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arXiv:math/0108224v1 [math.OC] 31 Aug 2001
On the Boundary Control of
Systems of Conservation Laws
Alberto Bressan and Giuseppe Maria Coclite
S.I.S.S.A., Via Beirut 4, Trieste 34014 Italy.
e-mail: bressan@sissa.it, coclite @sissa.it
Abstract. The pape r is concern ed with the bou ndary controllability of entropy weak solutions to
hype rbolic systems of conser vation laws. We prove a general result on the asymptotic stabilization
of a system near a constant state. On the oth er hand, we give an example showing that exact
controllability in finite time cannot be achieved, in general.
0
1 - Introduction
Consider an n × n system of c onservation laws on a bounde d interval:
u
t
+ f(u)
x
= 0 t ≥ 0, x ∈ ]a, b[ . (1.1)
The system is assumed to be strictly hyperbo lic, each charac teristic field being either linearly
degene rate or genuinely nonlinear in the sense of Lax [8]. We shall also assume that all characteristic
speed s are bounded away from zero. More precisely, let f : Ω 7→ IR
n
be a smooth map, defined on
an open set Ω ⊆ IR
n
. For each u ∈ Ω, call λ
1
(u) < · · · < λ
n
(u) the eigenvalues of the Jacobian
matrix Df (u). We assume that ther e exists a minimum speed c
0
> 0 and an inte ger p ∈ {1, . . . , n}
such that
λ
i
(u) < 0 if i ≤ p,
λ
i
(u) > 0 if i > p,
(1.2)
λ
i
(u)
≥ c
0
> 0 u ∈ Ω. (1.3)
By (1.2), for a solution defined on the strip t ≥ 0, x ∈ ]a, b[ , there will be n − p characteristics
entering at the boundary point x = a, and p characteristics entering at x = b. The initial-
bound ary value problem is thus well posed if we prescribe n − p scalar co nditions at x = a and p
scalar conditions at x = b [11]. See also [1, 2] for the case of general entropy-weak solution s taking
values in the space BV of functions with bounded variation.
In the present paper we study the e ffe ct of bou ndary conditions on the solution of (1.1) from
the point of view of control theory. Name ly, given an initial condition
u(0, x) = φ(x) x ∈ ]a, b[ (1.4)
with sma ll total variation, we regard the boundary data as control functions, and study the family
of configura tions
R(T )
.
=
u(T, ·)
⊂ L
1
[a, b] ; IR
n
(1.5)
which can be r eached by the syst em at a given time T > 0.
Beginning with the simplest case, c onsider a strictly hyperbolic system wit h constant coeffi-
cients:
u
t
+ Au
x
= 0, (1.6)
where A is a n × n constant matr ix, with real distinct eigenvalues
λ
1
< · · · < λ
p
< 0 < λ
p+1
< · · · < λ
n
.
Call
τ
.
= max
i
b − a
|λ
i
|
the maximum time taken by waves to cross the interval [a, b]. In t his case, it is easy to see that
the reachable se t in (1.5) is the entire space: R(T ) = L
1
for all T ≥ τ . In other words, the
system is completely controllable after time τ. Inde ed, for a ny T ≥ τ and initial and terminal data
φ, ψ ∈ L
1
[a, b]; IR
n
, one can always find a solution of (1.4), defined on the recta ngle [0, T ] × [a, b]
such that
u(0, x) = φ(x), u(T, x) = ψ(x) x ∈ [a, b].
1
Such solution can be const ructed as follows. Let l
1
, . . . , l
n
and r
1
, . . . , r
n
be dual bases of right a nd
left eigenvectors of A so that l
i
· r
j
= δ
ij
. For i = 1, . . . , n, let u
i
(t, x) be a so lution to the scalar
Cauchy problem
u
i,t
+ λ
i
u
i,x
= 0,
u
i
(0, x) =
(
l
i
· φ(x) if x ∈ [a, b],
l
i
· ψ(x + λ
i
T ) if x ∈ [a − λ
i
T, b − λ
i
T ],
0 otherwise.
Then the restriction of
u(t, x) =
X
i
u
i
(t, x)r
i
to th e interva l [0, T ] × [a, b] satisfies (1.6) and takes the required initial and terminal values. Of
course, this corresponds to the so lution of an initial-boundary value problem, d etermined by the
n boundary conditions
(
l
i
· u(t, a) = u
i
(t, a) i = p + 1, . . . , n,
l
i
· u(t, b) = u
i
(t, b) i = 1, . . . , p.
This re sult on exact boundar y co ntrollability ha s be en extended in [9, 1 0] to the case of general
quasilinear systems of the form
u
t
+ A(u)u
x
= 0.
In this case, the existence of a solution taking the prescribed initial and terminal values is obtained
for all suffic ie ntly small data φ, ψ ∈ C
1
.
Aim of the present paper is to study analogous controllability properties within the context of
entropy weak solutions t 7→ u(t, ·) ∈ BV . For the definitions and basic properties of weak solutions
we refe r to [4]. For general nonlinear systems, it is clear that a complete controllability result
within the space BV ca nnot hold. Indeed, already fo r a scalar conservation law, it was proved in
[3] that the profiles ψ ∈ BV which can be attained at a fixed time T > 0 are only those which
satisfy the Oleinik-type conditions
ψ
′
(x) ≤
f
′
ψ(x)
(x − a)f
′′
ψ(x)
for a.e. x ∈ [a, b].
