Lower compactness estimates for scalar balance laws

TL;DR: In this article, the compactness of the semigroup of the image through St of bounded sets C in L 1 \ L 1 which is denoted by L 1 is studied.

Abstract: We study the compactness in L 1 of the semigroup (St)t 0 of entropy weak solutions to strictly convex scalar conservation laws in one space dimension. The compactness of St for each t > 0 was established by P. D. Lax (1). Upper estimates for the Kolmogorov's "-entropy of the image through St of bounded sets C in L 1 \ L 1 which is denoted by

Summary

2.1 Proof of Theorem 2

• The authors obtain a proof of Theorem 2 as a consequence of the following two propositions that shall be established below.
• Proposition 1. Suppose that f : R → R is a twice continuously differentiable map satisfying (2), (3).

3.1 Proof of Theorem 3

• In order to establish Theorem 3, the authors will make use of a local Oleinik type estimate for balance laws (9).
• An inequality of this kind was established in [13, Theorem 1.2].
• For source terms of the form g = g(u), a global Oleinik type estimate was obtained in [7, Section 4].
• Proposition 3. Under the assumptions of Lemma 1, assume that f , g satisfy also (3) and (11), respectively.

4.2 Proof of Lemma 3

• The authors shall first establish the conclusion of the Lemma 3 for compactly supported functions v, that belong to C∞0 (R), and thus by (80) satisfy v′(x) ≤.
• Hence, in both cases the authors get the estimate (81) when v is smooth.

Figures

Figure 1: The function Fι for n = 10 and ι = (−1,−1, 1, 1, 1,−1, 1,−1,−1, 1)

Full text

Lower compactness estimates for scalar balance laws
Fabio Ancona
∗
, Olivier Glass
†
, Khai T. Nguyen
∗
Dedicated to Prof. Constantine Dafermos in the occasion of his 70
th
birthday
October 11, 2011
Abstract
In this pap er, we study the compactness in L
1
loc
of the semigroup (S
t
)
t≥0
of entropy weak solutions
to strictly convex scalar conservation laws in one space dimension. The compactness of S
t
for each
t > 0 was established by P. D. Lax [10]. Upper estimates for the Kolmogorov’s ε-entropy of the image
through S
t
of bounded sets in L
1
∩ L
∞
were given by C. De Lellis and F. Golse [5]. Here, we provide
lower estimates on this ε-entropy of the same order as the one established in [5], thus showing that
such an ε-entropy is of size ≈ (1/ε). Moreover, we extend these estimates of compactness to the case
of convex balance laws.
1 Introduction
Consider a scalar conservation law in one space dimension
u
t
+ f(u)
x
= 0, (1)
where u = u(t, x) is the state variable, and f : R → R is a twice continuously differentiable, (uniformly)
strictly convex function:
f
00
(u) ≥ c > 0 ∀ u ∈ R. (2)
Without loss of generality, we will suppose
f
0
(0) = 0, (3)
since one may always reduce the general case to this one by performing the space-variable and flux
transformations x → x + tf
0
(0) and f(u) → f(u) − uf
0
(0). We recall that problems of this type do not
possess classical solutions since discontinuities arise in finite time even if the initial data are smooth.
Hence, it is natural to consider weak solutions in the sense of distributions that, for sake of uniqueness,
satisfy an entropy criterion for admissibility [4]:
u(t, x−) ≥ u(t, x+) for a.e. t > 0, ∀ x ∈ R , (4)
where u(t, x±) denote the one-sided limits of u(t, ·) at x. The equation (1) generates an L
1
-contractive
semigroup of solutions (S
t
)
t≥0
that associates, to every given initial data u
0
∈ L
1
(R) ∩ L
∞
(R), the
unique entropy admissible weak solution S
t
u
0
.
= u(t, ·) of the corresponding Cauchy problem (cfr. [4, 9]).
This yields the existence of a continuous semigroup (S
t
)
t≥0
acting on the whole space L
1
(R). Such a
semigroup S
t
was shown by Lax [10] to be compact as a mapping from L
1
(R) to L
1
loc
(R), for every t > 0.
De Lellis and Golse [5], following a suggestion by Lax [10], used the Kolmogorov’s ε-entropy concept,
which is recalled below, to provide a quantitative version of this compactness effect.
∗
Dipartimento di Matematica Pura ed Applicata, Universit`a degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy
†
Ceremade, Universit´e Paris-Dauphine, CNRS UMR 7534, Place du Mar´echal de Lattre de Tassigny, 75775 Paris Cedex
16, France
1

