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HIGH FREQUENCY WAVES AND THE MAXIMAL SMOOTHING EFFECT

25 pages
HIGH FREQUENCY WAVES AND THE MAXIMAL SMOOTHING EFFECT FOR NONLINEAR SCALAR CONSERVATION LAWS STEPHANE JUNCA Abstract. The article first studies the propagation of well prepared high frequency waves with small amplitude ? near constant solutions for en- tropy solutions of multidimensional nonlinear scalar conservation laws. Sec- ond, such oscillating solutions are used to highlight a conjecture of Lions, Perthame, Tadmor, (1994), [34], about the maximal regularizing effect for nonlinear conservation laws. For this purpose, a new definition of nonlinear flux is stated and compared to classical definitions. Then it is proved that the smoothness expected by [34] in Sobolev spaces cannot be exceeded. Key-words: multidimensional conservation laws, nonlinear flux, geometric optics, Sobolev spaces, smoothing effect. Mathematics Subject Classification: Primary: 35L65, 35B65; Secondary: 35B10, 35B40, 35C20. Contents 1. Introduction 1 2. High frequency waves with small amplitude 5 3. Characterization of nonlinear flux 8 4. Sobolev estimates 14 5. Highlights about a Lions,Perthame,Tadmor conjecture 22 References 23 1. Introduction This paper deals with the maximal regularizing effects for nonlinear mul- tidimensional scalar conservation laws. The important point to note here is the definition of nonlinear flux. Indeed there are various definitions see [18, 34, 4, 11].

  • initial oscillating

  • highly oscillating

  • super-critical highly oscillating

  • sobolev exponent

  • maximal sobolev exponent

  • high frequency

  • classical definition

  • conservation laws


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HIGHFREQUENCYWAVESAND
THEMAXIMALSMOOTHINGEFFECT
FORNONLINEARSCALARCONSERVATIONLAWS
STE´PHANEJUNCA

Abstract.
Thearticlefirststudiesthepropagationofwellpreparedhigh
frequencywaveswithsmallamplitude
ε
nearconstantsolutionsforen-
tropysolutionsofmultidimensionalnonlinearscalarconservationlaws.Sec-
ond,suchoscillatingsolutionsareusedtohighlightaconjectureofLions,
Perthame,Tadmor,(1994),[34],aboutthemaximalregularizingeffectfor
nonlinearconservationlaws.Forthispurpose,anewdefinitionofnonlinear
fluxisstatedandcomparedtoclassicaldefinitions.Thenitisprovedthat
thesmoothnessexpectedby[34]inSobolevspacescannotbeexceeded.

Key-words
:multidimensionalconservationlaws,nonlinearflux,geometricoptics,
Sobolevspaces,smoothingeffect.
MathematicsSubjectClassification
:
Primary:35L65,35B65;Secondary:35B10,35B40,35C20.

Contents
1.Introduction
2.Highfrequencywaveswithsmallamplitude
3.Characterizationofnonlinearflux
4.Sobolevestimates
5.HighlightsaboutaLions,Perthame,Tadmorconjecture
References

158412232

1.
Introduction
Thispaperdealswiththemaximalregularizingeffectsfornonlinearmul-
tidimensionalscalarconservationlaws.Theimportantpointtonotehere
isthedefinitionofnonlinearflux.Indeedtherearevariousdefinitionssee
[18,34,4,11].In[34]theygivethewellknowndefinition1.1belowanda
conjectureaboutthemaximalsmoothingeffectinSobolevspacesrelatedto
theparameter“
α
“fromtheirdefinition.Thestudyofperiodicsolutionsleads
toanotherdefinitions[18,4].Weobtainnewdefinition3.1forsmoothflux.It
generalizesthedefinitionof[4].Forsmoothflux,ourdefinitionisequivalent
Date
:March15,2011.

1

2

STE´PHANEJUNCA

totheclassicaldefinition1.1andimpliesthestrictnon-linearityof[18].Fur-
thermore,itgivesaneasywaytocomputetheparameter“
α
”.Ourdefinition
showsthatsmoothingeffectsforscalarconservationlawsstronglydependon
thespacedimension.Ournewcharacterizationofnonlinearfluxcomesfrom
thestudyofthehighestoscillationswhichcanbepropagatedbythesemi-
group
S
t
associatedtotheconservationlaw.Indeedpropertiesof
S
t
arelinked
tothederivativesofthefluxasin[4,11,19].
Tobemoreprecise,welookforSobolevboundsforentropysolutions
u
(
.,.
)
fo(1.1)

