scalar conservation laws in several space dimensions Exact controllability to trajectories for entropy solutions to C.R. Acad. Sci. Paris, Ser.I

(1)

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Partial differential equations

Exact controllability to trajectories for entropy solutions to scalar conservation laws in several space dimensions

Contrôlabilité exacte aux trajectoires pour des lois de conservation scalaires multidimensionnelles

Carlotta Donadello

^a

, Vincent Perrollaz

^b

aUniversitédeBourgogneFranche-Comté,Laboratoiredemathématiques,CNRSUMR6623,16,routedeGray,25000Besançon,France bUniversitédeTours,InstitutDenis-Poisson,CNRSUMR7013,ParcdeGrandmont,37000Tours,France

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received13July2018

Acceptedafterrevision4February2019 Availableonline14February2019 PresentedbyJean-MichelCoron

We describeanew methodthat allowsus to obtainaresult ofexact controllabilityto trajectoriesofmultidimensionalconservationlawsinthecontextofentropysolutionsand underamerenon-degeneracyassumptionontheﬂuxandanaturalgeometriccondition.

©²⁰¹⁹Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.Âll^rights^reserved.

r é s um é

On décrit dans cet article une nouvelle méthode permettant d’obtenir un résultat de contrôlabilitéexacteauxtrajectoirespour desloisde conservationscalaires enplusieurs dimensionsd’espacedanslecadredessolutionsentropiquesetsousunesimplehypothèse denon-dégénérescenceduﬂuxetunehypothèsegéométriquenaturelle.

©²⁰¹⁹Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.Âll^rights^reserved.

1. Introduction

Inthispaper,weconsiderascalarconservationlawinseveralspacedimensions,i.e.apartialdifferentialequationofthe form

∂

tu

+

^divx

(

f

(

u

)) =

⁰

,

t

∈ R

⁺

,

x

∈ ⊂ R

^d

,

d

≥

¹

,

(1)

where

isanopensetwithsmoothboundary(C² îs^sufficient),ând ^f^,^the^flux^function,îsⁱⁿC¹

(

R

,

R^d

)

.

Weareinterestedinthefollowingcontrollabilityproblem.Givenaninitialdatumu0∈^L^∞

()

,asuitabletargetproﬁleu1, andapositivetime T,weaimatconstructinganentropyweaksolutionu∈^L^∞

(R

⁺×R^d;R)^of

E-mailaddresses:[email protected](C. Donadello),[email protected](V. Perrollaz).

https://doi.org/10.1016/j.crma.2019.01.012

1631-073X/©²⁰¹⁹Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.Âll^rights^reserved.

(2)

⎧ ⎪

⎨

⎪ ⎩

∂

tu

+

^divx

(

f

(

u

)) =

⁰

,

in

(

0

,

T

) × ,

u

(

0

,

x

) =

^u0

(

x

),

on

,

u

(

T

,

x

) =

^u1

(

x

),

on

,

(2)

byusingtheboundarydataon

(

0

,

T

)

×

∂

ascontrols.

Givenanyextensivequantityudeﬁnedonadomain

,suchasmassorenergy,aconservationlawforutranslatesinto apartialdifferentialequationthephysicalobservationthatthetotalamountofuin

changesataratethatcorrespondsto thenetﬂuxofu, f

(

u

)

,throughtheboundary

∂

.Thiskindofequationsiswidelyusedinmodelingphenomenasuchasgas dynamics(Eulerequations),electromagnetism,magneto-hydrodynamics,shallowwater,combustion,roadtraﬃc,population dynamics,andpetroleumengineering.

It iswell knownthat evenstarting frominitialdatain Cc^∞

(R

^d

)

,the classicalsolutions to(1) candevelop singularities (jumpdiscontinuities)inﬁnitetime,see[20] foraverycompleteintroductiontotheﬁeld.