For gen eral n × n systems, a complete characterization of the reachable set R(T ) does not
seem possible, due to the complexity of r epeated wave-front interactions.
Our first result is concerned with stabilization near a consta nt state. Assuming that all
characteristic speeds are bounded away from zero, we show that the system can be asymptotically
stabilized to any state u
∗
∈ Ω, with quadratic rate of convergence.
Theorem 1. Let K be a compact, connected subset of the open domain Ω ⊂ IR
n
. Then there exist
constants C
0
, δ, κ > 0 such that the following holds. For every constant state u
∗
∈ K and every
initial data u(0) = φ : [a, b] 7→ K w ith Tot.Var.{φ} < δ, there exists an entropy weak solution
u = u( t, x) of (1.1) such that, for all t > 0,
Tot.Var .
u(t)
≤ C
0
e
−2
κt
, (1.7)
u(t, x) − u
∗
L
∞
≤ C
0
e
−2
κt
. (1.8)
2
The proof will be given in Section 2. An interesting question is whether the constant state
u
∗
can be exactly reached, in a finite time T . By the results in [9], this is indeed the case if the
initial data has sma ll C
1
norm. On the contrary, in the final part of this paper, we show that exact
controllability in finite time cannot be attained in general, if the initial data is only assumed to be
small in BV .
Our counterexample is concerned with a class of strictly hyperbolic, genuinely nonlinear 2 × 2
systems of t he form (1.1). More precisely, we assume
(H) The eigenvalues λ
i
(u) of the Jacobian matrix A(u) = Df(u) satisfy
−λ
∗
< λ
1
(u) < −λ
∗
< 0 < λ
∗
< λ
2
(u) < λ
∗
. (1.9)
Moreover, the right eigenvectors r
1
(u), r
2
(u) satisfy th e inequalities
Dλ
1
· r
1
> 0, Dλ
2
· r
2
> 0, (1.10)
r
1
∧ r
2
< 0, r
1
∧ (Dr
1
· r
1
) < 0, r
2
∧ (Dr
2
· r
2
) < 0. (1.11)
A partucular system which satisfies the above assumptions is the one stud ie d by DiPerna [7]:
ρ
t
+ (uρ)
x
= 0 ,
u
t
+
u
2
2
+
K
2
γ − 1
ρ
γ−1
x
= 0 ,
with 1 < γ < 3. Here ρ > 0 and u denote the density and the velocity of a gas, respectively.
The last two inequalities in (1.11) imply that the rarefact ion cu rves (i.e. the integral curves of
the vector fields r
1
, r
2
) in the (u
1
, u
2
) plane turn clockwise (fig. 1). In such case, the interaction of
two shocks of the same family generate s a shock in the other family.
figure 1
3
Theorem 2. Consider a 2 × 2 system satisfying the assumption (H). Then there exist initial data
φ : [a, b] 7→ IR
2
having arbitrarily small total bounded variation for which the fo llowing holds. For
every entropy weak solution u of (1.1), (1.4), with Tot.Var.
u(t, ·)
remaining small for all t, the
set of shocks in u(t, ·) is dense on [a, b], for each t > 0. In particular, u(t, ·) cannot be constant.
As a preliminary, in Section 3 we establish an Oleinik-type estimate on the decay of positive
waves. This bound is of independent intere st, and sharpens the results in [5], for systems satisfying
the additional conditions (H).
As a con sequence, this implies that positive waves are “weak”, and cannot co mpletely cancel
a shock within finite time. The p roof of Theorem 2 is then achieved by an induction ar gument. We
show that, if the set of 1-shocks is d ense on [0, T ] × [a, b], t hen the set of points P
j
= ( t
j
, x
j
) where
two 1-shocks interact and create a new 2-shock is also dense on the sa me domain. Therefore, new
shocks are consta ntly generated, and t he solution can ne ver be reduce d to a constant. Details of
the pr oof will be given in Section 4.
As in [9], all of the above results refer to the case where total control on the bounda ry values
is available. As a con sequence, the pro blem is reduced to proving the existence (or nonexistence)
of a n entropy weak solution defined on the o pen strip t > 0, x ∈ ]a, b[ , satisfying the required
conditions. This is a first step toward the analysis of more general controllability problems, where
the c ontrol acts only on some o f the boundary conditions. We thus leave open the case where a
subset o f indices I ⊂ {1, . . . , n} is given, and one requires
l
i
· u(t, a) =
α
i
(t) if i ∈ I,
0 if i /∈ I,
i = p + 1, . . . , n,
l
i
· u(t, b) =
α
i
(t) if i ∈ I,
0 if i /∈ I,
i = 1, . . . , p,
for some control functions α
i
acting only on the components i ∈ I.
Throughout the fo llowing, we denote by r
i
(u), l
i
(u) the right and left i-eigenvectors of the
Jacobian matrix A(u)
.
= Df (u). As in [4], we write σ 7→ R
i
(σ)(u
0
) for the parametrize d i-
rarefaction curve through the sta te u
0
, so that
d
dσ
R
i
(σ) = r
i
R
i
(σ)
, R
i
(0) = u
0
.
The i-shock curve throug h u
0
is denoted by σ 7→ S
i
(σ)(u
0
). It satisfies the Rankine-Hugoniot
equations
f
S
i
(σ)
− f(u
0
) = λ
i
(σ)
S
i
(σ) − u
0
for some shock speed λ
i
. We recall (see [4], Chapter 5) that the general Riemann problem is so lved
in terms of the composite curves
Ψ
i
(u
0
)(σ) =
R
i
(u
0
)(σ), if σ ≥ 0,
S
i
(u
0
)(σ), if σ < 0.