t
u
+div
x
F
(
u
)=0
,
where
t

[0
,
+

[,
x

R
d
,
u
:[0
,
+

[
t
×
R
x
d

R
,
F
:
R

R
d
isasmoothflux
function,
F

C

(
R
,
R
d
),andtheinitialdataisonlyboundedin
L

(
R
x
d
,
R
):
(1.2)
u
(0
,
x
)=
u
0
(
x
)
.
Let
a
(
u
)be
F
0
(
u
).Obviously,if
F
islinear,
a
(
u
)=
a
aconstantvector,
u
(
t,
x
)=
u
0
(
x

t
a
),thereisnosmoothingeffect.In[34]wasfirstproveda
regularizingeffectiftheflux
F
isnonlinear.Thesharpmeasurementofthe
non-linearityplaysakeyroleinourstudy.Letusrecalltheclassicaldefinition
fornonlinearfluxfrom[34].
Definition1.1.[NonlinearFlux
[34]
]
Let
M
beapositiveconstant,
F
:
R

R
d
issaidtobe
nonlinear
on
[

M,M
]
ifthereexist
α>
0
and
C
=
C
α
>
0
suchthatforall
δ>
0
α(1.3)sup
τ
2
+
|
ξ
|
2
=1
|
W
δ
(
τ,ξ
)
|≤
Cδ,
where
(
τ,ξ
)

S
d

R
d
+1
,
i.e.
τ
2
+
|
ξ
|
2
=1
,and
|
W
δ
(
τ,ξ
)
|
istheonedimen-
sionalmeasureofthesingularset:
W
δ
(
τ,ξ
):=
{|
v
|≤
M,
|
τ
+
a
(
v
)

ξ
|≤
δ
}⊂
[

M,M
]
and
a
=
F
0
.
Indeed
W
δ
(
τ,ξ
)isaneighborhoodofthecricitalvalue
v
forthesymbolofthe
linearoperator
L
[
v
]intheFourierdirection(
τ,ξ
)where
L
[
v
]=

t
+
a
(
v
)

r
x
.
Thesymbolinthisdirectionis:
i
(
τ
+
a
(
v
)

ξ
).Thisoperatorissimplyrelated
withanysmoothsolution
u
ofequation(1.1)bythechainruleformula:

t
u
+div
x
F
(
u
)=

t
u
+
a
(
u
)

r
x
u
=
L
[
u
]
u.
α
isadegeneracymeasurementoftheoperator
L
parametrizedby
v
.
α
dependsonlyontheflux
F
andthecompactset[

M,M
]:
α
=
α
[
F
,M
].In
thesequelwedenoteby
(1.4)
α
sup=
α
sup[
F
,M
]
,
thesupremumofall
α
satisfying(1.3).
α
,ormoreprecisely
α
sup,isthekeyparametertodescribethesharpsmoothing
effectforentropysolutionsofnonlinearscalarconservationlaws.Forsmooth
fluxtheparameter
α
alwaysbelongsto[0
,
1],forinstance:
α
sup=0fora
linearflux,
α
=1forstrictlyconvexfluxindimensionone.Forthefirsttime
α
supischaracterizedbelowinsection3.Indeed,forsmoothnonlinearflux,

OSCILLATIONSANDSMOOTHINGEFFECTFORCONSERVATIONLAWS3

1isalwaysanintegergreaterorequaltothespacedimension.
αpusInallthesequelweassumethat
M
≥k
u
0
k

andtheflux
F
isnonlinearon
[

M,M
],so
(1.5)
α
sup
>
0
.
If(1.5)istruethentheentropysolutionoperatorassociatedwiththenonlinear
conservationlaw(1.1),(1.2),
S
t
:
L

(
R
x
d
,
R
)

L

(
R
x
d
,
R
)
u
0
(
.
)
7→
u
(
t,.
)
,
hasaregularizingeffectforall
t>
0,mapping
L

(
R
x
d
,
R
)into
W
lso,c
1
(
R
x
d
,
R
).
αIn[34],theyprovedthisregularizingeffectforall
s<
.
α+2αIn[39]theresultisimprovedforall
s<
underagenericassumption
α2+1on
a
0
=
F
00
.
P.L.Lions,B.PerthameandE.Tadmorconjecturedin1994abetterregu-
larizingeffect,see[34],(remark3,p.180,line14-17).In[34]theyproposed
anoptimalbound
s
supforSobolevexponentsofentropysolutions:
(1.6)
s
sup=
α
sup
.
Thatistosaythat
u
belongsinall
W
lso,c
1
(
R
d
,
R
)forall
s<α
sup.
Theshocksformationimplies
s<
1and
s
sup

