laws

Gui-Qiang ChenDepartment of Mathematics, Northwestern University

Evanston, IL 60208-2730, USA

Contents

1. Introduction 34

2. Compactness Methods and Conservation Laws 372.1. BV Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2. L1 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3. Compensated Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4. Compactness and Large-Time Behavior Problems . . . . . . . . . . . . . . . . . . . . 392.5. Compensated Compactness and Conservation Laws . . . . . . . . . . . . . . . . . . 422.6. Scalar Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.7. 2 × 2 Strictly Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3. Nonstrictly Hyperbolic Conservation Laws 503.1. Isentropic Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2. Hyperbolic Conservation Laws with Umbilic Degeneracy . . . . . . . . . . . . . 54

4. Compressible Euler Equations 564.1. Non-isentropic Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2. Multi-D Compressible Euler Equations with Geometric Structure . . . . 58

5. Hyperbolic Systems of Conservation Laws with Relaxation 63

Acknowledgments 69References 70

c©0000 (copyright holder)

34 Compactness Methods and Nonlinear Hyperbolic Conservation Laws

1. Introduction

We are concerned with a hyperbolic system of conservation laws:

∂tu+ ∇x · f(u) = 0, x ∈ Rd, u ∈ Rn, (1.1)

where f = (f1, · · · , fd) : Rn → (Rn)d is a nonlinear mapping. Since the conser-vation law is a fundamental law of nature, most of partial differential equationsarising from physical or engineering science can be formulated into form (1.1) or itsvariants (with dissipation, relaxation, memory, damping, dispersion, etc.).

The hyperbolicity of system (1.1) requires that, for all ξ = (ξ1, · · · , ξd) ∈ Sd−1,the matrix (

∑dj=1 ξj∇fj(u))n×n have n real eigenvalues λj(u, ξ), j = 1, 2, · · · , n, and

be diagonalizable. When d = n = 1, it is a scalar equation with λ = f ′(u), whichis alway hyperbolic. A system is called nonstrictly hyperbolic or resonant, if thereexist some ξ0 ∈ Sd−1 and u0 ∈ Rn such that λi(u0, ξ0) = λj(u0, ξ0) for some i 6= j.Such a degeneracy is quite generic. For example, when d = 3, n = 2 (mod4), Lax[La2] showed that any system must be nonstrictly hyperbolic. The same situationoccurs for n = ±2,±3,±4 (mod 8) (see [FRS]).

Since f(u) is a nonlinear mapping, solutions of the Cauchy problem (even start-ing from smooth initial data) generally develop singularities in a finite time:

∃T < +∞, |∇xu(t, x)| → ∞, when t→ T,

and then the solutions become discontinuous functions. This feature reflects thephysical phenomenon of breaking of waves and development of shock waves. Forthis reason, attention focuses on solutions in the space of discontinuous functions.Therefore, one can not directly use the classical analytic techniques that predomi-nate in the theory of partial differential equations of other types.

To overcome this difficulty in constructing global discontinuous solutions of theCauchy problem of system (1.1) with initial data:

u|t=0 = u0(x), (1.2)

one natural strategy, motivated from physics, is to construct approximate solutionsuε(t, x) via the singular perturbation methods as the following.

(1). Viscosity method: Add the term ε∇x · (D(u)∇xu) on the right hand sideof (1.1). Usually one chooses D(u) the unit matrix. Then uε(t, x) are generated bythe parabolic equations.

(2). Numerical methods: uε(t, x) are generated by the difference equations:

Dtuε +Dx · f(uε) ≈ 0, ε = ∆t,

or other efficient shock capturing numerical methods, associated with (1.1).Other perturbation methods include the relaxation methods and the kinetic

methods to construct approximate solutions uε(t, x).

As an example, we consider uε(t, x), generated by the following Cauchy problem:

∂tuε + ∇x · f(uε) = ε∆xu

ε, (1.3)

with Cauchy data:

uε|t=0 = uε0(x) → u0(x) ∈ L∞, a.e. ε→ 0, (1.4)

Gui-Qiang Chen 35

satisfying‖uε‖L∞ ≤ C, C independent of ε. (1.5)

Assume that uε(t, x) are compact, that is, there exists a subsequence uεk(t, x)∞k=1

such thatuε(t, x) → u(t, x), a.e. (1.6)

Let (η, q) : Rn → R×Rd be an entropy-entropy flux pair (an entropy pair, forshort), determined by the linear hyperbolic systems:

∇qk(u) = ∇η(u)∇fk(u), k = 1, 2 · · · , d. (1.7)

Taking any convex entropy η(u), i.e. ∇2η(u) ≥ 0, and multiplying ∇η(uε) bothsides of (1.3), we have

∂tη(uε) + ∇x · q(uε) = ε∆xη(uε) − ε(∇xuε)>∇2η(uε)∇xu

ε ≤ ε∆xη(uε). (1.8)

Then ∫ ∫(η(uε)∂tϕ +q(uε) · ∇xϕ)dxdt +

∫η(uε

0(x))ϕ(0, x)dx≥ −ε

∫ ∫η(uε)∆xϕdxdt,

(1.9)

for any nonnegative test function ϕ(t, x) ∈ C∞0 (Rd+1

+ ).Take ε = εk → 0, k → ∞, and use (1.6) together with (1.5). We conclude∫ ∫

(η(u)∂tϕ+ q(u) · ∇xϕ)dxdt +∫η(u0(x))ϕ(0, x)dx ≥ 0, (1.10)

for any convex entropy pair (η, q) and any nonnegative test function ϕ(t, x) ∈C∞

0 (Rd+1+ ). This implies that the limit function u(t, x) is an entropy solution of

(1.1)-(1.2).From the discussion above, it is clear that the essential problem in this strategy

is to study the compactness of approximate solutions with respect to ε → 0 inorder to obtain global entropy solutions. That is, whether one can obtain thestrong convergence of a subsequence of uε(t, x). The compactness problem is alsoclosely related to the large-time behavior problems for entropy solutions (see §2.4and Chen-Frid [CF1-5]).

From our further discussions, we will find that the compactness plays indeedan important role in hyperbolic conservation laws.

In these notes, we focus on some basic ideas, methods, and approaches via someprototypical examples for solving nonlinear hyperbolic conservation laws. Manyimportant results in compactness methods and various topics in conservation lawsare not covered; and some of them can be found in the references cited here.

In Section 2, we describe several important compactness frameworks and discusstheir important applications to the existence and asymptotic behavior of entropysolutions. As examples, scalar conservation laws and 2 × 2 strictly hyperbolic sys-tems are analyzed to show how the compactness of exact and approximate solutionsis established.

In Section 3, we discuss how the compactness methods can be applied to solv-ing important nonstrictly hyperbolic systems of conservation laws, including theisentropic Euler equations and the quadratic systems with umbilic degeneracy.

In Section 4, we analyze some further results on the compressible Euler equa-tions with the aid of the compactness methods, in addition to those in Section3.


In Section 5, we discuss the stability and behavior of singular limits of the zerorelaxation time for hyperbolic systems of conservation laws with relaxation.

Gui-Qiang Chen 37

2. Compactness Theories and Conservation Laws

In this section, we describe several important compactness frameworks anddiscuss their immediate applications to nonlinear hyperbolic conservation laws.

2.1. BV Compactness.

If a sequence of functions uε(t, x) satisfies

‖uε‖L∞ ≤ C ‖u0‖L∞, TV (uε) ≤ C TV (u0),

where C is independent of ε, then there exists a subsequence uεk such that

uεk(t, x) → u(t, x), a.e.,

when k → ∞, by the Helly theorem. The role of these norms is indicated byGlimm’s theorem [Gl] for hyperbolic systems (also see [Ol,Vo,CS]).

2.2. L1 Compactness. If a sequence of functions uε satisfies

(i). ‖uε‖L1loc

≤ C, C > 0 independent of ε;(ii). uε(t, x)ε>0 is equicontinuous in L1

loc(Rd+1+ ), that is, for any compact

subset Ω ⊂ Rd+1+ , ∫ ∫

Ω

|uε(t+ ∆t, x+ ∆x) − uε(t, x)|dxdt

uniformly tends to zero as ∆t,∆x → 0. Then there exists a subsequence uεk suchthat

uεk(t, x) → u(t, x), L1loc(R

d+1+ ).

The role of the L1 norm for the compactness is indicated by Kruzkov’s theorem [Kr]for scalar conservation laws.

2.3. Compensated Compactness.

Let Ω ⊂ Rd+1+ be a bounded open set. Denote y = (t, x) ∈ Ω. Assume that a

sequence of functions uε : Ω → Rn satisfies

uε ∗ u ∈ L∞(Ω), (2.1)

and

A0∂uε

∂t+

d∑j=1

Aj∂uε

∂xj= gε(t, x) is compact in H−1

loc (Ω), (2.2)

for some m× n matrices Aj , j = 0, 1, · · · , d.Then (2.2) can be regarded as m first order differential equations for the n

unknowns. A mathematical question is whether f(uε)f(u) subsequentially in thesense of distributions for any smooth nonlinear function f . If (2.2) were elliptic,then the weak convergence would be equivalent to the convergence in L2

loc. However,this is not the case in general. For example, uε(t, x) = sin

(t+x

ε

), f(u) = u2. It is

well known that uε0 and (uε)2 ∗ 1

2 6= 02. Nevertheless, we have the followingtheorem.


Theorem 2.1 [Ta1]. Let uε : Ω → R4 be measurable functions satisfying

w − limuε = u, in L24(Ω),

∂uε1

∂t+∂uε

2

∂x,∂uε

3

∂t+∂uε

4

∂xare compact in H−1

loc (Ω).

Then there exists a subsequence (still labeled) uε such that∣∣∣∣ uε1 uε

2

uε3 uε

4

∣∣∣∣ ∣∣∣∣ u1 u2

u3 u4

∣∣∣∣ in the sense of distributions.

Theorem 2.2 [Ta1]. Let K ⊂ Rn be a bounded open set and uε : Ω → K. Thenthere exists a family of probability measures νy(λ) ∈ prob.(Rn)y∈Ω such that

supp νy ⊂ K, a.e. y ∈ Ω,

and, for any continuous function f , there is a subsequence uεk satisfying

f(uεk) ∫Rn

f(λ)dνy(λ) =< νy, f > .

As a corollary, we haveTheorem 2.3. A uniformly bounded sequence uε converges to u a.e. in Ω sub-sequentially if and only if the corresponding family of Young measures νy reducesalmost all points y to a family of Dirac masses concentrated at u(y), i.e. νy = δu(y).Proof. Suppose that the uniformly bounded sequence uε in L∞, ‖uε‖L∞ ≤ M ,converges to u a.e. Then, for any f ∈ C(Rn), there exists a subsequence uεk∞k=1 ⊂uεε>0 and a family of Young measures νy ∈ Prob.(Rn)y∈Ω, supp νy ⊂ λ | |λ| ≤M, such that

f(uεk(y)) ∗〈νy(λ), f(λ)〉, f(uεk(y)) −→ f(u(y)), a.e. y ∈ Ω.

The uniqueness of the weak limits implies

〈νy(λ), f(λ)〉 = f(u(y)) = 〈δu(y)(λ), f(λ)〉, a.e. for any f ∈ C(Rn).

This means thatνy(λ) = δu(y)(λ), a.e.

On the other hand, if the Young measures νy(λ) = δu(y)(λ), then there exists asubsequence uεk∞k=1 such that

(uεk)2 ∗u2, uεk

∗u.

Then ∫Ω

(uεk − u)2dy =∫

Ω

((uεk)2 − 2uuεk + u2)dy −→ 0, k → ∞.

Using the uniform boundedness of uεk(y), one immediately has

‖uεk − u‖Lp(Ω) → 0, k → ∞.

Remark 1. The Young measure νy can be also understood as the limiting proba-bility distribution of the value of uε(y) near the point y as ε→ ∞ (see Ball [Ba]).Set

< νεy,r, f >

∆=1

|B(y, r)|

∫B(y,r)

f(uε(ξ))dξ, ∀ f ∈ Cc(Rn).

Gui-Qiang Chen 39

Then ‖νεy,r‖ ≤ 1. Therefore, there exist a subsequence εk∞k=1 and a measure νy,r

such that νεky,r νy,r, that is,

< νy,r, f >=1

|B(y, r)|

∫B(y,r)

< νξ, f > dξ.

Assume that y is an Lebesgue point. Then

< νy,r, f >→< νy, f >, r → 0.

Remark 2. The deviation between the weak and strong convergence is measuredby the spreading of the support of νy. If f is a Lipschitz continuous function, then

‖f(ω∗ − limuε) − ω∗ − lim f(uε)‖L∞

= ‖f(< νy, λ >)− < νy, f(λ) > ‖L∞

≤ ‖f‖Lip‖ < νy, λ− < νy, λ >> ‖L∞

≤ ‖f‖Lip supy

(diam(supp νy)).

The following compactness interpolation is very useful for obtaining the H−1loc

compactness needed in Theorem 2.1.

Theorem 2.4. Let q > 1 and r ∈ (q,∞] are constants, and p ∈ [q, r). Then

(compact set of W−1,qloc (Ω)) ∩ (bounded set of W−1,r

loc (Ω))⊂ (compact set of W−1,p

loc (Ω)).

This is a generalization of Murat’s lemma [Ta1,Mu1]. The proof is simple andcan be found in [DCL1,Ch4].

Some more general results and detailed proofs can be found in Tartar [Ta1-4] and Murat [Mu1-4]. Other related references include Ball [Ba], Chen [Ch4],Coifman-Lions-Meyer-Semmes [CLM], Dacorogna [Da], Evans [Ev1], Hormander[H1], Morrey [Mo1-2], Struwe [St], Young [Yo], and the references cited therein.

2.4. Compactness and Large-Time Behavior Problems.

It is clear from the discussions above that the compactness plays an essentialrole in constructing global entropy solutions. In fact, it is also very important forthe large-time behavior problems.

One of the main asymptotic problems is the decay of periodic entropy solutions,which is an important nonlinear phenomenon. For the linear case, initial oscillationspropagate, and there is no decay. However, this is not the case for systems withcertain nonlinearity. For one-dimensional convex scalar conservation laws (f ′′(u) 6=0), Lax [La1] first showed the decay of periodic solutions with period P :

|u(t, x) − u| ≤ C

t, with u =

1|P |

∫P

u0(x)dx.

