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Journal ArticleDOI

Sbv regularity of entropy solutions for a class of genuinely nonlinear scalar balance laws with non-convex flux function

Roger Robyr1
01 Jun 2008-Journal of Hyperbolic Differential Equations (World Scientific Publishing Company)-Vol. 05, Iss: 01, pp 449-475

AbstractIn this work we study the regularity of entropy solutions of the genuinely nonlinear scalar balance laws We assume that the source term g ∈ C1(ℝ × ℝ × ℝ+), that the flux function f ∈ C2(ℝ × ℝ × ℝ+) and that {ui ∈ ℝ : fuu(ui,x,t) = 0} is at most countable for every fixed (x,t) ∈ Ω. Our main result, which is a unification of two proposed intermediate theorems, states that BV entropy solutions of such equations belong to SBVloc(Ω). Moreover, using the theory of generalized characteristics we prove that for entropy solutions of balance laws with convex flux function, there exists a constant C > 0 such that: where C can be chosen uniformly for (x + h,t), (x,t) in any compact subset of Ω.

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Summary

  • In this work the authors study the regularity of entropy solutions of the genuinely nonlinear scalar balance laws SBV regularity of entropy solutions for a class of genuinely nonlinear scalar balance laws with non-convex flux function.
  • In their proofs the authors will have to deal with one-dimensional fun tions of bounded variation, therefore they olle t here some useful properties.
  • The authors shall make use of these Theorems and in parti ular of the Norossing property of Theorem 3.3, to prove the SBVloc regularity of u(x, t).
  • In here the authors state the Theorem for genuine hara teristi be ause under an appropriate normalization, the notions of "sho k-free" and "genuine" are equivalent.
  • Two genuine hara teristi s may interse t only at their end points.
  • In this part of the paper the authors analyze the regularity of the entropy solutions of the onservation laws (1).
  • Moreover, u has a better stru ture than any 2-dimensional BV -fun tion.

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Archive
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Strickhofstrasse39
CH-8057Zurich
www.zora.uzh.ch
Year:2008
SBVregularityofentropysolutionsforaclassofgenuinelynonlinearscalar
balancelawswithnon-convexuxfunction
Robyr,R
Abstract:Inthisworkwestudytheregularityofentropysolutionsofthegenuinelynonlinearscalar
balancelawsWeassumethatthesourcetermgC1(฀××฀+),thattheuxfunctionfC2(฀××฀+)
andthatui:fuu(ui,x,t)=0isatmostcountableforeveryxed(x,t)Ω.Ourmainresult,whichis
aunicationoftwoproposedintermediatetheorems,statesthatBVentropysolutionsofsuchequations
belongtoSBVloc(Ω).Moreover,usingthetheoryofgeneralizedcharacteristicsweprovethatforentropy
solutionsofbalancelawswithconvexuxfunction,thereexistsaconstantC>0suchthat:whereC
canbechosenuniformlyfor(x+h,t),(x,t)inanycompactsubsetofΩ.
DOI:https://doi.org/10.1142/S0219891608001544
PostedattheZurichOpenRepositoryandArchive,UniversityofZurich
ZORAURL:https://doi.org/10.5167/uzh-8522
JournalArticle
AcceptedVersion
Originallypublishedat:
Robyr,R(2008).SBVregularityofentropysolutionsforaclassofgenuinelynonlinearscalarbalance
lawswithnon-convexuxfunction.JournalofHyperbolicDierentialEquations,5(2):449-475.
DOI:https://doi.org/10.1142/S0219891608001544