ThemostgeneralwellposednessresultforclassicalsolutionstotheCauchyproblemstatesthat,foranyinitialdatumu0 in H^s,withs

>

1+^d₂^,^there^exists^a^solution^to^{(1) in}C⁰

([

⁰

,

T],^H^s

)

∩C¹

([

⁰

,

T],^H^s⁻¹

)

,whoselifespanT canbeestimated dependingon f andu0.

However, mostoftheliteraturedevoted toconservationlawsfocusesonaclass ofweak (distributional)solutionsthat satisﬁesan additionalselectioncriterium, necessarytoensureuniqueness, calledentropycondition.Inthecaseofascalar conservationlawinseveralspacedimensions,acompletewellposednesstheoryforentropysolutionstotheCauchyproblem isduetoKruzhkov[25].

The problemwe aim atsolving,see (2), can be addressedboth inthe frameworkof classicalor ofentropysolutions.

In the ﬁrst case, controls, besides driving the state to the target, are also responsible for preventing the formation of singularities.Severalresultsexistinthisframework,see[13],[28],and[16] forasurvey.Unfortunately,thisapproachdoes not allowone toattain manyphysicallyrelevantstatesinvolvingjumpdiscontinuities andleadsto controlstrategiesthat are ingeneralnot veryrobust.Indeedvery smallperturbations ofthecontrolmightleadto blowupofthederivativesof thesolutionbeforetimeT.

In the presentpaper,we are interested inthe controllabilityof entropysolutions. The literature inthis framework is less abundant also due to specific technical difficulties, even if we can notice a growing interest of researchers in this field. The classical methodology forexact controllabilityrelies heavily on linearization, whichis not possible (orat least horribly technical) anymore arounddiscontinuous solutions. Moreover,Bressan andCoclite showedin[12] that nonlinear conservationlawsmayfailthelineartest.Indeed,theyprovidedasystemforwhichthelinearizedapproximationarounda constantstateiscontrollable,whiletheoriginalnonlinearsystemcannotreachthatsameconstantstateinfinitetime.

A separate issueis related to theirreversibility of entropysolutions: the set ofadmissible target states in time T is reducedanditsdescription,ofteninvolvinga numberofhighly technicalconditions,isinitselfan openprobleminmost cases,see[3,4,6,7,14].

In theexisting literature, we candistinguish essentiallythree approachestoward thestudyofexact controllabilityfor conservationlawsinonespacedimension(considerequation(1) withd=^1).

Starting from the pioneering work by Ancona and Marson [4], several results have been obtained using the theory of generalizedcharacteristics introduced by Dafermos in[19], as[4,7,14,24,32] orthe explicitLax–Oleinik representation formula, as[1,6]. The latter technique is applicable only when the flux function f is strictly convex/concave, while the theory of generalizedcharacteristics coversalso the(slightly) more generalcaseof aflux function f with oneinflection point.

The returnmethodintroduced by Coron[16] isthebasis ofthe approachdevelopedby Horsinin[24] and, combined withtheclassicalvanishingviscositymethod,playsakeyrolein[23] andintheonlypapertoourknowledgeinwhichthe fluxfunction f isallowedtohaveafinitenumberofinflectionpoints[26].

Theasymptoticstabilizationofentropysolutionstoscalarconservationlawsisthetopicof[10,33,34].

Theonlyavailabletoolforinvestigatingtheexactcontrollabilityofsystemsofconservationlawsinonespacedimension isthewavefronttrackingalgorithm[11],whichhasbeensuccessfullyappliedin[3,12,21,22,29].

Theasymptoticstabilizationofentropysolutionstosystemshasbeenstudiedin[5,9,18].

Itseemsdifficulttoinvestigatetheexactcontrollabilityofentropysolutionsofscalarconservationlawsinseveralspace dimensionsusingthetechniquesdesignedfortheone-dimensionalcase.Inthepresentpaper,weproposeanewapproach, inspiredbytheworkonstabilizationbyCoron[15] andbyCoron,Bastin,andd’AndéaNovel[17];seealsothemonography [9] foracomprehensivepresentationofthemethod.Theconditionsweimposeonthefluxfunctionaretechnicalandwill bedetailedinthenextSection,butwestressthatinthespecialcased=^{1 they}âre^not^related^to^convexity^(orconcavity).