Glimm-Lax’s theory [GL] indicates the decay of periodic solutions obtained throughGlimm’s method for 2×2 strictly hyperbolic and genuinely nonlinear systems for ini-tial data of small oscillation. Using the generalized characteristics, Dafermos [Da2]showed for such systems that any periodic entropy solution, with local boundedvariation and small oscillation, asymptotically decays.


In this section, with the aid of compactness arguments, we discuss a new an-alytic approach to study the asymptotic decay of L∞ periodic solutions of hyper-bolic systems of conservation laws, without the restrictions of either smallness orlocal bounded variation on the L∞ periodic initial data and specific reference toany particular method for constructing the entropy solutions.

Let u(t, x) be a period entropy solution for the Cauchy problem (1.1)-(1.2).Then we have the following decay framework.

Theorem 2.5 [CF1-2]. Consider system (1.1) endowed with a strictly convexentropy η∗(u). Let u(t, x) ∈ L∞(Rd+1

+ ) be an entropy solution of (1.1)-(1.2), whichis periodic in x with period P , and the corresponding self-similar scaling sequence

uT (t, x) ≡ u(T t, Tx) is compact in L1loc(R

d+1+ ). (2.3)

Then the periodic solution u(t, x) asymptotically decays in Lp to the average u ofits initial data over the period:

esslimt→∞

∫P

|u(t, x) − u|pdx→ ∞, for any 1 ≤ p <∞, (2.4)

with u = 1|P |

∫Pu0(x)dx.

Sketch of Proof. We assume P = [0, 1]d for simplicity. We divide the proof intoseveral steps.Step 1. Since uT (t, x) is compact in L1

loc(Rd+1+ ), there exists a subsequence uTk(t, x)

converging to some function u(t, x) ∈ L∞(Rd+1+ ) in L1

loc(Rd+1+ ). Then we conclude

that u(t, x) = u(t) from the periodicity of uTk(t, x). Notice that

∂tuTk(t, x) + ∇x · f(uTk(t, x)) = 0 in the sense of distributions.

Setting k → ∞, we have∂tu(t) = 0,

that is, ∫ ∫u(t)∂tϕdxdt +

∫uϕdx = 0, ∀ ϕ ∈ C∞

0 (Rd+1+ ),

where we used thatu0(Tx)

∗

∫P

u0(x)dx ≡ u,

since u0(x) is periodic. This implies that u(t) = u =∫

Pu0(x)dx. Since the limit

is unique, the whole sequence uT (t, x) strongly converges to u in L1loc(R

d+1+ ) when

T → ∞. Therefore, we have

1T d+1

∫ T

0

∫|ξ|≤r

|u(t, ξt) − u|ptddξdt

=∫ 1

0

∫|x|≤rt

|uT (t, x) − u|pdxdt → 0, when T → ∞.

(2.5)

Step 2. The periodic entropy solution u(t, x) satisfies the entropy inequality (1.10)for the strictly convex entropy pair (η∗, q∗). In (1.10), we use

η∗(u, u) = η∗(u) − η∗(u) −∇η∗(u)(u− u) ≥ 0,q∗(u, u) = q∗(u) − q∗(u) −∇η∗(u)(f(u) − f(u)),

Gui-Qiang Chen 41

which is also a strictly convex entropy pair. Then we can deduce that, for anya ∈ Rd, ∫

a+P

η∗(u(t2, x), u)dx ≤∫

a+P

η∗(u(t1, x), u)dx, (2.6)

for all 0 ≤ t1 < t2, t1, t2 ∈ (0,∞) − T , where meas(T ) = 0.

Step 3. Given T > 0, T ∈ (0,∞) − T , we take all the right n-prisms given byx ∈ a+P , for a ∈ Zd, and t ∈ [[rT ]/(2r), T ], in the interior of the cone |x| ≤ rt | 0 ≤t ≤ T . The number of such prisms is larger than [rT ]d. Using the periodicity ofu(t, x), inequality (2.6) with t2 = T , which holds for a.e. t1 = t ∈ (0, T ) over theperiod P , and the strict convexity of the entropy η∗, we obtain that there existsc0 > 0, independent of T , such that

1T d+1

∫ T

0

∫|x|≤rt η∗(u(t, x), u)dxdt ≥ [rT ]d

T d+1

∫ T[rT ]2r

∫P η∗(u(t, x), u)dxdt

≥ [rT ]d

T d+1

∫ T[rT ]2r

∫P η∗(u(T, x), u)dxdt

≥ c0∫

P |u(T, x) − u|2dx,(2.7)

where we used [a] as the largest integer less than or equal a.Noting the uniform boundedness of the periodic solution, we have

1T d+1

∫ T

0

∫|x|≤rt

η∗(u(t, x), u)dxdt ≤ C1T d+1

∫ T

0

∫|x|≤rt

|u(t, x) − u|2dxdt≤ C2

T d+1

∫ T

0

∫|ξ|≤r

|u(t, ξt) − u|2tddξdt→ 0, T → +∞.

(2.8)

Combining (2.7) with (2.8), we obtain∫P

|u(T, x) − u|2dx→ 0, when T → ∞, T ∈ (0,∞) − T .

The boundedness of the periodic solution and the Holder inequality yield (2.4) forall p ∈ [1,∞). This completes the proof of Theorem 2.5.

Remark 1. In Theorem 2.5, we assume that the self-similar scaling sequenceuT (t, x) is compact in L1

loc(Rd+1+ ). This is a corollary of the compactness of solution

operators for hyperbolic conservation laws.

Remark 2. In Theorem 2.5, the assumption u(t, x) ∈ L∞(Rd+1+ ) can be

replaced by u(t, x) ∈ Lq(Rd+1+ ), q > 2. Then the asymptotic decay of an Lq periodic

entropy solution u(t, x) of (1.1)-(1.2), with period P , in the sense of (2.4) for p = 2implies (2.4) for any p ∈ [1, q).

Remark 3. The decay in (2.5) implies

d+ 1T d+1

∫ T

0

|u(t, ξt) − u|ptddt → 0, in L1loc(R

dξ), when T → ∞, (2.9)

which can be regarded as a decay definition in a weak sense (see [CF1]). Under thisweak definition, the periodic solution decays in the sense of long time-average and(2.9) means

< µT (·), |u(·, ξ·) − u|p >→ 0, in L1loc(R

dξ), when T → ∞,


where µT (t) = d+1T d+1X[0,T ](t)tddt, which is a family of probability measures. We

notice that limit (2.9) is also equivalent to

1T

∫ T

0

|u(t, ξt) − u|pdt→ 0, in L1loc(R

dξ), when T → ∞, (2.10)

for the L∞ periodic solution. In this case, the family of average measures is

µT (t) =1TX[0,T ](t)dt.

The compactness is also important in solving other asymptotic problems suchas the asymptotic stability of Riemann solutions. For the details, see Chen-Frid[CF3-5] with aid of the Gauss-Green formula for divergence-measure fields in L∞

or BV (see [CF6]).

2.5. Compensated Compactness and Conservation Laws.

The next problem is how the compactness required for constructing entropysolutions and their asymptotic behavior (Theorem 2.5) can be achieved. FromMurat’s lemma [Mu1] and an idea in [Ch5] (also [Ch2]), we first have

Theorem 2.6. Consider a hyperbolic system of conservation laws (1.1) witha convex entropy pair (η∗, q∗). Assume that the uniformly bounded sequenceuT (t, x) ∈ L∞(Rd+1

+ ) satisfies ∂tη(uT ) + ∇x · q(uT ) ≤ 0 in the sense of distri-butions, for any convex entropy pair (η, q) ∈ Λ, where Λ is a linear space of entropypairs of (1.1) including (η∗, q∗). Then

∂tη(uT ) + ∇x · q(uT ) is compact in W−1,ploc (Rd+1

+ ), p ∈ (1,∞), (2.11)

for any entropy pair (η, q) ∈ Λ satisfying |∇2η| ≤ C∇2η∗.This theorem is designed to prove the compactness of the self-similar scaling

sequence uT (t, x) or the solution operator in L1loc(R

d+1+ ) with the aid of the com-

pensated compactness methods.

Consider the one-dimensional case. Assume that a sequence of functions uε

satisfies‖uε‖L∞ ≤ C, (2.12)

and, for any entropy pair (η, q) ∈ C2,

∂tη(uε) + ∂xq(uε) is compact in H−1loc (R2

+), (2.13)

where ∇q = ∇η∇f . The question is whether uε(t, x) → u(t, x), a.e., ε→ 0.From Theorem 2.2, there exist a family of probability measures νt,x(λ)(t,x)∈R2

+

and a subsequence εk∞k=1 such that

(ηj(uεk(t, x)), qj(uεk(t, x))) ∗ (< νt,x, ηj(λ) >,< νt,x, qj(λ) >), j = 1, 2,∣∣∣∣ η1(uεk(t, x)) q1(uεk(t, x))

η2(uεk(t, x)) q2(uεk(t, x))

∣∣∣∣ < νt,x,

∣∣∣∣ η1(λ) q1(λ)η2(λ) q2(λ)

∣∣∣∣ >, k → ∞.

Therefore, we have from Theorem 2.1 that

< νt,x,

∣∣∣∣ η1(λ) q1(λ)η2(λ) q2(λ)

∣∣∣∣ >=∣∣∣∣ < νt,x, η1(λ) > < νt,x, q1(λ) >< νt,x, η2(λ) > < νt,x, q2(λ) >

∣∣∣∣ , (2.14)

for any entropy pairs (ηj , qj) ∈ C2, j = 1, 2.

Gui-Qiang Chen 43

To prove uε(t, x) → u(t, x), a.e., it suffices to reduce νt,x to the Dirac massδu(t,x)(λ), for fixed a.e. (t, x).

2.6. Scalar Conservation Laws.

As the simplest example, we now show how the compactness can be achievedfor scalar conservation laws

∂tu+ ∂xf(u) = 0. (2.15)Theorem 2.7. Assume that a sequence uε(t, x) satisfies (2.12)-(2.13) only for twoentropy pairs

(η1(u), q1(u)) = (u− k, f(u) − f(k)),

(η2(u), q2(u)) = (f(u) − f(k),∫ u

k

f ′(ξ)2dξ), k ∈ R.(2.16)

Then(i). f(u) is weakly continuous subsequentially with respect to the sequence uε.

That is, there exists a subsequence uεk∞k=1 ⊂ uεε>0 such that

uεk(t, x) ∗ u(t, x), f(uεk(t, x)) ∗

f(u(t, x)), k → ∞.

(ii). Furthermore, if there is no interval in which the flux function f(u) is linear,then the sequence uε(t, x) is compact in L1

loc(R2+).

Proof. From the framework discussed in §2.5, it suffices to study the Young mea-sures νt,x(λ) satisfying (2.14).

Since the sequence uε(t, x) satisfies (2.12), then there exists a subsequenceuεk∞k=1 ⊂ uεε>0 such that

uεk(t, x) ∗ u(t, x), f(uεk(t, x)) ∗

〈νt,x, f(λ)〉, k → ∞.

Then, for fixed (t, x) in (2.14), we take

(η1(λ), q1(λ)) = (λ− u(t, x), f(λ) − f(u(t, x))),

(η2(λ), q2(λ)) = (f(λ) − f(u(t, x)),∫ λ

u(t,x)

f ′(ξ)2dξ).

From (2.14), we have

< νt,x,

∣∣∣∣∣∣∣∣λ− u(t, x) f(λ) − f(u(t, x))

f(λ) − f(u(t, x))∫ λ

u(t,x)

f ′(ξ)2dξ

∣∣∣∣∣∣∣∣>

=

∣∣∣∣∣∣∣∣0 < νt,x, f(λ) − f(u(t, x)) >

< νt,x, f(λ) − f(u(t, x)) > < νt,x,

∫ λ

u(t,x)

f ′(ξ)2dξ >

∣∣∣∣∣∣∣∣.

Hence

< νt,x, (λ − u(t, x))∫ λ

u(t,x)

f ′(ξ)2dξ − (f(λ) − f(u(t, x)))2 >

+ < νt,x, f(λ) − f(u(t, x)) >2= 0.(2.17)


Since the first term of (2.17) is nonnegative from the Jensen inequality, we have

< νt,x, f(λ) >= f(u(t, x)) = f(< νt,x, λ >),

which implies Theorem 2.7(i). Furthermore, if there is no interval in which f(u)is linear, one obviously has from the first term of (2.17) that νt,x = δu(t,x), whichyields Theorem 2.7(ii).

Remark. The simple proof presented here is essentially from Chen-Lu [CLu] byusing only two entropy pairs (2.16). It is actually very important for applications touse only the two entropy pairs for the reduction, such as in the convergence proof ofhigh-order difference schemes (see [CL]) and zero relaxation limits (see [CLi,CLL]).A reduction proof by using infinite entropy pairs was first given by Tartar [Ta1].

Combining Theorem 2.7 with Theorems 2.5-2.6, we concludeTheorem 2.8. Consider scalar conservation laws (2.15) satisfying that there is nointerval in which the flux function f(u) is linear. Then

(i). The solution operator u(t, ·) = Stu0(·) : L∞ → L∞ exists and is compactin L1

loc.(ii). Given any periodic initial data, there exists a unique periodic entropy

solution. Moreover, any periodic entropy solution asymptotically decays.Proof. We divide the proof into three steps.Step 1. For any u0(x) ∈ L∞, set

um0 (x) = u0(x)Xm(x), Xm(x) =

1, |x| ≤ m,0, |x| ≥ m.

(2.18)

Thenum

0 (x) → u0(x), a.e.

Fix m ∈ Z+, the set of positive integers. Consider the Cauchy problem for theparabolic equations:

∂tu+ ∂xf(u) = ε∂xxu,

u|t=0 = um0 (x), a.e.

(2.19)

Then there exist global smooth solutions uεm(t, x) of (2.19) satisfying

‖uεm‖L∞ ≤ ‖um

0 ‖L∞ ≤ ‖u0‖L∞ , (2.20)

by the maximum principle.Multiply both sides of the equations in (2.19) by η′(uε

m) for any C2 functionη(u) (as an entropy). Then

∂tη(uεm) + ∂xq(uε

m) = ε∂xxη(uεm) − εη′′(uε

m)(∂xuεm)2, (2.21)

where q(u) =∫ u

η′(v)f ′(v)dv is the corresponding entropy flux.