(1.12)
4
2 - Proof of Theorem 1
The proof re lies on the two following two lemmas.
Lemma 1. In the setting of Theorem 1, there exists a time T > 0 such that the following holds.
For every pair of states ω, ω
′
∈ K there exists an entropic solution u = u(t, x) of (1.1) such that
u(0, x) ≡ ω, u(T, x) ≡ ω
′
for all x ∈ [a, b]. (2.1)
Proof. Consider the function
Φ(σ
1
, . . . , σ
n
; v, v
′
)
.
= Ψ
n
(σ
n
) ◦ · · · ◦ Ψ
p+1
(σ
p+1
)(v
′
) − Ψ
p
(σ
p
) ◦ · · · ◦ Ψ
1
(σ
1
)(v). (2.2)
Observe that, whenever v = v
′
, the n × n Jacobian matrix ∂Φ/∂σ
1
· · · σ
n
computed at σ
1
=
σ
2
= · · · = σ
n
= 0 has full r ank. Inde ed, the columns of this matrix are given by the linearly
independent vectors −r
1
(v), . . . , −r
p
(v), r
p+1
(v), . . . , r
n
(v). By the Implicit Function Theorem
and a compactness argument we can find δ > 0 such that the following holds. For every v, v
′
∈ K,
with |v − v
′
| ≤ δ, there exist unique values σ
1
, . . . , σ
n
such that
v
′′
.
= Ψ
n
(σ
n
) ◦ · · · ◦ Ψ
p+1
(σ
p+1
)(v
′
) = Ψ
p
(σ
p
) ◦ · · · ◦ Ψ
1
(σ
1
)(v) . (2.3)
Defining the time
τ
.
= max
1≤i≤n
sup
u∈Ω
b − a
λ
i
(u)
, (2.4)
we claim that there exists an entro py weak solution u : [0, 2τ] × [a, b] 7→ Ω such that
u(0, x) ≡ v, u(2τ, x) ≡ v
′
. (2.5)
figure 2
5
The function u is constructed as follows (fig. 2). For t ∈ [0, τ] we let u be the so lution of the
Riemann problem
u(0, x) =
n
v if x < b,
v
′′
if x > b.
(2.6)
Moreover, for t ∈ [τ, 2τ], we define u as t he solution of the Riemann problem
u(τ, x) =
n
v
′
if x < a,
v
′′
if x > a.
(2.7)
It is now clear that the restriction of u to the domain [0, 2τ] × [a, b] satisfies the conditions (2.5).
Indeed, by (2.3), on [0, τ ] the solution u conta ins only waves of families ≤ p, originating at the
point (0, b). By (2.4) these waves cross the whole interva l [a, b] and exit from the boundary point
a before time τ. Hence u(τ, x) ≡ v
′′
. Similarly, still by (2.3), for t ∈ [τ, 2τ] the function u contains
only waves of families ≥ p + 1, originating at the point (τ, a). By (2.4) these waves cross the whole
interval [a, b] and exit from the boundary point b before time 2τ . Hence u(2τ, x) ≡ v
′
.
Next, given any two states ω, ω
′
∈ K, by the connectedness a ssumption we can find a chain of
points ω
0
= ω, ω
1
, . . . , ω
N
= ω
′
in K such that |ω
i
−ω
i−1
| < δ for every i = 1, . . . , N. Repeating the
previous construction in connection with each pair of states (ω
i−1
, ω
i
), we thus obtain an entropy
weak solution u : [0, 2Nτ ] × [a, b] 7→ Ω that satisfies the conclusion of th e lemma, with T = 2Nτ .
In the following, we shall construct the desired solution u = u(t, x) as limit of a seque nce of
front tracking approximations. Roughly speaking, an ε-approximate front tracking solution is a
piecewise c onstant function u
ε
, having jumps along a finite set of straight lines in the t-x plane say
x = x
α
(t), w hich approximately satisfies the Ra nkine-Hugoniot equations:
X
α
f
u(t, x
α
+)
− f (u(t, x
α
−)
− ˙x
α
u(t, x
α
+) − u(t, x
α
−)
< ε
for all t > 0. For details, see [4], p.125.
Lemma 2. In the setting of Theorem 1, for every state u
∗
∈ Ω there exist constants C, δ
0
> 0
for which the following holds. For any ε > 0 and every piecewise constant function ¯u : [a, b] 7→ Ω
such tha t
ρ
.
= sup
x∈[a,b]
¯u(x) − u
∗
≤ δ
0
, δ
.
