Journal ArticleDOI

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

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.

### 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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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
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
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

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Abstract: Introduction Part I: Representation formulas for solutions: Four important linear partial differential equations Nonlinear first-order PDE Other ways to represent solutions Part II: Theory for linear partial differential equations: Sobolev spaces Second-order elliptic equations Linear evolution equations Part III: Theory for nonlinear partial differential equations: The calculus of variations Nonvariational techniques Hamilton-Jacobi equations Systems of conservation laws Appendices Bibliography Index.

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### "Sbv regularity of entropy solutions..." refers background in this paper

• ...Scalar conservation laws in one space dimension and Hamilton–Jacobi equations in one dimension are strictly connected: entropy solutions correspond to viscosity solutions (see [9])....

[...]

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Abstract: This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws. This new edition places increased emphasis on hyperbolic systems of balance laws with dissipative source, modeling relaxation phenomena. It also presents an account of recent developments on the Euler equations of compressible gas dynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised, expanded and brought up to date, and has been enriched with new applications to elasticity and differential geometry. The bibliography, also expanded and updated, now comprises close to two thousand titles. From the reviews of the 3rd edition: "This is the third edition of the famous book by C.M. Dafermos. His masterly written book is, surely, the most complete exposition in the subject." Evgeniy Panov, Zentralblatt MATH "A monumental book encompassing all aspects of the mathematical theory of hyperbolic conservation laws, widely recognized as the "Bible" on the subject." Philippe G. LeFloch, Math. Reviews

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### "Sbv regularity of entropy solutions..." refers background or methods in this paper

• ...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....

[...]

• ...In [8], the generalized characteristic of Theorem 3....

[...]

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TL;DR: By Luigi Ambrosio, Nicolo Fucso and Diego Pallara: 434 pp.
Abstract: By Luigi Ambrosio, Nicolo Fucso and Diego Pallara: 434 pp., £55.00, isbn 0-19-850254-1 (Clarendon Press, Oxford, 2000).

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.

880 citations