Take η(u) = u2

2 in (2.21) and integrate in [0,∞) × (−∞,∞). We have

ε

∫ ∞

0

∫ ∞

−∞(∂xu

εm)2dxdt =

12

∫ ∞

−∞um

0 (x)2dx ≤ Cm, (2.22)

where Cm > 0 is independent of ε.Notice that, for any φ(t, x) ∈ H1

0 (Ω), Ω ⊂⊂ R2+,

|ε∫ ∫

∂xxη(uεm)φdxdt| ≤ Cε

∫ ∫|∂xu

εm||∂xφ|dxdt

≤ Cm√ε‖√ε∂xu

εm‖L2(Ω)‖φ‖H1

0 (Ω) ≤ Cm√ε‖φ‖H1

0 (Ω),

Gui-Qiang Chen 45

where we used (2.22). This implies that

‖ε∂xxη(uεm)‖H−1

loc(R2+) ≤ Cm

√ε→ 0, ε→ 0.

Furthermore, we have

‖εη′′(uεm)(∂xu

εm)2‖L1

loc(R2+) ≤ C‖

√ε∂xu

εm‖L2

loc(R2+) ≤ Cm,

which implies

εη′′(uεm)(∂xu

εm)2 is compact in W−1,p

loc (R2+), p ∈ (1, 2),

from an embedding theorem, for fixed m ∈ Z+.Then we conclude that


m) is compact in W−1,ploc (R2

+), 1 < p < 2. (2.23)

On the other hand,


m) is bounded in W−1,∞loc (R2

+). (2.24)

Theorem 2.4 indicates from (2.23)-(2.24) that


m) is compact in H−1loc (R2

+). (2.25)

Then the compactness theorem (Theorem 2.7) implies that there exists a sub-sequence uεk

m∞k=1 ⊂ uεmε>0 such that, for fixed m ∈ Z+,

uεkm (t, x) → um(t, x), a.e. k → ∞.

Step 2. From Step 1, we conclude that the sequence um(t, x) satisfies(i). ‖um‖L∞ ≤ ‖u0‖L∞;(ii). For any convex entropy pair (η, q),

∂tη(um) + ∂xq(um) ≤ 0

in the sense of distributions. Theorem 2.6 implies that, for any C2 entropy pair(η, q),

∂tη(um) + ∂xq(um) is compact in H−1loc (R2

+).Then, from Theorem 2.7, there exists a subsequence umk

∞k=1 ⊂ um∞m=1

such thatumk

(t, x) → u(t, x), a.e. k → ∞,

and the limit function u(t, x) is an entropy solution. Kruzkov’s uniqueness theoremindicates that u(t, x) is the unique entropy solution for the Cauchy problem for(2.15), which uniquely defines the solution operator u(t, ·) = Stu0(·).Step 3. If the initial data are periodic, then we can follow the same arguments oneach period of the periodic viscous solutions for the compactness estimates in Step 1(instead of the cutoff of the initial data) to conclude that there is a unique periodicentropy solution with the same period as the initial data. The same arguments asin Step 2 yield the compactness of the solution operator. Then we conclude fromTheorem 2.5 that the periodic solution asymptotically decays in the sense (2.4).

This completes the proof.

Remark: For the convex case f ′′(u) 6= 0, the compactness of the solution operatorwas first obtained by Lax [La1]. If the reflection points u | f ′′(u) = 0 are iso-lated, Dafermos [Da1] showed the decay of periodic solutions. In Theorem 2.8, thereflection points u | f ′′(u) = 0 are allowed to be dense, even a nonzero measure


(such as the Cantor set). The similar results hold even for multidimensional scalarconservation laws (see Chen-Frid [CF1-2]).

Gui-Qiang Chen 47

2.7. 2 × 2 Strictly Hyperbolic Systems. System (1.1) (d = 1) is calledstrictly hyperbolic and genuinely nonlinear if λ1(u) < λ2(u) and ∇λj(u) · rj(u) 6=0, j = 1, 2, where rj(u) is the right eigenvector corresponding to the jth eigenvalueλj(u). We call wi(u), i = 1, 2, Riemann invariants if they satisfy

∇wi(u) · rj(u) = 0, i 6= j.

For a 2 × 2 hyperbolic system, we have

∂twj + λj∂xwj = 0, j = 1, 2,

and∇q · rj = λj∇η · rj ,

from (1.7). Thenqwj = λj(w1, w2)ηwj , j = 1, 2.

Therefore, the entropy η satisfies the following equation:

ηw1w2 +λ2w1(w1, w2)λ2 − λ1

ηw2 −λ1w2(w1, w2)λ2 − λ1

ηw1 = 0. (2.26)

From (2.26), we have a family of the Lax entropy pairs [La3] in the followingform:

ηk = ekwiV0 + V1k +O( 1

k2 ),qk = ekwiλiV0 + H1

k +O( 1k2 ), (2.27)

where V0 > 0, and

λiV1 −H1 = V0∇λi · ri∇wi · ri

= V0λiwi .

Thenqkηk

= λi +λiwi

k+O(

1k2

), k >> 1. (2.28)

As in the scalar case, an important step needed for the existence and asymptoticdecay of entropy solutions is the following compactness result.

Theorem 2.9 [Di1]. If a sequence of functions uε(t, x) satisfies(i). uε(t, x) ∈ K, a.e., where K ⊂⊂ R2 is a bounded set.(ii). For any C2 entropy pair (η, q),

∂tη(uε) + ∂xq(uε) is compact in H−1loc (R2

+).

Thenuε(t, x) is compact in L1

loc(R2+).

Proof. We need to prove only that νt,x, determined by uε(t, x)ε>0, is a familyof Dirac masses δu(t,x) for a.e. (t, x), where u(t, x) is the weak-star limit of uε(t, x).For simplicity, we drop the index (t, x) of νt,x in the proof.

We first define probability measures µ±k as follows:

< µ±k , h >=

< ν, hη±k >

< ν, η±k >, ∀ h ∈ C(R2), k >> 1,

where η±k are the Lax entropies in (2.27).Since | < µ±

k , h > | ≤ ‖h‖C , which implies ‖µ±k ‖M ≤ 1, there are subsequences

µ±kj

such that µ±kj

weakly converge to some measures µ±, i.e.,

< µ±, h >= limj→∞

< µ±kj, h >, ∀ h ∈ C(R2).


Consider the smallest rectangleQ containing supp ν in the (w1, w2)-coordinates.We claim that the following

(a). suppµ± ⊂ Q ∩ w = w±1 ,

(b). < µ+, q − λ1η >=< µ−, q − λ1η >, for any entropy pair (η, q),

imply that ν is a Dirac mass.In fact, take (η, q) = (ηk, qk) from (2.27) in (b). Then

| < µ+, qk − λ1ηk > | = | < µ+,−ekw1λ1w1

1k

+O(1k2

) > | ≥ C1

kekw+

1 ,

where we used the genuine nonlinearity and C1 > 0. We also have

| < µ−, qk − λ1ηk > | ≤ C2

kekw−

1 .

HenceC1e

kw+1 ≤ C2e

kw−1 , ∀ k >> 1.

That is,

ek(w+1 −w−

1 ) ≤ C2

C1, ∀ k >> 1,

which yields w−1 = w+

1 . Similarly, we have w−2 = w+

2 . Hence, νt,x must be the Diracmass δu(t,x).

Now the remainder is to prove (a) and (b).(a). For any h ∈ C(R2) satisfying h|Q+

ε= 0, where

Q+ε = Q ∩ w+

1 − ε < w1 < w+1 ,

we have

< µ+, h > = limk→∞

< ν, hηk >

< ν, ηk >≤ lim

k→∞

|∫

w−1 ≤w1≤w+

1 −εhekw1V0dν|

|∫

w+1 − ε

2≤w1≤w+1ekw1V0dν|

≤ limk→∞

e−ε2k

∫w−

1 ≤w1≤w+1 −ε |h|V0dν∫

w+1 − ε

2≤w1≤w+1V0dν

= 0.

Thus suppµ+ ⊂ Q ∩ w = w+1 . Using the similar method, we can also prove

µ− ⊂ Q ∩ w = w−1 . Then (a) is proved.

(b). From (2.14), we have

< ν, q > − < ν, η >< ν, q±k >

< ν, η±k >=< ν, η±kq − ηq±k >

< ν, η±k >.

Note that q±k = (λ1 +O( 1k ))η±k. Let k → ∞. Then

< ν, q > − < ν, η >< µ+, λ1 >=< µ+, q − λ1η >, (2.29)

and< ν, q > − < ν, η >< µ−, λ1 >=< µ−, q − λ1η > . (2.30)

It suffices for (b) from (2.29)-(2.30) to show < µ+, λ1 >=< µ−, λ1 >.Take η1 = ηk, η2 = η−k in (2.14), we have

< ν, qk >

< ν, ηk >− < ν, q−k >

< ν, η−k >=< ν, η−kqk − ηkq−k >

< ν, ηk >< ν, η−k >. (2.31)

Gui-Qiang Chen 49

We note that the right hand side of (2.31) tends to 0 as k → ∞. Hence

< µ+, λ1 >=< µ−, λ1 > .

Then (b) follows. This completes the proof.

Now we consider the p-system:∂tτ − ∂xv = 0,

∂tv + ∂xp(τ) = 0,(2.31)

where p(τ) ∈ C2, p′(τ) < 0, τp′′(τ) > 0, as τ 6= 0.The eigenvalues and Riemann invariants are

λ1 = −λ2 = −√−p′(τ),

wj = v + (−1)j+1

∫ τ

0

√−p′(τ)dτ, j = 1, 2,

and the right eigenvectors are

r1 =(1,

√−p′(τ)

)>, r2 =

(−1,

√−p′(τ)

)>.

From τp′′(τ) > 0, we know that ∇λj · rj 6= 0, j = 1, 2, except the points of theline τ = 0. Therefore, system (2.34) is genuinely nonlinear except the points of theline τ = 0. We denote (w+

1 , w+2 ), (w−

1 , w−2 ) as P,Q, respectively, in the previous

proof. Since (2.30) and (a)-(b) still hold, we can easily prove that w+1 = w+

2 , w−1 =

w−2 and

suppµ+ = P, suppµ− = Q,must be true, provided that w−

j < w+j , j = 1, 2.

From (b), we have

q(P ) − λj(P )η(P ) = q(Q) − λj(Q)η(Q).

Let (η, q) = (τ,−v). Since τ = 0 at the points P and Q, v(P ) = v(Q). Thusvt,x must be a Dirac mass δu(t,x).

Notice that, for this system, there are bounded invariant regions for the ap-proximate solutions generated from the viscosity method. Then we have

Theorem 2.10. The result of Theorem 2.9 still holds for the p-system (2.31), al-though the genuine nonlinearity fails at the points of line τ = 0. Furthermore, givenany initial data in L∞, there exists a global entropy solution, and the correspondingsolution operator is compact in L1

loc(R2+).

For the structure of Young measures for 2× 2 strictly hyperbolic systems withlinear degeneracy, see Serre [Se] for the important reduction and Chen [Ch2] for themethod of quasidecouping by using the initial information.


3. Nonstrictly Hyperbolic Conservation Laws

In this section we discuss how the compactness methods can be applied tosolving important nonstrictly hyperbolic systems of conservation laws, that is, someeigenvalues coincide at some points in the state space.

The study of nonstrictly hyperbolic equations has an extensive history, datingback at least two hundred years to the work of Euler [Eu], who posed a degenerateequation, the Euler-Poisson-Darboux equation. Its close relation with wave theory,fluid dynamics, and geometry has attracted great attention for two centuries fromsuch mathematicians as Poisson [Po], Darboux [Dar], Riemann [Ri], Volterra [Vol],and also Weinstein, Erdelyi, Lions, and others in the 1960’s (see [Ya]). On theother hand, the theory of linear equations with multiple characteristics is also welldeveloped. An important feature of such equations is the loss of differentiability[DeG], which leads to ill-posedness in Sobolev spaces but well-posedness in theGevrey classes [Ge]. Another feature is that sign conditions on the subprincipalsymbol play an important role (cf. [Fr,Oh,H2-3]).

Nonlinear nonstrictly hyperbolic systems of conservation laws arise from suchdisparate areas as gas dynamics, multiphase flow in porous media, elasticity, water-wave problems, and magnetohydrodynamics. Such systems appear naturally inmultidimensional systems of conservation laws. This means that, in general, plane-wave solutions for such multidimensional systems are governed by one-dimensionalnonstrictly hyperbolic systems.

There are two kinds of degeneracy: parabolic degeneracy and hyperbolic degen-eracy, which govern different behavior of solutions near the degenerate points. Asystem is called to have hyperbolic degeneracy at a degenerate point u0 if the matrix∇f(u0) is diagonalizable; otherwise, we call the system has parabolic degeneracy.In this section we discuss two prototypes: the system of isentropic Euler equations,which has parabolic degeneracy near the vacuum, and hyperbolic conservation lawswith umbilic degeneracy, which have hyperbolic degeneracy.

3.1. Isentropic Euler Equations.

The system of isentropic Euler equations for a compressible fluid reads∂tρ+ ∂xm = 0,∂tm+ ∂x(m2

ρ + p(ρ)) = 0,(3.1)

where the pressure p(ρ) is a given nonlinear function, determined by the fluid underconsideration, such as real gases and other complex fluids (see Courant-Friedrichs[CFr], Stoker [St], Truesdall [Tr], Pedlosky [Pe], and Lions [Lio]). The density ρ ≥ 0and mass m are physically restricted by |m| ≤ Cρ for some constant C > 0 sothat the function (ρ,m) → m2/ρ remains Lipschitz continuous, even at the vacuumρ = 0. For ρ > 0, the velocity v = m/ρ is uniquely defined. Strict hyperbolicityand genuine nonlinearity in the sense of Lax [La4] away from the vacuum for (3.1)require that the pressure law satisfy

p′(ρ) > 0, 2p′(ρ) + ρp′′(ρ) > 0, for ρ > 0. (3.2)

At the vacuum, the two characteristic speeds of (3.1) may coincide and the systemsbe nonstrictly hyperbolic. A simple calculation shows that the vacuum can not be

Gui-Qiang Chen 51

avoided for this system even for some Riemann solutions with large Riemann initialdata away from the vacuum, in general. This system is an archetype of nonlinearhyperbolic conservation laws in fluid mechanics (see [CFr,Sm,St,Wh,CM,Lio]).