1since
W
1
,
1
functionsdonot
haveshock.
Inonedimension(d=1)andforstrictlyconvexfluxitiswellknownfrom
LaxandOleinikthattheentropysolutionbecomes
BV
,see[33].(1.6)istrue
inthiscasesince
u
belongsin
W
lso,c
1
forall
s<
1:
s
sup=1=
α
sup.
Amainresultofthepaperistogiveaninsightoftheconjecture(1.6)by
provingtheinequality
(1.7)
s
sup

α
sup
.
Examplesoffamilyofsolutionsexactlyboundedin
W
lso,c
1
withtheconjectured
maximal
s
=
α
supandwithnoimprovementoftheSobolevexponentina
strip[0
,T
0
]
×
R
d
,
T
0
>
0,aregiveninthispaper.
Afirstproofof(1.7),forsomeinterestingexamples,canbefoundin[16]for
d
=1,andalsoin[11]for
d

1.
Itwillbeprovedthatforawellchosen
u

[

M,M
],thereexists
T
0
>
0,
suchthatforall
ρ>
0andforall0
<t<T
0
,
S
t
(
B

(
u,ρ
))isnotasub-
1,ssetof
W
loc
(
R
x
d
)forall
s>α
sup,where
B

(
u,ρ
))=
{
u

L

(
R
d
,
R
)
,
k
u

u
k
L

(
R
d
,
R
)

}
.

4

STE´PHANEJUNCA

Highoscillatingsolutionsof(1.1)areusedforthispurposeNearaconstant
stateandfor
L

data,acompletestudyofcriticalgeometricopticsforweak
entropysolutionsisdonein[4].Nearasmooth(nonconstant)solution,an-
otherfeaturesaregivenin[29].Here,resultsof[4]aresimplifiedandprovedfor
particularsuper-criticalhighlyoscillatingclassicalsolutions(withoutshocks
onastrip).Thisallowstogiveproofof(1.7).
Considertheproblem(1.1)withoscillatinginitialdata:
ε

v

x

(1.8)
u
ε
(0
,
x
)=
u
0
(
x
):=
u
+
εU
0
ε
γ
,
where
U
0
(
θ
)isaoneperiodicfunctionw.r.t.
θ
,
γ>
0,
u
isaconstantground
state,
u

[

M,M
],
v

R
d
.Thecase
γ
=1istheclassicalgeometricoptics
forquasilinearequations,see[35,17,25,27].Inthispaperwefocuson
critical
oscillations
when
γ>
1.
Oneofthetwofollowingasymptoticexpansions(1.9)or(1.10),isexpected
in
L
l
1
oc
(]0
,
+

[
×
R
d
,
R
)fortheentropy-solution
u
ε
ofconservationlaw(1.1)
withhighlyoscillatingdata(1.8)when
ε
goesto0,
φ
(
t,
x
)
(1.9)
u
ε
(
t,
x
)=
u
+
εUt,
γ
+
o
(
ε
)
ε(1.10)or
u
ε
(
t,
x
)=
u
+
εU
0
+
o
(
ε
)
,
where
Z
theprofile
U
(
t,θ
)satisfiesaconservationlawwithinitialdata
U
0
(
θ
),
1U
0
=
U
0
(
θ
)

andthephase
φ
satisfiestheeikonalequation:
0(1.11)

t
φ
+
a
(
u
)

r
x
φ
=0

(0
,
x
)=
v

x
,
i.e.
φ
(
t,
x
)=
v

(
x

t
a
(
u
))
.
Thebehaviordescribedby(1.9)isalackofcompactnessforthesemi-group
S
t
ofequation(1.1)sincethissemi-grouppropagatesdatauniformlybounded
1,γ/1in
W
loc
withoutimprovingtheSobolevexponent(aswewillseeattheend
ofthispaperinthesection5).
Otherwise,if
γ
istoobig(
γα
sup
>
1aswewillseebelow)andtheinitial
oscillatingdataarenotconstant,thenthehighoscillationsarecanceledfor
positivetime.Behavior(1.10)meansthatanonlinearsmoothingeffectis
associatedforthesemi-groupofequation(1.1).
Combiningthesetwopossiblebehaviorsofhighlyoscillatingsolutionsthe
articlehighlightstheconjecture.Indeedthisallowtoprovethatthemaximal
smoothingeffectconjecturedbyLions,PerthameandTadmorin[34]cannot
beexceeded:(1.7).Furthermore,Theorem5.1belowshowsthatthereexistsa
familyofwellchoseninitialdatain
W
lso,c
1
with
s
=
α
supsuchthattheassoci-
atedsolutionskeepthesameuniformboundwithoutanyimprovementofthe
Sobolevexponent
s
=
α
sup.Neverthelessthecompleteconjecture:equality
(1.6),isstillanopenproblem.

