boundary for compressible and incompressible fluids On the local existence for the Euler equations with free C.R. Acad. Sci. Paris, Ser.I

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Partial differential equations

On the local existence for the Euler equations with free boundary for compressible and incompressible fluids

Sur l’existence locale de solutions des équations d’Euler pour les fluides compressibles et incompressibles, avec frontière libre

Marcelo M. Disconzi

^a

, Igor Kukavica

^b

aDepartmentofMathematics,VanderbiltUniversity,Nashville,TN37240,USA

bDept.ofMathematics,UniversityofSouthernCalifornia,LosAngeles,CA90089-2532,USA

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received31July2017 Accepted5February2018 Availableonline21February2018 PresentedbytheEditorialBoard

We consider the free boundary compressible and incompressible Euler equations with surfacetension.Inbothcases,weprovidea prioriestimatesforthelocalexistencewiththe initialvelocityinH³,withthe H³conditiononthedensityinthecompressiblecase.An additionalconditionisrequiredonthefreeboundary.Comparedtotheexistingliterature, bothresults lower the regularityof initial data for the Lagrangian Euler equationwith surfacetension.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

ré s u m é

Nous considéronsles équations d’Euler compressibles et incompressibles avec frontière libreettension desurface.Danslesdeux cas, nousfournissonsdes estimationsapriori pourl’existencedesolutionslocalesavecvitesseinitialedans H³ etlacondition H³sur ladensité dansle cascompressible.Unecondition supplémentaireest nécessaire surla frontièrelibre.Parcomparaisonaveclalittérature,lesdeuxrésultatsabaissentlarégularité desdonnéesinitialespourleséquationsd’Eulerencoordonnéeslagrangiennes,avectension desurface.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

1. Introduction

In this note, we address the waterwave problem, which hasbeen studied extensively. Our settingis rather general:

we consider the Euler equations witha free surface and allow initial data to have nonzero curl, i.e. the initial data is rotational.Thedomainisassumedtobeoffinitedepth,buttheresultscanbeeasilyadaptedtotheinfinitedepthaswell.

E-mailaddresses:marcelo.disconzi@vanderbilt.edu(M.M. Disconzi),kukavica@usc.edu(I. Kukavica).

https://doi.org/10.1016/j.crma.2018.02.002

1631-073X/©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

We consider both thecompressible andincompressiblecases; forthe compressibleEuler equations,we assume that the densityisboundedfrombelow,i.e.weconsiderthecaseofaliquid.

Ouraim inthis note isto announce tworecent resultson the localexistence withnon-zero surface tension.In both maintheorems(thecompressibleandincompressiblecases,respectively),weassumethattheinitialdataisofH³regularity intheinterior.Thislowerstheregularity ofexistingresultsforthecompressibleequations.Intheincompressiblesetting, ourresultlowerstheknownregularityinLagrangian coordinates,albeititdoesnotimproveoverwhathasbeenobtained inEuleriancoordinates.

The historyof the Eulerequations withfree interface is rich—we refer the readerto [2,4,5] fora more complete ac- count. We only mentiona few important worksdealing withnon-zero surface tension.The first general well-posedness resultfortheincompressiblefree-boundaryEulerequationswithsurface tensionis[2],followedby[9,10] and[3].Earlier, well-posednesshadbeenestablishedundertheassumptionthattheinitial vorticityvanishesontheboundary[8].Forthe compressibleequationswithsurfacetension,thefirstresultwas[1].

Inthefirstpartofthenote,weaddressthecompressibleEulerequations,whileinthesecondpartwetreattheincom- pressibleversion.

2. Compressiblecase

IntheLagrangiansetting,thefree-surfacecompressibleEulerequationsread

R

∂

tv^α

+

^aμα

∂

μq

=

^{0 in}

[

⁰

,

T

) × ,

(1a)

∂

_tR

+

^Raμα

∂

_μv_α

=

^{0 in}

[

⁰

,

T

) × ,

(1b)

∂

_ta^αβ

+

^aαγ

∂

_μv_γa^μβ

=

^{0 in}

[

⁰

,

T

) × ,

(1c)

q

=

^q

(

R

)

in

[

⁰

,

T

) × ,

(1d)

a^μαN_μq

+ σ _|

a^TN

|

g

η

^α

₌

0 on

[

⁰

,

T

) ×

1

,

(1e)

v^μN_μ

=

^{0 on}

[

⁰

,

T

) ×

0

,

(1f)

η (

0

,·) =

id

,

R

(

0

,·) =

0

,

v

(

0

, ·) =

v0

.