The so-called polytropic perfect gas is described by the equation of state:

p∗(ρ) = κ ργ , γ > 1. (3.3)

One may assume κ = (γ− 1)2/(4γ), which is a convenient normalization. For earlyresults on the existence of global entropy solutions of (3.1), we refer to Riemann [Ri]for the Riemann problem, Zhang-Guo [ZG] and Ding-Chang-Wang-Hsiao-Li [DCW]for a special class of initial data with bounded variation, and Nishida-Smoller [NS]for large total variation with small γ − 1 or vice versa by using the Glimm scheme[Gl]. For the limit case γ = 1, Nishida [Ni] first established the existence of globalsolutions in BV for large BV initial data. For arbitrarily large L∞ initial data,the case γ = 1 + 2/N (N ≥ 5 odd) was first treated by DiPerna [Di2]. The caseγ ∈ (1, 5/3], which is the natural interval of γ for polytropic gases, was completedin Ding-Chen-Luo [DCL1] and Chen [Ch1].

As we discussed in §1-§2, for a construction of global entropy solutions for(3.1), one needs the compactness (i.e. the strong convergence) of a sequence ofapproximate solutions, say (ρε,mε). To achieve this, it suffices to prove that thecorresponding Young measures νt,x, governed by (2.14) for a.e. (t, x) and for anytwo weak entropy pairs (ηj , qj), j = 1, 2, are Dirac masses in the (ρ,m)-plane forour case. This is implied by that the support of any Young measure in the (ρ, v)-plane, still denoted by νt,x, is either a single point or a subset of the vacuum line(ρ, v) | ρ = 0, v ∈ R

. One of the main difficulties to reduce the Young measure

is that the commutation relation (2.14) holds only for weak entropy pairs, which isnot allowed for any C2 entropy pairs (different from those in §2.7), because of thedegeneracy of the system near the vacuum.

As we discussed in §1-§2, for a construction of global entropy solutions for(3.1), one needs the compactness (i.e. the strong convergence) of a sequence ofapproximate solutions, say (ρε,mε). To achieve this, it suffices to prove that thecorresponding Young measures νt,x, governed by (2.14) for a.e. (t, x) and for anytwo weak entropy pairs (ηj , qj), j = 1, 2, are Dirac masses in the (ρ,m)-plane forour case. This is implied by that the support of any Young measure in the (ρ, v)-plane, still denoted by νt,x, is either a single point or a subset of the vacuum line(ρ, v) | ρ = 0, v ∈ R

. One of the main difficulties to reduce the Young measure

is that the commutative relation (2.14) holds only for weak entropy pairs, which isnot allowed for any C2 entropy pairs (different from those in §2.7), because of thedegeneracy of the system near the vacuum.

When (3.3) holds, the so-called weak entropies of (3.1) can be expressed byconvolution of an arbitrary smooth function ψ(s) and the entropy kernel, χ∗(ρ, v, s),defined by

χ∗(ρ, v; s) = M∗ [ργ−1 − (v − s)2]λ+, λ =3 − γ

2(γ − 1). (3.4)

Here y+ = max(y, 0) and M∗ > 0 is the constant of normalization. Namely, onehas η(ρ, v) = η(ρ, ρv) :=

∫RI χ∗(ρ, v, s)ψ(s) ds. The entropy flux kernel has also an

explicit form: σ∗ :=(v + θ (s− v)

)χ∗.


In the proof of DiPerna [Di2], Ding-Chen-Luo [DCL1], and Chen [Ch1], theheart of the matter is to construct special functions ψ in order to exploit the setof constraints (2.14). These test functions are suitable approximations of high-order derivatives of the Dirac mass. In DiPerna [Di2], the case that λ ≥ 2 is aninteger was treated, in which all weak entropies are polynomial functions of theRiemann invariants. The idea of applying the technique of fractional derivativeswas introduced in Chen [Ch1] and Ding-Chen-Luo [DCL1] in order to deal withreal values of λ.

Lions-Perthame-Souganidis [LPS] and Lions-Perthame-Tadmor [LPT], moti-vated by a kinetic formulation for (3.1) and (3.3), made the observation that theuse of ψ could be bypassed and (2.14) may be directly expressed by the entropykernel χ∗(s) = χ∗(ρ, v; s) and the entropy flux kernel σ∗(s) = σ∗(ρ, v; s). That is,for all s1 and s2,

< νt,x, χ∗(s1)σ∗(s2) − χ∗(s2)σ∗(s1) >=< νt,x, χ∗(s1) >< νt,x, σ∗(s2) > − < νt,x, χ∗(s2) >< νt,x, σ∗(s1) > .

(3.5)A simpler proof of the reduction of the Young measures satisfying (3.5) was givenin [LPS] for the case γ ∈ (1, 3). The range γ ∈ [3,∞) was treated in [LPT]. Theprevious techniques of choosing suitable approximations of the derivatives of theDirac mass, in [Ch1, DCL1, Di2], corresponds in their approach to computing asuitable number of s-derivatives of the commutation equation (3.5), the numberof fractional derivatives being related to the exponent λ which characterizes thesingularities of the entropy kernel. This is technically delicate since such derivativesof the kernel generate Dirac masses, due to its limited regularity. It was observedthat the average < ν, χ∗ > is smoother (as a function of s) than χ∗(ρ, v, s) (as afunction of (ρ, v, s)).

Recently, we established a general compactness framework for approximate so-lutions of this system with a general pressure law for large initial data in [CL1-2].The pressure function p(ρ) ∈ C4(0,∞) satisfies that there exist γ ∈ (1, 3) and C > 0such that

p(ρ) = κργ(1 + P (ρ)), |P (k)(ρ)| ≤ Cρ1−k, 0 ≤ k ≤ 4, (3.6)

for sufficiently small ρ. The solutions under consideration will remain in a boundedsubset of ρ ≥ 0 so that the behavior of p(ρ) for large ρ is irrelevant. This meansthat the pressure law p(ρ) has the same principal singularity as the γ-law gas,but (3.6) allows additional singularities in the derivatives when ρ → 0. Indeedobserve that, for k > γ + 1, ργP (k)(ρ) is unbounded when ρ → 0. Observe thatp(0) = p′(0) = 0, but, for k > γ, the higher derivative p(k)(ρ) is unbounded nearthe vacuum.

Theorem 3.1 (Chen-LeFloch [CL2]). Consider the compressible Euler system(3.1) with the general pressure law (3.2) and (3.6) with Cauchy data

(ρ,m)|t=0 = (ρ0(x),m0(x)). (3.7)

Then(i). Given any measurable and bounded initial data (ρ0,m0) satisfying

0 ≤ ρ0(x) ≤ C0, |m0(x)| ≤ C0 ρ0(x), for a.e. x and some C0 > 0, (3.8)

Gui-Qiang Chen 53

there exists an entropy solution (ρ,m) of the Cauchy problem (3.1) and (3.7), glob-ally defined in time, satisfying

0 ≤ ρ(t, x) ≤ C, |m(t, x)| ≤ C ρ(t, x), for a.e. (t, x), (3.9)

where C > 0 depends only on C0 and the pressure function p(·).(ii). Let (ρε,mε) be a sequence of functions satisfying (3.9) uniformly in ε such

that, for any weak entropy pair (η, q),

∂tη(ρε,mε) + ∂xq(ρε,mε) is compact in H−1loc (R2

+). (3.10)

Then the sequence (ρε,mε) is compact in L1loc(R

2+).

Since (3.3) is not assumed, the explicit formula (3.4) are no longer available.In particular, the entropy flux kernel, denoted by σ, cannot be directly expressedfrom the entropy kernel, denoted by χ. It turns out that several crucial steps of theproofs in the previous references can be no longer carried out.

As a first step, we constructed the entropy kernel χ and the entropy flux kernelσ. The entropy kernel is governed by a highly singular equation of Euler-Poisson-Darboux type:

χwz −Λ(w − z)w − z

(χw − χz) = 0, (3.10)

or, equivalently,∂ρρχ− a(ρ)∂vvχ = 0, (3.11)

with initial data:χ|ρ=0 = 0, χρ|ρ=0 = δv=s, (3.12)

involving a Dirac mass, where a(ρ) = p′(ρ)ρ−2. Notice that the function Λ(w − z)is not smooth in w − z in general, and that each of its derivatives produces certainextra power of 1/(w − z), which is singular near w − z ≈ 0, that is, ρ ≈ 0. There-fore, the equation (3.10) or (3.11) is much more singular than the classical Euler-Poisson-Darboux equation that is the case for the γ-law (3.3). New techniques weredeveloped, and special care was made to study the singularities of their derivatives.One of our major observations is that the principal singularities of χ and σ can bedetermined (rather explicitly) from the ones in χ∗, modulo a nonlinear transforma-tion involving the pressure law p(ρ). In addition, we observed several properties of“cancellation of singularities” for the function E(s1, s2) := χ(s1)σ(s2) − χ(s2)σ(s1)and its derivatives of order λ+ 1, as s2 → s1.

Then we proved that any Young measure satisfying the Tartar commutationrelation (2.14) for all weak entropy pairs is a Dirac mass.

Then we proved that any Young measure satisfying the Tartar-Murat commu-tation relation (2.14) for all weak entropy pairs is a Dirac mass. As in [LPS], wecan write (2.14) directly in terms of the kernels χ and σ. Our proof relies on thefollowing identity:

< νt,x, χ(s1) >< νt,x, ∂λ+1s2

∂λ+1s3

E(s2, s3) >+ < νt,x, ∂

λ+1s2

χ(s2) >< νt,x, ∂λ+1s3

E(s3, s1) >+ < νt,x, ∂

λ+1s3

χ(s3) >< νt,x, ∂λ+1s2

E(s1, s2) >= 0(3.14)

in the sense of distributions, for all real variables sj , j = 1, 2, 3, where we used thefractional derivative operator ∂λ+1

s . Then we made s2 and s3 converge to s1 andjustified that the sum of the second and third terms converge to zero in a weak sense.


On the other hand, the first term is highly singular: the main term of interest hasthe form of the product of a function of bounded variation by a bounded measure,which is not defined in a classical sense and may actually have several rigorousmeanings (see [DLM]). We took advantage of this observation and showed that thefirst term generally converges weakly to a limit that is not zero, which involvedseveral careful analyses. Further analysis is also necessary to show that the othersingular terms vanish at the limit, in which some delicate techniques were neededsince there is no explicit formula available. Then the compactness result follows.

With the compactness framework, we can conclude

Theorem 3.2. Assume that the initial data (ρ0(x),m0(x)) satisfy (3.8). Then(i). There exists an entropy solution (ρ,m) of the Cauchy problem (3.1) and

(3.7), globally defined in time, such that

0 ≤ ρ(t, x) ≤ C, |m(t, x)| ≤ C ρ(t, x), a.e. (t, x),

where C depends only on C0 and the pressure function p(·).(ii). The solution operator (ρ,m)(t, ·) = St(ρ0,m0)(·) determined by (i) is

compact in L1loc(R

2+).

(iii). Let (ρ,m) ∈ L∞(R2+) be a periodic entropy solution of (3.1) and (3.7)

with period [α, β]. Then (ρ,m) asymptotically decays:

esslimt→∞

∫ β

α

(|ρ(t, x) − ρ|r + |m(t, x) − m|r

)dx = 0, for all 1 ≤ r <∞,

where (ρ, m) := 1β−α

∫ β

α(ρ0(x),m0(x))dx.

(iv). The approximate solutions, generated from the Lax-Friedrichs scheme orthe Godunov scheme, strongly converge (subsequentially) to an entropy solution of(3.1) and (3.7).

Remark. The results in Theorem 3.2 are somewhat surprising, since the fluxfunction of (3.1) is only Lipschitz continuous. The example found by Greenberg-Rascle [GR] demonstrates that there exist certain systems with only C1 (but not C2)flux functions admitting time-periodic and space-periodic solutions. This exampleindicates that the compactness and asymptotic decay of entropy solutions are verysensitive with respect to the smoothness of flux functions.

By the Euler-Lagrange transformation, the results established here for (3.1) canbe reformulated to those for the Lagrangian system (see Wagner [Wa] and Chen[Ch2]).

3.2. Hyperbolic Conservation Laws with Umbilic Degeneracy.

We are now concerned with hyperbolic systems of conservation laws with hy-perbolic umbilic degeneracy. A point u0 ∈ Rn in the state space is called an umbilicpoint for a 2 × 2 hyperbolic system of conservation laws:

∂tu+ ∂xf(u) = 0, u ∈ R2,

if λ1(u0) = λ2(u0). Such a umbilic degeneracy allows a degree of interaction, ornonlinear resonance, between distinct modes, and leads to high singularities missingin the strictly hyperbolic case.

Gui-Qiang Chen 55

Consider a hyperbolic system of conservation laws:∂tu+ ∂x(∇uC(u)) = 0,

C(u) = 12 (1

3au31 + bu2

1u2 + u1u22),

(3.15)

with a unique umbilic point (0, 0).System (3.15) can be reduced from the following system:

∂tu+ ∂xf(u) = 0, u ∈ R2, (3.16)

with an isolated hyperbolic umbilic point u0. That is, ∇f(u0) is diagonalizable,and there is a neighborhood N of u0 such that ∇f(u) has distinct eigenvalues forall u ∈ N − u0.

Take the Taylor expansion for f(u) about u = u0:

f(u) = f(u0) + ∇f(u0)(u − u0) +12(u− u0)>∇2f(u0)(u − u0) + high order terms.

(3.17)Since ∇f(u0) is diagonalizable, we can make a coordinate transformation to

eliminate the linear term from (3.17) and obtain the the quadratic flux systems:

∂tu+ ∂xQ(u) = 0, (3.18)

omitting the high order terms, where Q(u) = 12u

>∇2f(u0)u. Such quadratic fluxfunctions determine the local behavior of the hyperbolic singularity near the umbilicpoint. Moreover, for any arbitrary system with umbilic hyperbolic degeneracy, ateach umbilic degenerate point, the wave curves of each family of elementary wavesare tangential to the corresponding curves of one of the systems in (3.18) or (3.15)under a suitable linear transformation. In fact, any quadratic flux system (3.18)can be transformed into the corresponding system in (3.15) by a linear coordinatetransformation by using the normal form theorem in [SS1].

The Riemann problem, a special Cauchy problem, for 2× 2 quadratic flux sys-tems was discussed by Isaacson, Marchesin, Paes-Lemes, Plohr, Schaeffer, Shearer,Temple, Zumbrum, and many others (1985-1996). For the simplest nontrivial sys-tem with an isolated umbilic point, the Cauchy problem was solved by Kan [Ka],and a different independent proof of this result was given by Lu [Lu].