Thismeans that,eveninthe one-dimensionalcase,ourresultisnew,although forthiscasestrongerresultsareavailable undermorerestrictivehypotheses.

Theﬁrstofourconditions,calledlaternondegeneracycondition,saysthattherangeofudoesnotcontainanyintervalon which f isaﬃne.Thisconditionisnecessarytoensuretheexistenceoftracesattheboundaryof

,see[35].

The secondcondition, calledlaterreplacementcondition,involves f togetherwithT and

.Roughlyspeaking,oncewe reducetotheone-dimensionalcase,itsaysthatallgeneralizedcharacteristicsissuedfrompoints

(

t

,

x

)

in{⁰}×

leavethe cylinder

(

0

,

T

)

×

beforetime T,sothatthedynamicsinthedomainonlydependsontheboundarydataandnotonthe initialdatafort largeenough.

(3)

2. Preliminarydeﬁnitionsandnotations

Inthefollowing,u→^sign

(

u

)

isthefunctiongivenby

∀

u

∈ R,

sign

(

u

) :=

⎧ ⎪

⎨

⎪ ⎩

1 ifu

>

0

,

0 ifu

=

⁰

,

−

^{1 if}^u

<

0

,

·|·^denotes^the^scalar^product^between^two^vectors^and

χ

E istheindicatorfunctionoftheset E.

Deﬁnition2.1.Given f∈C¹ R;R^d

andu0∈^L^∞

()

,weconsidertheequation

∂

tu

+

^div

(

f

(

u

)) =

⁰

,

for

(

t

,

x

) ∈ (

0

, +∞ ) × ,

(3)

supplementedwiththeinitialcondition

u

(

0

,

x

) =

u0

(

x

),

forx

∈ .

(4)

A function u∈^L^∞

(

[⁰

,

+∞

)

×

)

is an entropy solution to (3)–(4) in [⁰

,

T

)

×

if, for any real number k and any non-negativefunction

φ

inC¹c

([

⁰

,

+∞)×

)

,wehave

(0,T)×

|

^u

(

t

,

x

) −

^k

| ∂

_t

φ (

t

,

x

) +

^sign

(

u

(

t

,

x

) −

^k

)

^f

(

u

(

t

,

x

)) −

^f

(

k

) |∇ φ (

t

,

x

)

^dt^dx

+

sign

(

u0

(

x

) −

k

)φ (

0

,

x

)

dx

≥

0

.

(5)

Wewillalsosaythatafunctionuisanentropysolution(withoutreferringtoanyinitialdata)in

(

0

,

+∞)×

whenit satisﬁes(5) foranynon-negative

φ

∈C¹c

((

0

,

+∞

)

×

)

.

Wenowintroduceasimplegeometricconditionwhichissuﬃcient(thoughclearlynotnecessary)toobtainourcontrol- labilityresult.

Deﬁnition2.2. Let

be a smoothopen set ofR^d^, Î ^beâ ^segmentôf Rând ^f :R→R^d â C¹ ^flux ^function.^We ^say^that thetriple

(

f

, ,

I

)

satisﬁesthereplacementconditionintimet

>

0 ifthereexistsavector w∈R^d^and^a^positive^number^c suchthat

L

:=

^sup

x∈

^w

|

^x

−

^inf

x∈

^w

|

^x

< +∞ ,

(6)

∀

^u

∈

^I

,

^f

(

u

)|

^w

≥

^c

,

and

t =

^L

c

.

(7)

Deﬁnition2.3.Wesaythattheﬂux f isnon-degenerateif,foranycouple

( τ , ζ )

∈R×R^d\ {

(

0

,

0

)

}^,^we^have L

{

^z

∈ R : τ ₊ ζ |

^f

(

z

) =

⁰

}

=

⁰

,

whereL^is^the^Lebesgue^measure.

