HIGH FREQUENCY WAVES AND THE MAXIMAL SMOOTHING EFFECT

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
)