Definition 1. Let (X, d) be a metric space and K a totally bounded subset of X. For ε > 0, let N
ε
(K)
be the minimal number of sets in a cover of K by subsets of X having diameter no larger than 2ε. Then
the ε-entropy of K is defined as
H
ε
(K | X)
.
= log
2
N
ε
(K).
Throughout the paper, we will call an ε-cover, a cover of K by subsets of X having diameter no larger
than 2ε.
De Lellis and Golse obtained an upper bound for the ε-entropy of the set of solutions to (1) at any
given time t > 0, as ε → 0
+
; that is to say, they showed how strong is the compactifying effect. Precisely,
they established the following result.
Theorem 1 ([5]). Consider a twice continuously differentiable f : R → R, satisfying (2), (3). Given
any L, m, M > 0, define the set of bounded, compactly supported initial data
C
[L,m,M]
.
=
n
u
0
∈ L
1
(R) ∩ L
∞
(R) | Supp (u
0
) ⊂ [−L, L], ku
0
k
L
1
≤ m, ku
0
k
L
∞
≤ M
o
. (5)
Then for ε > 0 sufficiently small, one has
H
ε

S
T
(C
[L,m,M]
) | L
1
(R)

≤
4
ε
4L(T )
2
cT
+ 4L(T )
r
2m
cT
!
∀ T > 0 , (6)
with
L(T )
.
= L + 2 sup
|z|≤M
|f
00
(z)|
p
2mT/c. (7)
The aim of this paper is to show that the ε-entropy estimates provided by Theorem 1 turn out to
be optimal, since we shall establish a lower bound on such an ε-entropy which is of the same order
as De Lellis and Golse’s upper bounds. Hence, we deduce that H
ε
(S
T
(C
[L,m,M]
) | L
1
(R)) is exactly of
size ≈ ε
−1
. Precisely, we prove the following.
Theorem 2. Under the assumptions and in the same setting of Theorem 1, for any T > 0, and for
ε > 0 sufficiently small, one has
H
ε

S
T
(C
[L,m,M]
) | L
1
(R)

≥
1
ε
·
L
2
48 · ln(2) · |f
00
(0)|T
. (8)
As suggested in [10], the knowledge of the ε-entropy magnitude of the solution set of (1) may play
an important role to provide estimates on the accuracy and resolution of numerical methods for (1).
The main steps of the proof of the lower bound (8) consist in:
1. Introducing a suitable class of piecewise affine functions and showing that any element of such
a class can be obtained, at any given time t, as the value u(t, ·) of an entropy admissible weak
solution of (1), with initial data in C
[L,m,M]
.
2. Providing an optimal estimate of the maximum number of functions in such a class that can
be contained in a subset of S
T
(C
[L,m,M]
) having diameter no larger than 2ε. This estimate is
established with a similar combinatorial argument as the one used in [1].
Remark 1. Since by (2), (7), we have L(T)
q
2m
cT
≤
L(T )
2
2cT
, one derives from (6) the estimate
H
ε

S
T
(C
[L,m,M]
) | L
1
(R)

≤
1
ε
·
24 L(T)
2
c T
.
Therefore, the size
1
ε
·
L
2
|f
00
(0)| T
of the lower bound (8) turns out to be the same as the one of the upper
bound on the ε-entropy of S
T
(C
[L,m,M]
) provided by Theorem 1, upon replacing L with L(T ), and |f
00
(0)|
with c.
2

Next, we address the more general case of convex balance laws. Namely, given a twice continuously
differentiable map f : R → R satisfying (2), (3), we will analyze the compactifying effect of the balance
law
u
t
+ f(u)
x
= g(t, x, u). (9)
As for (1) we will consider weak solutions of (9) that satisfy the entropy admissibility condition (4). The
source term is assumed to be a continuously differentiable map g : R
+
× R × R → R, that satisfies the
following assumptions:
g(t, x, 0) = 0 ∀ (t, x) ∈ R
+
× R, (10)
∃ C > 0 s.t. |g
x
(t, x, u)| ≤ C|u| ∀ (t, x, u) ∈ R
+
× R × R , (11)
∃ ω ∈ L
1
loc
(R
+
) s.t. |g
u
(t, x, u)| ≤ ω(t) for a.e. t ∈ R
+
, ∀ (x, u) ∈ R
2
. (12)
In particular, (10), (12) together imply
∃ ω ∈ L
1
loc
(R
+
) s.t. |g(t, x, u)| ≤ ω(t) · |u| for a.e. t ∈ R
+
, ∀ (x, u) ∈ R
2
. (13)
Under assumptions (12) or (13), for each u
0
∈ L
1
(R) ∩ L
∞
(R), there exists a unique entropy admissible
solution u(t, x) of (9) with initial condition u(0, ·) = u
0
, see [4, 7, 9].
Remark 2. Condition (10) in particular implies the fact that the source term g, if not zero, does depend
on u, since otherwise one would have g = g(t, x) = 0, for all t, x. Because of (10), all solutions u(t, ·)
to (9) with initial data of compact support remain compactly supported for all times t > 0.
We shall denote by E
t
the evolution operator that associates, to every initial data u
0
∈ L
1
(R)∩L
∞
(R),
the entropy admissible solution E
t
u
0
.
= u(t, ·) of the corresponding Cauchy problem for (9). We establish
the following.
Theorem 3. Let f : R → R be a twice continuously differentiable map that satisfies (2), (3), and
g : R
+
× R × R → R be a continuously differentiable map that satisfies (10), (11), (12). Then, in the
same setting of Theorem 1, for any T > 0 and for ε > 0 sufficiently small, one has
H
ε