SBV regularity of entropy solutions for a lass
of genuinely nonlinear salar balane laws with
non-onvex ux funtion.
R.Robyr - UNI Zurih - Preprint
January 17, 2008
Abstrat
In this work we study the regularity of entropy solutions of the genuinely nonlinear
salar balane laws
D
t
u(x, t) + D
x
[f(u(x, t), x, t)] + g(u(x, t), x, t) = 0
in an op en set
R
2
.
We assume that the soure term
g C
1
(R × R × R
+
)
, that the ux funtion
f
C
2
(R × R × R
+
)
and that
{u
i
R : f
uu
(u
i
, x, t) = 0}
is at most ountable for every
xed
(x, t)
. Our main result, whih is a uniation of two prop osed intermediate
theorems, states that
BV
entropy solutions of suh equations belong to
SBV
loc
(Ω)
.
Moreover, using the theory of generalized harateristis we prove that for entropy
solutions of balane laws with onvex ux funtion, there exists a onstant
C > 0
suh that:
u([x + h]+, t) u(x, t) Ch, (h > 0)
where
C
an b e hosen uniformly for
(x + h, t), (x, t)
in any ompat subset of
.
1 Intro dution
In [2℄ the authors have shown that entropy solutions
u(x, t)
of salar onservation laws
D
t
u(x, t) + D
x
[f(u(x, t))] = 0
in an op en set
R
2
(1)
with lo ally uniformly onvex ux funtion
f C
2
(R)
and
f
′′
> 0
, are funtions of
lo ally sp eial b ounded variation, i.e. the distributional derivative
Du
has no Cantor
part. In the proof proposed by Ambrosio and De Lellis the go o d geometri struture
of the harateristis eld orrelated to the entropy solution play an imp ortant role and
allows to dene a geometri funtional whih jumps every time when a Cantor part of
the distributional derivative
Du(., t)
app ears in the solution. In partiular we reall here
two signiant prop erties of the harateristis: they are straight lines and two dierent
1

bakward harateristis an ross only at
t = 0
(the so-alled no rossing prop erty). We
note also that for equations (1) we an take the well-known Oleinik estimate as entropy
riterion, i.e. a distributional solution
u(x, t)
of (1) is an entropy solution provided that:
u(x + z, t) u(x, t)
˘
C
t
z,
for a
C > 0
(2)
holds for all
t > 0
,
x, z R
where
z > 0
. In [2℄ the one-sided estimate (2) is used as
entropy riterion and it is used to get the pro of.
In this note we extend this regularity result to a bigger lass of hyp erb oli onservation
laws. At rst we again onsider salar onservation laws (1) but allowing the hange of
onvexity of the ux funtion
f
at a ountable set of p oints. One of the diulties in
dealing with these equations is that rarefation waves may app ear even for
t > 0
and
onsequently the no rossing prop erty used in [2℄ do es not hold. For instane, it is p ossible
to onstrut a Riemann problem where the ux funtion has two inetions p oints and
a sho k splits into two ontat disontinuities (see [11℄). As we will see the strategy of
the pro of is not as ompliated as one an exp et: using an appropriate overing of
and working lo ally we redue the problem to the onvex or onave ase. Thus, our rst
extension theorem states:
Theorem 1.1.
Let
f C
2
(R)
be a ux funtion, suh that
{u
i
R : f
′′
(u
i
) = 0}
is at
most ountable. Let
u BV (Ω)
be an entropy solution of the salar onservation law (1).
Then there exists a set
S R
at most ountable suh that
τ R\S
the fol lowing holds:
u(., τ) SBV
loc
(Ω
τ
)
with
τ
:= {x R : (x, τ) }.
(3)
In the seond part of this pap er we fo us our attention on genuinely nonlinear salar
balane laws
D
t
u(x, t) + D
x
[f(u(x, t), x, t)] + g(u(x, t), x, t) = 0
in
R
2
(4)
where the soure term
g
b elongs to
C
1
(R × R × R
+
)
, the ux funtions
f
b elongs to
C
2
(R × R × R
+
)
and
f
uu
(., x, t) > 0
for any xed
(x, t)
. Again the geometri struture
of the harateristis is not as easy as in the ase treated in [2℄: now the harateristis
are Lipshitz urves and in general are not straight lines. The dierent shap e of the
harateristis are due to the presene of the soure term and to the dep endene of
f
on the
p oints
(x, t)
. Fortunately, we an make use of the theory of generalized harateristis
intro dued by Dafermos (see [6℄,[7℄,[8℄) to analyze the b ehavior of the harateristis for
entropy solutions of (4). Imp ortant for our analysis is the no-rossing prop erty b etween
genuine harateristis. Thanks to this prop erty we an exp et to repro due the geometri
pro of prop osed in [2℄. All the denitions and propositions ab out the theory of generalized
harateristis, whih are helpful in our work, are listed in setion 3. Another problem, due
to the presene of the soure term and of the
(x, t)
dep endene, is that for equations (4)
the Oleinik estimate (2) stop to b e true. Moreover, the Oleinik estimate annot b e taken
as entropy riterion. What we an do, is to nd a suitable generalization of this estimate,
i.e. we will prove using the generalized harateristis that:
2

