Content uploaded by Alberto Bressan

Author content

All content in this area was uploaded by Alberto Bressan on Jun 27, 2015

Content may be subject to copyright.

arXiv:math/0212392v1 [math.AP] 1 Dec 2002

ICM 2002 · Vol. I · 159–178

Hyperbolic Systems of Conservation Laws

in One Space Dimension

Alberto Bressan*

Abstract

Aim of this paper is to review some basic ideas and recent developments

in the theory of strictly hyperbolic systems of conservation laws in one space

dimension. The main focus will be on the uniqueness and stability of entropy

weak solutions and on the convergence of vanishing viscosity approximations.

2000 Mathematics Subject Classiﬁcation: 35L60, 35L65.

Keywords and Phrases: Hyperbolic system of conservation laws, Entropy

weak solution, Vanishing viscosity.

1. Introduction

By a system of conservation laws in m space dimensions we mean a ﬁrst order

system of partial diﬀere ntial equations in divergence for m:

∂

∂t

U +

m

X

α=1

∂

∂x

α

F

α

(U) = 0 , U ∈ IR

n

, (t, x) ∈ IR × IR

m

.

The components of the vector U = (U

1

, . . . , U

n

) are the conserved quantities. Sys-

tems of this type express the bala nce equations of continuum physics, when small

dissipation eﬀects are neglected. A basic example is provided by the eq uations of

non-viscous gases, acc ounting for the conse rvation o f mass , momentum and ener gy.

The s ubject is thus very classical, having a long tradition which can be traced back

to Euler (1755) and includes contributions by Stokes, Riemann, Weyl and Von Neu-

mann, among several others. The continued attention of analysts and mathematical

physicists during the span of over two centuries, however, has not accounted for a

comprehensive mathematical theory. On the contrary, as remar ked in [Lx2], [D2],

[S2], the ﬁeld is still replenished with challenging open problems. In several space

dimensions, not even the global existence of solutions is presently known, in any

* S.I.S.S.A., Via Beirut 4, Trieste 34014, Italy. E-mail: bressan@sissa.it

160 A. Bressan

signiﬁcant degree of generality. Until now, most of the analysis has been concerned

with the one-dimensional case, and it is only here that basic questions could b e

settled. In the re mainder of this pape r we shall thus consider systems in one space

dimension, referring to the books of Majda [M], Serre [S1] or Dafermos [D3] for a

discussion of the multidimensional case.

Toward a rigorous mathematical analysis of solutions, the main diﬃculty that

one encounters is the lack of regularity. Due to the strong nonlinearity of the

equations and the absence of diﬀusion terms with s moothing eﬀect, solutions which

are initially smooth may become discontinuous within ﬁnite time. In the presence

of discontinuities, most of the classical tools of diﬀere ntial calculus do not a pply.

Moreover, for general n ×n systems, the powe rful techniques of functional analysis

cannot be used. In particular, solutions cannot be re presented as ﬁxed points of

a nonlinea r transformation, or in variational form as critical points of a suitable

functional. Dealing with vector valued functions, comparison arg uments based on

upper and lower solutions do not apply either. Up to now, the theory of conservation

laws has progre ssed largely by ad hoc methods. A survey of these techniques is the

object of the prese nt paper.

The Cauchy problem for a system of conservatio n laws in one space dimension

takes the form

u

t

+ f(u)

x

= 0, (1.1)

u(0, x) = ¯u(x). (1.2)

Here u = (u

1

, . . . , u

n

) is the vector of conserved quantities, while the c omponents

of f = (f

1

, . . . , f

n

) are the ﬂuxes. We shall always assume that the ﬂux function

f : IR

n

7→ IR

n

is smooth and that the system is strictly hyperbolic, i. e., at ea ch

point u the Jacobian matrix A(u) = Df(u) has n real, distinct eigenvalues

λ

1

(u) < ··· < λ

n

(u). (1.3)

As already mentioned, a distinguished feature of nonlinear hyperbolic systems is

the po ssible loss of regularity. Even with smooth initial data, it is well known that

the solution ca n develop shocks in ﬁnite time. Therefore, solutions deﬁned globally

in time c an o nly be found within a space of discontinuous functions. The equation

(1.1) must then be interpreted in distributional sense. A vector valued function

u = u(t, x) is a weak solution of (1.1) if

ZZ

u φ

t

+ f(u) φ

x

dxdt = 0 (1.4)

for every test function φ ∈ C

1

c

, continuously diﬀerentiable with compact suppor t.

In particular, the piecewise constant function

u(t, x)

.

=

u

−

if x < λt ,

u

+

if x > λt ,

(1.5)

Hyperbolic Systems of Conservation Laws 161

is a weak solution of (1.1) if and only if the left and right states u

−

, u

+

and the

sp e e d λ satisfy the famous Rankine-Hugoniot equations

f(u

+

) −f(u

−

) = λ (u

+

− u

−

) . (1.6)

When discontinuities are prese nt, the weak so lution of a Cauchy problem may

not be unique. To single out a unique “good” solution, additional entropy conditions

are usually imposed along shocks [Lx1], [L3]. These conditions often have a physical

motivation, characterizing those solutions which can be recovered from higher order

models, letting the diﬀusion or dispersion coeﬃcients approach zero (see [D3]).

In one space dimension, the mathematical theory of hyper bolic systems of

conservatio n laws has developed along two main lines.

1. The BV setting, pioneered by Glimm (1965). Solutions are here constructed

within a space of functions with bounded variation, controlling the BV norm by a

wave interaction potential.

2. The L

∞

setting, introduced by DiPerna (1983), ba sed on weak convergence and

a compensated compactness argument.

Both appro aches yield results on the global existence of weak solutions. How-

ever, it is only in the BV setting that the well posedness of the Cauchy problem

could recently be proved, as well as the stability and c onverge nce of vanishing vis-

cosity approximations. On the other hand, a counterexample in [BS] indicates that

similar results cannot be expected, in general, for solutions in L

∞

. In the remainder

of this paper we thus concentrate on the theory of BV solutions, referring to [DP2]

or [S1] for the alternative approach based on compensated compactness.