E
T
(C
[L,m,M]
) | L
1
(R)

≥
1
ε
·
L
2
· exp

−kωk
L
1
(0,T )
)
48 · ln(2) · |f
00
(0)|T
. (14)
Since balance laws are not considered in [5], following the same lines of the proof in [5] we also establish
the same type of upper bound for H
ε
(E
T
(C
[L,m,M]
) | L
1
(R)) as the one given in Theorem 1.
Let us introduce the following notations. Given t > 0, M > 0 and a, b ∈ R with a < b, we set
∆
a,b,t
(M)
.
=
n
(s, x) |s ∈ [0, t], a − (t − s) · kf
0
k
L
∞
(−M,M)
≤ x ≤ b + (t − s) · kf
0
k
L
∞
(−M,M)
o
, (15)
and
k
a,b,t
(M)
.
= max

|g
x
(s, x, u)|; (s, x) ∈ ∆
a,b,t
(M) , u ∈ [−M, M]

. (16)
We obtain the following result.
Theorem 4. In the same setting of Theorem 1, assume that f : R → R is a twice continuously differ-
entiable map that satisfies (2), (3), and g : R
+
× R × R → R is a continuously differentiable map that
satisfies (10), (12). Then, for ε > 0 sufficiently small, one has
H
ε

E
T
(C
[L,m,M]
) | L
1
(R)

≤
1
ε
·
8 L(T )
2
·

1 + 2(1 + c T
2
K
L,T
) exp

kωk
L
1
(0,T )


c T
∀ T > 0,
where
L(T )
.
= L + 2kf
00
k
L
∞
(−M
T
, M
T
)
r
2mT
c
h
1 + T
p
c K
L,T
i
· exp(kωk
L
1
(0,T )
),
K
L,T
.
= k
−L
T
,L
T
,T
(M
T
),
3

with
L
T
.
= L + kf
00
k
L
∞
(−M
T
, M
T
)
· M
T
T,
and
M
T
.
= exp

kωk
L
1
(0,T )

· M .
Remark 3. Theorems 3 and 4 remain true if the source term has the form g = g(t, u), and satisfies
only the condition (12), together with g(·, 0) ∈ L
1
loc
. Clearly, in this case, the solution u(t, ·) of (9) will
not be in general compactly supported, but instead the difference between u(t, ·) and the solution of (9)
with zero initial data has always compact support. So, it will be convenient to compute the ε-entropy of
the translated set E
T
(C
[L,m,M]
) − E
T
0, which obviously coincides with the one of E
T
(C
[L,m,M]
). In this
way we will see in Subsection 4.3 that one can establish, for ε > 0 sufficiently small, the estimate
H
ε

E
T
(C
[L,m,M]
) | L
1
(R)

≤
1
ε
·
8 L(T )
2
· (1 + 2 exp(kωk
L
1
(0,T )
))
c T
∀ T > 0 , (17)
where
L(T )
.
=L + 2kf
00
k
L
∞
(−M
g
T
, M
g
T
)
r
2mT
c
· exp(kωk
L
1
(0,T )
),
with
M
g
T
.
=exp(kωk
L
1
(0,T )
)(M + kg(·, 0)k
L
1
(0,T )
).
Moreover, for any T > 0, and for ε > 0 sufficiently small, we derive the estimate
H
ε

E
T
(C
[L,m,M]
) | L
1
(R)

≥
1
ε
·
L
2
· exp

−kωk
L
1
(0,T )