Theorem 1.2.
Let
f C
2
(R × R × R
+
)
be a ux funtion suh that
f
uu
(.) > 0
. Let
g C
1
(R × R × R
+
)
be a soure term and let
u L
(Ω)
be an entropy solution of the
balane law (4). In any xed ompat set
K
there exists a positive onstant
C > 0
suh that:
u([x + z]+, t) u(x, t) Cz, (z > 0).
(5)
for every
(x, t), (x + z, t) K
.
However, for balane laws it is imp ossible to reover a onstant of the form
C =
˘
C/t
,
where
C
dep ends only on the time and on the seond derivative of
f
, estimate (5) is
suient to obtain all the regularity-results stated in this paper. The seond theorem on
the
SBV
regularity proposed is:
Theorem 1.3.
Let
f C
2
(R × R × R
+
)
be a ux funtion suh that
f
uu
(.) > 0
. Let
g C
1
(R × R × R
+
)
be a soure term and let
u L
(Ω)
be an entropy solution of the
balane law (4). Then there exists a set
S R
at most ountable suh that
τ R\S
the
fol lowing holds:
u(., τ) SBV
loc
(Ω
τ
)
with
τ
:= {x R : (x, τ) }.
(6)
Combining the two Theorems on the
SBV
regularity we get a generalized Theorem,
whih says that also for balane laws with a ux funtion whih hanges onvexity at
most ountable many times, the entropy solution is a lo ally
SBV
funtion. Thus, as a
onsequene of Theorem 1.3 and 1.1 and of the sliing theory of
BV
funtions, we state:
Theorem 1.4.
Let
f C
2
(R × R × R
+
)
be a ux funtion, suh that
{u
i
R : f
uu
(u
i
, x, t) = 0}
is at most ountable for any xed
(x, t)
. Let
g C
1
(R × R × R
+
)
be a soure term and let
u BV (Ω)
be an entropy solution of the balane law (4):
D
t
u(x, t) + D
x
[f(u(x, t), x, t)] + g(u(x, t), x, t) = 0
in
R
2
.
(7)
Then there exists a set
S R
at most ountable suh that
τ R\S
the fol lowing holds:
u(., τ) SBV
loc
(Ω
τ
)
with
τ
:= {x R : (x, τ) }.
(8)
Moreover,
u(x, t) SBV
loc
(Ω)
.
Salar onservation laws in one spae dimension and Hamilton-Jaobi equations in one
dimension are stritly onneted: entropy solutions orresp ond to visosity solutions (see
[9℄). Thus, at the end of the pap er using Theorem 1.3 we obtain also a regularity statement
for visosity solutions
u
of a lass of Hamilton-Jaobi equations: we prove that the gradient
Du
of suh solutions belongs to
SBV
loc
(Ω)
.
3

