Uniqueness in a class of Hamilton–Jacobi equations with constraints

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Mathematical analysis/Partial differential equations

Uniqueness in a class of Hamilton–Jacobi equations with constraints

Unicité pour une classe d’équations de Hamilton–Jacobi avec contraintes Sepideh Mirrahimi, Jean-Michel Roquejoffre

Institutdemathématiques(UMRCNRS5219),UniversitéPaul-Sabatier,118,routedeNarbonne,31062Toulousecedex,France

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

Articlehistory:

Received13February2015 Accepted18March2015 Availableonlinexxxx PresentedbyHaïmBrézis

Inthisnote,wediscussaclassoftime-dependentHamilton–Jacobiequations depending onafunctionoftime,thisfunctionbeingchoseninordertokeepthemaximum ofthe solution ofthe constant value 0. The main result of the note is that the full problem hasauniqueclassicalsolution.Themotivationisaselection–mutationmodelthat,inthe limitofsmalldiffusion,exhibitsconcentrationonthezero-levelsetofthesolutiontothe Hamilton–Jacobiequation.Theuniquenessresultthatweproveimpliesstrongconvergence anderrorestimatesfortheselection–mutationmodel.

©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

r é s um é

Danscettenote,ondiscuteuneclassed’équationsdeHamilton–Jacobidépendantdutemps et une fonction inconnue du temps choisie pour que le maximum de la solution de l’équation deHamilton–Jacobi prennetout letempslavaleur0.Le résultatprincipalde cettenoteestqueleproblèmecompletadmetuneuniquesolutionclassique.Lamotivation est unmodèledesélection–mutationqui, danslalimited’unediffusiviténulle,présente une concentration sur la ligne de niveau 0 de la solution de l’équation de Hamilton–

Jacobi. Le résultatd’unicité que nousdémontrons implique uneconvergence forte, avec estimationsd’erreurpourlemodèledesélection–mutation.

©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

Versionfrançaiseabrégée

Onprésentedanscettenoteunrésultatd’unicitépourleproblèmedeHamilton–Jacobisuivant,d’inconnues

(

u

(

t

,

x

),

I

(

t

))

:

∂

tu

= |∇

^u

|

²

+

^R

(

x

,

I

), (

t

>

0

,

x

∈ R

^d

),

max

x u

(

t

,

x

) =

⁰

,

I

(

0

) =

^I0

>

0

,

u

(

0

,

x

) =

^u0

(

x

).

(1)

E-mailaddresses:[email protected](S. Mirrahimi),[email protected](J.-M. Roquejoffre).

http://dx.doi.org/10.1016/j.crma.2015.03.005

1631-073X/©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

La donnée R

(

x

,

I

)

vérifiedeshypothèsesde stricte concavitépar rapportà xetde monotonieparrapport à I,explicitées plus bas.La donnéeinitiale u₀ vérifieaussi deshypothèsesspéciales de concavité.Ainsi, lafonctionI

(

t

)

doitêtrechoisie pourquelasolutionu

(

t

,

x

)

del’équationdeHamilton–Jacobiait,àchaqueinstant,unmaximumégalà0.

Nousavonsalorsle

Theorème0.1.OnchoisitR∈^{C2,etonsuppose}l’existencedeIM

>

0telquemax

x∈R^dR

(

x

,

IM

)

=⁰=^R

(

0

,

IM

)

.Deplus,R estsupposée strictementconcaveet,pour|^x|^{grand,compriseentredeuxparaboles.Ladonnéeinitialeu}0estégalementàdécroissancequadratique etstrictementconcave.

Le problème(1)auneuniquesolution

(

u

,

I

)

,où u estunesolution classiquedel’équationdeHamilton–Jacobi.Deplus,u∈ L^∞_loc

R⁺;^W_loc^3,∞

(

R^d

)

∩^W_loc^1,∞

R⁺;^L∞_loc

(

R^d

)

×^W1,∞

(

R

)

.

L’existencepour(1)aétédémontréeenplusieursendroits,voirparexemple[8]ou[1].L’unicitéestdoncnotrerésultat principal ;c’étaitunproblèmeouvert.L’unicitéétaiteneffetconnueseulementpouruncastrèsparticulier(voir[8]).