We shall ﬁrst review the main ideas involved in the construction of weak so-

lutions, based on the Riemann problem and the wave interaction functional. We

then present more recent results on stability, uniqueness and characterization of

entropy weak solutions. All this material can be found in the monograph [B3]. The

last section contains an outline of the latest work o n stability and convergence of

vanishing viscosity approximations.

2. Existence of weak solutions

Toward the construction of more general solutions of (1.1), the basic building

block is the Riemann problem, i.e. the initial value problem where the data are

piecewise constant, with a single jump at the origin:

u(0, x) =

u

−

if x < 0 ,

u

+

if x > 0 .

(2.1)

Assuming that the amplitude |u

+

− u

−

| of the jump is small, this problem was

solved in a classical paper of Lax [Lx1], under the additional hypothesis

162 A. Bressan

(H) For each i = 1, . . . , n, the i- th ﬁeld is either genuinely nonlinear, so that

Dλ

i

(u) · r

i

(u) > 0 for all u, or linearly degenerate, with Dλ

i

(u) · r

i

(u) = 0 for

all u.

The solution is self-similar: u(t, x) = U(x/t). It consists of n + 1 constant sta tes

ω

0

= u

−

, ω

1

, . . . , ω

n

= u

+

(see Fig. 1). Each couple of adiacent states ω

i−1

, ω

i

is separated either by a shock (the thick lines in Fig. 1) satisfying the Rankine

Hugoniot equations, or else by a centered rarefaction. In this second case, the

solution u varies continuously between ω

i−1

and ω

i

in a sector of the t-x-plane (the

shaded region in Fig. 1) where the gradient u

x

coincides with an i-eigenvector of

the matrix A(u).

2∆

∆

∆ 2∆

ω =

0

u

−

ω =

3

u

+

x

0

t

t

x x

ω

1

t

ω

2

Figure 1 Figure 2

Approximate solutions to a more general Cauchy problem can be constructed

by patching together several solutions of Riema nn problems. In the Glimm scheme

(Fig. 2), one works with a ﬁxed grid in the x-t plane, with mesh sizes ∆x, ∆t. At

time t = 0 the initial data is approximated by a piecewise constant function, with

jumps at grid points. Solving the correspo nding Riemann problems, a solution is

constructed up to a time ∆t suﬃciently s mall so that waves generated by diﬀerent

Riemann problems do not interact. By a r andom sampling procedure, the solution

u(∆t, ·) is then approximated by a piecewise constant function having jumps only

at grid points. Solving the new Riemann problems at every o ne of these points, one

can prolong the solution to the next time interval [∆t, 2∆t], etc. . .

’

t

1

t

2

x

x

σ

σ

’

t

Figure 3

An alternative technique for co ntructing approximate solutions is by wave-

Hyperbolic Systems of Conservation Laws 163

front tracking (Fig. 3). This method was introduced by Dafermos [D1] in the scalar

case and later developed by various authors [DP1], [B1], [R], [BJ]. It now provides

an eﬃcient tool in the study of general n ×n systems of c onservation laws, both for

theoretical and numerical purpo ses [B3], [HR].

The initial data is here approximated with a piecewise constant function, and

each Riemann problem is solved approximately, within the cla ss of piecewise con-

stant functions. In particula r, if the exact solution contains a centered rarefaction,

this must be approximated by a rarefaction fan, containing several small jumps. At

the ﬁr st time t

1

where two fronts interact, the new Riemann problem is again ap-

proximately solved by a piecewise constant function. The solution is then prolonged

up to the second interaction time t

2

, where the new Riemann problem is solved,

etc. . . The main diﬀerence is that in the Glimm scheme one speciﬁes a priori the

nodal points where the the Riema nn problems are to be solved. On the other hand,

in a solution constructed by wave-front tracking the locations of the jumps and of

the interaction points depend on the solution itself, and no restarting proce dure is

needed.

In the end, both algo rithms produce a sequence of approximate solutions,

whose convergence relies on a compactness argument based on uniform bounds on

the total variation. We sketch the main idea involved in these a priori BV bounds.

Consider a piecewise constant function u : IR 7→ IR

n

, say with jumps at points

x

1

< x

2

< ··· < x

N

. C all σ

α

the amplitude of the jump at x

α

. T he total s trength

of waves is then deﬁned as

V (u)

.

=

X

α

|σ

α

|. (2.2)

Clearly, this is an equivalent way to measur e the total variation. Along a solution

u = u(t, x) constructed by front tracking, the quantity V (t) = V

u(t, ·)

may well

increase at interaction times. To provide global a priori bounds, following [G] one

introduces a wave interaction potential, deﬁned as

Q(u) =

X

(α,β)∈A

|σ

α

σ

β

|, (2.3)

where the summation runs over the set A of a ll couples of approaching waves.

Roughly speaking, we say that two wave-fronts located at x

α

< x

β

are approaching

if the one at x

α

has a faster speed than the one at x

β

(hence the two fronts are

exp ected to collide at a future time). Now consider a time τ where two incoming

wave-fronts interact, say with strengths σ, σ

′

(for exa mple, take τ = t

1

in Fig . 3).

The diﬀere nce betwee n the outgoing waves emerging from the interaction and the

two incoming waves σ, σ

′

is of magnitude O(1) · |σσ

′

|. On the other hand, after

time τ the two incoming waves are no longer approaching. This accounts for the

decrease of the functional Q in (2.3) by the amount |σσ

′

|. Observing that the new

waves generated by the interaction could approach all other fronts, the change in

the functionals V, Q acr oss the interaction time τ is estimated as

∆V (τ) = O(1) · |σσ

′

|, ∆Q(τ) = −|σσ

′

| + O(1 ) · |σσ

′

|V (τ−).

164 A. Bressan

If the initial data has small total variation, for a suitable constant C

0

the quantity

Υ(t)

.

= V

u(t, ·)

+ C

0

Q

u(t, ·)

is monotone decreasing in time. This argument provides the uniform BV bounds on

all approximate solutions. Using Helly’s compactness theorem, one obtains the con-

vergence of a subsequence of approximate solutions, and hence the global existence

of a weak solution.

Theorem 1. Let the system (1.1) be strictly hyperbolic and satisfy the assumptions

(H). Then, for a suﬃciently small δ > 0 the following holds. For every initial

condition ¯u with

k¯uk

L

∞

< δ , Tot.Var.{¯u} < δ , (2.4)

the Cauchy problem has a weak solution, deﬁned for all times t ≥ 0.