Wecannowstateourmaintheoremonexactcontrollabilitytotrajectoriesforaconservationlawinanyspacedimension.

Theorem1.Letv∈C⁰

((

0

,

+∞

)

;^L¹

())

∩^L^∞

((

0

,

+∞

)

×

)

beanentropysolutionto(3)andu0beafunctioninL^∞

()

. Wesupposethatbothu0andv takevaluesinasegmentI suchthat

(

f

, ,

I

)

satisﬁesthereplacementconditionintimet.Wealso supposethattheﬂuxf isnon-degenerate.

Then,foranytimesT1andT2largerthant,thereexistsanentropysolutionu to(3)satisfying

u

(

0

,

x

) =

^u0

(

x

),

u

(

T₁

,

x

) =

^v

(

T₂

,

x

)

for almost every x

∈ .

Remark 1.Forthesakeofsimplicity,weomittowriteheretheexactformofthecontrolsweuse.InthenextSection,we preciseinwhichsensetheboundaryconditionson

∂

aretakenintoaccountbyentropysolutionsandinthelastSection, intheproofofTheorem1,wewriteourcontrolsinafullyexplicitway.

(4)

Remark 2. Acharacterization ofthe setofadmissible target profilesat fixedtime T ≥^{0 for}â ^scalar conservationlawin several space dimensionsis not available inthe literature. We stress, however,that in the statement of Theorem 1, we reallyneedtoassumethat v isanentropysolutiononthecylinder

(

0

,

+∞

)

×

becausethecompleteknowledgeof v is necessaryinourproof.

3. Boundaryconditionsandentropysolutions

We have sofar avoided theprecise formulationof boundaryconditions for(3). In general, giventhe initial boundary valueproblem

⎧ ⎪

⎨

⎪ ⎩

∂

tu

+

^div

(

f

(

u

)) =

⁰

, (

0

, +∞) × ,

u

(

t

,

x

) =

^ub

(

t

,

x

), (

t

,

x

) ∈ (

0

, +∞) × ∂ ,

u

(

0

,

x

) =

^u0

(

x

),

x

∈ ,

(8)

itsentropysolutionudoesnotsatisfytheboundaryconditionintheusualsense,asthetraceofuon

∂

doesnotcoincide withtheprescribedDirichletdatum.Thesituationiseasiertovisualizeintheone-dimensionalcase,aswe canseeinthe followingexample.

Example 1. Assume d=^1,

=

(

0

,

+∞

)

, f

(

u

)

= ^u²

2 andimpose in (8) constant initial andboundary data,u0= −^{1 and} ub= −¹

2. Then the initial condition is transportedalong characteristic curves with negative slope up to the boundary, while no characteristiccan spring out ofthe boundary itself.The trace of the solutionat x=^{0 can} ^only^take ^the^value u

(

t

,

0⁺

)

=^u0,andu_bcannotbeattained.

For thisreason, the boundaryconditions should be interpreted ina broader sense, madeprecise by Leroux[27], and Bardos,Leroux,andNédélec[8].Inthesettingoftheexampleabove,wesaythattheboundaryconditionisfulfilledinthe sense ofBardos–Leroux–NédélecassoonasthesolutiontotheRiemannproblemwithdatauL=ûb anduR=û

(

t

,

0⁺

)

only containswavesofnon-positivespeed(i.e.waveswhichdonotenterthedomain).Inthegeneralmultidimensionalcase,this takesthefollowingform.

Deﬁnition3.1.Let I

(

a

,

b

)

denotethe intervalofextrema a andb,andlet

η (

x

)

bethe outerunit normalof

∂

at

(

t

,

x

)

∈

(

0

,

T

)

×

∂

. Then we say that the boundary condition u_b in the IBVP (8) is fulﬁlled at

(

t

,

x

)

∈

(

0

,

T

)

×

∂

if for any k∈^I

(

ub

(

t

,

x

),

u

(

t

,

x

))

sign

(

u

(

t

,

x

) −

^ub

(

t

,

x

)) (

f

(

u

(

t

,

x

)) · η (

x

) −

^f

(

k

) · η (

x

)) ≥

⁰

.