Lemodèle(1)intervientdanslalimite

ε

→^{0 dessolutionsde}

∂

tn_ε

− ε

n_ε

=

^nε

ε

^R

x

,

I_ε

(

t

)

(

t

>

0

,

x

∈ R

^d

),

I_ε

(

t

) =

R^d

ψ (

x

)

n_ε

(

t

,

x

)

dx

,

(2)

oùnε

(

t

,

x

)

estladensitéd’unepopulationcaractériséeparun traitbiologiquex d-dimensionnel. Lacompétitionpourune ressource unique est représentée par Iε

(

t

)

,

ψ >

0, régulière donnée. Le terme R

(

x

,

I

)

est le taux de reproduction. Les hypoth`ses de concavité sontdes hypothèses techniques,mais pertinentes au plan biologique.Par une transformationde Hopf–Colenε=^exp

(

uε

/ ε )

,onseramèneàl’équationsuruε suivante :

∂

tu_ε

= ε

u_ε

+ |∇

u_ε

|

²

+

R

(

x

,

I_ε

)

(3)

qui, danslalimite

ε

→^{0,donnel’équationsur u.Ons’attendalorsà cequen}ε seconcentreaux pointsoù u estproche de 0.Et,danscettelimite,Iε apparaîtcommeunesortedemultiplicateurdeLagrange.

La convergence de (3) vers (1) à une sous-suite près est connue depuis[8]. Le Théorème 0.1 donne la convergence de toute la famille, ainsique desestimations d’erreur. Soit xε

(

t

)

lepoint où uε

(

t

, .)

atteint son maximum. On suppose l’existencedeI⁰ telque0

<

I⁰≤^Iε

(

0

)

:=

R^d

ψ(

x

)

n⁰ε

(

x

)

dx

<

IM,etonsuppose :

n⁰_ε

=

e^u0ε/ε

=

^r

ε

^d/2e

u₀/ε

,

avec u0

∈

C²

(R

^d

)

and max

x∈R^du0

(

x

) =

0

.

Lerésultatestalorsle

Theorème0.2.Soitnεlasolutionde(2)etuεdéfiniepar(3).Nousavonslesdéveloppementsasymptotiquessuivants :

I_ε

=

^I

+ ε

I1

+

^o

(

1

),

x_ε

=

^x

+ ε

x1

+

^o

(

1

),

u_ε

=

^u

+ ε

log

(

^r

ε

^d2

) + ε

u1

+

^o

(

1

).

Lestermes I₁,x₁ etu₁vontêtreprésentésdans[7].Cerésultatimpliquelecorollaire :

Corollaire0.3.Nousavonsl’approximationsuivantepournε:

n_ε

(

t

,

x

) =

^r

ε

^d2

exp

(

u₁

+

^u

ε ^{) +}

^o

⁽

¹

⁾

.

Enparticulier,lorsque

ε

→^{0,toutelasuite}

(

nε

)

εconverge : n_ε

(

t

,

x

) −→ ¯ ρ (

t

) δ

x

− ¯

^x

(

t

)

au sens des mesures

,

avec

ρ (

t

)

= ^I

(

t

) ψ(

x

(

t

))

^.

End’autrestermes,lapopulationseconcentre suruntrait dominant,quiévolue avecletemps.Onnote quelaconver- gencedenεàunesous-suiteprésétaitdéjàétabliedans[5].

Touscesrésultatsserontdétaillésdans[6]et[7].

1. Introduction

Thepurposeofthisnoteistodiscussuniquenessinthefollowingproblem,withunknowns

(

I

(

t

),

u

(

t

,

x

))

:

∂

tu

= |∇

^u

|

²

+

^R

(

x

,

I

) (

t

>

0

,

x

∈ R

^d

),

max

x u

(

t

,

x

) =

⁰

,

I

(

0

) =

^I0

>

0

,

u

(

0

,

x

) =

^u0

(

x

),

⁽⁴⁾

whereI₀

>

0 andu₀isaconcave,quadraticfunction:

−

^L₀

−

^L₁

|

^x

|

²

≤

^u0

(

x

) ≤

^L0

−

^L1

|

^x

|

²

, −

^2L₁

≤

^D2u0

≤ −

^2L1

,

D³u₀

∈

^L∞

(R

^d

).

Theconstraintonthemaximumofu

(

t

, .)

makestheproblemnonstandard.Ourmainresultis

Theorem1.1.ChooseR∈^{C2,andsupposethatthereisI}M

>

0suchthatmax

x∈R^dR

(

x

,

IM

)

=⁰=^R

(

0

,

IM

)

.Alsoassumethefollowing concavityandregularitypropertiesforR:

−

^K₁

|

^x

|

²

≤

^R

(

x

,

I

) ≤

^K0

−

^K1

|

^x

|

²

,

for 0

≤

^I

≤

^IM

,

−

^2K1

≤

^D2R

(

x

,

I

) ≤ −

^2K1

<

0and D³R

( · ,

I

) ∈

^L∞

( R

^d

)

for0

≤

^I

≤

^IM

,

−

^K₂

≤ ∂

IR

≤ −

^K2

,

| ∂

_{I x}²

iR

(

x

,

I

) | + | ∂

³RI x_ix_j

(

x

,

I

) | ≤

^K3

,

for0

≤

^I

≤

^IM

,

and i

,

j

=

¹

,

2

, · · · ,

d

.