OSCILLATIONSANDSMOOTHINGEFFECTFORCONSERVATIONLAWS5

OnotherhandthismaximalSobolevexponentisnotsufficienttogetsome
tracesforentropicsolutionsonsetswithco-dimension-one.Thisseemsto
contradictthestructureofaBVfunctionofentropysolutionsobtainedbyDe
Lellis,OttoandWestdickenbergin[14].Indeed,thisBVstructureforone
solutionandSobolevboundsforasetofsolutionsaredifferentapproachof
thesmoothingeffects.Seeforinstance[6,11,14,15,16,31,41]wherethe
tracespropertiesortheSobolevexponentofentropysolutionsarestudied.In
fact,thismeansthatthemaximalSobolevexponentforentropysolutionsdoes
notgiveenoughinformationaboutthefinestructureforentropysolutionsof
conservationlaws.

Thepaperisorganizedasfollows.Insection2examplesofsuper-critical
highlyoscillatingsolutionsareexpounded.Insection3,theconceptofflux
non-linearityisclarifiedandcharacterized.Section4isdevotedtogetoptimal
Sobolevestimatesonoscillatingsolutionsbuiltinsection2.Finally,thesection
5highlightsconjecture(1.6).

2.
Highfrequencywaveswithsmallamplitude
Thesection2dealswithhighlyoscillatinginitialdatanearaconstantstate
(1.8)onlyuniformlyboundedin
W
1
/γ,
1
(seesection4below).Thepropagation
ofsuchoscillatingdataisobtainedunderthecrucialcompatibilitycondition
(2.1)below.Otherwise,ifthethecompatibilitycondition(2.1)isnowhere
satisfied,thenonlinearsemi-groupassociatedtoequation(1.1)cancelsthis
toohighoscillations,seeTheorem2.2.Thevalidityorinvalidityofassumption
(2.1)isakeypointtocharacterizenonlinearfluxinsection3.

Theorem2.1.[Propagationofsmoothhighoscillations]
Let
γ
belongsto
]1
,
+

[
andlet
q
betheintegersuchthat
q

1


q
.
Assume
F
belongsto
C
q
+3
(
R
,
R
d
)
,
U
0

C
1
(
R
/
Z
,
R
)
,
v
6
=(0
,
∙∙∙
,
0)
and
(2.1)
a
(
k
)
(
u
)

v
=0
,k
=1
,
∙∙∙
,q

1
thenthereexists
T
0
>
0
suchthat,forall
ε

]0
,
1]
,thesolutionsofconservation
law
(1.1)
withinitialoscillatingdata
(1.8)
aresmoothon
[0
,T
0
]
×
R
and
φ
(
t,
x
)
u
ε
(
t,
x
)=
u
+
εUt,
γ
+
O
(
ε
1+
r
)
in
L

([0
,T
0
]
×
R
d
)
,
εwhere
0
<r
=1
ifγ
=
q,
andthesmoothprofile
U
isuniquely
q

γelse,
determinedbytheCauchyproblem
(2.2)
,
(2.3)
,
φ
isgivenbytheeikonalequa-
tion
(1.11)
:
q
+1

1

(
q
)

(2.2)

t
U
+
b∂U
=0
,b
=
(
q
+1)!
a
(
u
)

v
ifγ
=
q,
∂θ
0
else.
(2.3)
U
(0

)=
U
0
(
θ
)
.

6

STE´PHANEJUNCA

WedealwithsmoothsolutionstocomputelaterSobolevbounds.Indeed
theasymptoticstaysvalidaftershocksformationandforallpositivetimebut
in
L
l
1
oc
insteadof
L

,see[4].
When
γ
=1,wedonotneedassumption(2.1).Itistheclassiccasefor
geometricoptics,see[35,17,25,26,27,28].
Indimension
d

2,itisalwayspossibletofindanontrivialvector
v
satisfying(2.1).Atleastfor
γ
=2,(2.1)reducestofind
v
6
=0suchthat
a
0
(
u
)

v
=0.Thus,suchsingularsolutionsalwaysexistsindimensiongreater
thanone.But,forgenuinenonlinearonedimensionalconservationlaw,there
isneversuchsolution.Ofcourse,weassume
U
0
nonconstantand
F
nota
linearfunctionnear
u
,elsetheTheoremisobvious.If
U
0
isconstant
u
ε
too.
If
F
islinearon[
u