= Tot.Var.{¯u} ≤ δ
0
, (2.8)
there exists an ε-approximate front tra cking solution u = u(t, x) of (1.1), with u(0, x) = ¯u(x), such
that
sup
x∈[a,b]
u(3τ, x) − u
∗
≤ Cδ
2
, Tot.Var.
u(3τ)
≤ Cδ
2
. (2.9)
Proof. On the doma in (t, x) ∈ [0, τ ] × [a, b], we construct u as an ε-approximate front tracking
solution in such a way t hat, whenever a front hits one of the bou ndaries x = a or x = b, no reflected
front is ever create d (fig. 3). Since all fronts emerging from the initial data ¯u at time t = 0 exit
from [a, b] within time τ, it is clear that u(τ ) c an contain only fronts of second or higher generation
order. In other words, the only fro nts t hat can be prese nt in u(τ, ·) are the new ones, generated
by intera ctions at times t > 0 (the dotted lines in fig. 3). Therefore , using the interaction estimate
(7.69) in [4] we obtain
sup
x∈[a,b]
u(τ, x) − u
∗
= O(1) · (ρ + δ) Tot.Var.
u(τ)
= O(1) · δ
2
. (2.10)
6
figure 3
We now apply a similar proced ure as in th e proof of Lemma 1, and construct a solution on the
interval [τ, 3τ] in such a way that u(3τ ) ≈ u
∗
. More precisely, to construc t u on the domain
[τ, 2τ] × [a, b], consider the state v
′′
implicitly defined by (2.2), with v
.
= u(τ, b−), v
′
.
= u
∗
. On
a forward neighborhood of the point (τ, b) we let u coincide with (a front-tracking a pproximation
of) the so lution to the Riemann problem
u(τ, x) =
n
u(τ, b−) if x < b,
v
′′
if x > b.
This pro cedure will introduce at the point (τ, b) a family of wave-fronts of families i = 1, . . . , p,
whose total strength is O(1) · (ρ + δ). Because of (2.4), all these fr onts will exit from the boundar y
x = a within t ime 2τ. Of course, they can interact wit h the other fronts present in u(τ, ·). In any
case, the to tal strength of fronts in u(2τ, ·) is still estimated as
Tot.Var .
u(2τ)
= O(1) · δ
2
. (2.11)
Next, to define u for t ∈ [2τ, 3τ ], consider the state v
′′′
implicitly de fined by
(
u(2τ, a+) = Ψ
n
(σ
n
) ◦ · · · ◦ Ψ
p+1
(σ
p+1
)(v
′′′
),
u
∗
= Ψ
p
(σ
p
) ◦ · · · ◦ Ψ
1
(σ
1
)(v
′′′
).
(2.12)
7
On a forward neighborh ood of the po int (2τ, a) we let u coincide with (a front-tracking approxi-
mation of) the solution to the Riemann problem
u(2τ, x) =
n
u(2τ, a+) if x > a,
v
′′′
if x < a.
This procedure introduces at the point (2τ, a) a family of wave-fronts of families i = p + 1, . . . , n,
whose total strength is O(1) · (ρ + δ). Because of (2.4), all these fr onts will exit from the boundar y
x = b within time 3τ. Of co urse, they can interact with the other fronts present in u(2τ, ·) . In any
case, the to tal strength of fronts in u(3τ, ·) is still estimated as
Tot.Var .
u(3τ)
= O(1) · δ
2
. (2.13)
Moreover, the d ifference between the values u( 3τ, x) and u
∗
will be of the same order of the total
strength of waves in u(τ, ·), so that the first inequality in (2.9) will also hold.
Proof of Theorem 1. Using the sa me a rguments as in the proof of Lemma 1.1, for every ε > 0
we can construc t an ε-approximate front tracking solution u = u(t, x) on [0, 2Nτ] × [a, b] such
that
sup
x∈[a,b]
u(2Nτ, x) − u
∗
= O(1) · δ, Tot.Var.
u(2Nτ)} = O(1) · δ . (2.14)
Choosing δ > 0 sufficiently small, we can assume that, in (2.14), O(1) · δ < δ
0
< 1/C, the constant
in Lemma 2. Calling T
.
= 2N τ, we can now repeat the c onstruction described in Lemma 2 on each
interval
T + 3kτ, T + 3(k + 1)τ
. This yields
sup
x∈[a,b]
u(T + 3kτ, x) − u
∗
≤ δ
k
, Tot.Var.
u(T + 3kτ)
≤ δ
k
, (2.15)
where the constants δ
k
satisfy the inductive relations
δ
k+1
≤ Cδ
2
k
. (2.16)
Choosing a sequence of ε-approximate front tracking solution s u
ε
satisfying (2.15 )-(2.16) and tak-
ing the limit as ε → 0, we obtain an entropy weak solution u which still satisfies the same estimates.
The bounds (1.7)-(1.8) are now a consequence of (2.15)-(2.16), with a suitable choice of the con-
stants C
0
, κ.
3 - Decay of positive waves
Throughout the fo llowing, we consider a 2 × 2 system of conservation laws
u
t
+ f (u)
x
= 0 , (3.1)
satisfying the a ssumptions (H). Following [6], p. 128, we construct a se t of Riemann coordinates
(w
1
, w
2
) . One can then choose the right eigenve ctors of Df(u) so that
r
i
(u) =
∂u
∂w
i
,
∂λ
i
∂w
i
= Dλ
i
· r
i
> 0 i = 1, 2. (3.2)
8
It will be convenient to perform mo st of the analysis on a special class of solutions: piecewise
Lipschitz functions with finitely many shocks and no compr ession waves. Due to the geometric
structu re of the system, this set of functions turns out to be positively invariant for the flow
generated by the hyperbolic system. We first derive several a prior i estimate concerning these
solutions, in particular on the strength and location of the shocks. We then observe that any BV
solution can be ob tained as limit of a sequence of piecewise Lipschitz solut ions in our special class.
Our estimates c an thus be extended to gen eral BV solutions.