This result is based on careful analysis of solutions of the Riemann pr oblem

and on the use of a quadratic interaction functional (2 .3) to control the creation of

new waves. Thes e techniques als o provided the basis for subsequent inves tigations

of Glimm and L ax [GL] and Liu [L2] on the asymptotic behavior of weak s olutions

as t → ∞.

3. Stability

The pre vious existence result relied on a compactness argument which, by

itself, does not provide informations on the uniqueness of solutions. A ﬁrst under-

standing of the dependence of weak solutions on the initial data was provided by

the analysis of front tracking approximations. The idea is to perturb the initial

data by s hifting the position of one of the jumps, say from x to a nearby po int x

′

(see Fig. 3). B y carefully estimating the corresponding shifts in the positions of

all wave-fronts at a later time t, one obtains a bound on the L

1

distance between

the original and the per turbed approximate solution. After much technical work,

this appr oach yielded a proof of the Lipschitz continuous dependence of solutions

on the initial data, ﬁrst in [BC1] for 2 ×2 systems, then in [BCP] for general n ×n

systems.

Theorem 2. Let the system (1.1) be strictly hyperbolic and satisfy the assumptions

(H). Then, for every initial data ¯u satisfying (2.4) the weak solution obtained as

limit of Glimm or front tracking approximations is unique and depends Lipschitz

continuously on the initial data, in the L

1

distance.

These weak solutions can thus be written in the fo rm u(t, ·) = S

t

¯u, as tra-

jecories of a semigroup S : D × [0, ∞[ 7→ D on some domain D containing all func-

tions with suﬃciently small total variatio n. For some Lipschitz constants L, L

′

one

has

S

t

¯u − S

s

¯v

L

1

≤ L k¯u − ¯vk

L

1

+ L

′

|t − s|, (3.1)

Hyperbolic Systems of Conservation Laws 165

for all t, s ≥ 0 and initial data ¯u, ¯v ∈ D.

An alternative proof of Theorem 2 was later achieved by a technique introduced

by Liu and Yang in [LY] and pr e sented in [BLY] in its ﬁnal form. The heart of the

matter is to construct a nonlinear functional, equivalent to the L

1

distance, which

is decreasing in time along every pair of solutions. We thus seek Φ = Φ(u, v) and a

constant C such that

1

C

·

v − u

L

1

≤ Φ(u, v) ≤ C ·

v − u

L

1

, (3.2)

d

dt

Φ

u(t), v(t)

≤ 0. (3.3)

v

u

= u(x)

0

ω

ω

1

2

ω

3

ω

= v(x)

x

x

α

q

1

3

q

α

σ

Figure 4

In connection with piecewise constant functions u, v : IR 7→ IR

n

generated by

a front tracking algorithm, this functional can be deﬁned as follows (Fig. 4). At

each point x, we connect the states u(x), v(x) by mea ns of n shock curves. In

other words, we construct intermediate states ω

0

= u(x), ω

1

, . . . , ω

n

= v(x) such

that each pair ω

i−1

, ω

i

is connected by an i-shock. These states can b e uniquely

determined by the implicit function theorem. Call q

1

, . . . , q

n

, the strengths of these

shocks. We rega rd q

i

(x) as the i-th scalar component of the jump

u(x), v(x)

. For

some constant C

′

, one clearly has

1

C

′

·

v(x) − u(x)

≤

n

X

i=1

q

i

(x)

≤ C

′

·

v(x) − u (x)

. (3.4)

The functional Φ is now deﬁned as

Φ(u, v)

.

=

n

X

i=1

Z

∞

−∞

W

i

(x)

q

i

(x)

dx, (3.5)

where the weights W

i

take the form

W

i

(x)

.

= 1 + κ

1

·

total strength o f waves in u and in v

which approa ch the i- wave q

i

(x)

+ κ

2

·

wave interaction potentials of u and of v

.

= 1 + κ

1

V

i

(x) + κ

2

Q(u) + Q(v)

(3.6)

166 A. Bressan

for suitable constants κ

1

, κ

2

. Notice that, by construction, q

i

(x) re presents the

strength of a ﬁctitious shock wave located at x, travelling with a speed λ

i

(x) de-

termined by the Rankine-Hugoniot equations. In (3.6), it is thus meaningful to

consider the quantity

V

i

(x)

.

=

X

α∈A

i

(x)

|σ

α

|,

where the summation extends to all wave-fronts σ

α

in u and in v which are ap-

proaching the i-shock q

i

(x). From (3.4) and the boundedness of the weights W

i

,

one ea sily derives (3.2). By ca reful estimates on the Riemann problem, one can

prove that also (3.3) is approximately satisﬁed. In the end, by taking a limit of

front tracking approximations, one obta ins Theor e m 2.

For general n × n systems, in (3.1) one ﬁnds a Lipschitz cons tant L > 1.

Indeed, it is only in the scalar case that the semigro up is contractive and the theor y

of accretive operators and abstract evolution equations in Banach spaces can be

applied, see [K], [C]. We refer to the ﬂow generated by a system of conservation

laws as a Riemann semigroup, beca use it is entirely determined by specifying how

Riemann problems are solved. As proved in [B2], if two s emigroups S, S

′

yield the

same solutions to all Riemann problems, then they coincide, up to the choice of

their domains.

From (3.1) one can deduce the error bound

w(T ) − S

T

w(0)

L

1

≤ L ·

Z

T

0

(

lim inf

h→0+

w(t + h) − S

h

w(t)

L

1

h

)

dt , (3.7)

valid for every Lipschitz continuous map w : [0, T ] 7→ D taking values inside the

domain of the semigroup. We can think of t 7→ w(t) as an approximate s olution of

(1.1), while t 7→ S

t

w(0) is the exact solution having the same initial data. According

to (3.7), the distance at time T is bounded by the integral of an instantaneous error

rate, ampliﬁed by the Lipschitz constant L of the semigr oup.

Using (3.7), one can estimate the distance between a front tracking approxima-

tion and the corresponding exact solution. Fo r approximate solutions constructed

by the Glimm scheme, a direct application of this same formula is not possible

because of the additional erro rs introduced by the restarting procedures a t times

t

k

.