(1g) Above, v, R,andq denote theLagrangian velocity, density,andthe pressure,respectively; N is theunit outer normalto

∂,aistheinverseof∇

η

,

σ

=^constant>0 isthecoefficientofsurfacetension,andg istheLaplacianofthemetricg_{i j} inducedon∂(t)bytheembedding

η

,i.e.gi j=∂i

η

_·∂j

η

₌∂i

η

^μ∂j

η

_μ.

Weconsiderthedomain0≡=T²×(0,1).Wenotethatusingthestraighteningmapin[7,Remark 4.2],itiseasyto modifytheapproachto considerageneralcurveddomain=R²×(0,h(x1,x2)).Applyingthechangeofvariablein[7], weget

R

∂

tv^α

+

^aμαbβμ

∂

_βq

=

⁰

insteadof(1a);herebisthecofactorinversematrixofthestraighteningmap.Theother equationsinthesystem(1a)–(1g) aremodifiedsimilarly.Themethodsoutlinedheretheneasilycarryoverforthenewsystemaswell,providedisatleast H⁵regular.

Denotingthecoordinateson by(x¹,x²,x³),wehave1=T²× {^x3=¹}^{asthefreeboundaryand}0=T²× {^x3=⁰} asthestationaryone.Onthepressurefunction,weassume

q

(

R

)

R

≥

^Aq

=

^constant

>

0

,

(2)

whichissatisfiedbyalargeclassofequationsofstate.

We denoteby thecanonical projection,on

η

(1),fromthetangent bundle of

η

() toits normalbundle, whichis givenby ^α_β=δ^α_β−^gkl∂k

η

^α∂l

η

β.Werecall thatinitial datafor(2) isrequiredto satisfycompatibilityconditions(cf. [5]).

Thefollowingisthemainresultinthecompressiblecase.

Theorem2.1.Letv₀beasmoothvectorfieldonand ₀asmoothpositivefunctiononboundedawayfromzerofrombelow.

Assumethatv0and 0satisfythecompatibilityconditions.Letq:(0,∞)→(0,∞)beasmoothfunctionsatisfying(2)inaneigh- borhoodof ₀.Then,thereexista T_∗>0andaconstant C_∗,dependingonlyon

σ

>0, v_{0 3}, v_{0 3,1},

0 3,

0 3,1,and (divv0)1 −¹,1 suchthatanysmoothsolution(v,R)to(1),withinitialcondition(v0,

0)anddefinedonthetimeinterval [⁰,T_∗),satisfies

N

(

t

) =

^{v 2}₃

+ ∂

tv ²₂

+ ∂

_t^{2v 2}₁

+ ∂

_t^{3v 2}₀

+

^{R 2}₃

+ ∂

tR ²₂

+ ∂

_t^{2R 2}₁

+ ∂

_t^{3R 2}₀

+ ∂∂

_t^{2v 2}_0,1

+ ∂

²

∂

tv ²_0,

1

≤

^C_∗

,

⁽³⁾

where∂standsforthetangentialderivative.

Next,wedescribe thestrategy oftheproof anddiscussthetreatmentofdifficultterms.Themainpartisbasedonthe energyestimateforthreetimederivatives,

1 2

d dt

R

(

0

)∂

_t³v^β

∂

_t³vβ

+

¹ 2

d dt

R

(

0

)

R q

¯

(

R

)(∂

_t³R

)

²

+

1

∂

_t³

(

Ja^αβq

)∂

_t³vβN_α

= −

R

(

0

)

R

∂

_t³

(

Ra^αβ

∂

_αv_β

) −

^Raαβ

∂

_t³

∂

_αv_β

∂

_t³

_q

R

+

R

(

0

)

∂

_t³

a^αβq R

−

^aαβ

∂

_t³

_q

R

∂

_t³

∂

_αv_β

−

3

R

(

0

)

q

¯

(

R

)

R

∂

_t⁴R

∂

_t²R

∂

tR

−

R

(

0

)

q

¯

(

R

)

R

∂

_t⁴R

(∂

tR

)

³

+

¹ 2

R

(

0

)∂

t

q

¯

(

R

)

R

(∂

_t³R

)

²

(4)

where

¯

q

(

R

) =

^q

(

R

)

R

and J=^det(∇

η

). The first two terms on the left provide a coercive term ∂t³v ²₀+ ∂t³R ²₀. In the third term I₁ =

1∂_t³(J aαβq)∂_t³v_βNα ontheleftside of(4),weuse(1e) forthepressure,leadingto I1=_t