Problem (4)has a unique solution

(

u

,

I

)

, where u solves the Hamilton–Jacobi equation in the classical sense. Moreover u∈ L^∞_loc

R⁺;^W3_loc^,∞

(

R^d

)

∩^W_loc^1,∞

R⁺;^L∞_loc

(

R^d

)

×^W1,∞

(

R

)

.

Existencefor(4)hasbeenprovedinvariouscontexts(see[8,1,5]).Thus,ourcontributionisuniqueness,whichhasupto nowbeenanopenproblem.Theuniquenesshasindeedbeenknownonlyforaveryparticularcase(see[8]).

Therestofthenote isorganizedasfollows.InSection 2,weexplain themotivationand,inparticular,themeaning of the various assumptions.In Section 3, we revisit existence for(4),which will entail an unconventionalODE formulation foruniqueness.Section4,whichisthemainpartofthenote,providesafairlycompletesketchoftheuniquenessproof.In Section5,wegiveanapplication.

2. Backgroundandmotivation

Model(4)arisesinthelimit

ε

→^{0 ofthesolutionstotheproblem}

∂

tn_ε

− ε

n_ε

=

^nε

ε

^R

x

,

I_ε

(

t

)

(

t

>

0

,

x

∈ R

^d

),

I_ε

(

t

) =

R^d

ψ (

x

)

n_ε

(

t

,

x

)

dx

,

(5)

wherenε

(

t

,

x

)

isthedensityofapopulation characterizedbya d-dimensionalbiologicaltrait x.Thepopulationcompetes for a single resource, thisis represented by Iε

(

t

)

, where

ψ

is a givenpositive smooth function. The term R

(

x

,

I

)

isthe reproduction rate; it is, ascan be expected, very negative for large x anddecreases asthe competition increases. Such modelscanbederived fromindividualbasedstochasticprocessesinthelimitoflargepopulations(see[2]).Theconcavity assumptionon Risatechnicalone,althoughbiologicallyrelevant.TheHopf–Coletransformationnε=^exp

(

uε

/ ε )

yieldsthe equation

∂

tu_ε

= ε

u_ε

+ |∇

u_ε

|

²

+

R

(

x

,

I_ε

)

(6)

which,in thelimit

ε

_→0, yields theequation for u. Now, Iε beinguniformlypositive andboundedin

ε

,the Hopf–Cole transformationleadstotheconstraintonu.Moreover,oneexpectsthatnε concentratesatthepointswhereuiscloseto0 andthefunctionIε appears,inthelimit,asasortofLagrangemultiplier.

Thisapproach,basedon theHopf–Coletransformation,to study(5) hasbeenintroducedin[4]andthendeveloped in differentcontexts (see forinstance[8,1,3,5]).Long-timeasymptotics ofsuch modelshasalsobeen studiedin[9]andthe referencestherein.

3. Existence

Existenceofasolutionto(4)isobtainedbyletting

ε

→^{0 in(6).Themainstepisthefollowingtheorem.}

Theorem3.1(Uniformestimatesforuε,[5]).Thereexists Im

>

0suchthat0

<

Im≤^Iε

(

t

)

≤^IM+^C

ε

².Moreover,wehavethe followingestimatesonuε

−

^L0

−

^L1

|

^x

|

²

− ε

2dL₁t

≤

^uε

(

t

,

x

) ≤

^L0

−

^L1

|

^x

|

²

+

K₀

+

^2d

ε

L₁

t

,

L₁

−

^{2t K}1

≤

^D2uε

(

t

,

x

) ≤ −

^2L1

,

^D3uε

(

t

, ·)

L^∞

≤

^C

(

T

),

for t

∈ [

⁰

,

T

].

⁽⁷⁾

Theboundsforuε canbeobtainedforanyuniformlyboundedfunctionIε,notonlyforthatof(5).Thisremarkwillbe animportantingredientoftheuniquenessproof.