Corollary 1.1
(Hamilton-Jaobi)
.
Let
H(u, x, t) C
2
(R × R × R
+
)
be loal ly uniformly
onvex in
u
, i.e.
D
uu
H > 0
. If
w W
1,
(Ω)
is a visosity solution of
w
t
(x, t) + H(w
x
(x, t), x, t) = 0,
(9)
then
Dw SBV
loc
(Ω)
.
It would b e interesting to nd the same regularity for entropy
BV
solutions of genuinely
nonlinear system of onservation laws in one spae dimension. We note that there are
analogies b etween the struture of the generalized harateristis of systems and the one
of the balane laws 4 of Theorem 1.4 prop osed in here: in b oth ases the harateristis
an interset at
t 6= 0
and in general they are not straight lines but Lipshitz urves,
whih are a.e. dierentiable. Although the geometry of the harateristis eld of these
two problems seems to b e similar, the ase of systems lo oks muh more diult. Another
op en question is the lo al
SBV
regularity for gradients of visosity solutions of uniformly
onvex Hamilton-Jaobi PDEs in higher spae dimensions. In [4℄ the authors have shown
that under strong regularity assumptions on the initial funtions
u
0
, the visosity solution
u
has a gradient
Du
, whih b elongs to the lass
SBV
, i.e.
D
2
u
is a measure with no Cantor
part (in fat the regularity theory of [4℄ and [5℄ gives stronger onlusions).
2 Funtions with b ounded variation and sp eial fun-
tions of b ounded variation
It is well-known that in general we annot nd lassial smo oth solutions for equations (1)
and (4): sho ks app ear in nite time even for smo oth initial data
u(x, 0) = u
0
(x)
. In order
to study all the p ossible solutions with jump disontinuities, we take the spae of funtions
of b ounded variation
BV
as working spae. We then ollet some denitions and theorems
ab out
BV
and
SBV
funtions.
Denition 2.1.
Let
u L
1
(Ω)
; we say that
u
is a funtion of bounded variation in
if the distributional derivative of
u
, denoted by
Du
, is representable by a nite Radon
measure on
. A funtion
u L
1
loc
(Ω)
has loal ly bounded variation in
if for eah
open set
V ⊂⊂
,
u
is a funtion of bounded variation in
V
. We write
u BV (Ω)
and
u BV
loc
(Ω)
respetively.
In our pro ofs we will have to deal with one-dimensional funtions of b ounded variation,
therefore we ollet here some useful prop erties. Using the Radon-Niko dym Theorem we
split the Radon measure
Du
into the absolute ontinuous part
D
a
u
(with resp et to
L
1
)
and the singular part
D
s
u
:
Du = D
a
u+D
s
u = Du (Ω\S)+Du S
where
S :=
n
x : lim
ρ0
|Du|(B
ρ
(x))
ρ
=
o
.
4

Citations
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Journal ArticleDOI
Abstract: We prove that if $${t \mapsto u(t) \in BV(\mathbb{R})}$$ is the entropy solution to a N × N strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields $$u_t + f(u)_x = 0,$$ then up to a countable set of times $${\{t_n\}_{n \in \mathbb{N} }}$$ the function u(t) is in SBV, i.e. its distributional derivative u x is a measure with no Cantorian part. The proof is based on the decomposition of u x (t) into waves belonging to the characteristic families $$u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal{M}(\mathbb{R}), \, \tilde r_i(t) \in \mathbb{R}^N,$$ and the balance of the continuous/jump part of the measures v i in regions bounded by characteristics. To this aim, a new interaction measure μ i,jump is introduced, controlling the creation of atoms in the measure v i (t). The main argument of the proof is that for all t where the Cantorian part of v i is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure μ i,jump is positive.

26 citations


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Abstract: We prove that if $t \mapsto u(t) \in \mathrm {BV}(\R)$ is the entropy solution to a $N \times N$ strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields \[ u_t + f(u)_x = 0, \] then up to a countable set of times $\{t_n\}_{n \in \mathbb N}$ the function $u(t)$ is in $\mathrm {SBV}$, i.e. its distributional derivative $u_x$ is a measure with no Cantorian part. The proof is based on the decomposition of $u_x(t)$ into waves belonging to the characteristic families \[ u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal M(\R), \ \tilde r_i(t) \in \mathrm R^N, \] and the balance of the continuous/jump part of the measures $v_i$ in regions bounded by characteristics. To this aim, a new interaction measure $\mu_{i,\jump}$ is introduced, controlling the creation of atoms in the measure $v_i(t)$. The main argument of the proof is that for all $t$ where the Cantorian part of $v_i$ is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure $\mu_{i,\mathrm{jump}}$ is positive.