Wehavetoprecise,however,thattheabovedefinitionisnotexactlytheoneweadoptinthepresentworkasexistence of tracesisnot guaranteed forthesolution to(8) in the L^∞ setting.The firstresultsdealing withthis problemwereby Otto [31],seealso[30].WeusemorerecentresultsbyAmmar,Carillo,andWittbold[2],whichbuilduponthoseideas.We alsorecalla(simplifiedversionofa)resultbyVasseur[35],showingthatifthefluxsatisfiesthenon-degeneracycondition, thenanyentropysolutionu∈^L^∞âdmitsâ^traceât^the^boundary.

Letusrecallsomedeﬁnitionsandresultsin[2].Weusethefollowingnotations.Foranyrealnumbers

α

andk,andany point x∈

∂

,wecall

η (

x

)

theouterunitnormalatxandintroduce

ω

⁺

(

x

,

k

, α ) :=

max

k≤^r,s≤^max(α,k)

|

f

(

r

) −

f

(

s

)| η (

x

)|, ω

⁻

(

x

,

k

, α ) :=

^max

min(α,k)≤r,s≤k

|

^f

(

r

) −

^f

(

s

) | η (

x

) | .

Forintegralsontheboundary,wedenotethesurfacemeasureatx∈

∂

byd

σ (

x

)

.

Deﬁnition3.2. Givena boundary condition u_b∈^L^∞

((

0

,

+∞)×

∂)

andan initial data u₀∈^L^∞

()

we saythat u isan entropysolutionto(8) whenthefollowingholdforanyk∈R^and^anynon-negativefunction

ζ

∈Cc^∞

(

[⁰

,

+∞

)

×R^d

)

T 0

(

u

(

t

,

x

) −

^k

)

⁺

∂

t

ζ (

t

,

x

) + χ

_{_u₍_t_,_x_)>_k_}

f

(

u

(

t

,

x

)) −

^f

(

k

)) |∇ ζ (

t

,

x

)

^dx^dt

+

∂

T 0

ζ (

t

,

x

) ω

⁺

(

x

,

k

,

u_b

(

t

,

x

))

dtd

σ (

x

) +

(

u₀

(

x

) −

^k

)

⁺

ζ (

0

,

x

)

dx

≥

⁰

,

(9)

(5)

T 0

(

k

−

^u

(

t

,

x

))

⁺

∂

t

ζ (

t

,

x

) + χ

_{u(t,x)<k}

^f

(

u

(

t

,

x

)) −

^f

(

k

))|∇ζ (

^t

,

x

)

^dx^dt

+

∂

T 0

ζ (

t

,

x

) ω

⁻

(

x

,

k

,

u_b

(

t

,

x

))

dtd

σ (

x

) +

(

k

−

^u0

(

x

))

⁺

ζ (

0

,

x

)

dx

≥

⁰

.

(10)

Thefollowingtwotheoremswereprovenin[2] (see[2],Theorem2.3and2.4).

Theorem2.Giveninitialandboundarydatau0∈^L^∞

()

andub∈^L^∞

((

0

,

+∞

)

×

∂)

,thereexistsauniqueentropysolution to(8).

Theorem3.Giveninitialdatau0,v0inL^∞

()

andboundarydatau_b,v_binL^∞

((

0

,

+∞

)

×

∂)

,thecorrespondingentropysolutions u andv satisfy

T 0

(

u

(

t

,

x

) −

^v

(

t

,

x

))

⁺

∂

_t

ζ (

t

,

x

) + χ

_{_u₍_t_,_x_)>_v₍_t_,_x_)}

f

(

u

(

t

,

x

)) −

^f

(

v

(

t

,

x

))) |∇ ζ (

t

,

x

)

^dx^dt

+

∂

T 0

ω

⁻

(

x

,

u_b

(

t

,

x

),

v_b

(

t

,

x

))ζ (

t

,

x

)

dtd

σ (

x

) +

(

u₀

(

x

) −

^v0

(

x

))

⁺

ζ (

0

,

x

)

dx

≥

⁰

,

(11)

foranynon-negativefunction

ζ

∈Cc^∞

([

0

,

+∞)×R^d

)

.