= k ∆t. However, relying on a careful a nalysis of Liu [L1], one can construct a

front tracking approximate solution having the same initial and terminal values a s

the Glimm solution. By this technique, in [BM] the authors prove d the estimate

lim

∆x→0

u

Glimm

(T, ·) − u

exact

(T, ·)

L

1

√

∆x · |ln ∆x|

= 0 . (3.8)

In other words, letting the mesh sizes ∆x, ∆t → 0 while keeping their ratio ∆x/∆t

constant, the L

1

norm of the error in the Glimm approximate s olution tends to zero

at a rate slightly slower than

√

∆x.

Hyperbolic Systems of Conservation Laws 167

4. Uniqueness

The uniquenes s and stability results stated in Theorem 2 refer to a special

class of weak solutions: those obtained as limits of Glimm or front tracking ap-

proximations. For several applications, it is desirable to have a uniqueness theorem

valid for general weak solutions, without reference to any particular constructive

procedure. Results in this direction were proved in [BLF], [BG], [BLe]. They are

all based on the error formula (3.7). In the proo fs, one considers a weak solution

u = u (t, x) of the Cauchy problem (1.1)–(1.2). Assuming that u satisﬁes suitable

entropy and regularity conditions, one shows that

lim inf

h→0+

u(t + h) − S

h

u(t)

L

1

h

= 0 (4.1)

at almost every time t. By (3.7), u thus coincides with the semig roup tr ajectory

t 7→ S

t

u(0) = S

t

¯u. Of course, this implies uniqueness. As an ex ample, we state

below the result of [BLe]. Consider the following assumptions:

(A1) (Conservation Equations) The function u = u(t, x) is a weak solution of

the Cauchy problem (1.1)–(1.2), taking va lues within the domain D of the

semigroup S. More precisely, u : [0, T ] 7→ D is co ntinuous w.r.t. the L

1

distance. The initial condition (1.2) holds, together with

ZZ

u φ

t

+ f (u) φ

x

dxdt = 0

for every C

1

function φ with compact support contained inside the open strip

]0, T [ ×IR.

(A2) (Lax Entropy Condition) Let u have an approximate jump discontinuity at

some point (τ, ξ) ∈]0, T [×IR. In other words, assume that there exists states

u

−

, u

+

∈ Ω and a speed λ ∈ IR such that, calling

U(t, x)

.

=

u

−

if x < ξ + λ(t − τ),

u

+

if x > ξ + λ(t − τ),

(4.2)

there holds

lim

ρ→0+

1

ρ

2

Z

τ +ρ

τ −ρ

Z

ξ+ρ

ξ−ρ

u(t, x) − U (t, x)

dxdt = 0. (4.3)

Then, for some i ∈ {1, . . . , n}, o ne has the entropy inequality:

λ

i

(u

−

) ≥ λ ≥ λ

i

(u

+

). (4.4)

(A3) (Bounded Variation Condition) The function x 7→ u

τ(x), x) has bounded

variation along every Lipschitz continuous space-like curve

t = τ(x)

, which

satisﬁes |dτ/dx| < δ a.e., for some constant δ > 0 small enough.

168 A. Bressan

Theorem 3. Let u = u(t, x) be a weak solution of the Cauchy problem (1.1)–(1.2)

satisfying the assumptions (A1), (A2) and (A3). Then

u(t, ·) = S

t

¯u (4.5)

for all t. In particular, the solution that satisﬁes the three above conditions is unique.

An additional characterization o f these unique solutions, based on local integral

estimates, was given in [B2]. The underlying idea is as follows. In a forward

neighborhood of a point (τ, ξ) where u has a jump, the weak solution u behaves

much in the same way as the solution of the corresponding Riemann problem. On

the other hand, on a region where its total variation is small, our solution u can be

accurately approximated by the solution of a linear hyperbolic system with constant

coeﬃcients.

To state the result more precisely, we introduce some notations. Given a

function u = u(t, x) and a point (τ, ξ), we denote by U

♯

(u;τ,ξ)

the solution of the

Riemann problem with initial da ta

u

−

= lim

x→ξ−

u(τ, x), u

+

= lim

x→ξ+

u(τ, x). (4.6)

In addition, we deﬁne U

♭

(u;τ,ξ)

as the solution of the linear hype rbolic Cauchy prob-

lem with constant coe ﬃcients

w

t

+

b

Aw

x

= 0, w(0, x) = u(τ, x). (4.7)

Here

b

A

.

= A

u(τ, ξ)

. Observe that (4.7) is obtained from the quasilinear system

u

t

+ A(u)u

x

= 0 (A = Df ) (4.8)

by “freezing” the coeﬃcients of the matrix A(u) at the point (τ, ξ) a nd choosing

u(τ) as initial data. A new notion o f “good solution” can now be introduced, by

locally comparing a function u with the self-similar solution of a Riemann problem

and with the solution of a linear hyperbolic system with constant coeﬃcients. More

precisely, we say that a function u = u(t, x) is a viscosity solution of the system

(1.1) if t 7→ u(t, ·) is continuous as a map with values into L

1

lo c

, and moreover the

following integral estimates hold.

(i) At e very point (τ, ξ), for every β

′

> 0 one has

lim

h→0+

1

h

Z

ξ+β

′

h

ξ−β

′

h

u(τ + h, x) − U

♯

(u;τ,ξ)

(h, x − ξ)

dx = 0. (4.9)

(ii) There exist c onstants C, β > 0 such that, for every τ ≥ 0 and a < ξ < b, one

has

lim sup

h→0+

1

h

Z

b−βh

a+βh

u(τ + h, x) − U

♭

(u;τ,ξ)

(h, x)

dx ≤ C ·

Tot.Var.

u(τ); ]a, b[

2

.

(4.10)

Hyperbolic Systems of Conservation Laws 169

As proved in [B2], this concept of viscosity solution completely characterizes

semigroup trajectories.

Theorem 4. Let S : D × [0, ∞[×D be a semigroup generated by the system of

conservation laws (1.1). A function u : [0, T ] 7→ D is a viscosity solution of (1.1) if

and only if u(t) = S

t

u(0) for all t ∈ [0, T ].