23 citations


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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

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Abstract: Consider a scalar conservation law in one space dimension with initial data in LI : If the flux f is in C 2 and locally uniformly convex, then for all t > 0, the entropy solution is locally in BV (functions of bounded variation) in space variable. In this case it was shown in [5], that for all most every t > 0, locally, the solution is in SBV (Special functions of bounded variations). Furthermore it was shown with an example that for almost everywhere in t > 0 cannot be removed. This paper deals with the regularity of the entropy solutions of the strict convex C 1 flux f which need not be in C 2 and locally uniformly convex. In this case, the entropy solution need not be locally in BV in space variable, but the composition with the derivative of the flux function is locally in BV. Here we prove that, this composition is locally is in SBV on all most every t > 0. Furthermore we show that this is optimal.

17 citations


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Abstract: In this paper we study the regularity of viscosity solutions to the following Hamilton–Jacobi equations $$\partial_{t}u+H(D_{x}u)=0\quad\hbox{in }\Omega\subset{\mathbb R}\times{\mathbb R}^{n}.$$ In particular, under the assumption that the Hamiltonian $${H\in C^2({\mathbb R}^n)}$$ is uniformly convex, we prove that D x u and ∂ t u belong to the class SBV loc (Ω).

16 citations


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  • ...This change of strategy is also motivated by the fact that for system of conservation laws the Hopf–Lax does not exists, whereas there is a suitable concept of generalized characteristics (see [8])....

    [...]

  • ...Moreover, we will restrict our analysis on good representative of solutions and then, under our initial hypothesis we follow the works of Dafermos [6–8] giving an introduction to the theory of generalized characteristics and recalling here some results, which we shall use in the sequel....

    [...]

  • ...Fortunately, we can make use of the theory of generalized characteristics introduced by Dafermos (see [6–8]) to analyze the behavior of the characteristics for entropy solutions of (1....

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1,802 citations


01 Jan 2000
Abstract: This book provides a self-contained introduction to the mathematical theory of hyperbolic systems of conservation laws, with particular emphasis on the study of discontinuous solutions, characterized by the appearance of shock waves. This area has experienced substantial progress in very recent years thanks to the introduction of new techniques, in particular the front tracking algorithm and the semigroup approach. These techniques provide a solution to the long standing open problems of uniqueness and stability of entropy weak solutions. This monograph is the first to present a comprehensive account of these new, fundamental advances, mainly obtained by the author together with several collaborators. It also includes a detailed analysis of the stability and convergence of the front tracking algorithm. The book is addressed to graduate students as well as researchers. Both the elementary and the more advanced material are carefully explained, helping the reader's visual intuition with over 70 figures. A set of problems, with varying difficulty, is given at the end of each chapter. These exercises are designed to verify and expand a student's understanding of the concepts and techniques previously discussed. For researchers, this book will provide an indispensable reference for the state of the art, in the field of hyperbolic systems of conservation laws. The last chapter contains a large, up to date list of references, preceded by extensive bibliographical notes.