δ,u
+
δ
]forsome
δ>
0,highoscillationspropagatefor
alltimewhen
ε
.
InfactTheorem2.1expressesakindofdegeneracyof
multi
dimensional
scalarconservationlaws.Thisdegeneracy(periodsmallerthantheamplitude)
appearsforquasilinearsystemswithalinearlydegenerateeigenvalue[7,8,9,3],
andforlinear[32]andsemi-linearsystems[25,26,28].
Noticethatfor
γ>
1,smoothsolutionsexistforlargertimethanitiscur-
rentlyknown[13,33]:
T
ε

1
/
|r
x
u
0
ε
|∼
ε
γ

1
.Furthermoreequation(2.2)is
nonlinearonlyif
γ

N
and
a
q
(
u
)

v
6
=0.
Proof:
FirstoneperformsaWKBcomputationswithfollowingansatz:
(2.4)
u
ε
(
t,
x
)=
u
+
εU
ε
t,φ
(
t,
x
)
,
γεq
X
+1
U
k
F
(
u
ε
)=
ε

F
(
k
)
(
u
)+
ε
q
+2
G
εq
(
U
ε
)
,
!k0=kZ
1
(1

s
)
q
+1
G

(
U
)=
U
q
+2
F
(
q
+2)
(
u
+
sεU
)
ds,
0
(
q
+1)!
g

(
U
)=
v
.G

(
U
)
,

t
U
ε
t,φ
(
t,
x
)=

t
U
ε

ε

γ
(
a
(
u
)

v
)

θ
U
ε
γεq1+kXdiv
x
F
(
u
ε
)=
ε
k
+1

γ

θ
U
ε
a
(
k
)
(
u
)

v
+
ε
q
+2
div
x
G
ε
(
U
ε
)
qk
=0
(
k
+1)!
=
ε
1

γ
(
a
(
u
)

v
)

θ
U
ε
+
ε
q
+1

γ
c
q

θ
U
εq
+1
+
ε
q
+2

γ

θ
g

(
U
ε
)
,
)q(where
c
q
=
a
(
q
+(
u
1))!

v
.Thensimplificationyields
(2.5)

t
u
ε
+div
x
F
(
u
ε
)=
ε


t
U
ε
+
ε
q

γ
c
q

θ
U
εq
+1
+
ε
1+
q

γ

θ
g

(
U
ε
)

.
Itsufficestotake
U
ε
solutionoftheonedimensionalscalarconservationlaws
with
ψ
ε
(
U
)=
ε
q

γ
c
q
U
q
+1
+
ε
1+
q

γ

θ
g

(
U
ε
)
(2.6)

t
U
ε
+

θ
ψ
ε
(
U
ε
)=0
,U
ε
(0

)=
U
0
(
θ
)
.

OSCILLATIONSANDSMOOTHINGEFFECTFORCONSERVATIONLAWS7

Noticethat
ψ
ε
=
O
(1)

C
l
1
oc
.If
γ
6
=
q
,
ψ
ε
isevensmaller:
ψ
ε
=
O
(
ε
r
)

C
l
1
oc
.
Bythemethodofcharacteristics,foreach
ε
,theexistenceofasmoothsolution
isobviousfortimeororder1
/
|
U
0
0
|
,morepreciselyasmoothsolutionsexists
onamaximalinterval
T
ε
,whereafirstshocklocatesatthetime
t
=
T
ε
(whichalwaysoccursiftheinitialperiodicdataisnonconstantand
c
q
6
=0).
Furthermore,
ψ
ε
dependssmoothlyof
ε
sowecantakeanuniformtimefor
all
ε

[0
,
1].Whichprovetheexistenceof0
<T
0
<T

=min
{
T
ε


]0
,
1]
}
for
u
ε
.Indeed,
T
ε
=

1
/
min
{
ψ
ε
00
(
U
0
(
θ
))
U
0
0
(
θ
)


[0
,
1]
}
.Let0
<T
0
<T

,
theoneperiodicfunctionw.r.t.
θ
,
U
ε
belongsto
C
1
([0
,T
0
]
×
R
/
Z
)and
u
ε
iswelldefinedby(2.4).Byconstruction
u
ε
satisfies(1.1),(1.8)and
u
ε

C
1
([0
,T
0
]
×
R
d
)forall0


1.
Therearetwocases:
γ
isanintegerornot.
q
=
γ
:From(2.5)and(1.1)weget

t
U
ε
+

θ
c
q
U
εq
+1
+
εg

(
U
ε
)=0
,∂
t
U
+
c
q

θ
U
q
+1
=0
,
U
ε
(0

)=
U
0
(
θ
)
,U
(0

)=
U
0
(
θ
)
.
Theclassicalmethodofcharacteristics,gives
C
1
characteristics,
C
1
solutions
dnak
U
ε