5. Vanishing viscosity approximations

A natural conjecture is tha t the entropic solutions of the hyperbolic system

(1.1) actually coincide with the limits of solutions to the parabolic s ystem

u

ε

t

+ f (u

ε

)

x

= ε u

ε

xx

, (5.1)

letting the viscosity coeﬃcient ε → 0. In view of the previous uniqueness results,

one expects that the vanishing viscosity limit should single out the unique “good”

solution of the Cauchy problem, satisfying the appropriate entropy c onditions. In

earlier literature, results in this directio n were based on three main techniques:

1 - Comparison principles for parabolic equations. For a scalar conservation

law, the existence, uniqueness and global s tability of vanishing visc osity solutions

was ﬁrst established by Oleinik [O] in one space dimension. The famous paper by

Kruzhkov [K] covers the more general class of L

∞

solutions and is also valid in

several space dimensions.

2 - Singular perturbations. Let u be a piecewise smooth solution of the n × n

system (1 .1), with ﬁnitely many non-interacting, entropy admissible shocks. In

this special case, using a singular perturbation technique, Goodman and Xin [GX]

constructed a family of solutions u

ε

to (5.1), with u

ε

→ u as ε → 0.

3 - Compensated compactness. If, instea d of a BV bound, only a uniform

bound on the L

∞

norm of solutions of (5.1) is available, one can still construct a

weakly convergent subsequence u

ε

⇀ u. In general, we canno t expect that this weak

limit sa tisﬁes the nonlinear equations (1.1). However, for a class of 2 × 2 systems,

in [DP2] DiPerna showed that this limit u is indeed a weak solution of (1.1). The

proof relies on a compensated compac tness argument, based on the re presentation

of the weak limit in terms of Young measures, which must reduce to a Dirac mass

due to the presence of a lar ge family of entropies.

Since the main existence and uniqueness results for hyperbolic systems of

conservatio n laws are valid within the space of B V functions, it is natural to seek

uniform BV bounds also for the vis c ous approximations u

ε

in (5.1). This is indeed

the main goal accomplished in [BB]. As soon as these BV bounds are established, the

existence of a vanis hing viscosity limit follows by a standard compactness argument.

The uniqueness of the limit can then be deduced from the uniqueness theorem in

[BG]. By further analysis , o ne can also prove the continuous dependence on the

170 A. Bressan

initial data for the viscous approximations u

ε

, in the L

1

norm. Remarkably, these

results a re valid for general n × n strictly hyperbolic systems, not necess arily in

conservatio n form.

Theorem 5. Consider the Cauchy problem for a strictly hyperbolic system with

viscosity

u

ε

t

+ A(u

ε

)u

ε

x

= ε u

ε

xx

, u

ε

(0, x) = ¯u(x) . (5.2)

Then there exist constants C, L , L

′

and δ > 0 such that the following holds. If

Tot.Var.{¯u} < δ ,

¯u(x)k

L

∞

< δ , (5.3)

then for each ε > 0 the Cauchy problem (5.2) has a unique solution u

ε

, deﬁned for

all t ≥ 0. Adopting a semigroup notation, this will be written as t 7→ u

ε

(t, ·)

.

= S

ε

t

¯u.

In addition, one has:

BV bounds : Tot.Var.

S

ε

t

¯u

≤ C Tot.Var.{¯u}. (5.4)

L

1

stability :

S

ε

t

¯u − S

ε

t

¯v

L

1

≤ L

¯u − ¯v

L

1

, (5.5)

S

ε

t

¯u − S

ε

s

¯u

L

1

≤ L

′

|t − s| +

√

εt −

√

εs

. (5.6)

Convergence. As ε → 0+, the solutions u

ε

converge to the trajectories of a

semigroup S such that

S

t

¯u − S

s

¯v

L

1

≤ L k¯u − ¯vk

L

1

+ L

′

|t − s|. (5.7)

These vanishing viscosity limits can be regarded as the unique vanishing viscosity

solutions of the hyperbolic Cauchy problems

u

t

+ A(u)u

x

= 0, u(0, x) = ¯u(x) . (5.8 )

In the conservative case where A(u) = Df(u) for some ﬂux function f, the

vanishing viscosity solution is a weak solution of

u

t

+ f (u)

x

= 0, u(0, x) = ¯u(x) , (5.9)

satisfying the Liu admissibility conditions [L3]. Moreover, the vanishing viscosity

solutions are precisely the same as the viscosity solutions deﬁned at (4.9)–(4.10) in

terms of local int egral estimates.

The key step in the proo f is to establish a priori bounds on the total variation

of solutions of

u

t

+ A(u)u

x

= u

xx

(5.10)

uniformly valid for all times t ∈ [0, ∞[ . We outline here the main ideas.

Hyperbolic Systems of Conservation Laws 171

(i) At each point (t, x) we decompose the gradient along a suitable basis of unit

vectors ˜r

i

, say

u

x

=

X

v

i

˜r

i

. (5.11)

(ii) We then derive an equation describing the evolution of these gradient compo-

nents

v

i,t

+ (

˜

λ

i

v

i

)

x

− v

i,xx

= φ

i

. (5.12)

(iii) Finally, we show that all source terms φ

i

= φ

i

(t, x) are integrable. Hence, for

all τ > 0,

v

i

(τ, ·)

L

1

≤

v

i

(0, ·)

L

1

+

Z

∞

0

Z

IR

φ

i

(t, x)

dxdt < ∞. (5.13)

In this connection, it seems natural to decompose the gradient u

x

along the

eigenvecto rs of the hyperbolic matrix A(u). This approach however does NOT work.

In the c ase where the solution u is a travelling viscous shock proﬁle, we would obtain

source terms which are not identically zer o. Hence they are certainly no t integrable

over the domain

t > 0 , x ∈ IR

.

An alternative approa ch, proposed by S. Bianchini, is to decompose u

x

as a

sum of gradients of viscou s t ravelling waves. By a viscous trave lling i-wave we mean

a solution of (5.10) having the form

w(t, x) = U(x − σt) , (5.14)

where the sp e e d σ is clo se to the i-th eigenvalue λ

i

of the hyper bolic matrix A.

Clearly, the function U must provide a solution to the second order O.D.E.