In Chen-Kan [CK1-2], a compactness framework was established for approxi-mate solutions to such systems by combining compensated compactness ideas withclassical methods, and by making a detailed analysis of a highly singular equationof Euler-Poisson-Darboux type. New techniques were developed to carry out thisstrategy, since the entropy equation has different type of singularities from (3.10),much more singular. Then this framework was successfully applied to proving theconvergence of the Lax-Friedrichs scheme, the Godunov scheme, and the viscositymethod and the existence of global entropy solutions for the Cauchy problem withlarge initial data for a canonical class of the quadratic flux systems. As a corollary,the compactness of the solution operator in L1

loc and the asymptotic decay of peri-odic entropy solutions were established, important nonlinear phenomena, especiallyfor these nonstrictly hyperbolic systems.


4. Compressible Euler Equations

In this section, we discuss some more results on the compressible Euler equationswith the aid of the compactness methods, in addition to those in §3.1.

4.1. Non-isentropic Euler Equations.

We consider the compressible non-isentropic Euler equations in Lagrangian co-ordinates

∂tτ − ∂xv = 0,

∂tv + ∂xp = 0,

∂t(e+ 12v

2) + ∂x(vp) = 0,

(4.1)

where τ, v, p, and e denote respectively the deformation gradient, the velocity, thepressure, and the internal energy. Other relevant fields are the entropy s and thetemperature θ.

The above system of conservation laws is complemented by the Clausius in-equality:

∂ts ≥ 0,which expresses the second law of thermodynamics.

Selecting (τ, v, s) as the state vector, we have the constitutive relations:

(e, p, θ) = (e(τ, s), p(τ, s), θ(τ, s)) (4.2)

satisfying the conditions:p = −eτ , θ = es.

Under the standard assumptions pv < 0 and θ > 0, system (4.1) is strictlyhyperbolic.

In this section, we first exhibit a class of constitutive relations:

p = h(τ − αs), e = βs−∫ w

h(y)dy, θ = αh(τ − αs) + β > 0, (4.3)

where α, β, w = τ − αs, and h(w) is a smooth function with h′(w) < 0 satisfying

h′′(w) − 4αh′(w)2

αh(w) + β

> 0, if w < w,

< 0, if w > w.(4.4)

The model (4.3) can be regarded as a “first-order correction” to the general consti-tutive equation (see Chen-Dafermos [CD1]).

Consider the Cauchy problem for (4.1) with periodic initial data:

(w, v, s)|t=0 = (w0(x), v0(x), s0(x)), (4.5)

satisfying|w0(x)| ≤ C0, |v0(x)| ≤ C0, s0(x) ∈ Mloc(R),

and

(w0(x), v0(x)) ∈ ΣC1 ≡ (w, v) | − C1 ≤ v ±∫ w

w

√−h′(ω)dω ≤ C1,

which contains only physical admissible states. In particular, if (w, v) ∈ ΣC1 , thenθ = αh(w) + β > 0.

Gui-Qiang Chen 57

Theorem 4.1 [CD1]. (i). There exists a global distributional solution(w(t, x), v(t, x), s(t, x)) for the Cauchy problem (4.1) and (4.3)-(4.5), satisfying

(w, v) ∈ L∞(R2+), (s, st) ∈ Mloc(R2

+), θ(w) ≥ 0,|s|[0, T0] × [−cT0, cT0] ≤ CT 2

0 ,(4.6)

for any c > 0, T0 > 0, with C > 0 independent of T0, where |s| denotes the variationmeasure associated with the signed measure s. Moreover, (w(t, x), v(t, x), s(t, x))satisfies the entropy inequality:

∂tη(w, v) + ∂xq(w, v) ≤ 0, st ≥ 0, (4.7)

in the sense of distributions for any C2 entropy pair (η(w, v), q(w, v)) of the sub-system

∂tw − ∂xv = 0, ∂tv + ∂xh(w) = 0, (4.8)

for which the strong convexity condition holds:

θηww −αh′(w)ηw ≥ 0, θηvv +αηw ≥ 0, (θηww − αh′(w)ηw)(θηvv +αηw)− η2wv ≥ 0.

(4.9)Furthermore, if the initial date are periodic, then the global distributional solution(w(t, x), v(t, x), s(t, x)) is also periodic with the same period.

(ii). Assume that the sequence (wT (t, x), vT (t, x)) satisfies the following:(a). There exists a constant C > 0 such that

‖(wT , vT )‖L∞ ≤ C; (4.10)

(b). The sequence

∂tη(wT , vT ) + ∂xq(wT , vT ) ≤ 0 in the sense of distributions, (4.11)

for any C2 convex entropy pair (η(w, v), q(w, v)) of subsystem (4.8) satisfying (4.9).Then the sequence (wT (t, x), vT (t, x)) is compact in L1

loc(R2+).

This theorem is due to Chen-Dafermos [CD1]. The existence is established byusing the vanishing viscosity method via the following equations:

∂tτ − ∂xv = ε∂x(κ∂x(τ − αs)),

∂tv + ∂xp = ε∂x(κ∂xv),

∂t(e+ 12v

2) + ∂x(vp) = ε∂x(κv∂xv) + ε∂x(ν∂θ),

and the compactness arguments as in §2.The key point is to study the compactness of (wε, vε) first and then the limit

behavior of sε. The main observation is that (wε, vε) satisfy the following 2 × 2subsystem, quasidecoupled from sε, reduced from (4.1):

∂tw − ∂xv = −αεκθ (∂xv)2 + αεκh′

θ (∂xw)2 + ε∂x(κ∂xw),

∂tv + ∂xh(w) = ε∂x(κ∂xv).

Combining our careful energy estimates with the previous compactness framework(Theorem 2.10), we concluded the strong compactness of (wε, uε) in L1

loc and theweak compactness of sε in the measure space M.


Condition (4.9) is a strong version of the convexity of entropy functions. Inparticular, the pair

(η∗(w, v), q∗(w, v)) = (12v2 −

∫ w

h(ω)dω, vh(w))

is a convex entropy pair satisfying (4.9).

Theorem 4.2 [CF2]. Let (τ(t, x), v(t, x), s(t, x)), |v(t, x)|+ |τ(t, x)−αs(t, x)| ≤ C,be a periodic entropy solution of (4.1) and (4.3)-(4.5) with period P satisfying (4.6)-(4.7). Then the velocity v(t, x) asymptotically decays to v = 1

|P |∫

Pv0(x)dx in Lp,

1 ≤ p <∞. Moreover, the pressure p(w(t, x)) and the temperature θ(w(t, x)) decayto

p = p(Θ−1(1|P |

∫P

Θ(w0(x))dx)), and θ = θ(Θ−1(1|P |

∫P

Θ(w0(x))dx)),

in Lp, 1 ≤ p <∞, respectively, where Θ(w) = βw + α∫ w

0 h(ω)dω.

This theorem indicates that, although the periodic solutions do not decay be-cause the linear degeneracy of the system, several important physical quantities,including velocity, pressure, and temperature, do asymptotically decay.

The uniqueness and asymptotic stability of Riemann solutions in the class ofentropy solutions in BV of the compressible Euler equations with a general consti-tutive relation were also established in Chen-Frid [CF5].

4.2. Multidimensional Compressible Euler Equations with GeometricStructure.

We are concerned with the global solutions with geometrical structure for thecompressible Euler equations of isentropic gas dynamics:

∂tρ+ ∇x · ~m = 0,

∂t ~m+ ∇x · ( ~m⊗~mρ ) + ∇xp = 0,

(4.12)

where ρ, ~m, and p are the density, the momentum, and the pressure of the gas,respectively. For polytropic gases, p(ρ) = κργ .

Consider the spherically symmetric solutions outside a solid core (|x| ≥ 1):

ρ(t, x) = ρ(t, r), ~m(t, x) = m(t, r)x

r, r = |x|. (4.13)

Then (ρ(t, r),m(t, r)) are determined by the equations:

∂tρ+ ∂rm = −A′(r)A(r) m,

∂tm+ ∂r(m2

ρ + p(ρ)) = −A′(r)A(r)

m2

ρ , r > 1,(4.14a)

with initial-boundary value problem:m|r=1 = 0,

(ρ,m)|t=0 = (ρ0(r),m0(r)), r > 1,(4.14b)

where A(r) = 2πd2

Γ( d2 )rd−1 is the surface area of d-dimensional sphere.

Gui-Qiang Chen 59

Although we derived (4.14a) through a spherically symmetric flow, the model(4.14a) also describe many important physical flows, such as transonic nozzle flowswith variable cross-sectional area A(r) ≥ c0 > 0.

The eigenvalues of (4.14a) are

λ± = u± c = c(M ± 1),

where c =√p′(ρ) is the sound speed, and M = u

c is the Mach number. We noticethat λ+ − λ− = 2c(ρ) = 2ρ

γ−12 → 0 as ρ → 0. On the other hand, the geometric

source speed is zero, and the eigenvalues λ± are also zero near M ≈ ±1, whichindicates that there is also a nonlinear resonance between the geometrical sourceterms and the characteristic modes.

The insights we sought for this problem are: (a) whether the solution has thesame geometrical structure globally; (b) whether the solution blows up to infinityin a finite time, especially the density. These questions are not easily understood inphysical experiments and numerical simulations, especially for the second question,due to a limited capacity of available instruments and computers. The centraldifficulty of this problem in the unbounded domain is the reflection of waves frominfinity and their strengthening as they move radially inward. Another difficulty isthat the associated steady-state equations change type from elliptic to hyperbolicat the sonic point; such steady-state solutions are fundamental building blocks inour approach.

Consider the steady-state solutions:

mr = −A′(r)A(r) m,

(m2

ρ + p(ρ))r = −A′(r)A(r)

m2

ρ ,

(ρ,m)|r=r0 = (ρ0,m0).

(4.15)

The first equation can be directly integrated to get

A(r)m = A(r0)m0. (4.16)

The second equation can be rewritten as

(A(r)m2

ρ)r +A(r)p(ρ)r = 0.

Hence, using (4.16), we have

(θu2 + ρ2θ)r = 0,

where θ = γ−12 , which implies

ρ2θ(θM2 + 1) = ρ2θ0 (θM2

0 + 1). (4.17)

Then (4.16)-(4.17) become(ρ

ρ0

)θ+1

=A(r0)M0

A(r)M,

(ρ

ρ0

)2θ

=θM2

0 + 1θM2 + 1

. (4.18)

Eliminating ρ in (4.18), we have

M

(1 + θM2)θ+12θ

A(r) =M0

(1 + θM20 )

θ+12θ

A(r0), (4.19)


and rewrite (4.19) as

F (M) =A(r0)A(r)

F (M0), (4.20)

where

F (M) = M

(1 + θ

1 + θM2

) θ+12θ

satisfies

F (0) = 0, F (1) = 1, F (M) → 0, when M → ∞,

F ′(M)(1 −M) > 0, when M ∈ [0,∞),

F ′(M)(1 +M) > 0, when M ∈ (−∞, 0].

Thus we see that, if A(r) < A(r0)|F (M0)|, no smooth solution exists be-cause the right hand side of (4.20) exceeds the maximum values of |F |. If A(r) >A(r0)|F (M0)|, there are two solutions of (4.20), one with |M | > 1 and the otherwith |M | < 1 since the line F = A(r0)

A(r) F (M0) intersects the graph of F (M) at twopoints.

For A′(r) = 0, the system becomes the one-dimensional isentropic Euler equa-tions, which has been discussed in §3.1.

For A′(r) 6= 0, the existence of global solutions for the transonic nozzle flowproblem was obtained in Liu [Li1] by first incorporating the steady-state buildingblocks with the random choice method [Gl], provided that the initial data havesmall total variation and are bounded away from both sonic and vacuum states. Ageneralized random choice method was introduced to compute transient gas flowsin a Laval nozzle in [GLi, GMP]. A global weak entropy solution with sphericalsymmetry was constructed in [MU] for γ = 1 and the local existence of such aweak solution for 1 < γ ≤ 5

3 was also discussed in [MT]. In Chen-Glimm [CG],we developed numerical shock capturing schemes and applied them to constructingglobal solutions of (4.12) with geometrical structure and large initial data in L∞

for 1 < γ ≤ 5/3, including both the spherical symmetric flows and the transonicnozzle flow. The case γ ≥ 5/3 was treated in [CW1]. We proved that the solutionsdo not blow up to infinity in a finite time. More precisely, we have

Theorem 4.3 (Chen-Glimm [CG]; also [CW1]). There are numerical approx-imate solutions (ρε,mε) of (4.14) such that

(i). 0 ≤ ρε(t, r) ≤ C, |mε(t,r)

ρε(t,r) | ≤ C;(ii). ∂tη(ρε,mε)+ ∂rq(ρε,mε) is compact in H−1

loc (Ω) for any weak entropy pair(η, q), where Ω ⊂ R2

+ or Ω ⊂ R+ × (1,∞).Furthermore, there is a convergent subsequence (ρεk ,mεk) in the approximate

solutions (ρε,mε) such that

(ρεk(t, r),mεk (t, r)) → (ρ(t, r),m(t, r)), a.e.

which is a global entropy solution of the initial-boundary problem (4.14) and satisfies

0 ≤ ρ(t, r) ≤ C, |m(t, r)ρ(t, r)

| ≤ C.

Then (ρ(t, x), ~m(t, x)), defined in (4.13) through (ρ(t, r),m(t, r)), is a global entropysolution of (4.12).

Gui-Qiang Chen 61

The approach in Chen-Glimm [CG] is to combine the shock capturing ideas with thefractional step techniques to develop first order Godunov shock capturing schemes,with piecewise constant building blocks replaced by piecewise smooth ones. Thisapproach was motivated by earlier important works (e.g. Liu [Li1-3], Glaz-Liu[GL], Glimm-Marshall-Plohr [GMP], Embid-Goodman-Majda [EGM], and Fok [F]).The main point is to use the steady-state solutions, which incorporate the maingeometrical source terms, to modify the wave strengths in the Riemann solutions.This construction yields better approximate solutions and permits a uniform L∞

bound. There are two technical difficulties to achieve this, both due to transonicphenomena. One is that no smooth steady-state solution exists in each cell ingeneral. This problem was easily solved by introducing a standing shock. Theother is that the constructed steady-state solution in each cell must satisfy thefollowing requirements:

(a). The oscillation of the steady-state solution around the Godunov valuemust be of the same order as the cell length to obtain the L∞ estimate for theconvergence arguments;

(b). The difference between the average of the steady-state solution over eachcell and the Godunov value must be higher than first order in the cell length toensure the consistency of the corresponding approximate solutions with the Eulerequations. That is,

1∆r

∫ (j+ 12 )∆r

(j− 12 )∆r

u(m∆t− 0, r)dr = umj (1 +O(|∆r|1+δ)), δ > 0.