4. Uniqueness

Foragivencontinuousfunction I

(

t

)

suchthat0

<

I

(

t

) <

IM,onemayconstructasolutionto

∂

tu= |∇^u|²+^R

(

x

,

I

)

with initial datumu0.JustasinTheorem 3.1,thissolutionsatisfiesestimates(7).Andso,foru

(

t

, .)

beingstrictlyconcaveand quadratically decreasing,there existsa unique function x¯

(

t

)

such that u

(

t

,

x¯

(

t

))

=^max

x∈R^du

(

t

,

x

)

.Assume that I

(

t

)

ischosen such thatu

(

t

,

^x¯

(

t

))

=^{0.Then,fromtheequationonu,wededucethat R}

(¯

^x

(

t

),

I

(

t

))

=^{0.Noticealsothat,because}

∂

IR

<

0, we have R

(

¯^x

(

t

),

0

) >

0.Finally,differentiating ∇^u

(

t

,

^x¯

(

t

))

=^{0 andpluggingintheequationforu,weobtainan ODEfor}¯^x:

˙¯

x

(

t

)

=

−^D2u

t

,

¯x

(

t

)

₋1

∇xR

¯ x

(

t

),

¯I

(

t

)

.

Theideaisthustochangetheconstrainedproblem(4)bythefollowingslightlynonstandarddifferentialsystem:

⎧ ⎨

⎩

R

x

(

t

),

I

(

t

)

=

⁰

,

fort

∈ [

⁰

,

T

] ,

˙

x

(

t

) =

−

D²u

t

,

x

¯ (

t

)

₋1

∇

xR

¯

x

(

t

), ¯

I

(

t

)

,

fort

∈ [

0

,

T

],

∂

tu

= |∇

^u

|

²

+

^R

(

x

,

I

),

in

[

⁰

,

T

] × R

^d

,

(8)

withinitialconditions

I

(

0

) =

I0

,

u

(

0

, ·) =

u0

(·),

x

(

0

) =

x0

,

such thatR

(

x0

,

I0

) =

0

.

(9) Note that (8)isreally adifferential systembecausethe assumptions on R implythat I

(

t

)

canimplicitly beexpressed in termsofx¯

(

t

)

.Anditisslightlynonstandard becausex¯ solves anODEwhosenonlinearitydependsonu.Finally,notethat, assoonasusatisfiestheconcavityandregularityestimates(7),system(8)isequivalenttotheconstrainedproblem(4).

Thissuggeststouseasimplefixed-point argumenttoproveuniquenessof(8)(andso,of(4)).Whichinturnsuggests to set up thefollowing scheme: starting fromx

(

t

)

∈^C

(

[⁰

,

T]

,

R^d

)

,such that x

(

0

)

=^x0, where R

(

x0

,

0

) >

0. Let I

(

t

)

solve R

(

x

(

t

),

I

(

t

))

=^{0 on}[⁰

,

T]^withR

(

x0

,

I0

)

=^0.Letv

(

t

, .)

betheuniquesolutionto

∂

tv

= |∇

v

|

²

+

R

(

x

,

I

),

v

(

0

,

x

) =

u0

(

x

).

(10)

Let y

(

t

)

solve ˙y

(

t

)

=

−^D2v

t

,

x

(

t

)

₋1

∇xR

x

(

t

),

I

(

t

)

on[⁰

,

T] ^{withinitial datum y}

(

0

)

=^x0.Setting y:=

(

x

)

,wenotice that uniqueness is proved assoon as we have proved that

has a unique fixed point. One additional feature about a solution

(¯

I

,

u

,

¯^x

)

to(8):

Lemma4.1.Thefunction¯I

(

t

)

isincreasing.

Weclaimthatourproblemreducestoprovingthefollowingtheorem.

Theorem4.2.ThereexistsC

>

0universaland

δ >

0,whichissmallas R

(

x₀

,

0

)

tendsto0,suchthat

isacontractionfrom C

(

[⁰

, δ

]

,

BC

(

x0

))

toitself;hereBr

(

a

)

denotestheballofcentrea∈R^dandradiusr

>

0.

Noteindeedthat,byLemma 4.1,wehave,because

∂

IR

<

0:

R

(¯

^x

(δ),

0

) =

^R

(¯

^x

(δ),

0

) −

^R

(¯

^x

(δ), ¯

I

(δ)) ≥

^c

¯

I

(δ) ≥

^{c I}0

,

forsomeuniversalc

>

0.HenceTheorem 4.2canbeiteratedtoyieldglobalexistenceanduniqueness.

Letusgive anoverviewoftheproofofTheorem 4.2.ForI∈^C

(

[⁰

, δ

]; [⁰

,

IM]

)

,let V

(

I

)

bethe(unique)solutionto(10).

Themainstepisthefollowing.