U
k
C
1
([0
,T
0
]
×
R
d
)
=
O
(
ε
)
,
erehwk
U
k
C
1
([0
,T
0
]
×
R
d
)
=
k
U
k
L

([0
,T
0
]
×
R
d
)
+
k

t
U
k
L

([0
,T
0
]
×
R
d
)
+
k

θ
U
k
L

([0
,T
0
]
×
R
d
)
.
integer
q>γ
:Theproofissimilarexcepttheterm
ε
r
c
q

θ
(
c
q
U
q
+1
)becomes
aremainder,with
r
=
q

γ
and
U
(
t,θ
)=
U
0
(
θ
),thus
k
U
ε
(
.,.
)

U
0
(
.
)
k
C
1
([0
,T
0
]
×
R
d
)
=
O
(
ε
r
)
.

Ifcondition(2.1)isviolated,oscillationsareimmediatelycanceled.
Theorem2.2.[Cancellationofhighoscillations,
[4]
]
Let
F
belongsto
C
q
+1
and
U
0

L

(
R
/
Z
,
R
)
,where
q

1


q
isdefined
inTheorem2.1.Ifforsome
0
<j<q
(2.7)
a
(
j
)
(
u
)

v
6
=0
thenthesolutions
u
ε
ofconservationlaw
(1.1)
withinitialoscillatingdata
(1.8)
for
ε

]0
,
1]
satisfywhen
ε

0
u
ε
(
t,
x
)=
u
+
εU
0
+
o
(
ε
)
in
L
l
1
oc
(]0
,
+

[
×
R
d
)
.
Obviouslytheinterestingcaseiswhen
U
0
isnonconstant.Inthiscontext,
when
U
0
issmoothandnonconstantthefirsttimewhenashockoccurs
T
ε

0
when
ε

0.Thussolutionsareweakentropysolutions.
Theproofisinthespiritof[4]andusesaveraginglemmas.Anotherproofis
possiblewithanotherargument:thedecayofperiodicsolutionsforlargetime,
see[5,10,12,13,18,22]aboutthisdecay.

8

STE´PHANEJUNCA

Proof:
Fornonconstantinitialdataitisimpossibletoavoidshockwaves
onanyfixedstrip[0
,T
0
]
×
R
d
with
T
0
>
0asinthepreviousproofofTheorem
2.1sincethetimespanofsmoothsolutionsis
ε
β
where
β
=
γ

j>
0.
First,withachangeofspacevariable
x

x

t.
a
(
u
),assumethat
a
(
u
)=0.
jLet
W
ε
(
t,θ
)=
U
ε
(
t,εθ
)where
U
ε
isdefinedin(2.4).Then
W
ε
satisfiesthe
onedimensionalconservationlaws:
(2.8)

t
W
ε
+

θ
c
j
W
εj
+1
+
εg

(
W
ε
)=0
,W
ε
(0

)=
U
0
(
ε

β
θ
)
,
Since
W
ε
(0
,.
)convergesweaklytowards
U
0
,and
W
ε
isrelativelycompactin
1L
loc
thankstoaveraginglemmas,
W
ε
convergestowardstheuniqueentropy
solution
W
of

t
W
+
c
j

θ
W
j
+1
=0
,W
(0

)=
U
0
.
Thatistosaythat
W
(
t,θ
)

U
0
.Then
v
ε
(
t,
x
)=
W
ε
(
t,ε

β
v

x
)converges
towards
U
0
in
L
l
1
oc
whichconcludestheproof.

3.
Characterizationofnonlinearflux
Theregularizingeffectgivenin[34]isrelatedwiththesharpexponent
α
=
α
supquantifyingpreciselythenon-linearityoftheflux.Therearesome
exampleswhere
α
iscomputedindimensiontwoin[34,39]andsomeremarks
in[2,23,24].Forthefirsttime,weobtain“thesharp
α
”forallsmoothflux
inTheorem3.1.
Anothergenuinenonlineardefinitionforthefluxdependingonthespace
dimension
d
,relatedtoweaklynonlineargeometricoptics,isgivenin[4].We
generalizethedefinitionfrom[4].ThankstoTheorem3.1below,thisnew
definitionisequivalenttotheclassicaldefinition1.1.
Furthermorethedefinition3.1impliesthestrictnonlinearitydefinedin[18].
Thiswillbeprovedattheendofthissectionwithotherrelatedresults.
Definition3.1.[Nonlinearflux]
Lettheflux
F
belongsto
C