LetusfinallyrecallasimplifiedversionoftheresultobtainedbyVasseurin[35],whichissufficientforouruse.

Theorem4.Assumethattheﬂux f isnon-degenerateandthatthedomain

isC²^.^Then^if^u∈^L^∞

((

0

,

+∞

)

×

)

isanentropy solutionof(3)inthesenseofDeﬁnition2.1,i.e.(5)issatisﬁedforanyk andanynon-negativefunction

φ

inCc¹

((

0

,

+∞

)

×

)

,then thereexistsboundarydatau_b∈^L^∞

((

0

,

T

)

×

∂)

andinitialdatau₀∈^L^∞

()

suchthatu istheuniqueentropysolutiontothemixed problem(8)inthesenseofDeﬁnition3.2.

4. Proofofthemainresult

Lemma4.1.Consider J:= [^A

,

B]⊂R^and^suppose^that

u0

(

x

) ∈

J

,

for a.e. x

∈ ,

u_b

(

t

) ∈

^J

,

for a.e. t

≥

⁰

.

ThentheuniqueentropysolutiontotheIBVP(8)withinitialandboundarydatau0andu_b,u satisﬁes

u

(

t

,

x

) ∈

^J ^{for a.e.}

(

t

,

x

)

in

(

0

, +∞ ) × .

Proof. Weprovehereinfulldetailsthatu

(

t

,

x

)

≤^B^for^a.e.

(

t

,

x

)

in

(

0

,

+∞)×

.Theinequality A≤^u

(

t

,

x

)

canbeobtained analogously.

Byhypothesis,wehaveforalmosteveryxin

and

(

t

,

y

)

in

(

0

,

+∞

)

×

∂

u0

(

x

) ≤

B

,

u_b

(

t

,

y

) ≤

B

,

thenforanyﬁxedtimet˜≥^0,^taking^a^sequence

ζ

n∈Cc^∞

(

R

)

→

χ

₍_−∞_,˜_t_] andusingk=^B,^from^{(9) we}^obtain

∂

˜

t 0

ω

⁺

(

y

,

B

,

u_b

(

t

,

y

))

dtd

σ (

y

) +

(

u₀

(

x

) −

^B

)

⁺

− (

u

(˜

t

,

x

) −

^B

)

⁺dx

≥

⁰

.

(12)

Itisclearthatfora.e.xin

and

(

t

,

y

)

in

(

0

,

˜t

)

×

∂

wehave

ω

⁺

(

y

,

B

,

u_b

(

t

,

y

)) =

⁰

, (

u₀

(

x

) −

^B

)

⁺

=

⁰

,

(6)

then(12) implies

(

u

(˜

t

,

x

) −

^B

)

⁺

=

⁰

,

for a.e.xin

whichisindeed

u

(˜

t

,

x

) ≤

B

,

for a.e.xin

. 2

Proposition4.2.Letu andv beentropysolutionsto(8)withrespectiveinitialdatau0andv0andthesameboundarydatumu_b.Let usalsosupposethatalldatatakevaluesinanintervalI thatsatisﬁesthereplacementconditionintimet=^L_c ^(with^a^direction^w).

Thenwecanconcludethat

∀

^t

≥ t,

u

(

t

,

x

) =

^v

(

t

,

x

),

for almost every x in

.

Proof. Letusdeﬁnefor

θ >

0 thefunctional Jθ by

∀

^t

≥

⁰

,

J_θ

(

t

) :=

|

^u

(

t

,

x

) −

^v

(

t

,

x

)|

^e^−θ^w^|^x^dx

.