U

′′

=

A(U) − σ

U

′

. (5.15)

The underlying idea for the decomposition is as follows. At each point (t, x), given

(u, u

x

, u

xx

), we seek travelling wave proﬁles U

1

, . . . , U

n

such that

U

i

(x) = u(x), i = 1, . . . , n , (5.16)

X

i

U

′

i

(x) = u

x

(x) ,

X

i

U

′′

i

(x) = u

xx

(x) . (5.17)

In ge neral, the system of algebraic e quations (5.16)–(5.17) admits inﬁnitely many

solutions. A unique so lution is singled out by considering only those travelling

proﬁles U

i

that lie on a suitable center m anifold M

i

. We now call ˜r

i

the unit vector

parallel to U

′

i

, so that U

′

i

= v

i

˜r

i

for some scalar v

i

. The decomposition (5.11) is

then obtained from the ﬁr st equation in (5.17).

Toward the BV estimate, the second part of the proof consists in deriving the

equation (5.12) and estimating the integrals of the source terms φ

i

. Here the main

172 A. Bressan

idea is that these source terms can be regarded as ge nerated by wave interactions.

In analogy with the hyperbolic case considered by Glimm [G], the total amount of

these interactions can be controlled by suitable Lyapunov functionals. We describe

here the main ones.

1. Consider ﬁrst two independent, scalar diﬀusion equations with stric tly diﬀerent

drifts:

(

z

t

+

λ(t, x)z

x

− z

xx

= 0 ,

z

∗

t

+

λ

∗

(t, x)z

∗

x

− z

∗

xx

= 0 ,

assuming that

inf

t,x

λ

∗

(t, x) − sup

t,x

λ(t, x) ≥ c > 0 .

We regard z as the density of waves with a slow speed λ a nd z

∗

as the density of

waves with a fast sp e e d λ

∗

. A transversal interaction potential is deﬁned as

Q(z, z

∗

)

.

=

1

c

ZZ

IR

2

K(x

2

− x

1

)

z(x

1

)

z

∗

(x

2

)

dx

1

dx

2

, (5.18)

K(y)

.

=

e

−cy/2

if y > 0 ,

1 if y ≤ 0 .

(5.19)

One can show that this functional Q is monoto nically decreasing along every couple

of solutions z, z

∗

. The total amount of interaction between fast and slow waves can

now be e stimated as

Z

∞

0

Z

IR

z(t, x)

z

∗

(t, x)

dxdt ≤ −

Z

∞

0

d

dt

Q

z(t), z

∗

(t)

dt

≤ Q

z(0 ), z

∗

(0)

≤

1

c

Z

IR

z(0 , x)

dx ·

Z

IR

z

∗

(0, x)

dx .

By means of Lyapunov functionals of this type one can control all s ource terms in

(5.12) due to the interaction of waves of diﬀerent families.

2. To control the interactions between waves of the same family, we seek functionals

which are decreasing along every solution of a scalar viscous conservation law

u

t

+ g(u)

x

= u

xx

. (5.20)

For this purpose, to a scalar function x 7→ u(x) we a ssociate the curve in the plane

γ

.

=

u

g(u) − u

x

=

conserved quantity

ﬂux

. (5.21)

In connection with a s olution u = u(t, x) of (5.20), the curve γ evolves according to

γ

t

+ g

′

(u)γ

x

= γ

xx

. (5.22)

Hyperbolic Systems of Conservation Laws 173

Notice that the vector g

′

(u)γ

x

is parallel to γ, hence the presence of this term in

(5.22) only amounts to a re parametrization of the curve, and does not aﬀect its

shape. The curve thus evolves in the direction of curvature. An obvious Lyapunov

functional is the length of the curve. In terms of the variables

γ

x

=

v

w

.

=

u

x

−u

t

, (5.23)

this length is given by

L(γ)

.

=

Z

|γ

x

|dx =

Z

p

v

2

+ w

2

dx . (5.24)

We can estimate the rate of decrease in the length as

−

d

dt

L

γ(t)

=

Z

IR

|v|

(w/v)

x

2

1 + (w/v)

2

3/2

dx ≥

1

(1 + δ

2

)

3/2

Z

|w/v|≤δ

|v|

(w/v)

x

2

dx ,

(5.25)

for any given constant δ > 0. This yields a useful a priori estimate on the integral

on the right hand side of (5.25).

3. In connection with the same cur ve γ in (5.21), we now introduce another func-

tional, deﬁned in terms of a wedge product.

Q(γ)

.

=

1

2

ZZ

x<x

′

γ

x

(x) ∧ γ

x

(x

′

)

dx dx

′

. (5.26)

For any curve that moves in the plane in the direction o f curvature, one can show

that this functional is monotone decreasing and its decrease bounds the area swept

by the curve: |dA| ≤ −dQ.

Using (5.22)–(5.23) we now compute

−

dQ

dt

≥

dA

dt

=

Z

|γ

t

∧ γ

x

|dx =

Z

|γ

xx

∧γ

x

|dx =

Z

|v

x

w − vw

x

|dx .

Integrating w.r.t. time, we thus obtain another useful a priori bound:

Z

∞

0

Z

|v

x

w − vw

x

|dx dt ≤

Z

∞

0

dQ

γ(t)

dt

dt ≤ Q

γ(0)

.

Together, the functionals in (5.24) and (5.26) allow us to estimate all source terms

in (5.12) due to the interaction of waves of the same family.

This yields the L

1

estimates on the source terms φ

i

, in (5.12), proving the

uniform bounds on the total variation of a solution u of (5.10). See [BB] for details.

Next, to prove the uniform stability of all solutions of the parabolic system

(5.10) having small total variation, we consider the linearized system describing the

174 A. Bressan

evolution of a ﬁrst order var iation. Inserting the formal expansion u = u

0

+ǫz+O(ǫ

2

)

in (5.10), we obta in

z

t

+

DA(u) · z

u

x

+ A(u)z

x

= z

xx

. (5.27)

Our basic goal is to prove the bound

z(t)

L

1

≤ L

z(0 )

L

1

, (5.28)

for some constant L and all t ≥ 0 and every so lution z of (5.27). By a sta ndard

homotopy argument, from (5.28) one easily deduces the Lipschitz continuity of the

solution of (5.8) on the initial data. Namely, for every couple of solutions u, ˜u with

small total variation one ha s

u(t) − ˜u(t)

L

1

≤ L

u(0) − ˜u(0)

L

1

. (5.29)

To prove (5.28) we decompose the vector z as a sum o f scalar components: z =

P

i

h

i

˜r

i

, write an evolution equation for these components:

h

i,t

+ (

˜

λ

i

h

i

)

x

− h

i,xx

=

ˆ

φ

i

,

and show that the source ter ms

ˆ

φ

i

are integrable on the doma in {t > 0 , x ∈ IR }.