Definition 1. We call U the set of all piecewise Lipschitz functions u : IR 7→ IR
2
with finitely
many jum ps, such that:
(i) at every jump, the corresponding Riemann problem is solved o nly in terms o f shocks (no
centered rarefactions);
(ii) no compression waves are present, i.e.: w
i,x
(x) ≥ 0 at almost every x ∈ IR, i = 1, 2 .
The next lemma establishes the forward inva riance of the set U.
Lemma 3. Consider the 2 × 2 system of conservation laws (3.1), satisfying the assumptions
(H). Let u = u(t, x) be the solution to a Cauchy problem, with small total variation, satisfying
u(0, ·) ∈ U. Then
u(t, ·) ∈ U for all t ≥ 0. (3.3)
Proof. We have to show that, as time progresses, the total number of shocks does not increase
and no compression wave is ever formed . This will be the case provided that
(i) The interaction of two shocks of the same family produ ces an outgoing shock of the other
family.
(ii) The interaction of a shock with an infinitesimal rarefaction wave of the same family produces
a rarefact ion wave in the other family.
Both of the above co nditions ca n be easily checked by analysing the relative positions of shocks
and rarefaction curves. We will do this for the first family, leaving the verification of the other case
to the reader.
Call σ 7→ R
1
(σ) the rarefaction curve through a st ate u
0
,parametrized so that
λ
1
R
1
(σ)
= λ
1
(u
0
) + σ .
It is well known that the shock curve through u
0
has a second order tangency with this rarefaction
curve. Hence there exists a smooth func tion c
1
(σ) such that the point
S
1
(σ)
.
= R
1
(σ) + c
1
(σ)
σ
3
6
r
2
(u
0
)
lies on this shock curve, for all σ in a neighborhood of zero. From the Rankine-Hugoniot equations
it now fo llows
χ(σ)
.
=
f
R
1
(σ) + c
1
(σ)(σ
3
/6) r
2
(u
0
)
− f(u
0
)
∧
R
1
(σ) + c
1
(σ)(σ
3
/6) r
2
(u
0
) − u
0
= 0 . (3.4)
9
Differentiating the wedge product (3.4) four times at σ = 0 and de noting derivatives with upper
dots, we obtain
d
4
χ
dσ
4
(0) = 4
λ
1
(u
0
)
...
R
1
(0) + 2
¨
R
1
(0) + λ
2
(u
0
) c
1
(0)r
2
(u
0
)
∧
˙
R
1
(0)
+ 6
λ
1
(u
0
)
¨
R
1
(0) +
˙
R
1
(0)
∧
¨
R
1
(0) + 4 λ
1
(u
0
)
˙
R
1
(0) ∧
...
R
1
(0) + c(0)r
2
(u
0
)
= 4
λ
2
(u
0
) − λ
1
(u
0
)
c
1
(0) r
2
(u
0
) ∧ r
1
(u
0
) + 2(Dr
1
· r
1
)(u
0
) ∧ r
1
(u
0
)
= 0 .
Hence
c
1
(0) =
(Dr
1
· r
1
) ∧ r
1
2(λ
2
− λ
1
)(r
1
∧ r
2
)
< 0 . (3.5)
By (3.5), the relative position of 1-shock and 1-rarefaction curves is as depicted in fig. 1. By the
geometr y of wave curves, the properties (i) and (ii) are now clear. Figure 4a illustrate s the inter-
action of two 1-shocks, while fig. 4b shows the interaction between a 1-shock and a 1-rarefaction.
By u
l
, u
m
, u
r
we denote the left, middle and right states before t he interaction, while u
′
m
is the
middle state after the interaction. In the two cases, the solution of the Riemann problem contains
a 2 -shock and a 2-rare f action, respectively.
figure 4a figure 4b
The next lemma shows t he decay of positive waves for solutions with small total variation,
taking values inside U.
Lemma 4. Let u = u(t, x) be a solution of the Cauchy problem for the 2 × 2 system (3.1)
satisfying (H). Assume that
u(t, ·) ∈ U t ≥ 0. (3.6)
Then there exist κ, δ > 0 such that if Tot.Va r.(u(t, ·)) < δ for a ll t, then its Riemann coordinates
(w
1
, w
2
) satisfy
0 ≤ w
i,x
(t, x) ≤
κ
t
, t > 0, i = 1, 2. (3.7)
Proof. We consider th e case i = 1. Fix any point (
¯
t, ¯x) . Since centered rarefac tion waves
are not present, there exists a unique 1-charac teristic through this point, which we denote as
t 7→ x
1
(t;
¯
t, ¯x). It is the solution of t he Cauchy problem
˙x(t) = λ
1
u(t, x(t))
, x(
¯
t) = ¯x. (3.8)
10
The evolution of w
1,x
along this characteristic is described by
d
dt
w
1,x
t, x
1
(t)
= w
1,xt
+ λ
1
w
1,xx
= −(λ
1
w
1,x
)
x
+ λ
1
w
1,xx
= −
∂λ
1
∂w
1
w
2
1,x
−
∂λ
1
∂ω
2
w
1,x
w
2,x
.
Since the system is genuinely nonlinear th ere exists k
1
> 0 su ch that ∂λ
1
/∂w
1
≥ k
1
> 0, h ence
d
dt
w
1,x
t, x
1
(t)
≤ −k
1
w
2
1,x
+ O(1) · w
1,x
w
2,x
. (3.9)
Moreover, at each time t
α
where the characteristic crosses a 2-shock of strength |σ
α
| we have the
estimate
w
1,x
(t
α
+) ≤
1 + O(1) · |σ
α
|
w
1,x
(t
α
−). (3.10)
Let Q(t) be the total interaction potential at time t (see for example [4], p. 202 ) and let V
2
(t) be
the total amount of 2-waves approaching our 1-wave located at x
1
(t). Re peating the arguments in
[4], p.139, we ca n find a constant C
0
> 0 such that the quantity
Υ(t)
.