Givent¯≥^0,^we^apply^{(11) to}^the^ordered^couples

(

u

,

v

)

and

(

v

,

u

)

,thenaddingtheinequalitiesweget

¯

t 0

(

v

(

t

,

x

) −

^u

(

t

,

x

))

⁺

+ (

u

(

t

,

x

) −

^v

(

t

,

x

))

⁺

∂

t

ζ (

t

,

x

)

+ ( χ

_{v(t,x)>u(t,x)}f

(

v

(

t

,

x

)) −

f

(

u

(

t

,

x

))) + ( χ

_{u(t,x)>v(t,x)}f

(

u

(

t

,

x

)) −

f

(

v

(

t

,

x

)))|∇ζ (

t

,

x

)

dxdt

+

∂

¯

t 0

2

ω

⁻

(

x

,

u_b

(

t

,

x

),

u_b

(

t

,

x

))ζ (

t

,

x

)

dtd

σ (

x

) +

((

v₀

(

x

) −

^u0

(

x

))

⁺

+ (

u₀

(

x

) −

^v0

(

x

))

⁺

)ζ (

0

,

x

)

dx

≥

⁰

,

whichisactually

0

≤

¯

t 0

|

^u

(

t

,

x

) −

^v

(

t

,

x

) | ∂

_t

ζ (

t

,

x

) +

^sign

(

u

(

t

,

x

) −

^v

(

t

,

x

))

^f

(

u

(

t

,

x

)) −

^f

(

v

(

t

,

x

)) |∇ ζ (

t

,

x

)

^dx^dt

+

|

^u0

(

x

) −

^v0

(

x

) | ζ (

0

,

x

)

dx

.

Weconsiderasequence

(ζ

n

)

n⊂Cc^∞

(R)

^convergingⁱⁿ^L¹ ^to

χ

_(−∞,¯_t_]e^−θ^w^|^x,sothatinthelimitn→ ∞^,^we^get

J_θ

(¯

t

) ≤

^Jθ

(

0

) +

¯

t 0

sign

(

u

(

t

,

x

) −

^v

(

t

,

x

))

^f

(

u

(

t

,

x

)) −

^f

(

v

(

t

,

x

)))| −

^w

θ

e⁻^θ^w^|^x

^dx^dt

.

(13)

Butsince

∀(

^a

,

b

) ∈

^I²

,

sign

(

a

−

^b

)

^f

(

a

) −

^f

(

b

)|

^w

=

^sign

(

a

−

^b

)

1 0

f

(

b

+

^s

(

a

−

^b

))

ds

(

a

−

^b

)|

^w

=

^sign

(

a

−

^b

) (

a

−

^b

)

1 0

^f

(

b

+

^s

(

a

−

^b

)) |

^w

^ds

≥ |

^a

−

^b

|

1 0

cds

≥

c

|

a

−

b

|,

from(13),Lemma4.1andthereplacementcondition,weobtain

(7)

Jθ

(¯

t

) ≤

Jθ

(

0

) − θ

¯

t 0

Jθ

(

s

)

ds

.

ThankstotheclassicalGronwall’slemma,weendupwith

J_θ

(¯

t

) ≤

^e⁻^c^θ^¯^t^Jθ

(

0

).

As¯twasarbitrarilychosen,ifM:=^sup

x∈^w|^x^and^m= ^inf

x∈^w|^x^,^we^can^write^that,^for^all^t≥^0,

u

(

t

) −

v

(

t

)

L¹()e⁻^θ^M

≤

Jθ

(

t

) ≤

u

(

t

) −

v

(

t

)

L¹()e⁻^θ^m

.

So,wecancompute

^u

(

t

) −

^v

(

t

)

L¹()

≤

^e^θ^M^Jθ

(

t

)

≤

^e^θ^M^−θ^ct^Jθ

(

0

)

≤

^e^−θ^c⁽^M⁻^c^m⁻^t⁾

^u0

−

^v0

L¹()

≤

e⁻^θ^c⁽^L^c⁻^t⁾

u0

−

v0

L¹()

.