24 · ln(2) · kf
00
k
L
∞
(−G
T
,G
T
)
T
. (18)
where
G
T
.
= 1 + kg(·, 0)k
L
1
(0,T )
exp(kωk
L
1
(0,T )
). (19)
As a final remark, we observe that it would be interesting to provide upper and lower quantitative
compactness estimates for the solution set of genuinely nonlinear 2 × 2 systems of conservation laws
(whose L
1
loc
compactness follows from the estimates provided in [6], as observed in [11]), while it remains
a completely open problem whether such a compactness property continues to hold (and possibly derive
similar quantitative estimates) for general systems of N conservation laws with genuinely nonlinear
characteristic fields.
The paper is organized as follows. In Section 2 we provide a tight lower bound for the ε-entropy of the
solution set of a convex conservation law, establishing Theorem 2. In Section 3 we derive an Oleinik type
inequality for convex balance laws with smooth source term, and then extend the results of Section 2 to
the case of convex balance laws, proving Theorem 3. Finally, in Section 4 we derive an upper bound for
the ε-entropy of the solution set of a convex balance law, proving Theorem 4; also, we prove Remark 3.
Acknowledgements. The authors would like to warmly thank the anonymous referee for a comment
and a suggestion that has contributed to simplify some proofs and to obtain slightly more general results
for balance laws. They wish to thank Institut Henri Poincar´e (Paris, France) for providing a very
stimulating environment at the “Control of Partial Differential Equations and Applications” program in
the Fall 2010, during which a part of this work was written. FA and KTN are partially supported by
the European Union Seventh Framework Programme [FP7-PEOPLE-2010-ITN] under grant agreement
n.264735-SADCO. OG is partially supported by the Agence Nationale de la Recherche, Project CISIFS,
grant ANR-09-BLAN-0213-02.
2 Lower compactness estimates for conservation laws
2.1 Proof of Theorem 2
For arbitrary positive constants L, M, m and b, let us consider the set
A
[L,m,M,b]
.
=
n
u
T
∈ BV(R) | Supp (u
T
) ⊂ [−L, L], ku
T
k
L
1
≤ m, ku
T
k
L
∞
≤ M, Du
T
≤ b
o
,
4

where the last inequality has to be understood in the sense of measures, i.e. the Radon measure Du
T
satisfies Du
T
(J) ≤ b · |J| for every Borel set J ⊂ R, |J| being the Lebesgue measure of J. We obtain a
proof of Theorem 2 as a consequence of the following two propositions that shall be established below.
Proposition 1. Suppose that f : R → R is a twice continuously differentiable map satisfying (2), (3).
Then, given any L, m, M, T > 0, for
0 ≤ h ≤ min

M,
m
2L
,
L
8T |f
00
(0)|

, (20)
sufficiently small, one has
A
[L
T
, Lh, h, (2T |f
00
(0)|)
−1
]
⊂ S
T
(C
[L,m,M]
), (21)
with
L
T
.
= L − 2T |f
00
(0)| · h. (22)
Proposition 2. Given L, m, M, b > 0, for any ε > 0 satisfying
ε ≤
min(m, LM)
6
, (23)
one has
H
ε
(A
[L,m,M,b]
| L
1
(R)) ≥
1
ε
·
2bL
2
27 ln(2)
. (24)
Notice that the lower bound (24) is independent on m and M , which appear only in the constraint (23).
Moreover, because of (20), the constant L
T
given by (22) satisfies L
T
≥ (3/4)L. Hence, applying (24),
with L = L
T
, b = (2T |f
00
(0)|)
−1
, and relying on (21), we recover the estimate (8), which proves Theo-
rem 2.
2.2 Proof of Proposition 1
1. We shall first prove the inclusion
A
[L
T
, 2Lh, h, (2T |f
00
(0)|)
−1
]
∩ C
1
(R) ⊂ S
T
(C
[L,m,M]
), (25)
(C
1
(R) denoting the set of continuously differentiable maps on R with values in R). More precisely, we
will show that any element u
T
∈ C
1
(R) of the set on the left-hand side of (21) can be obtained as the
value at time T of a weak admissible solution to (1), which is backward constructed starting from u
T
by
reversing the direction of time. Namely, given
u
T
∈ A
[L
T
, 2Lh, h, (2T |f
00
(0)|)
−1
]
∩ C
1
(R), (26)
set
w
0
(x)
.
= u
T
(−x) ∀ x ∈ R, (27)
and consider the entropy weak solution w(t, x)
.
= S
t
w
0
of (1) with initial data w
0
. By well-known
properties of solutions to scalar conservation laws, and because of (20), (26), w verifies the L
1
and L
∞
bounds (cfr. [4, Theorem 6.2.3, Theorem 6.2.6]):
kw(t, ·)k
L
1
≤ kw
0
k
L
1
≤ 2Lh ≤ m ,
kw(t, ·)k
L
∞
≤ kw
0
k
L
∞
≤ h ≤ M,
∀ t > 0. (28)
Next, observe that the function
u(t, x)
.
= w(T − t, −x), (t, x) ∈ [0, T ] × R, (29)
is a weak solution of (1) in the sense of distribution, which, by (27), clearly satisfies
u(T, ·) = u
T
.
5

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Lower compactness estimates for scalar balance laws" ?