(
R
,
R
d
)
and
I
=[
c,d
]
,
c<d
.Thefluxissaidto
benonlinearon
I
if,forall
u

I
,thereexists
m

N

suchthat
(3.1)
rank
{
a
0
(
u
)
,
∙∙∙
,
a
(
m
)
(
u
)
}
=
d.
Furthermore,thefluxissaidtobegenuinenonlinearif
m
=
d
isenoughin
(3.1)
forall
u

I
.
Asusual,thenon-linearityisamatterofthesecondderivativesof
F
:
F
00
=
a
0
.
Noticethat
m

d
,thusthegenuinenonlinearcaseisthestrongestnonlinear
case.Thegenuinenonlinearcasewasfirststatedin[4],seecondition(2.8)and
Lemma2.5p.447therein.Thegenuinenonlinearconditiondet(
a
0
,
∙∙∙
,
a
(
d
)
)
6
=
0formultidimensionalconservationslawswasalsoin[11],seecondition(16)
p.84therein.

OSCILLATIONSANDSMOOTHINGEFFECTFORCONSERVATIONLAWS9

Thesimplestexampleofvelocity
a
associatedwithagenuinenonlinearflux
F
isgivenin[4,11,2]:

2
d
+1

a
(
u
)=(
u,u
2
,
∙∙∙
,u
d
)with
F
(
u
)=
u,
∙∙∙
,u.
1+d2Thedefinition3.1ismoreexplicitwithfollowingintegerswith
I
=[

M,M
].
(3.2)
d
F
[
u
]=inf
{
k

1
,rank
{
F
00
(
u
)
,
∙∙∙
,
F
(
k
+1)
(
u
)
}
=
d
}≥
d,
(3.3)
d
F
=sup
|
u
|≤
M
d
F
[
u
]
∈{
d,d
+1
,
∙∙∙}∪{
+
∞}
.
Indeedthedefinition3.1statesthatthefluxisgenuinenonlinearwhen
d
F
reachesisminimalvalue,
d
F
=
d
.
Conversely,iftheflux
F
islinear,then
a
isaconstantvectorin
R
d
and
d
F
reachesismaximalvalue,
d
F
=+

.
Between
d
F
=
d
and
d
F
=+

,thereisalargevarietyofnonlinearflux.
Thefollowingtheoremgivestheoptimal
α
,(1.3),forsmoothflux.
Theorem3.1.[Sharpmeasurementofthefluxnon-linearity]
Let
F
beasmoothflux,
F

C

([

M,M
]
,
R
d
)
,themeasurementoftheflux
non-linearity
α
sup
isgivenby
11α
sup=
d

d.
FFurthermore,if
α
sup
>
0
thereexists
u

[

M,M
]
suchthat
d
F
=
d
F
[
u
]
.
Asimilarresultforthegenuinenonlinearcase:
d
F
=
d
,canbefoundin[2].
ThisTheoremisapowerfultooltocomputetheparameter
α
sup,forin-
stance:

if
F
(
u
)=(cos(
u
)
,
sin(
u
))then
F
isgenuinenonlinearand
α
sup=1
/
2
sincedet(
F
00
(
u
)
,F
000
(
u
))=1.

if
F
ispolynomialwithdegreelessorequaltothespacedimension
d
then
α
sup=0and
F
doesnotsatisfydefinition3.1.
Itiswellknownthatthe“Burgersmulti-D”flux
F
(
u
)=(
u
2
,
∙∙∙
,u
2
)is
notnonlinearwhen
d

2.Thesequence

ofosc

illationswithlargeam-
plitude(
u
ε
)
0


1
givenby
u
ε
(
t,
x
)=sin
x
1
ε

x
2
givesusglobalsmooth
solutionswhilethesequence(
u
ε
)
0


1
blowsupinany
W
lso,c
1
forall
.0>s•
If
F
ispolynomialsuchthat
deg
(
F
i
)

2forall
i
andalldegreesare
1distinct,then
F
isnonlinearand
α
sup=
0
.
m
i
ax
deg
(
F
i
)
ForsmoothFluxtheoptimal
α
istheinverseofaninteger.Notallvalue
of
α
arepossiblefor
F

C

.Withlesssmoothflux,othervaluesof
α
are
possible,see[34,39,24].
Ageometricapproach,likeMorsetheoryispossibletoproveTheorem3.1,
seeasuggestionin[20].Wechooseanotherapproachsimilartosomeproofs
ofphasestationnarylemmas,see[38,2,24].