These requirements are naturally satisfied by smooth steady-state solutions thatare bounded away from the sonic state in the cell. The general case must includethe transonic steady-state solutions. The sonic difficulty was overcome, as in exper-imental physics, by introducing an additional standing shock with continuous massand by adjusting its left state and right state in the density and its location to con-trol the growth of the density. These requirements can yield the H−1 compactnessestimates for entropy dissipation measures

∂tη(ρε,mε) + ∂rq(ρε,mε)

and the strong convergence of approximate solutions (ρε,mε) with the aid of acompactness framework discussed in §3.1.

Besides, the above method can be also applied to studying the Euler-Poissonequations for compressible flows:

∂tρ+ ∇x · ~m = 0,

∂t ~m+ ∇x · ( ~m⊗~mρ ) + ∇xp(ρ) = ρ∇xφ− ~m

τ ,

∆xφ = ρ−D(x),

(4.21)

where ρ ∈ R, ~m ∈ Rd, and φ ∈ R denote the density, the mass, and the poten-tial of the fluid flows, respectively. This system describes the dynamic behavior ofmany important physical flows including the propagation of electrons in submicronsemiconductor devices and the biological transport of ions for channel proteins.


System (4.21) with spherical symmetry can be rewritten as

∂tρ+ ∂rm = a(r)m,

∂tm+ ∂r(m2

ρ + p(ρ)) = a(r)m2

ρ + ρ∂rφ− mτ ,

∂rrφ = a(r)∂rφ+ ρ−D(r),

(4.22)

where a(r) = −N−1r and the force term ∂rφ is nonlocal involving the global behavior

of solutions.In the case N = 1, a(r) = 0, system (4.22) becomes the one-dimensional Euler-

Poisson system. Such a case was studied by Degong-Markowich [DM], Gamba [Ga],Marcati-Natalini [MN], and others. For the multidimensional case with large initialdata of geometrical structure, the nonlinear resonance between the characteristicmodes and the geometrical source terms occurs at the sonic state. Another dif-ficulty is that some source terms are nonlocal and the equations involve integralterms. We developed Chen-Glimm’s shock capturing difference scheme to tacklethese difficulties and obtained an L∞ estimate and the convergence of the shockcapturing scheme. Then we established the existence of weak entropy solutions withgeometrical structure of the Euler-Poisson equations in [CW2].

Gui-Qiang Chen 63

5. Hyperbolic Conservation Laws with Relaxation

There are two basic theories to describe the nonequilibrium phenomena in me-chanics: kinetic theory from microscopic level and continuum theory from macro-scopic level. Since the pioneering work of Hilbert [Hi] and Chapman-Enskog [Ce,CE],there have been many activities in studying the kinetic limits from the kineticnonequilibrium processes to the continuum (equilibrium or nonequilibrium) pro-cesses with the aid of the moment closure techniques from kinetic theory and the ki-netic formulation techniques from continuum theory (cf. [BG,CP,Ce,KM,Ke,Le,LPT,MRS,MR,VK,Wh] and the references cited therein).

We are concerned with the relaxation limit of hyperbolic systems of conservationlaws with stiff relaxation terms to the corresponding local systems, which modelsdynamic limit from the continuum and kinetic nonequilibrium processes to theequilibrium processes, as the relaxation time tends to zero. Typical examples forthe limit include gas flow near thermo-equilibrium, viscoelasticity with vanishingmemory, kinetic theory with small Knudsen number, and phase transition withsmall transition time. In general, such relaxations can be modeled as having afunctional dependence on the basic dependent variables. An important case is thatthe relaxation depends only on the local values of the basic dependent variablesand can be modeled by the following hyperbolic system of conservation laws in theform:

∂tU + ∇x · F (U) +1εR(U) = 0, x ∈ RD, U ∈ RN , (5.1)

where U = U(t, x) ∈ RN represents the density vector of basic physical variablesand ε is the relaxation time, which is very small. We assume that the system ishyperbolic, and the relaxation term R(U) is endowed with an n×N constant matrixQ with rank n < N such that QR(U) = 0. This yields n independent conservedquantities u = QU . In addition, we assume that each u uniquely determines a localequilibrium value U = E(u) satisfying R(E(u)) = 0 and such that QE(u) = u, forall u.

The simplest examples are 2 × 2 systems:

∂tu+ ∂xf(u, v) = 0,

∂tv + ∂xg(u, v) + 1εh(u, v) = 0, (5.2)

where h(u, v) = a(u, v)(v − e(u)), a(u, e(u)) 6= 0. For such systems, D = 1, N =2, n = 1, U = (u, v)>, E(u) = (u, e(u))>, and Q = (1, 0).

The local equilibrium limit turns out to be highly singular because of shock andinitial layers and to involve many challenging problems in nonlinear analysis and ap-plied sciences. Roughly speaking, the relaxation time measures how far the nonequi-librium states are away from the corresponding equilibrium states; understandingits limit behavior is equivalent to understanding the stability of the equilibriumstates. It connects nonlinear integral partial differential equations with nonlinearpartial differential equations. This limit also involves the singular limit problemfrom nonlinear strictly hyperbolic systems to mixed hyperbolic-elliptic ones, evenpurely elliptic ones in some cases (see [CLL]). The basic issue for such a limit is itsstability.


As a first example, we consider the p-system:∂tu+ ∂xv = 0,

∂tv + ∂xp(u) + 1ε (v − f(u)) = 0.

(5.3)

It is in the form (5.2) with Λ1 = −√p′(u) < Λ2 =

√p′(u). If f(u) = λu, p(u) =

Λ2u, then u satisfies

∂tu+ λ∂xu+ ε(∂ttu− Λ2∂xxu) = 0.

The limit ε→ 0 is stable if and only if the characteristics satisfy −Λ < λ < Λ.A more interesting example is the compressible Euler equations:

∂tρ+ ∇x · (ρ~v) = 0,

∂t(ρ~v) + ∇x · (ρ~v ⊗ ~v) + ∇xp = 0,

∂t(ρE) + ∇x · (ρE~v + p~v) = 0.

(5.4)

In local thermodynamic equilibrium as we discussed in §4, the system is closedby the constitutive relation:

p = p(ρ, e), E =12|~v|2 + e.

When the temperature varies over a wide range, the gas may not be in localthermodynamic equilibrium, and the pressure p may then be regarded as a functionof only a part e of the specific internal energy, while another part q is governed bya rate equation:

∂t(ρq) + ∇x · (ρq~v) =Q(ρ, e) − q

τ(ρ, e), (5.5)

p = p(ρ, e), E =12|~v|2 + e+ q. (5.6)

Another example is the 3 × 3 semilinear system in one spatial dimension

∂tu+ ∂xv = 0,∂tv + ∂xσ = 0,∂tσ + E2∂xv + σ−p(u)

ε = 0, E > 0.(5.7)

This is a strictly hyperbolic system with eigenvalues 0,±E.Then the local equilibrium approximation is obtained by setting σ = p(u) in

the first two equations, which yields∂tu+ ∂xv = 0,∂tv + ∂xp(u) = 0. (5.8)

In particular, if p′(u) changes sign as u changes, then the 2 × 2 system (5.8) is anelliptic-hyperbolic mixed system, although (5.7) is always hyperbolic.

To understand the stability of the zero relaxation limit, we first analyze thep-system (5.3) to see the points. Notice that vε = f(uε) − ε(∂tv

ε + ∂xp(uε)). If wecan show

(uε(t, x), vε(t, x)) → (u(t, x), v(t, x)), a.e.,

Gui-Qiang Chen 65

then the zero relaxation limit of (uε, vε) is a weak solution of the local equilibriumv = f(u),

∂tu+ ∂xf(u) = 0.(5.9)

Consider a formal expansion of vε in the form

vε ≈ f(uε) + εv1(uε) + ε2v2(uε) + · · · . (5.10)

Then in the ε0-level, we have

∂tuε + ∂xf(uε) ≈ 0,

∂tf(uε) + ∂xp(uε) + v1(uε) ≈ 0,(5.11)

which impliesv1(uε) ≈ −[p′(uε) − f ′(uε)2]∂xu

ε. (5.12)Dropping all the higher-order terms in the expansion leads to a first-order cor-

rection to the local equilibrium approximation in the form:

∂tuε + ∂xf(uε) ≈ ε∂x[(p′(uε) − f ′(uε)2)∂xu

ε]. (5.13)

This evolution equation will be dissipative provided that the following stabilitycriterion holds:

Λ1 < λ < Λ2,

where λ = f ′(uε),Λ1,2 = ±√p′(uε).

For a general system (5.1), the similar arguments yield that the first correctionisU = E(u) − ε(∇UR(E(u)))−1(I − P (u))∇x · F (E(u)),

∂tu+ ∇x · f(u) = ε∇x · [Q∇UF (E(u))(∇UR(E(u)))−1(I − P (u))∇x · F (E(u))],(5.14)

where P (u) = ∇uE(u)Q is a projection (P 2 = P ) onto the tangent space of theimage of E(u).

Definition 5.1. A twice-differential function Φ(U) is called an entropy for system(5.1) provided that

(i). ∇2Φ(U)∇F (U) · ξ is symmetric.(ii). ∇Φ(U)R(U) ≥ 0.(iii). The following are equivalent, i.e.,

(a). R(U) = 0 ⇐⇒ U = E(u),(b). ∇Φ(U)R(U) = 0,(c). ∇Φ(U) = ν>Q, for some ν ∈ Rn.

An entropy Φ is called convex if(d). ∇2Φ(U) ≥ 0.

If inequality (d) is strict, the entropy is called strictly convex.

Such a strictly convex entropy exists for most of physical systems. For example,under certain conditions, system (5.4)-(5.6) has a global strictly convex entropy (seeCoquel-Perthame [CP]).

Furthermore, we have

Theorem 5.1 [CLL]. Suppose that system (5.1) is endowed with a strictly convexentropy pair (Φ,Ψ). Then


(i). The local equilibrium approximation

∂tu+ ∇x · f(u) = 0 (5.15)

is hyperbolic with the strictly convex entropy pair:

(ϕ(u), ψ(u)) = (Φ,Ψ)|U=E(u). (5.16)

(ii). The characteristic speeds of (5.15) associated with any wave number ξ ∈RD are determined as the critical values of the restricted Rayleigh quotient:

w → W>∇2UΦ(E(u))∇UF (E(u)) · ξWW>∇2

UΦ(E(u))W, (5.17)

where W = ∇uE(u)w for w ∈ Rn. The characteristic speeds of (5.15) are interlacedwith the characteristic speeds of (5.1).

(iii). The first correction (5.14) is locally dissipative with respect to the entropyϕ(u). For a 2 × 2 system (5.2), this implies that the subcharacteristic stabilitycondition:

Λ− < λ < Λ+, on v = e(u). (5.18)On the other hand, for a 2×2 system (5.2) satisfying the strictly subcharacter-

istic stability condition (5.18), the existence of a strictly convex entropy pair (φ, ψ)for the local equilibrium equation implies the existence of a strictly convex entropypair (Φ,Ψ) for system (5.2) over an open set Oφ containing the local equilibriacurve v = e(u), along which it satisfies (5.16).

The next issue is how the strong convergence of the zero relaxation limit to thelocal equilibrium equations can be achieved for the systems with a strictly convexentropy. For this, we consider a 2 × 2 system (5.2).

Assume Uε(t, x) = (uε(t, x), vε(t, x)) ⊂ K, bounded open convex set, are solu-tions of (5.2) satisfy the entropy condition: For any convex entropy pair (Φ,Ψ),

∂tΦ(Uε) + ∂xΨ(Uε) +1εΦv(Uε)h(Uε) ≤ 0, ∀ ∇2

UΦ(U) ≥ 0, (5.19)

in the sense of distributions. For simplicity, we assume that there exist two convexand dissipative entropy pairs (Φi,Ψi), i = 1, 2, on K such that

φ2(u) − φ1(u) = Cf(u),

where φi(u) = Φi|v=e(u), f(u) = f1(u, e(u)), and

C < sup(u,v)∈K

f ′′(u)/( inf(u,v)∈K

φ′′1 (u)).

The existence of such entropy functions is related to the stability theory (Theorem5.1) (see [CLi,CLL,Na,CF2]).

Theorem 5.2 [CLL]. Assume that there is no interval in which f(u) is linear. Let

‖(uε0 − u, vε

0 − v)‖L2 ≤ C,

with v = e(u). Then there is a subsequence (still denoted) Uε strongly convergingalmost everywhere:

Uε(t, x) → U(t, x) = (u(t, x), e(u(t, x)))>, a.e.

The limit functions (u, v) satisfy(i). v(t, x) = e(u(t, x)) almost everywhere for t > 0;

Gui-Qiang Chen 67

(ii). u(t, x) is the unique entropy solution of the Cauchy problem∂tu+ ∂xf(u) = 0,

u|t=0 = w∗ − limuε0(x).

This limit is the one of compressible Euler type. Theorem 5.2 shows thatthe solutions of the relaxation system indeed tend to those of the local relaxationapproximation, which are inviscid conservation laws, when the stability conditionis satisfied. The main difficulty is that the solutions of the full systems are only themeasurable functions with certain boundedness. We remark:

(a). Notice that the initial data may even be far from equilibrium. The con-vergence result indicates that the limit functions (u, v) indeed come into local equi-librium as soon as t > 0. This shows that the limit is highly singular. In fact, thislimit consists of two processes simultaneously: one is the initial layer limit, and theother is the shock layer limit.

(b). The compactness of the zero relaxation limit indicates that the sequence Uε

is compact no matter how oscillatory the initial data are. Note that the relaxationsystems are allowed to be linearly degenerate; and the initial oscillations can prop-agate along the linearly degenerate fields for the homogeneous systems (cf. [Ch2]).This fact shows that the relaxation mechanism coupling with the nonlinearity ofthe equilibrium equations can kill the initial oscillations, just as the nonlinearity forthe homogeneous system can kill the initial oscillations.

(c). The above discussions are based on the L∞ apriori estimate. In many phys-ical systems, the estimate can be achieved. Such examples include the p-system andmodels in viscoelasticity, chromatography, and combustion (see [CLi,CLL,Na,CF2,RAA,TW,Wh]), which have natural invariant regions. For some special models,even uniform BV bound of relaxation solutions (uε, vε) can be achieved [TW],which ensures the convergence of the zero relaxation limit via the Helly principle.

