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

TLDR: In this paper, the authors studied the regularity of entropy solutions of nonlinear scalar balance laws with convex flux functions and showed that BV entropy solutions belong to SBVloc(Ω).

Abstract: In 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.

Figures

Figure 2: Chara teristi ones and bases.Lemma 6.1. Let τ > 0. If y ∈ Sτ , then for all θ ∈ [0, τ ] there exists a positive onstant cj su h that L1(Iθy,τ ) = χ+(θ; y, τ)− χ−(θ; y, τ) ≤ −cjντ ({y}). (55)Lemma 6.2. Let τ0 > 0. Then for µτ0-a.e. x there exists η := η(x, τ0, τ) > 0 su h that

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Year:2008
SBVregularityofentropysolutionsforaclassofgenuinelynonlinearscalar
balancelawswithnon-convexuxfunction
Robyr,R
Abstract:Inthisworkwestudytheregularityofentropysolutionsofthegenuinelynonlinearscalar
balancelawsWeassumethatthesourcetermg฀C1(฀×฀×฀+),thattheuxfunctionf฀C2(฀×฀×฀+)
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
Ω ⊂ 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) ∈ Ω
. 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
Ω
.
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
Ω ⊂ 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
Ω
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
Ω
τ
:= {x ∈ R : (x, τ) ∈ Ω}.
(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
Ω ⊂ 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) ∈ Ω
. 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) ∈ Ω
. 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 soure term and let
u ∈ L
∞
(Ω)
be an entropy solution of the
balane law (4). In any xed ompat set
K ⊂ Ω
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 soure 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
Ω
τ
:= {x ∈ R : (x, τ) ∈ Ω}.
(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 soure 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
Ω ⊂ 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
Ω
τ
:= {x ∈ R : (x, τ) ∈ Ω}.
(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 loal 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
Ω
if the distributional derivative of
u
, denoted by
Du
, is representable by a nite Radon
measure on
Ω
. A funtion
u ∈ L
1
loc
(Ω)
has loal ly bounded variation in
Ω
if for eah
open set
V ⊂⊂ Ω
,
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 ∈ Ω : lim
ρ↓0
|Du|(B
ρ
(x))
ρ
= ∞
o
.
4

Frequently asked questions

Q1. What are the contributions in this paper?

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, ∞ ).