Lemma4.3.LetI1

,

I2∈^C

(

[⁰

, δ

]; [⁰

,

IM]

)

.Then

V

(

I1

) −

V

(

I2

)

_W2,∞([⁰,δ]×R^d)

≤

C

I1

−

I2

L^∞([⁰,δ])

δ.

Thislemma,onceproved,opensthewaytoTheorem 4.2.Indeedtheequation R

(

x

,

I

)

=^{0 yieldsasmoothmappingx}→^I, and I→^{V is a Lipschitz mappingthanks to Lemma 4.3. Moreover, theequation y}˙

(

t

)

=

−^D2v

t

,

x

(

t

)

−1

∇xR

x

(

t

),

I

(

t

)

yieldsaLipschitzmappingv→^{ybytheestimatesforv givenby}Theorem 3.1.

Lemma 4.3ismoreinvolved.IfI1andI2areasintheassumptionsofthelemma,thefunctionr=^V

(

I1

)

−^V

(

I2

)

solves

∂

tr

= (∇

v1

+ ∇

v2

) · ∇

r

+

R

(

x

,

I2

) −

R

(

x

,

I1

),

in

[

0

, δ] × R

^d

,

r

(

0

,

x

) =

⁰

,

for allx

∈ R

^{d (11)}

with v_i=^V

(

I_i

)

. Note that the above equation has a unique classical solution that can be computed by the method of characteristics.Thecharacteristicssolve

˙

γ (

t

) = −∇

v1

(

t

, γ ) − ∇

v2

(

t

, γ ),

(12)

and,duetotheestimatesofTheorem 3.1,theyexistglobally.So,onemaysuccessivelyexpressr givenbyintegrationalong characteristics,andestimateitsderivativesrecursively.

5. Application

Convergence of (6)to (4)had alreadybeen proved in[8,1],along subsequences. The uniqueness partof Theorem 1.1 yields theconvergence ofthefull familyofsolutions uε to (6),insteadofconvergence alonga subsequence. Moreoverit allowsanexpansionofIε,uε andxε (themaximumpointofuεateachtime)intermsof

ε

.Herearetheresults.

AssumethatthereisI⁰suchthat0

<

I⁰≤^Iε

(

0

)

:=

R^d

ψ(

x

)

n⁰ε

(

x

)

dx

<

IM,andthat

n⁰_ε

=

^eu0ε/ε

=

^r

ε

^d/2e

u⁰/ε

,

with u⁰

∈

^C2

(R

^d

)

and max

x∈R^du⁰

(

x

) =

⁰

.

Theresultisthefollowingtheorem.

Theorem5.1.Letnεbethesolutionto(5)anduεbedefinedby(6).Wehavethefollowingasymptoticexpansions

I_ε

=

^I

+ ε

I₁

+

^o

(

1

),

x_ε

=

^x

+ ε

x₁

+

^o

(

1

),

u_ε

=

^u

+ ε

log

(

^r

ε

^d2

) + ε

u₁

+

^o

(

1

).

ThetermsI₁,x₁ andu₁ willbeprovidedin[7].Thisyieldsthefollowingcorollary.

Corollary5.2.Wehavethefollowingapproximationfornε:

n_ε

(

t

,

x

) =

^r

ε

^d2

exp

u1

(

t

,

x

) +

^u

(

t

,

x

) ε

+

^o

(

1

)

.

Inparticular,as

ε

→^{0,thewholesequence}

(

nε

)

_εconverges:

n_ε

(

t

,

x

) −→ ¯ ρ (

t

) δ

x

− ¯

x

(

t

)

,

weakly in the sense of measures

,

with

ρ (

t

)

= ^I

(

t

) ψ(

x

(

t

))

^.

Inotherwords,thepopulationdensityconcentratesonadominanttraitthat evolvesovertime.We notethatthecon- vergenceofnε alongsubsequenceswasalreadyestablishedin[5].

Theaboveresultswillbedetailedin[6]and[7].

Acknowledgements

S.MirrahimiwaspartiallyfundedbytheFrenchANRprojectsKIBORDANR-13-BS01-0004andMODEVOLANR-13-JS01- 0009. J.-M. Roquejoffre has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreementn. 321186 – ReaDi –“Reaction-Diffusion Equations, Propa- gationandModelling”held byHenri Berestycki.He wasalso partiallysupported bythe FrenchNationalResearchAgency (ANR),withintheprojectNONLOCALANR-14-CE25-0013.BothauthorsaregratefultotheLabexCIMIfororganizingaPDE- probabilityquarterinToulouse,whichprovidedquiteastimulatingenvironmenttothiscollaboration.

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