01

STE´PHANEJUNCA

TheproofofTheorem3.1needssomelemmas.FirstwerecallaLemma
from[2,30]givingtheoptimal
α
forfunctionson
R
.
Let
ϕ

C

([
c,d
]
,
R
)and
v

[
c,d
],themultiplicityof
ϕ
on
v
isdefinedby
m
ϕ
[
v
]=inf
{
k

N

(
k
)
(
v
)
6
=0
}∈
N
=
N
∪{
+
∞}
.
Itmeansthatif
k
=
m
ϕ
then
ϕ
(
k
)
(
v
)
6
=0and
ϕ
(
j
)
(
v
)=0for
j
=0
,
1
,
∙∙∙
,k

1.
Forinstance
m
ϕ
[
v
]=0means
ϕ
(
v
)
6
=0;
m
ϕ
[
v
]=1means
ϕ
(
v
)=0,
ϕ
0
(
v
)
6
=0
and
m
ϕ
[
v
]=+

means
ϕ
(
j
)
(
v
)=0forall
j

N
.
Setthemultiplicityof
ϕ
on[
c,d
]by
m
ϕ
=sup
v

[
c,d
]
m
ϕ
[
v
]

N
.
Lemma3.1
([2,30])
.
Let
ϕ

C

([
c,d
]
,
R
)
with
c<d
,and
Z
(
ϕ,ε
)=
{
v

[
c,d
]
,
|
ϕ
(
v
)
|≤
ε
}
.
If
0
<m
ϕ
<
+

thenthereexists
C>
1
suchthat,forall
ε

]0
,
1]
,
(3.4)
C

1
ε
α

meas
(
Z
(
ϕ,ε
))


α
with
α
=1
.
mϕThecase
m
ϕ
=0isnotinterestingsincethereisnozerointhissituation.
Quantity
m
ϕ
ispositivesimplymeansthattheset
Z
(
ϕ,
0)ofrootsof
ϕ
isnot
empty.
Proof.
Sinceanyrootof
ϕ
hasafinitemultiplicity,thecompactset
Z
(
ϕ,
0)
isdiscreteandthenfinite:
Z
(
ϕ,
0)=
{
z
1
,
∙∙∙
,z
ν
}
.Foreach
z
i
and
h>
0,let
V
i
(
h
)be]
z
i

h,z
i
+
h
[

[
c,d
].Forany0
<h<
|
b

a
|
,wehave
h

meas(
V
i
(
h
))

2
h.
Foranyroot
z
i
,thereexists
h
i

]0
,
|
b

a
|
[,
A
i
>
0and
δ
i
>
0suchthat
(3.5)
δ
i
|
h
|
k
i
≤|
ϕ
(
z
i
+
h
)
|≤
A
i
|
h
|
k
i
forall
h

V
i
(
h
i
)
,
with
k
i
=
[
m
ϕ
[
z
i
].Thisisadirec

tconsequenceofTa

ylor-Lagrangeformula.
Let
V
be
V
i
(
h
i
)and
ε
0
=min1
,
min
v

[
c,d
]
\
V
|
ϕ
(
v
)
|
.Bythecontinuityof

onthecompactset[
c,d
]
\
V
,
ε
0
ispositive.Thenforall0
<ε<ε
0
,we
have
Z
(
ϕ,ε
)

V
.If
ε
≥|
ϕ
(
z
i
+
h
)
|
for
|
h
|
<h
i
,thenfrom(3.5),wehave
(
ε/δ
i
)
1
/k
i
≥|
[
h
|
.Thislastinequalityimpliesfor0
<ε<ε
0

1that
Z
(
ϕ,ε
)is
asubsetof
V
i
((
ε/δ
i
)
1
/k
i
)andthen
iX
ν
X
ν
!
meas(
Z
(
ϕ,ε
))

2(
ε/δ
i
)
1
/k
i

2
δ
i

1
/k
i
ε
1
/m
ϕ
.
i
=1
i
=1
Itgivesinequality(3.4).Toobtaintheoptimalityof
α
,let
z
j
bearootof
ϕ
with
maximalmultiplicityi.e.
m
ϕ
[
z
j
]=
m
ϕ
=
k
.Againfrom(3.5),
V
j
((
ε/A
j
)
1
/k
)is
asubsetof
Z
(
ϕ,ε
)forall
ε

]0

0
[.Thenwehave(
ε/A
j
)
1
/k

meas(
Z
(
ϕ,ε
)),
whichisenoughtogettheoptimalityof
α
=1
/k
andconcludestheproof.