= V
1
(t) + C
0
Q(t), t > 0,
is non-increasing. Moreover, for a.e. t one has
˙
Υ(t) ≤ −
λ
2
− λ
1
|w
2,x
|
t, x
1
(t)
,
while at times t
α
where x
1
crosses a 2-shock of st rength |σ
α
| there holds
Υ(t
α
−) ≤ Υ(t
α
+) − |σ
α
| .
Call W (t)
.
= w
1,x
t, x
1
(t)
. By the previous estimat es, from (3.9) an d (3.10) it follows form
˙
W (t) ≤ −k
1
W
2
(t) − C
˙
Υ(t)W (t), (3.9)
W (t
α
+) − W (t
α
−) ≤ C
Υ(t
α
+) − Υ(t
α
−)
W (t
α
−), (3.11)
for a suitable constant C. We now observe that
y(t)
.
=
e
−CΥ(t )
Z
t
0
k
1
e
−CΥ(s)
ds
is a distributional solution of t he equation
˙y = −k
1
y
2
− C
˙
Υ(t)y ,
with y(t) → ∞ as t → 0+. A compa rison argument now yields W (t) ≤ y(t). Since Υ is positive
and decreasing, we have
W (t) ≤
¯
W (t) ≤
1
k
1
1
Z
t
0
e
−CΥ(s)
ds
≤
e
CΥ(0)
k
1
t
,
for all t > 0. This establishes (3.7) for i = 1, with κ
.
= e
CΥ(0)
/k
1
. The case i = 2 is identical.
11
We conclude this section by proving a decay estimate for positive waves, valid for ge neral BV
solutions of the system (3.1). For this purpose, we need to recall some definitions introduced in
[5]. See also p . 201 in [4].
Let u : IR 7→ IR
2
have bounded variation. By possibly changing the values of u at countably
many points, we can assume that u is right continuous. The distribut ional deriva tive µ
.
= D
x
u is
a vector measure, which c an be decomposed into a continuous and an atomic part: µ = µ
c
+ µ
a
.
For i = 1, 2, the scalar measures µ
i
= µ
i
c
+ µ
i
a
are defined as follows. The continuous part of µ
i
is
the Radon me asure µ
i
c
such that
Z
φ dµ
i
c
=
Z
φ l
i
(u) · dµ
c
(3.12)
for every scalar continuous function φ with co mpact su pport. The atomic part of µ
i
is the measure
µ
i
a
concentrated o n the countable set {x
α
; α = 1, 2, . . .} where u has a jump, such that
µ
i
a
{x
α
}
= σ
α,i
.
= E
i
u(x
α
−), u(x
α
+)
(3.13)
is th e size of th e i-th wave in the solution of the c orresponding Riemann problem with data u(x
α
±).
We regard µ
i
as the measure of i-waves in the solution u. It can be decomposed in a positive and
a negative part, so th at
µ
i
= µ
i+
− µ
i−
, |µ
i
| = µ
i+
+ µ
i−
. (3.14)
The decay estimate in (3.7) can now be extende d to general BV solutions. Indeed, we show that
the density of positive i-waves decays as κ/t. By meas(J) we denote here the Lebesgue measure
of a set J.
Lemma 5. Let u = u(t, x) be a solution of the Cauch y problem for the 2×2 system (3.1) satisfying
(H). Then there exist κ, δ > 0 such that if Tot.Var.(u(t, ·)) < δ for all t, then the measures µ
1+
t
,
µ
2+
t
of positive waves in u(t, ·) satisfy
µ
i+
t
(J) ≤
κ
t
meas (J) (3.15)
for ever y Borel set J ⊂ IR and every t > 0, i = 1, 2.
Proof. For every BV solution u of (3.1) we can construc t a sequence of solutions u
ν
with u
ν
→ u
as ν → ∞ and such that u
ν
(t, ·) ∈ U for all t. Calling (w
ν
1
, w
ν
2
) the Riemann coordina tes of u
ν
, by
Lemma 4 we have
0 ≤ w
ν
i,x
(t, x) ≤
κ
t
, t > 0 , i = 1, 2, ν ≥ 1 . (3.16)
For a fixed t > 0, observe that the map x 7→ w
ν
1
(t, x) ha s upward jumps precisely at the points x
α
where u(t, ·) has a 2-shock. Define ˜µ
ν
as the positive, p urely atomic measure, conc entrated on the
finitely ma ny points x
α
where u(t, ·) has a 2-shock, such that
˜µ
ν
{x
α
}
= w
ν
1
(t, x
α
+) − w
ν
1
(t, x
α
−) ≤ C |σ
α
|
3
(3.17)
for some constant C. By possibly taking a subsequence, we can assume the existence of a weak limit
˜µ
ν
⇀ ˜µ. Because of the estimate in (3.17), the measure ˜µ is purely atomic, and is concentrated
on the set of points x
β
which are limits as ν → ∞ of a se quence of po ints x
ν
α
where u
ν
(t, ·) has
a 2-shock of uniformly positive strength |σ
ν
| ≥ δ > 0. Therefore, ˜µ is concentrated on the set of
points where the limit solution u(t, ·) has a 2-shock, and makes no contribution to the positive part
of µ
1+
t
. We thus conc lude that the posit ive part of µ
1+
t
is absolutely continuous w.r.t. Lebesgue
measure, with density ≤ κ/t. An analogo us argument holds for µ
2+
t
.