0

1∂_t³(√

gg

η

^α)∂_t³vα,where weset

σ

=^{1 for}simplicity.Thismayberewrittenas

I1

= −

t

0

1

√

g g^{i j}

^α_μ

∂

j

∂

_t²v^μ

∂

i

∂

_t³v_α

−

t

0

1

∂

i

( √

g g^{i j}

^α_μ

)∂

j

∂

_t²v^μ

∂

_t³v_α

= −

¹ 2

1

√

g g^{i j}

^α_μ

∂

j

∂

_t²v^μ

∂

i

∂

_t²v_α

+

¹ 2

t

0

1

∂

t

√

g g^{i j}

^α_μ

∂

j

∂

_t²v^μ

∂

i

∂

_t²v_α

−

t

0

1

∂

i

( √

g g^{i j}

^α_μ

)∂

j

∂

_t²v^μ

∂

_t³v_α

+

¹ 2

1

√

g g^{i j}

^α_μ

∂

j

∂

_t²v^μ

∂

i

∂

_t²v_α

|

0

=

^I11

+

^I12

+

^I13

+

^I14

.

(5)

Theintegral I11leadstoacoerciveterm ¹₄ ∂∂_t²v ²_0,

1.Thehighest-ordertermresultswhen∂ifallson√

g g^{i j},whichgives theintegral

I₁₃₁

= −

t

0

1

∂

i

( √

g g^{i j}

)ˆ

^nαn

ˆ

μ

∂

j

∂

_t²v^μ

∂

_t³v_α

,

forwhichweneedtoexploreitsspecialstructure.Notethatweusedtheidentity^α_μ= ˆ^nαnˆμ,ⁿˆ ^{beingtheunitouternormal} tothemovingboundary.ItturnsoutthatthisintegralcancelsapartoftheintegralresultingfromI14(cf. [5] forcomplete details).

Itremains todiscussthetreatmentoftherightsideof(4).Denotethetermsontherightsideof(4) byJ1–J5.Allthe integralsresultingfromJ1,J3,J4,andJ5 canbeestimatedusingintegrationby partsintime andspaceandemploying theHölderandSobolevinequalities.Ontheotherhand,whenexpandingJ2,thereisatrickytermT=_t

0

∂_t³Aμα∂_t³∂_μvαq where A= ^{J a,whichcannotbetreatedinthisway,andspecial}cancellationsareneededtocontrolit.Wenamelyclaim

|

T

|

N

+

P0

+

P

t

0

P

,

(6)

whereP^{denotesagenericpolynomialin}N^,P0agenericpolynomialofthenormsoftheinitialdata.

Herewesketchthemainideasusedinproving(6).UsingtheformulaforthecofactormatrixA= ^{J a,werewrite}

T

=

t

0

q

^αλτ

∂

2

∂

_t²v_λ

∂

3

η

τ

∂

1

∂

_t³v_α

+

t

0

q

^αλτ

∂

2

η

λ

∂

3

∂

_t²v_τ

∂

1

∂

_t³v_α

−

t

0

q

^αλτ

∂

1

∂

_t²v_λ

∂

3

η

_τ

∂

2

∂

_t³v_α

−

t

0

q

^αλτ

∂

1

η

_λ

∂

3

∂

_t²v_τ

∂

2

∂

_t³v_α

+

t

0

q

^αλτ

∂

1

∂

_t²v_λ

∂

2

η

_τ

∂

3

∂

_t³v_α

+

t

0

q

^αλτ

∂

1

η

λ

∂

2

∂

_t²v_τ

∂

3

∂

_t³v_α

+

^L

=

T1

+ · · · +

T6

+

L

(7)

where A¹α =

^αλτ∂2

η

λ∂3

η

_τ, A²α= −

^αλτ∂1

η

λ∂3

η

_τ, A³α=

^αλτ∂1

η

λ∂2

η

_τ, and L stands for the lower-order terms. Also,

ε

^αβγ isthetotallyanti-symmetricsymbolwith

ε

¹²³=^{1.Toexplorethe}cancellation,wegroupthetermsin(7) asT₁+^T3, T2+^T5,andT4+^T6.Allthreepairsaretreatedbyintegratingbypartsintime.WhenwritingoutT1+^T3,thehighest-order termsare

t

0

q

^αλτ

∂

2

∂

_t²v_λ

∂

3

η

_τ

∂

1

∂

_t³v_α

+

t

0

q

^αλτ

∂

1

∂

_t³v_λ

∂

3

η

_τ

∂

2

∂

_t²v_α

andtheleadingtermscancelbyrelabelingtheindices

α

↔λ inthesecond integral.Thesamecancellationoccursalsoin theothergroupingsT2+^T5andT4+^T6.However,thereweintegratethederivatives∂2and∂3;thelastsumproducesthe boundaryterm

1q∂_t²v₂∂2∂_t²v₃.Toboundthisterm,weusetheprojection identity^α_β=δ^α_β−^gkl∂k

η

^α∂l

η

β,whichallows ustorelate∂∂_t²vand∂∂_t²v³andwrite

1

q

∂

_t²v2

∂

2

∂

_t²v3

=

1

q

∂

_t²v2

(

³_λ

∂

2

∂

_t²v^λ

+

^gkl

∂

_k

η

3

∂

_l

η

_λ

∂

2

∂

_t²v^λ

).