In this paper, the authors study the compactness in Lloc of the semigroup ( St ) t≥0 of entropy weak solutions to strictly convex scalar conservation laws in one space dimension. Here, the authors provide lower estimates on this ε-entropy of the same order as the one established in [ 5 ], thus showing that such an ε-entropy is of size ≈ ( 1/ε ). 

Q2. How does one obtain the estimate of v?

(90)Since vν → v in L1(R) as ν → 0+, and because there holds ‖vν‖∞ → ‖v‖∞ as ν → 0+, the authors then recover from (90) the estimate (81) for v, thus completing the proof of the lemma. 

Q3. what is the entropy admissible solution of a Cauchy problem?

Because of (10), all solutions u(t, ·) to (9) with initial data of compact support remain compactly supported for all times t > 0.The authors shall denote by Et the evolution operator that associates, to every initial data u0 ∈ L1(R)∩L∞(R), the entropy admissible solution Etu0 . = u(t, ·) of the corresponding Cauchy problem for (9). 

Q4. what is the determinacy of the set a,b,t(Mt?

Relying on (56), and because of (52), the authors deduce that the set ∆a,b,t(Mt) defined in (15) is a backward domain of determinacy relative to the interval [a, b] and to the time t, since it contains all backward generalized characteristics emanating from points (t, x), x ∈ [a, b].2. Fix t > 0, a, b ∈ R, a < b, and consider x < y two points of continuity of u(t, ·) inside [a, b]. Let ξx(·) and ξy(·) be the (unique) backward generalized characteristics (cfr. [4, Theorems 11.9.5]) emanating from (t, x) and (t, y), respectively. 

Q5. What is the author's reaction to the anonymous referee?

The authors would like to warmly thank the anonymous referee for a comment and a suggestion that has contributed to simplify some proofs and to obtain slightly more general results for balance laws. 

Q6. What is the entropy of the solution w?

Let wn(t, ·) . = St(w n 0 ) and w(t, ·) . = St(w0) be the entropy weak solutions of (1) with initial data, respectively, wn0 (·) . = unT (−·) and w0(·) . = uT (−·). 

Q7. What is the entropy admissible for the solution set of conservation laws?

for any T > 0, and for ε > 0 sufficiently small, the authors derive the estimateHε ( ET (C[L,m,M ]) | L1(R) ) ≥ 1 ε ·L2 · exp ( −‖ω‖L1(0,T ) ) 24 · ln(2) · ‖f ′′‖L∞(−GT ,GT ) T . (18)where GT . = 1 + ‖g(·, 0)‖L1(0,T ) exp(‖ω‖L1(0,T )). (19)As a final remark, the authors observe that it would be interesting to provide upper and lower quantitative compactness estimates for the solution set of genuinely nonlinear 2 × 2 systems of conservation laws (whose L1loc compactness follows from the estimates provided in [6], as observed in [11]), while it remains a completely open problem whether such a compactness property continues to hold (and possibly derive similar quantitative estimates) for general systems of N conservation laws with genuinely nonlinear characteristic fields. 

Q8. What is the entropy of the solution set of (1)?

given a twice continuously differentiable map f : R→ R satisfying (2), (3), the authors will analyze the compactifying effect of the balance lawut + f(u)x = g(t, x, u). (9)As for (1) the authors will consider weak solutions of (9) that satisfy the entropy admissibility condition (4). 

Q9. What is the -entropy of the lower bound?

Remark 1. Since by (2), (7), the authors have L(T ) √2m cT ≤L(T )22cT , one derives from (6) the estimateHε ( ST (C[L,m,M ]) | L1(R) ) ≤ 1 ε · 24L(T ) 2 c T .Therefore, the size 1ε · L2 |f ′′(0)|T of the lower bound (8) turns out to be the same as the one of the upper bound on the ε-entropy of ST (C[L,m,M ]) provided by Theorem 1, upon replacing L with L(T ), and |f ′′(0)| with c. 

Q10. What is the uT A[LT, Lh, h,?

for any given uT ∈ A[LT , Lh, h, b0], the authors may consider a sequence {unT } ⊂ A[LT , 2Lh, h, b0] ∩ C1(R,R) such that limn→∞ ‖unT − uT ‖L1 = 0.