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In this work the authors study the regularity of entropy solutions of the genuinely nonlinear scalar balance laws Moreover, using the theory of generalized characteristics the authors prove that for entropy solutions of balance laws with convex flux function, there exists a constant C > 0 such that: where C can be chosen uniformly for ( x +h, t ), ( x, t ) in any compact subset of Ω. DOI: https: //doi. org/10. In this work the authors study the regularity of entropy solutions of the genuinely nonlinear s alar balan e laws Dtu ( x, t ) + Moreover, using the theory of generalized hara teristi s the authors prove that for entropy solutions of balan e laws with onvex ux fun tion, there exists a onstant C > 0 su h that: u ( [ x + h ] +, t ) − u ( x−, t ) ≤ Ch, ( h > 0 ) where C an be hosen uniformly for ( x + h, t ), ( x, t ) in any ompa t subset of Ω. 1 Introdu tion In [ 2℄ the authors have shown that entropy solutions u ( x, t ) of s alar onservation laws Dtu ( x, t ) +Dx [ f ( u ( x, t ) ) ] = 0 in an open set Ω ⊂ R ( 1 ) with lo ally uniformly onvex ux fun tion f ∈ C ( R ) and f ′′ > 0, are fun tions of lo ally spe ial bounded variation, i. e. the distributional derivative Du has no Cantor part. The authors note also that for equations ( 1 ) they an take the well-known Oleinik estimate as entropy riterion, i. e. a distributional solution u ( x, t ) of ( 1 ) is an entropy solution provided that: u ( x+ z, t ) − u ( x, t ) ≤ Then there exists a set S ⊂ R at most ountable su h that ∀τ ∈ R\\S the following holds: u (., τ ) ∈ SBVloc ( Ωτ ) with Ωτ: = { x ∈ R: ( x, τ ) ∈ Ω }. ( 3 ) In the se ond part of this paper the authors fo us their attention on genuinely nonlinear s alar balan e laws Dtu ( x, t ) +Dx [ f ( u ( x, t ), x, t ) ] + g ( u ( x, t ), x, t ) = 0 in Ω ⊂ R ( 4 ) where the sour e term g belongs to C ( R × R × R+ ), the ux fun tions f belongs to C ( R×R×R ) and fuu (., x, t ) > 0 for any xed ( x, t ) ∈ Ω. Fortunately, the authors an make use of the theory of generalized hara teristi s introdu ed by Dafermos ( see [ 6℄, [ 7℄, [ 8℄ ) to analyze the behavior of the hara teristi s for entropy solutions of ( 4 ). Thanks to this property the authors an expe t to reprodu e the geometri proof proposed in [ 2℄. What the authors an do, is to nd a suitable generalization of this estimate, i. e. they will prove using the generalized hara teristi s that: 2 Theorem 1. 2. Let f ∈ C ( R × R × R ) be a ux fun tion su h that fuu (. ) > However, for balan e laws it is impossible to re over a onstant of the form C = C̆/t, where C depends only on the time and on the se ond derivative of f, estimate ( 5 ) is su ient to obtain all the regularity-results stated in this paper. Then there exists a set S ⊂ R at most ountable su h that ∀τ ∈ R\\S the following holds: u (., τ ) ∈ SBVloc ( Ωτ ) with Ωτ: = { x ∈ R: ( x, τ ) ∈ Ω }. ( 6 ) Combining the two Theorems on the SBV regularity the authors get a generalized Theorem, whi h says that also for balan e laws with a ux fun tion whi h hanges onvexity at most ountable many times, the entropy solution is a lo ally SBV fun tion. Then there exists a set S ⊂ R at most ountable su h that ∀τ ∈ R\\S the following holds: u (., τ ) ∈ SBVloc ( Ωτ ) with Ωτ: = { x ∈ R: ( x, τ ) ∈ Ω }. ( 8 ) Moreover, u ( x, t ) ∈ SBVloc ( Ω ). Thus, at the end of the paper using Theorem 1. 3 the authors obtain also a regularity statement for vis osity solutions u of a lass of Hamilton-Ja obi equations: they prove that the gradient Du of su h solutions belongs to SBVloc ( Ω ). The authors note that there are analogies between the stru ture of the generalized hara teristi s of systems and the one of the balan e laws 4 of Theorem 1. 4 proposed in here: in both ases the hara teristi s an interse t at t 6= 0 and in general they are not straight lines but Lips hitz urves, whi h are a. In [ 4℄ the authors have shown that under strong regularity assumptions on the initial fun tions u0, the vis osity solution u has a gradient De nition 2. 