Soforanyt≥t= ^L

c,letting

θ

→ +∞^,^we^obtain u

(

t

,

x

) =

^v

(

t

,

x

)

for almost everyxin

. 2

WearereadytoproveTheorem1.

Proof. Weaimatprovingthatthereexistsanentropysolutiontotheproblem

⎧ ⎪

⎨

⎪ ⎩

∂

_tu

+

^divx

(

f

(

u

)) =

⁰

,

in

(

0

,

T₁

) × ,

u

(

0

,

x

) =

^u0

(

x

),

on

,

u

(

T1

,

x

) =

^v

(

T2

,

x

),

on

.

In view of the well-posedness resultstated inTheorem 2, our goalis achievedonce we constructsuitable boundary conditions,whichcanbeinterpretedascontrolsinoursetting.

CaseT₂

>

T₁.

ThankstoTheorem4,itmakessensetoconsider

w0

(

x

) =

^v

(

T2

−

^T1

,

x

),

for a.e.x

∈ ,

w_b

(

s

,

x

) =

^v

(

T₂

−

^T1

+

^s

,

x

),

for a.e.x

∈ ∂

ands

≥

⁰

.

WecallwtheuniqueentropysolutiontotheIBVP(8) withdataw0,wbon

(

0

,

T1

)

×

.Theformoftheequationimplies that,foralmostevery

(

s

,

x

)

in

(

0

,

T₁

)

×

,w

(

s

,

x

)

=^v

(

T₂−^T1+^s

,

x

)

.

Byhypothesis, T1≥t, so,asadirectapplicationofProposition 4.2,we canconcludethat theentropysolutions tothe mixedproblemsoftheform(8) withinitialdatau0 andw0,respectively,andcommonboundarydatum wb satisfy

u

(

T1

,

x

) =

w

(

T1

,

x

)

for a.e. x

∈ ,

whichmeans

u

(

T1

,

x

) =

^v

(

T2

,

x

)

for a.e. x

∈ .

CaseT₁

>

T₂. Wedeﬁne

w0

(

x

) =

^v

(

T2

− t,

x

),

for a.e.x

∈ ,

w_b

(

s

,

x

) =

^v

(

T₂

− t +

^s

,

x

),

for a.e.x

∈ ∂

ands

≥

⁰

,

wheretisthetimegivenbythereplacementcondition.

WecallwtheuniqueentropysolutiontotheIBVP(8) withdataw₀,w_bon

(

0

,

t)×

.Theformoftheequationimplies that,foralmostevery

(

s

,

x

)

in

(

0

,

t)×

,w

(

s

,

x

)

=^v

(

T2−t+^s

,

x

)

.

(8)

Weconsidernowaboundaryconditionofthefollowingform

u_b

(

t

,

x

) = b,

fort

∈ (

0

,

T₁

− t),

x

∈ ∂ ,

w_b

(

t

− (

T₁

− t),

x

)

fort

∈ (

T₁

− t, +∞ ),

x

∈ ∂ ,

wherebisanyconstantstateintheinterval I.

TheIBVP(8) withdatau0,ub admitsauniqueentropysolutionu in

(

0

,

+∞

)

×

. Wecall^u˜0 theproﬁleofuattimet=^T1−t.

NowitisclearthatifweapplyProposition4.2totheentropysolutionsu˜ etw to(8) withrespectiveinitialdatau˜0and w0andcommonboundarydatawb weobtain

˜

u

(t,

x

) =

w

(t,

x

),

for a.e.x

∈ ,

whichmeans

u

(

T1

,

x

) =

^v

(

T2

,

x

),

for a.e.x

∈ . 2

Acknowledgements

The first authoracknowledges thefinancial support oftheRégion BourgogneFranche-Comté, project “Analyse mathé- matique etsimulationnumériqued’EDPissusdeproblèmesde contrôleetdu traficroutier”(OPE-2017-0067).Thesecond authorwassupportedbyANRFinite4SOS(15-CE23-0007).

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