Thelastexpressioncanthenbe controlledusingthecoercivetermsandthetrace inequality.Collectingalltheinequalities leadsto

∂

_t³v ²₀

+ ∂

_t³R ²₀

+ ∂∂

_t²v ²_0,1

≤

N

+

P0

+

P

t

0

P

.

Theenergyestimatesfortwotimederivativesandonetimederivativearesimilar.Forinstance,usinganalogousmethods, weobtain

∂∂

t²v ²₀

+ ∂∂

t²R ²₀

+ ∂

²

∂

tv ²_0,

1

≤

N

+

P0

+

P

t

0

P ⁽⁸⁾

and

∂

²

∂

tv ²₀

+ ∂

²

∂

tR ²₀

+ ∂

³v ²_0,

1

≤

N

+

P0

+

P

t

0

P

.

(9)

We emphasize however, that unlike [1], we cannot continue and perform the same estimate for ∂³v ²₀+ ∂³R ²₀. The reasonisthattheanalogoftheterm(7) cannotbetreatedthesamewayduetolackoftimederivatives.

Instead,weobtainthefullcontrolofN ^{byusinganewCauchyinvariancepropertyforthe}compressibleEulerequations, statednext.NotethattheCauchyinvarianceprovidesa3Danalogofthe2Dvorticitypreservation.

Theorem2.2.(CompressibleCauchyInvarianceFormula)Let(v,R)beasmoothsolutionto(1)definedon[⁰,T).Then

ε

^αβγ

∂

βv^μ

∂

γ

η

μ

= ω

^α₀

+

t

0

ε

^αβγa^λμ

∂

λq

∂

γ

η

μ

∂

βR

R²

,

(10)

for0≤^t<T ,where

ω

0isthevorticityattimezero.

Here,weprovideasketchoftheproof(cf. [5] forcompletedetails).First,wehave

∂

t

( ε

^αβγ

∂

βv^μ

∂

γ

η

μ

) = ε

^αβγ

∂

βv^μ

∂

γv_μ

+ ε

^αβγ

∂

β

∂

tv^μ

∂

γ

η

μ

= −

¹

R

ε

^αβγ

∂

β

(

a^λμ

∂

λq

)∂

γ

η

μ

+

¹

R²

ε

^αβγa^λμ

∂

λq

∂

βR

∂

γ

η

μ

,

by theanti-symmetryof

ε

^αβγ and(1a).Sincea=(∇

η

)⁻¹,weget∂β(a^λμ∂γ

η

μ)=∂βa^λμ∂γ

η

μ+^aλμ∂γ∂β

η

μ=^0,andafter somesimplealgebra,∂t(

ε

^αβγ∂βvμ∂γ

η

μ)= _R¹2

ε

^αβγa^λμ∂λq∂βR∂γ

η

μ.Theformula(10) thenfollowsbyintegratingintime.

TherestoftheproofofTheorem2.1maynowbeperformedsimilarlyto [6].Namely,controlofthecurlisprovidedby theCauchyinvarianceformula(10).Thenthedivergenceofvanditstimederivativesareobtainedfrom(1b) inaninductive fashion, usingthe factthatthethird timederivativesareindependentlyestimatedfrom(4).Finally,theboundaryintegral isestimatedusing(5),(8),and(9).Theexception isthemissingestimate forv₃,whichisobtainedfromtheequation(1e) (cf. [5] fordetails).

Collectingalltheestimatesleadsto

N

(

t

) ≤

C0P

(

N

(

0

)) +

P

(

N

(

t

))

t

0

P

(

N

(

s

))

ds

,

where P isapolynomial,andtherestfollowsbyastandardGronwallargument.

3. Incompressiblecase

In this section, we describe the a priori estimates forthe local existence forthe incompressible Eulerequations. We addresstheequationsintheLagrangiancoordinates,wheretheytaketheform

∂

tv^α

+

^aμα

∂

μq

=

^{0 in}

[

⁰

,

T

) × ,

(11a)

a^αβ

∂

_αv_β

=

^{0 in}

[

⁰

,

T

) × ,

(11b)

∂

ta^αβ

+

a^αγ

∂

_μv_γa^μβ

=

0 in

[

0

,

T

) × ,

(11c)

a^μαN_μq

+ σ |

a^TN

|

g

η

^α

=

0 on

[

0

,

T

) ×

1

,

(11d)

v^μN_μ

=

^{0 on}

[

⁰

,

T

) ×

0

,

(11e)

η (

0

, · ) =

^id

,

v

(

0

, · ) =

^v0

.

