HIGH FREQUENCY WAVES AND THE MAXIMAL SMOOTHING EFFECT

Description

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

Sujets

• High frequency
• Conservation law

Informations

• Publié par: pefav
• Nombre de lectures: 73
• Langue: English

Extrait

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
)