1. Let u ∈ L1 ( Ω ) ; the authors say that u is a fun tion of bounded variation in Ω if the distributional derivative of u, denoted by Du, is representable by a nite Radon measure on Ω. A fun tion u ∈ Lloc ( Ω ) has lo ally bounded variation in Ω if for ea h open set V ⊂⊂ Ω, u is a fun tion of bounded variation in V. The authors are now ready to re all ( see for instan e Theorem 3. 28 and Proposition 3. 92 of [ 1℄ ): Proposition 2. 1. Let u ∈ BV ( Ω ) and let Ω ⊂ R. Let A be the set of atoms of Du. Then: ( i ) Any good representative u is ontinuous in Ω\\A and has a jump dis ontinuity at any point of A. De nition 3. 1. The entropy solution u ( x, t ) of the equation ( 4 ) is a lo ally integrable fun tion whi h satis es the following properties: 1. For almost all t ∈ [ 0, ∞ ) the one-sided limits u ( x+, t ) and u ( x−, t ) exist for all x ∈ R. 2. u ( x, t ) solves the balan e equation ( 4 ) in the sense of distributions. ( 11 ) Throughout the paper the authors shall denote the entropy solution by u ( x, t ) and they shall write u ( x+, t ) and u ( x−, t ) for the one-sided limits of u (., t ) ( also denoted by u and uL ). Moreover, the authors will restri t their analysis on good representative of solutions and then, under their initial hypothesis they follow the works of Dafermos ( [ 6℄, [ 7℄, [ 8℄ ) giving an introdu tion to the theory of generalized hara teristi s and re alling here some results, whi h they shall use in the sequel. The authors shall make use of these Theorems and in parti ular of the Norossing property of Theorem 3. 3, to prove the SBVloc regularity of u ( x, t ). The hara teristi s χ−, χ+: [ 0, τ ] → R are genuine and are the solutions of the ODEs ( 14 ) with the following initial onditions: χ− ( τ ) = y, v− ( τ ) = u ( y−, τ ) and χ+ ( τ ) = y, v+ ( τ ) = u ( y+, τ ). In this part of the paper the authors analyze the regularity of the entropy solutions of the onservation laws ( 1 ). The authors re all that in [ 2℄ the ux fun tion f ∈ C was sele ted to be stri tly onvex and it was proved that entropy solutions are lo ally SBV. Then there exists a set S ⊂ R at most ountable su h that ∀τ ∈ R\\S the following holds: u (., τ ) ∈ SBVloc ( Ωτ ) with Ωτ: = { x ∈ R: ( x, τ ) ∈ Ω }. ( 15 ) Sin e the arguments are quite standard, the authors propose only a sket h of the proof of the above Lemma: Proof. If f ′′ < 0, i. e. the ux fun tion is stri tly on ave and the authors may prove the lemma dire tly using the onvex ase. In parti ular by the onvex ase and the de nition of ũ, there exists S̃ = S ⊂ R at most ountable su h that ∀τ ∈ R\\S̃ the following holds: ũ (., τ ) ∈ SBVloc ( Ω̃τ ) with Ω̃τ: = { x ∈ R: ( x, τ ) ∈ Ω̃ }. ( 17 ) 8 4. 2 Proof of Theorem 1. 1 Step 1: Preliminary remarks. For this two sets the authors state: Claim 4. 1. For any τ they have that |D xuτ | ( Cτ ) = 0. Proof. For every τ one has |D xuτ | ( Cτ ) = |D c xuτ | ( Jτ ∪ Fτ ) ≤ |D c xuτ | ( Jτ ) + |D c xuτ | ( Fτ ). ( 19 ) Observe that for all τ the Cantor part is zero on the jump sets Jτ, sin e by ( 10 ): D xuτ = Dxuτ ( S\\Jτ ) ⇒ D c xuτ ( Jτ ) = 0 ⇒ |D c xuτ | ( Jτ ) = 0. ( 20 ) Using the se ond statement of Proposition 2. 1, the authors show that even the se ond term of inequality ( 19 ) vanishes. This on ludes the proof of the laim, i. e. |D xuτ | ( Cτ ) = 0. 9 The authors on lude with: De nition 2. 2. Let u ∈ BV ( Ω ), then u is a spe ial fun tion of bounded variation and they write u ∈ SBV ( Ω ) if Du = 0, i. e. if the Cantor part is zero. Furthermore, ( χ (. ), v (. ) ) satisfy the lassi al hara teristi equations { χ̇ ( t ) = fu ( v ( t ), χ ( t ), t ) v̇ ( t ) = −fx ( v ( t ), χ ( t ), t ) − g ( v ( t ), χ ( t ), t ) ( 14 ) on ( a, b ). Furthermore, if u ( y+, τ ) < u ( y−, τ ), then u ( χ ( t ) +, t ) < u ( χ ( t ) −, t ) for all t ∈ [ τ, ∞ ).