12
Corollary 1. Let u = u(t, x) be a solution of the 2 × 2 system (1.1). Let the assumptions (H)
hold. Fix ε > 0 and consider the subinterval [a
′
, b
′
]
.
= [a +ε, b −ε]. Assume that, at time t = 0, the
measures µ
1+
, µ
2+
of positive waves in u(0, ·) o n [a, b] vanish identically. Then, for every t > 0
one has
µ
i+
t
(J) ≤
κλ
∗
ε
meas (J) (3.18)
for ever y Borel set J ⊂ [a
′
, b
′
] and every t > 0, i = 1, 2.
Indeed, recalling (1.9), the values of u(t, ·) restricted to the interval [a
′
, b
′
] can be obtained by
solving a Cauchy problem, with initial data assigned on t he whole interval [a, b] at time t − ε/λ
∗
.
4 - Proof of Theorem 2
Lemma 6. In the same setting as Lemma 4, assume that there exists κ
′
> 0 such that
0 ≤ w
i,x
(t, x) ≤ κ
′
t ∈ [0, T ], i = 1, 2 . (4.1)
Let t 7→ x(t) be the location of a shock, with strength
σ(t)
. There exists a constant 0 < c < 1
such tha t
σ(t)
≥ c
σ(s)
, 0 ≤ s < t ≤ T . (4.2)
Proof. To fix the ideas, le t u(t, ·) have a 1-shock located at x(t), with strength
σ(t)
. Outside
points of interaction with other shocks, the strength satisfies an inequality of the form
d
dt
σ(t)
≥ −C ·
w
1,x
t, x(t) +
+ w
1,x
t, x(t) −
w
2,x
t, x(t) +
+ w
2,x
t, x(t) −
σ(t)
. (4.3)
At times where o ur 1-shock interacts with other 1-shocks, it s strength incr eases. Moreover, at each
time t
α
where our 1-shock interact s with a 2-shock, say of strength |σ
α
|, one has
σ(t
α
+)
≥
σ(t
α
−)
1 − C
′
|σ
α
|
. (4.4)
for some constant C
′
. Assuming that t he total variation remains small, the total amount of 2-
shocks which cross any given 1-shock is uniformly small. Hence, (4.3)-(4.4) together imply (4.2).
Lemma 7. Let t 7→ u(t, ·) ∈ U be a solution of the Cauchy problem for a genuinely nonlinear
2 × 2 system satisfying (1.11). Assume that there exists κ
′
> 0 such that
w
i,x
(t, x) ≤ κ
′
t ∈ [0, T ], i = 1, 2 . (4.5)
Since no centered rarefactions are present, any two i-characteristics, say x(t) < y(t), can uniquely
be traced ba ckward up to time t = 0. There exists a constant L > 0 such that
y(t) − x(t) ≤ L
y(s) − x(s)
0 ≤ s < t ≤ T . (4.6)
Proof. Consider the case i = 2. By definition, the characteristics are solution s of
˙x(t) = λ
2
u(t, x(t))
, ˙y(t) = λ
2
u(t, y(t))
.
13
Since the characteristic speed λ
2
decreases across 2-shocks, we can write
˙y(t) − ˙x(t) ≤ C
Z
y(t)
x(t)
w
1,x
(t, ξ)
+
w
2,x
(t, ξ)
dξ + C
X
α∈S
1
[x,y]
σ
α
(t)
, (4.7)
where S
1
[x, y] denotes the se t of all 1-shocks located inside th e interval
x(t), y(t)
. Intr oduce the
function
φ(t, x)
.
=
0 if x ≤ x( t),
x−x(t)
y(t)−x(t)
if x(t) < x < y(t),
1 if x ≥ y(t).
Moreover, de fine the functional
Φ(t)
.
=
X
α∈S
1
φ
t, x
α
(t)
σ
α
(t)
+ C
0
Q(t) ,
where the summation now refers to all 1-shocks in u(t, ·) and Q is the usual interaction potential.
Observe that the map t 7→ Φ(t) is non-increasing. By (4.5) and (4.7) we can now write
˙y(t) − ˙x(t) ≤ C
′
1 −
˙
Φ(t)
y(t) − x(t)
for some constant C
′
. This implies (4.6) with L = exp
C
′
T + C
′
Φ(0)
.
The next result is the key ingredient toward the proof of Theorem 2. It p rovides the density
of the set of interaction points where new shocks a re generated.
Lemma 8. Fix ε > 0 and define a
′′
= a + 2ε, b
′′
= b − 2ε. Consider a 2 × 2 system of
the form (1.1), satisfying (H). Let u be an entropy weak solution defined on [0, τ] × [a, b], with
τ
.
= ε/4λ
∗
. Let (3.18) hold for all t ∈ [0, τ ], and assume that u(0, ·) has a dense set of 1-shocks on
the interval [a
′′
, b
′′
]. Then, for 0 ≤ t ≤ τ, the solution u(t, ·) has a set of 1-shocks wh ich is dense
on [a
′′
, b
′
− λ
∗
t] and a set of 2-shocks which is dense on [a
′′
, b
′′
].