For every initial data u(0, ·) = ¯u with small total variation, the previous argu-

ments yield the existence of a unique global solution to the para bolic system (5.8),

depending Lipschitz continuously on the initial data, in the L

1

norm. Perform-

ing the rescaling t 7→ t/ε, x 7→ x/ε, we immediately obtain the same results for

the Cauchy problem (5.2). Adopting a semigroup notation, this solution can be

written as u

ε

(t, ·) = S

ε

t

¯u. Thanks to the uniform bounds on the total variation, a

compactness argument yields the e xistence of a stro ng limit in L

1

lo c

u = lim

ε

m

→0

u

ε

m

(5.30)

at least for some s ubse quence ε

m

→ 0 . Since the u

ε

depend continuously on the

initial data, with a unifor m Lipschitz constant, the same is true of the limit solution

u(t, ·) = S

t

¯u. In the conservative case where A(u) = Df(u), it is not diﬃcult to

show that this limit u actually provides a weak solution to the Cauchy problem

(1.1)–(1.2).

The o nly remaining issue is to show that the limit in (5.30) is unique, i.e. it

does not depend on the s ubsequence {ε

m

}. In the standard conservative case, this

fact can already be deduced from the uniqueness result in [BG]. In the general case,

uniqueness is pr oved in two steps. First we show that, in the special case of a

Riemann problem, the solution obta ined as vanishing viscosity limit is unique and

can be completely characterized. To conclude the proof, we then rely on the same

Hyperbolic Systems of Conservation Laws 175

general argument a s in [B2]: if two Lipschitz semigroups S, S

′

provide the same

solutions to all Riemann problems, then they must coincide. See [BB] for details.

6. Concluding remarks

1. A classical tool in the a nalysis of ﬁrst order hyperbolic systems is the method of

characteristics. To study the system

u

t

+ A(u)u

x

= 0 ,

one decomposes the solution along the e igenspaces of the matrix A(u). The evo-

lution of these components is then described by a family of O.D.E’s along the

characteristic curves. In the t-x plane, these are the curves which satisfy dx/dt =

λ

i

u(t, x)

. The lo c al dec omposition (5.16)–(5.17 ) in terms of viscous travelling

waves makes it possible to implement this “hyperbolic” approach also in connection

with the parabolic system (5.10). In this case, the projections are taken along the

vectors ˜r

i

, while the characteristic curves are deﬁned as dx/dt = σ

i

, where σ

i

is the

sp e e d of the i-th travelling wave. Notice that in the hyperbolic case the projections

and the wave speeds depend only on the sta te u , through the eigenvectors r

i

(u) and

the eigenvalues λ

i

(u) of the matrix A(u). On the o ther hand, in the parabolic case

the construction involves the derivatives u

x

, u

xx

as well.

2. In nearly all previous works on BV solutions for systems of conservation laws,

following [G] the basic estimates on the total variation were obtained by a careful

study of the Riemann problem and of elementary wave interactions. The Riemann

problem also takes the center stage in all earlier proofs of the stability of solutions

[BC1], [BCP], [BLY]. In this connection, the hypothesis (H) introduced by Lax [Lx 1]

is widely adopted in the literature. It guarantees that solutions of the Riemann

problem have a simple structure, co nsisting of at most n elementary waves (shocks,

centered rarefactions or contact discontinuities). If the assumption (H) is dropped,

some res ults on global existence [L3], and continuous dependence [AM] are still

available, but their proofs become far more technical. On the other hand, the

approach introduced in [BB ] marks the ﬁrst time where uniform BV estimates are

obtained without a ny reference to Riemann problems. Global exis tence and stability

of weak solutions are obtained for the whole class of strictly hyperbolic s ystems,

regardless of the hypothesis (H).

3. For the viscous system of conservation laws

u

t

+ f (u)

x

= u

xx

,

previous re sults in [L4], [SX], [SZ], [Yu] have established the stability of special types

of solutions, for example travelling viscous shocks or viscous r arefactions. Taking

ε = 1 in (5.2), from Theorem 5 we obtain the uniform Lipschitz stability (w..r.t. the

L

1

distance) of ALL viscous solutions with suﬃciently small total variation. An

176 A. Bressan

interesting alternative technique for proving stability of visc ous solutions, based on

sp e c tral methods, was recently developed in [HZ].

4. In the present survey we only considered initial data with small total variation.

This is a convenient setting, adopted in much of the curre nt literature, which guar-

antees the global existence of BV s olutions of (1.1) and captures the main features

of the problem. A rece nt example cons tructed by Jenssen [J] shows that, for initial

data with large total variation, the L

∞

norm of the solution can blow up in ﬁnite

time. In this more general setting, o ne expects tha t the existence and uniqueness

of weak solutions, together with the convergence of vanishing viscosity approxima-

tions, should hold locally in time as long as the total variation remains bounded.

For the hyperbolic system (1.1), results on the local existence and stability of solu-

tions with large BV data can be found in [Sc] and [BC2], re spectively. Because of

the counterexample in [BS], on the other hand, similar well posedness results are

not expected in the general L

∞

case.

References

[AM] F. Ancona and A. Marson, Well p osedness for general 2 × 2 systems of

conservatio n laws, Amer. Math. Soc. Memoir, to appear.

[BaJ] P. Baiti and H. K. Jens sen, On the front tracking algorithm, J. Math. Anal.

Appl. 217 (1998), 395–404.

[BB] S. Bianchini and A. Bressan, Vanishing viscosity so lutions of nonlinear hy-

perbolic systems, preprint S.I.S.S.A., Trieste 2001.

[B1] A. Bressan, Global solutions to systems of conservation laws by wave-front

tracking, J. Math. Anal. Appl. 170 (1992), 414–432.

[B2] A. Bressan, The unique limit of the Glimm scheme, Arch. Rational Mech.

Anal. 130 (1995), 205–230.

[B3] A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimen-

sional Cauchy Problem. Oxfo rd University Pre ss, 200 0.