Now we are concerned with the weakly nonlinear limit for (5.2). Let

uε = u+ εwε, vε = v + εzε, (5.20)

where (u, v) = (u, e(v)) is an equilibrium state.Upon rescaling the time variable t and translating the space variable x as the

slow time variable εt and the moving space variable x− λ(u)t, respectively,

(t, x) → (t, x− λ(u)t),

the flux functions in equations (5.2) with the stability condition satisfy

λ(u) = 0, Λ1(u)Λ2(u) < 0.

The limit process as ε → 0 is a weakly nonlinear limit corresponding to the limitfrom the Boltzmann equations to the incompressible Navier-Stokes equations. Themain observation is that the linearization of the local relaxation approximationabout an equilibrium gives a simple advection dynamics with the equilibrium char-acteristic speed. This can be understood in a formal fashion. If one applies thesame asymptotic scaling to the first correction to the local equilibrium approxima-tion, one again arrives at the weakly nonlinear approximation. This shows thatthe latter is a distinguished limit of the former and makes clear why it inheritsthe good features of the former. Its advantage is that the solutions of the Burgers


equation are smooth even for the case that the initial data are not smooth. Thusthe solutions remain globally consistent with all the assumptions that were used toderive the weakly nonlinear approximation.

We now justify this approximation using the stability theory and the energyestimate techniques. Linearized version of the limit is well understood which relatesto the “random walk” in Brownian motion (cf. [Ev2,Pi,Ku]). From (5.2) and (5.20),we obtain that (wε, zε) satisfy

ε2∂tw

ε + ∂xf(u+ εwε, v + εzε) = 0,

ε2∂tzε + ∂xg(u+ εwε, v + εzε) + 1

εh(u+ εwε, v + εzε) = 0,

(wε, zε)|t=0 = (wε0(x), z

ε0(x)).

(5.21)

Theorem 5.3 [CLL]. If there exist ε0 > 0, C0 > 0, 0 < ε ≤ ε0, such that

‖(wε0, z

ε0)‖H3 ≤ C0, ‖zε

0 −e(u+ εwε

0) − e(u)ε

‖L2 ≤ C0, (5.22)

then there exists a unique global solution (wε, zε) ∈ H3 of (5.21) for each ε > 0such that

(wε(t, x), zε(t, x)) → (w(t, x), z(t, x)) ∈ L2, ε→ 0,

z(t, x) = e′(u)w(t, x),

∂tw + λ′(u)∂x(w2

2 ) +Λ1(u)Λ2(u)hv(u, e(u))

∂xxw = 0.

Since the proof of Theorem 5.3 is very technical, we list only the main stepsbelow.Step 1. We replace (Φ,Ψ) by (Φ∗,Ψ∗), where

Φ∗ = Φ(U) − Φ(U) −∇Φ(U)(U − U),

Ψ∗ = Ψ(U) − Ψ(U) −∇Ψ(U)(F (U) − F (U)).(5.21)

Then Φ∗v(u, e(u)) ≥ c0(v − e(u))2 for some c0 > 0.From ∇2Φ∗(U) > 0, we obtain

ε

∫ ∞

−∞Φ∗(uε, vε)dx + c0

∫ t

0

∫ ∞

−∞

(vε − e(uε))2

εdxdt

≤ ε

∫ ∞

−∞Φ∗(uε

0(x), vε0(x))dx ≤ Cε3

∫ ∞

−∞(wε

0(x)2 + zε

0(x)2)dx.

(5.22)

Therefore,

‖zε − e(u+ εwε) − e(u)ε

‖L2 ≤ Cε. (5.23)

Step 2. Eliminating zε, we obtain

∂twε + ∂x(

λ′(u)2

(wε)2) +Λ1(u)Λ2(u)hv(u, e(u))

∂xxwε +

ε2

hv(u, v(u))∂ttw

ε

= Eε(t, x,D2wε, D2zε).(5.24)

Gui-Qiang Chen 69

Using the energy estimates, we get∑i,j=1

i+j≤3

ε2(i−1)

∫ t

0

∫ ∞

−∞|∂i

τ∂jx(wε, εizε)|2(τ, x)dxdτ ≤ C, (5.25)

where C is a constant independent of ε.

Step 3. Then we prove Eε(t, x,D2wε, D2zε) → 0, when ε→ 0.

Step 4. Since ‖wε‖H1 ≤ C, using the Sobolev embedding theorem, we obtain thatthere exists a subsequence (still denoted) wε converging strongly in L2. That is

wε(t, x) → w(t, x).

Using the estimate (5.23), we imply that zε strongly converges in L2

zε(t, x) → e′(u)w(t, x).

Then Theorem 5.3 follows.

More details for the proof can be found in Chen-Levermore-Liu [CLL].

Some recent developments in hyperbolic systems of conservation laws can befound in [JL,JX,Na,CF2,TW,Tz,,Ch4] and the references cited therein.

Some recent ideas and approaches in attacking hyperbolic conservation lawswith memory can be found in Dafermos [D5], Nohel-Rogers-Tzavaras [NRT], andChen-Dafermos [CD2] with the aid of the compactness methods.

Acknowledgments.

These notes are based on a series of lectures delivered in the Morningside Centerof Mathematics of Academia Sinica in Beijing and the Workshop on Geometryand Analysis held in Taiwan, July-August, 1997. The author would like to thankProfessor Shing-Tung Yau for invitation, and Professors Ling Hsiao and Shu-ChengChang for warm hospitality during my stay in Beijing and Taiwan, respectively.Feiming Huang deserves my thanks for his effective assistance for these notes. Gui-Qiang Chen’s research was supported in part by NSF grants DMS-9623203, DMS-9708261, and DMS-9722855, and by an Alfred P. Sloan Foundation Fellowship.


References

[Ba] J. M. Ball, A version of the fundamental theorem for Young measures. In: PDEs and Con-tinuum Models of Phase Transitions (Nice, 1988), Lecture Notes in Phys. 344 (1989),207-215, Springer, Berlin-New York.

[BG] C. Bardos, F. Golse, and C. D. Levermore, Fluid dynamic limits of discrete velocity ki-netic equations, In: Advances in Kinetic Theory and Continuum Mechanics, R. Gatignoland Soubbaramayer, eds., Springer-Verlag, Berlin-New York (1991), 57-91.

[BL] A. Bressan and P. LeFloch, Uniqueness of weak solutions to systems of conservation laws,Arch. Rational Mech. Anal. 140 (1970), 301-317.

[BLY] A. Bressan, T.-P. Liu, and T. Yang, L1 stability estimates for 2 × 2 conservation laws,Arch. Rational Mech. Anal. (to appear).

[CP] R. Caflisch and G. C. Papanicolaou, The fluid dynamic limit of a nonlinear model Boltz-mann equations, Comm. Pure Appl. Appl. 32 (1979), 521–554.

[Ce] C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag: NewYork, 1988.

[CE] J. S. Chapman and T. Cowling, The Mathematical Theory of Non-Uniform Gases, Cam-bridge University Press, 1970.

[Ch1] G.-Q. Chen, Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III),Acta Math. Sci. 6 (1986), 75-120 (in English); 8 (1988), 243-276 (in Chinese).

[Ch2] G.-Q. Chen, The method of quasidecouping for discontinuous solutions to conservationlaws, Arch. Rational Mech. Anal. 121 (1992), 131-185.

[Ch3] G.-Q. Chen, Relaxation limit for conservation laws, Z. Angew. Math. Mech. 76 (1996),381-384.

[Ch4] G.-Q. Chen, Entropy, Compactness, and Conservation Laws, Lecture notes, NorthwesternUniversity (1999).

[Ch5] G.-Q. Chen, Hyperbolic systems of conservation laws with a symmetry, Comm. PartialDiff. Eqs. 16 (1991), 1461-1487.

[CD1] G.-Q. Chen and C. M. Dafermos, The vanishing viscosity method in one-dimensionalthermoelasticity, Trans. Amer. Math. Soc. 347 (1995), 531-541.

[CD2] G.-Q. Chen and C. M. Dafermos, Global solutions in L∞ for a system of conservationlaws of viscoelastic materials with memory, J. Partial Diff. Eqs. 10 (1997), 369-383.

[CF1] G.-Q. Chen and H. Frid, Asymptotic decay of solutions of conservation laws, C. R. Acad.Sci. Paris, 323 (1996), 257-262.

[CF2] G.-Q. Chen and H. Frid, Decay of entropy solutions of nonlinear conservation laws, Arch.Rational Mech. Anal. 1998 (to appear).

[CF3] G.-Q. Chen and H. Frid, Large-time behavior of entropy solutions of conservation laws,J. Diff. Eqs. 1998 (to appear).

[CF4] G.-Q. Chen and H. Frid, Uniqueness and asymptotic stability of entropy solutions in BVfor the compressible Euler equations, Preprint 1998, Northwestern University.

[CF5] G.-Q. Chen and H. Frid, Large-time behavior of entropy solutions in L∞ for multidi-mensional conservation laws, In: Advances in Nonlinear Partial Differential Equationsand Related Areas, pp. 28-44, G.-Q. Chen et al (eds.), World Scientific: Singapore, 1998.

[CF6] G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws,Preprint 1997 (submitted for publication on April 1998).

[CG] G.-Q. Chen and J. Glimm, Global solution to the compressible Euler equations withgeometrical structure, Commun. Math. Phys. 179 (1996), 153-193.

[CK1] G.-Q. Chen and P. T. Kan, Hyperbolic conservation laws with umbilic degeneracy (I),Arch. Rational Mech. Anal. 130 (1995), 231-276.

[CK2] G.-Q. Chen and P. T. Kan, Hyperbolic conservation laws with umbilic degeneracy (II),Preprint, Northwestern University, 1998.

[CL1] G.-Q. Chen and P. LeFloch, Entropies and weak solutions of the compressible isentropicEuler equations, C. R. Acad. Sci. Paris, 324 (1997), 1105-1110.

[CL2] G.-Q. Chen and P. LeFloch, Compressible Euler equations with general pressure law andrelated equations, Preprint, January 1998, Northwestern University.

[CL] G.-Q. Chen and J.-G. Liu, Convergence of difference schemes with high resolution forconservation laws, Math. Comp. 66 (1997), 1027–1053.

Gui-Qiang Chen 71

[CLi] G.-Q. Chen and T.-P. Liu, Zero relaxation and dissipation limits for hyperbolic conser-vation laws, Comm. Pure. Appl. Math. 46 (1993), 255-281.

[CLL] G.-Q. Chen, C. D. Levermore, and T.-P. Liu, Hyperbolic conservation laws with stiffrelaxation terms and entropy, Comm. Pure Appl. Math. 47 (1994), 787-830.

[CLu] G.-Q. Chen and Y.-G. Lu, A study to applications of the theory of compensated com-pactness, Chinese Science Bulletin 34 (1989), 15-19.

[CW1] G.-Q. Chen and D. Wang, Shock capturing approximations to the compressible Eulerequations with geometric structure and related equations, Z. Angew. Math. Phys. 49(1998), 341-362.

[CW2] G.-Q. Chen and D. Wang, Convergence of shock capturing schemes for the compressibleEuler-Poisson equations, Commun. Math. Phys. 179 (1996), 333-364.

[CM] A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics,Springer-Verlag: Berlin-Heidelberg-New York, 1979.

[CLM] R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and

Hardy spaces, J. Math. Pures Appl. 72 (1993), 247–286.[CS] E. Conway and J. Smoller, Global solutions of the Cauchy problem for quasilinear first-

order equations in several space variable, Comm. Pure Appl. Math. 19 (1966), 95-105.[CP] F. Coquel and B. Perthame, Relaxation of energy and approximate Riemann solvers for

general pressure laws in fluid dynamics, Preprint, 1997.[CFr] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag:

New York, 1962.[Da] B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity of Nonlinear Func-

tionals, Lecture Notes in Math. 922, Springer-Verlag: New York, 1982.[Da1] C. M. Dafermos, Applications of the invariance principle for compact processes II: As-

ymptotic behavior of solutions of a hyperbolic conservation laws, J. Diff. Eqs. 11 (1972),416-424.

[Da2] C. M. Dafermos, Large time behavior of periodic solutions of hyperbolic systems ofconservation laws, J. Diff. Eqs. 121 (1995), 183–202.

[Da3] C. M. Dafermos, Hyperbolic systems of conservation laws, Proceedings of the Inter-national Congress of Mathematicians, Vols.1-2 (Zurich, 1994), 1096–1107, Birkhuser,Basel, 1995.

[Da4] C. M. Dafermos, Generalized characteristics in hyperbolic systems of conservation laws,Arch. Rational Mech. Anal. 107 (1989), 127–155.

[Da5] C. M. Dafermos, Solutions in L∞ for a conservation law with memory, AnalyseMathematique et Applications, Gauthier-Villars, Paris, 1988, 117-128.

[DML] G. Dal Maso, F. Murat, and P. LeFloch, Definition and weak stability of nonconservativeproducts, J. Math. Pure Appl. 74 (1995), 483-548.

[Dar] G. Darboux, Legons Sur La Theorie Generale des Surfaces, T. II, Paris, 1914-1915.[DeG] E. De Giorgi, Un esempio di non-unicita della soluzione del problema di Cauchy, Univer-

sita di Roma, Rendiconti di Matematica, 14 (1955), 382-387.[DM] P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model

for semiconductors, Appl. Math. Letters 3, 25-29.[D] X. Ding, Theory of conservation laws in China, In: Hyperbolic Problems: Theory, Nu-

merics, Applications, pp. 110-119, J. Glimm et al (eds.), World Scientific: Singapore,1996.

[DCW] X. Ding, T. Zhang, C.-H. Wang, L. Hsiao, and T.-C. Li, A study of the global solutions forquasilinear hyperbolic systems of conservation laws, Scientica Sinica 16 (1973), 317-335.

[DCL1] X. Ding, G.-Q. Chen, and P. Luo, Convergence of the Lax-Friedrichs scheme for isentropicgas dynamics (I)-(II), Acta Math. Sci. 5 (1985), 483-500, 501-540 (in English); 7 (1987),467-480, 8 (1988), 61-94 (in Chinese).

[DCL2] X. Ding, G.-Q. Chen and P. Luo, Convergence of the fractional step Lax-Friedrichsscheme and Godunov scheme for isentropic gas dynamics, Commun. Math. Phys. 121(1989), 63-84.