Proof. By the assumptions of the lemma, there exists a sequence of piecewise Lipschitz solutions
t 7→ u
ν
(t) ∈ U such that u
ν
→ u in L
1
,
0 ≤ w
ν
i,x
(t, x) ≤
2κλ
∗
ε
i = 1, 2, ν ≥ 1 ,
and moreover the following holds. For every ρ > 0, t here exists δ > 0 such that each u
ν
(0, ·)
(with ν large enough) contains at least one 1-shock of strength
σ
ν
(0)
≥ δ on every subinterval
J ⊂ [a
′′
, b
′′
] having length ≥ ρ.
To prove t he first statement in Lemma 8, fix t ∈ [0, τ ] and c onsider any non-trivial interval
[p, q] ⊆ [a
′′
, b
′′
−tλ
∗
]. Call s 7→ p
ν
(s), s 7→ q
ν
(s) the backward characteristics through these points,
relative to the solution u
ν
. We thus have
(
˙p
ν
(s) = λ
1
u
ν
(s, p
ν
(s))
,
˙q
ν
(s) = λ
1
u
ν
(s, q
ν
(s))
,
(
p
ν
(t) = p,
q
ν
(t) = q.
By Lemma 7, q
ν
(0) − p
ν
(0) ≥ ρ for some ρ > 0 independ ent of ν. Hence, each solution u
ν
contains
a shock of strength
σ
ν
(s)
≥ δ located inside the interval
p
ν
(0), q
ν
(0)
. Lemma 5 now yields
14
σ
ν
(t)
≥ cδ. By possibly ta king a subsequence, we conclude that the limit solution u(t, ·) contains
a 1 -shock of positive strength at the point x(t) = lim x
ν
(t) ∈ [p, q].
To prove the second statement, we will show th at the set of points where two 1-shocks in u
interact and produce a new 2 -shock is dense on the triangle
∆
.
=
(t, x) ; t ∈ [0, τ ], a
′′
< x < b
′′
− λ
∗
t
.
Indeed, let t ∈ [0, τ] and p < q be as before. For each ν sufficiently large, let t 7→ x
ν
(t) be the
location of a 1-shock in u
ν
, with strength
σ
ν
(t)
≥ δ > 0. Assume x
ν
(·) → x(·) as ν → ∞, and
x
ν
(t) ∈ [p, q], so that x(t) is the location of a 1-shock of th e limit solution u, say with strength
σ(t)
> 0.
t
t
’’
x
~
ν
x
ν
z
y
ν
ν
ν
’
t
a
b
’
’
figure 5
We claim that the set of times
ˆ
t where some ot her 1-shock σ
′
impinges on σ and generates
a new 2-shock is dense on [0, t]. To see th is, fix 0 < t
′
< t
′′
< t. For each ν sufficiently large,
consider the ba ckward 1-characteristics y
ν
, z
ν
impinging fr om the left on the shock x
ν
at times
t
′′
, t
′
respectively (fig . 5). These provide solutions to the Cau chy problems
˙y
ν
(t) = λ
1
u
ν
t, y
ν
(t))
, y
ν
(t
′′
) = x
ν
(t
′′
),
˙z
ν
(t) = λ
1
u
ν
(t, z
ν
(t))
, z
ν
(t
′
) = x
ν
(t
′
),
respectively. Observe that
z
ν
(0) − y
ν
(0) ≥ ρ
for some ρ > 0 independent of ν. In deed, the genuine nonlinearity of the system implies
λ
1
u
ν
(t, x
ν
(t)−)
− ˙x
ν
(t) ≥ κ
u
ν
t, x
ν
(t)+)
− u
ν
t, x
ν
(t)−)
≥ κδ.
Therefore,
x
ν
(t
′
) − y
ν
(t
′
) ≥ ρ
′
> 0,
for some constant ρ
′
> 0 independent of ν. By Lemma 6, the inte rval
y
ν
(0), z
ν
(0)
has uniformly
positive len gth. Hence it contains a 1-shock of u
ν
(0, ·) with uniformly positive streng th
σ
ν
(0)
≥
δ > 0. By Lemma 5, every u
ν
has a 1-shock with streng th
σ
ν
(t)
≥ cδ located along some curve
t 7→ ˜x
ν
(t) with
y
ν
(t) < ˜x
ν
(t) < z
ν
(t) t ∈ [0, t
′
] .
Clearly, this second 1-shock impinges on the sh ock x
ν
at some time t
ν
∈ [t
′
, t
′′
], creating a new
2-shock with uniformly large stre ngth. Letting ν → ∞ we obt ain the result.
15
Proof of Theorem 2. Let δ
0
> 0 be given. We can then construct an initial cond it ion u(0, ·) = φ,
with To t.Var.{φ} < δ
0
, having a dense set of 1-shocks on the interval [a, b], and no ot her waves. As
a consequence, for any ε > 0 by Corollary 1 we have the estimate (3.18) on the density of positive
waves away from the boundary.
Fix τ = ε/4λ
∗
, and consider again the subinterval [a
′′
, b
′′
] = [a + 2ε, b − 2ε]. We can
apply Lemma 8 fi rst on the time interval [0, τ ], obtaining the density of 2-shocks on th e region
[0, τ] × [a
′′
, b
′′
]. Then, by induction on m, the same argument is repeated on each time interval
t ∈
mτ, (m + 1)τ
, proving the theorem.
Acknowledgment. The second author warmly thanks p rofessor Benedetto Piccoli for stimulating
conversations.
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16