[BC1] A. Bressan and R. M. Colombo, The semigroup generated by 2 × 2 conser-

vation laws, Arch. Rational Mech. Anal. 133 (1995), 1–75.

[BC2] A. Bressan and R. M. Colombo, Unique solutions of 2 ×2 conservation laws

with large da ta, Indiana Univ. Math. J. 44 (1995), 677–725.

[BCP] A. Bressan, G. Crasta and B. Piccoli, Well posednes s of the Cauchy problem

for n × n cons e rvation laws, Amer. Math. Soc. Memoir 694 (2000).

[BG] A. Bressan and P. Goatin, Oleinik type e stimates and uniqueness for n ×n

conservatio n laws, J. Diﬀ. Equat. 156 (1999), 2 6–49.

[BLF] A. Bressan and P. LeFloch, Uniqueness of weak solutions to systems of

conservatio n laws, Arch. Rat. Mech. Anal. 140 (19 97), 301–317.

[BLe] A. Bressa n and M. Lew icka, A uniqueness condition for hyperbolic systems

of conservation laws, Discr. Cont. Dynam. Syst. 6 (2000), 67 3–682.

[BLY] A. Bressan, T. P. Liu and T. Yang, L

1

stability estimates for n ×n conser-

vation laws, Arch. Rational Mech. Anal. 149 (1999), 1–22.

Hyperbolic Systems of Conservation Laws 177

[BM] A. Bressan and A. Mars on, Error bounds for a deterministic version of the

Glimm scheme, Arch. Rat. Mech. Anal. 142 (1998), 155–176.

[BS] A. Bressan and W. Shen, Uniqueness for discontinuous O.D.E. and cons e r-

vation laws, Nonlinear Analysis, T. M. A. 34 (1998), 637–652.

[C] M. Crandall, The semigroup a pproach to ﬁrst-order quasilinear equations

in several space variables, Israel J. Math. 12 (1972), 108–132.

[D1] C . Dafermos, Polygonal approximations of solutions of the initial value prob-

lem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33–41.

[D2] C . Dafermos, Hyperbolic systems of conservation laws, Proceedings of the

International Congress of Mathematicians, Z¨urich 1994, Birch´auser (1995),

1096–1107.

[D3] C . Dafermos , Hyperbolic Conservation Laws in Continuum Physics, Springer-

Verlag, Berlin 2000.

[DP1] R. DiPerna, Global existence of solutions to nonlinear hype rbolic systems

of conservation laws, J. Diﬀ. Equat. 20 (1976), 187–212.

[DP2] R. DiPerna, Convergence of approximate solutions to conservation laws,

Arch. Rational Mech. Anal. 82 (1983), 27–70.

[G] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equa-

tions, Comm. Pure Appl. Math. 18 (1965), 697–715.

[GL] J. Glimm and P. L ax, Decay of solutions o f systems of nonlinear hyperbolic

conservatio n laws, Amer. Math. Soc. Memoir 101 (1970).

[GX] J. Goodman and Z . Xin, Viscous limits for piecewise smooth solutions to

systems of conservation laws, Arch. Rational Mech. Anal. 121 (1992),

235–265.

[HR] H. Holden and N. H. Risebro Front Tracking for Hyperbolic Conservation

Laws, Springer Verlag , New York 2002.

[HZ] P. Howard and K. Zumbrun, Pointwise semigroup methods for stability of

viscous shock waves, Indiana Univ. Math. J. 47 (1998), 727–841.

[K] S. Kruzhkov, First order quasilinear equations with several space variables,

Math. USSR Sbornik 10 (1970), 217–243.

[J] H. K . Jenssen, Blowup for systems of conservatio n laws, SIAM J. Math.

Anal. 31 (2000), 894–908.

[Lx1] P. Lax, Hyp e rbolic sy stems of conservation laws II, Comm. Pure Appl.

Math. 10 (19 57), 537–566.

[Lx2] P. Lax, Problems solved and unsolved conce rning nonlinear P.D.E., Proc-

cedings of the International Congress of Mathematicians, Warszawa 1983.

Elsevier Science Pub. (1984), 119–138.

[L1] T. P. Liu, The deterministic version of the Glimm scheme, Comm. Math.

Phys. 57 (1977), 135–148.

[L2] T. P. Liu, Linear and nonlinear large time behavior of solutions of general

systems of hype rbolic conservation laws, Comm. Pure Appl. Math. 30

(1977), 767–796.

178 A. Bressan

[L3] T. P. Liu, Admissible solutions of hyperbolic conservation laws, Amer.

Math. Soc. Memoir 240 (1981).

[L4] T. P. Liu, Nonlinear stability of shock waves, Amer. Math. Soc. Memoir

328 (1986).

[LY] T. P. Liu and T. Yang, L

1

stability for 2 × 2 systems of hyper bolic conser-

vation laws, J. Amer. Math. Soc. 12 (1999), 729–774.

[M] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in

Several Space Variables, Springer -Verlag, New York, 1984.

[O] O. Oleinik, Discontinuous solutions of nonlinear diﬀerential e quations (1957),

Amer. Math. Soc. Translations 26, 95–172.

[R] N. H. Risebro, A front-tracking alternative to the random choice method,

Proc. Amer. Math. Soc. 117 (1993), 1125–1139.

[Sc] S. Schochet, Suﬃcient conditions for local existence via Glimm’s scheme fo r

large BV data, J. Diﬀerential Equations 89 (1991), 317–354.

[S1] D. Serre, Systems of Conservation Laws I, II, C ambridge University Press ,

2000.

[S2] D. Serre, Systems of conse rvation laws : A challenge for the XXIst century,

Mathematics Unlimited - 2001 and beyond, B. Engquist and W. Schmid

eds., Springer-Verlag, 2001.

[SX] A. Szepessy and Z. Xin, Nonlinear s tability abd viscous shocks, Arch. Ra-

tional Mech. Anal. 122 (1993), 53–103.

[SZ] A. Sze pess y and K. Zumbrun, Stability of r arefaction waves in v iscous me-

dia, Arch. Rational Mech. Anal. 133 (1996), 249–298.

[Yu] S. H. Yu, Zero-dissipation limit of solutions with shocks for s ystems of hyper-

bolic co ns ervation laws, Arch. Rational Mech. Anal. 146 (1999), 275–370.