[Di1] R. DiPerna, Convergence of approximate solutions to conservation laws, Arch. RationalMech. Anal. 88 (1985), 223-270.

[Di2] R. DiPerna, Convergence of viscosity method for isentropic gas dynamics, Commun.Math. Phys. 91 (1983), 1-30.


[EGM] P. Embid, J. Goodman, and A. Majda, Multiple steady state for 1-D transonic flow,SIAM J. Sci. Stat. Comp. 5, 21-41.

[Eu] L. Euler, Institutiones Calculi Integralis, III, Petropoli, 1790.[Ev1] L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations,

CBMS, Vol. 72, Providence, Rhode Island, 1990.[Ev2] L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear

PDE, Proc. Royal Soc. Edinburgh 111A (1989), 141-172.[F] S. K. Fok, Extensions of Glimm’s method to the problem of gas flow in a duct of variable

cross-section, Ph. D. Thesis, Department of Mathematics, University of California atBerkeley, 1981.

[FRS] S. Friedlands, J. W. Robbin, and J. Sylvester, On the crossing rule, Comm. Pure Appl.Math. 37 (1984), 19-38.

[Fr] M. Froissart, Hyperbolic Equations and Waves, Battelle Seattle 1968 Recontres, Springer-Verlag: Berlin-Heidelberg, 1970.

[Ga] I. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model forsemiconductors, Comm. Partial Diff. Eqs., 17 (1992), 553-577.

[Ge] M. Gevrey, Sur la nature analytique des solutions des equations aux derivavees partielles,

Annales Ecole Norm. Sup. 35 (1917), 129-189.[GLi] H. Glaz and T.-P. Liu, The asymptotic analysis of wave interactions and numerical

calculation of transonic nozzle flow, Adv. Appl. Math. 5 (1984), 111-146.[Gl] J. Glimm, Solutions in the large for nonlinear hyperbolic system of equations, Comm.

Pure Appl. Math. 18 (1965), 95-105.[GL] J. Glimm and P. D. Lax, Decay of Solutions of Nonlinear Hyperbolic Conservation Laws,

Mem. Amer. Math. Soc. 101 (1970).

[GM] J. Glimm and A. Majda, Multidimensional Hyperbolic Problems and Computations, IMAVolumes in Mathematics and its Applications, 29, Springer-Verlag: New York, 1991.

[GMP] J. Glimm, G. Marshall and B. Plohr, A generalized Riemann problem for quasi-onedimensional gas flow, Adv. Appl. Math. 5 (1984), 1-30.

[GR] J. Greenberg and M. Rascle, Time periodic solutions of conservation laws, Arch. RationalMech. Anal. 115 (1991), 395-407.

[GLY] C. Gu, D. Li, W. Yu, and Z. Hou, Discontinuous initial-value problem for hyperbolicsystems of quasilinear equations (I)-(III), Acta Math. Sinica 11 (1961), 314-323, 324-327; 12 (1962), 132-143 (in Chinese).

[Hi] D. Hilbert, Grundzuge einer allgemeinen Theorie de linearen Integralgleichungen, Teub-ner 1912.

[H1] L. Hormander, Lectures on Nonlinear Hyperbolic Differential Equations, Springer-Verlag:Berlin, 1997.

[H2] L. Hormander, The Analysis of Linear Partial Differential Operators II, Springer-Verlag:Berlin-Heidelberg, 1983.

[H3] L. Hormander, Hyperbolic systems with double characteristics, Comm. Pure Appl. Math.46 (1993), 261-301.

[JL] S. Jin and C. Levermore, Numerical schemes for hyperbolic conservation laws with stiffrelaxation terms. J. Comput. Phys. 126 (1996), 449–467.

[JX] S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitraryspace dimensions, Comm. Pure Appl. Math. 48 (1995), 235-277.

[Ka] P.-T. Kan, On the Cauchy problem of a 2×2 system of nonstrictly hyperbolic conservationlaws, Ph. D. thesis, Courant Institute of Mathematical Sciences, NYU, 1989.

[KM] S. Kawashima, A. Matsumura, and T. Nishida, On the fluid dynamical approximationto the Boltzmann equation at the level of the Navier-Stokes equations, Commun. Math.Phys. 70 (1979), 97-124.

[Ke] J. Keizer, Statistical Thermodynamics of Nonequilibrium Processes, Springer-Verlag:New York 1987.

[Kr] S. Kruzkov, First-order quasilinear equations with several space variables, Mat. Sb. 123(1970), 228-255.

[Ku] T. G. Kurtz, Convergence of sequences of semigroups of nonlinear operators with anapplication to gas kinetics, Trans. Amer. Math. Soc. 186 (1974), 259-272.

Gui-Qiang Chen 73

[La1] P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10(1957), 537-566.

[La2] P. D. Lax, The multiplicity of eigenvalues. Bull. Amer. Math. Soc. (N.S.) 6 (1982),213–214.

[La3] P. D. Lax, Shock waves and entropy, In: Contributions to Nonlinear Functional Analysis,ed. E. A. Zarantonello, Academic Press, pp. 603-634, 1971.

[La4] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory ofShock Waves, SIAM: Philadelphia, 1973.

[La5] P. D. Lax, Problems solved and unsolved concerning linear and nonlinear partial differ-ential equations, Proceedings of the International Congress of Mathematicians, Vols.1-2(Warsaw, 1983), 119–137, PWN, Warsaw, 1984.

[Le] C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys. 83(1996), 1021-1065.

[Li1] T.-P. Liu, Quasilinear hyperbolic systems, Commun. Math. Phys. 68 (1979), 141-172.

[Li2] T.-P. Liu, Nonlinear stability and instability of transonic gas flow through a nozzle,Commun. Math. Phys. 83 (1983), 243-260.

[Li3] T.-P. Liu, Nonlinear resonance for quasilinear hyperbolic equation, J. Math. Phys. 28(1987), 2593-2602.

[Li4] T.-P. Liu, Hyperbolic conservation laws with relaxation, Commun. Math. Phys. 108(1987), 153-175.

[Li5] T.-P. Liu, The deterministic version of the Glimm scheme, Commun. Math. Phys. 57(1977), 135–148.

[LY] T.-P. Liu and T. Yang, L1 stability for systems of hyperbolic conservation laws, In: Non-linear Partial Differential Equations and Applications, G.-Q. Chen and E. DiBenedetto(eds.), Contemporary Mathematics, AMS: Providence, 1999.

[Lio] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vols. 1-2, Oxford University Press:Oxford, 1998.

[LPT] P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gasdynamics and p-systems, Commun. Math. Phys. 163 (1994), 169-172.

[LPS] P. L. Lions, B. Perthame and P. Souganidis, Existence and stability of entropy solu-tions for the hyperbolic systems of isentropic gas dynamics in Eulerian and LagrangianCoordinates, Preprint.

[Lu] Y.-G. Lu, Convergence of the viscosity method for a nonstrictly hyperbolic system, ActaMath. Sinica 12 (1992), 230-239.

[LS] T. Luo and D. Serre, Linear stability of shock profiles for a rate-type viscoelastic systemwith relaxation, Quart. Appl. Math. 56 (1998), 569–586.

[MMU] T. Makino, K. Mizohata, and S. Ukai, Global weak solutions of the compressible Eulerequations with spherical symmetry (I)-(II), Japan J. Industrial Appl. Math. 9 (1992),431-449.

[MN] P. Marcati, and R. Natalini, Weak solutions to a hydrodynamic model for semiconductorsand relaxation to the drift-diffusion equation, Arch. Rational Mech. Anal. 129 (1995),129-145.

[MRS] P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor Equations,Springer-Verlag: Wien-New York, 1990.

[MR] I. Muller, and Y. Ruggeri, Extended Thermodynamics, Springer-Verlag: New York, 1993.[Mo1] C. B. Morrey, Quasiconvexity and the lower semicontinuity of multiple integrals, Pacific

J. Math. 2 (1952), 25-53.[Mo2] C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag: Berlin,

1966.[Mu1] F. Murat, The injection of the positive cone of H−1 in W−1, q is completely continuous

for all q < 2, J. Math. Pures Appl. (9) 60 (1981), 309–322.[Mu2] F. Murat, Compacite par compensation, Ann. Suola Norm. Pisa (4) 5 (1978), 489-507.[Mu3] F. Murat, Compacite par compensation: condition necessarie et suffisante de continuite

faible sous une hypothese de range constant, Ann. Scuola Norm. Sup. Pisa (4) 8 (1981),69-102.


[Mu4] F. Murat, A survey on compensated compactness, In: Contributions to Modern Calculusof Variations (Bologna, 1985), 145–183, Pitman Res. Notes Math. Ser. 148, LongmanSci. Tech.: Harlow, 1987.

[Na] R. Natalini, Convergence to equilibrium for the relaxation approximations of conservationlaws, Comm. Pure Appl. Math. 49 (1996), 795–823.

[Ni] T. Nishida, Global solution for an initial-boundary value problem of a quasilinear hyper-bolic systems, Proc. Jap. Acad. 44 (1968), 642-646.

[NRT] J. A. Nohel, R. C. Rogers, and A. E. Tzavaras, Weak solutions for a nonlinear system inviscoelasticity, Comm. Partial Diff. Eqs. 13 (1988), 97-127.

[Oh] Y. Ohya, On E. E. Levi’s functions for hyperbolic equations with triple characteristics,Comm. Pure Appl. Math. 25 (1972), 257-263.

[Ol] O. Oleinik, Discontinuous solutions of nonlinear differential equations, Usp. Mat. Nauk.(N.S.) 12 (1957), 3-73.

[Pe] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag: New York.

[Pi] M. A. Pinsky, Lectures on Random Evolutions, World Scientific: Singapore, 1991.[Po] S. D. Poisson, Memoire sur lintegration des equations lineaires aux derivees partielles, J.

L’ecole Polytechnique, 12 (1823), 19.[RAA] H. K. Rhee, R. Aris, and N. R. Amundsen, First-Order Differential Equations: Theory

and Application of Hyperbolic Systems of Quasilinear Equations, Prentice Hall Interna-tional Series, I, 1986; II, 1989.

[Ri] B. Riemann, Uber die Fortpflanzung ebener Luftwellen von endlier Schwingungsweite,Abhandl. Koenig. Gesell. Wiss., Goettingen, Vol. 8 (1860), 43.

[SS1] D. Schaeffer and M. Shearer, The classification of 2×2 systems of nonstrictly hyperbolicconservation laws, with application to oil recovery, Comm. Pure Appl. Math. 40 (1987),141-178.

[SS2] D. Schaeffer and M. Shearer, Riemann problems for nonstrictly hyperbolic 2 × 2 systemsof conservation laws, Trans. Amer. Math. Soc. 304 (1987), 267-306.

[SY] R. Schoen and S.-T. Yau, Lectures on Differential Geometry, Conference Proceedingsand Lecture Notes in Geometry and Topology I, International Press, Cambridge, MA,1994.

[Se] D. Serre, La compacite par compensation pour les systemes hyperboliques non lineairesde deux equations une dimension d’espace, J. Math. Pure Appl. (9) 65 (1986), 423-468.

[Sm] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag: New York,1982.

[St] J. J. Stokes, Water Waves: The Mathematical Theory with Applications, John Wiley &Sons, Inc.: New York, 1958.

[Str] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equa-tions and Hamiltonian Systems, Springer-Verlag: Berlin, 1990.

[Ta1] L. Tartar, Compensated compactness and applications to partial differential equations.In: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, pp. 136–212,Res. Notes in Math. 39, Pitman, Boston, Mass.-London, 1979.

[Ta2] L. Tartar, Une nouvelle methode de resolution d’equations aux derivees partielles non-lineaires, Lecture Notes in Math. 665, 228-241, Springer-Verlag, 1977.

[Ta3] L. Tartar, The compensated compactness method applied to systems of conservationlaws, In: Systems of Nonlinear P.D.E., J. M. Ball (eds.), pp. 263–285, NATO series, C.Reidel publishing Co. 1983.

[Ta4] L. Tartar, H-measures, a new approaches for studying homogenization, oscillations andconcentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh 115A(1990), 193–230.

[T] K. Trivisa, A priori estimates in hyperbolic systems of conservation laws via generalizedcharacteristics, Comm. Partial Diff. Eqs. 22 (1997), 235-267.

[Tr] C. Truesdall, Rational Thermodynamics, 2nd Edition, Springer-Verlag: New York, 1984.[TW] A. Tveito and R. Winther, On the rate of convergence to equilibrium for a system of

conservation laws with a relaxation term, SIAM J. Math. Anal. 28 (1997), 136–161.[Tz] A. Tzavaras, Materials with internal variables and relaxation to conservation laws,

Preprint, University of Wisconsin, 1998.

Gui-Qiang Chen 75

[VK] W. G. Vincenti and C. H. Kruger, Jr., Introduction to Physical Gasdynamics, R. E.Krieger Publication Co. 1982.

[Vo] A. Volpert, The Space BV and quasilinear equations, Mat. Sb. 73 (1967), 255-302.[Vol] V. Volterra, Sulle vibrazioni luminose nei mezzi isotropi, Rend. Accad. Nazl. Lincel, 1

(1892), 161-170.[Wa] D. H. Wagner, Equivalence of Euler and Lagrangian equations of gas dynamics for weak

solutions, J. Diff. Eqs. 68 (1987), 118-136.[WX] W.-C. Wang and Z. Xin, Asymptotic limit of initial boundary value problems for conser-

vation laws with relaxational extensions, Comm. Pure Appl. Math. 51 (1998), 505-535.[Wh] G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience: New York, 1973.[Wu] X.-M. Wu, Equations of Mathematical Physics (in Chinese), Chinese Advanced Educa-

tion Publishing Co.: Beijing, 1956.[Ya] G.-J. Yang, The Euler-Poisson-Darboux Equations (in Chinese), Yuannan University

Press: Yuannan, 1989.

[YZ] L. Ying and Z. Teng, Hyperbolic Systems of Conservation Laws and Difference Methods(in Chinese), Chinese Scientific Publishing Co.: Beijing, 1991 (in Chinese).

[Yo] L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theories,W. B. Sauders: Philadelphia, 1969.

[Wh] T. Zhang and Y.-F. Guo, A class of initial-value problem for systems of aerodynamicequations, Acta Math. Sinica 15 (1965), 386-396.

Compactness Methods and Nonlinear Hyperbolic Conservation Laws

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