(11f)

All thequantitiesare asinthecompressiblecase, exceptforq,whichisnow aLagrange multiplier enforcingtheincom- pressibilityconstraint.Thenwehavethefollowinga prioriestimatessupportingthelocalexistencewiththevelocity H³ in theinterior,withaconditiononthetraceatthefreeboundary.

Theorem3.1.Assumethat

σ

>0in(11).Letv0beasmoothdivergence-freevectorfieldon.ThenthereexistT_∗>0andaconstant C₀,dependingonlyon v_{0 3}, v_{0 4,1},and

σ

>0,suchthatanysmoothsolution(v,q)to(11)withinitialconditionv₀anddefined onthetimeinterval[⁰,T_∗),satisfies

v 3

+ ∂

tv 2.5

+ ∂

_t²v 1.5

+ ∂

_t³v 0

+

^q 3

+ ∂

tq 2

+ ∂

_t²q 1

≤

^C0

.

(12)

Observethat theSobolevexponentsin(3) and(12) differ.Theproof ofTheorem3.1isdifferentthan theoneofTheo- rem2.1;however,themainstepsalsorelyontheenergyinequalityforthethirdtimederivativeofthevelocity

1

2

∂

_t³v ²₀

=

¹

2

∂

_t³v

(

0

)

²₀

−

t

0

∂

_t³

∂

_μ

(

a^μαq

)∂

_t³v_α

.

Themaindifferencewiththecompressiblecaseareinthetreatmentofthepressureandinboundingtheboundaryintegral

−

t

0

1

√

g

(

g^{i j}g^kl

−

^gljgik

)∂

_j

η

^α

∂

_k

η

_λ

∂

_t²

∂

_lv^λ

∂

_t³

∂

_iv_α

.

(13)

Forthepressure,themaininequalitiesaresummarizedinthefollowingstatement.

Lemma3.2.Wehavetheestimates

q 3

≤

^C

∇

^v 2 v 2

+

^C

∂

tv 1.5,1

+

^C

,

∂

tq 2

≤

C

∇

v 1.5+δ

(

q 2.5

+ ∂

tv 1.5

) +

C

( ∇

v 1.5

∇

v L^∞

+ ∇∂

tv 1.5

)

v 1.5+δ

+

C

∂

_t²v 1,1

+

C v 2.5

+

C

∂

tv 2.5

,

∂

_t^2q 1

≤

^C

(

^v 1.5

∇

^v L^∞

+ ∂

tv _1.5

)(

^q 2

+ ∂

tv ₁

) +

^C

∇

^v L^∞

( ∂

tq ₁

+ ∂

_t^2v 0

) +

^C

(

^v 1.5

∇

^{v 2}_L∞

+ ∂

tv _1.5

∇

^v L^∞

+ ∂

_t^2v 1.5

)

^v 1

+

^C

∂

_t^3v 0

+

^C

( ∂

tv _2.5

+

^v 3

∂

tq ₁

) +

^C

(

1

+

^{v 2}₃

+ ∂

tv ₂

)( ∂

tv ₂

+

^{v 2}₃

),

fort∈(0,T)andδ >0asmallnumber.

Thelemmaisobtainedbysolvingtheellipticboundaryvalueproblemsforq,∂tq,and∂_t²q.Forq andq_t,weconsiderthe Neumannproblem,whileforthesecondderivative,weusetheDirichletproblem,togetherwithanestimate

∂

_t^2q 0,1

≤ ∂

_t^2q 1

+

^C

( ∂

tv _2.5

+

^v 3

∂

tq ₁

) +

^C

(

1

+

^{v 2}₃

+ ∂

tv ₂

)( ∂

tv ₂

+

^{v 2}₃

)

(cf. [4] for details). The treatment for the boundary integral (13) is involved and it relies on its determinant structure (cf. [2]). The restof the proof is obtained by the H¹ estimate for ∂_t²v. However, in order to control v ₃, we use the additionalboundaryregularityprovidedbythesurfacetension,togetherwiththediv-curlestimatesprovidedbytheCauchy invariance[7].

Acknowledgements

We thank the referee for corrections and useful suggestions. MMD was partially supported by the NSF grant DMS-1305705,whileIKwaspartiallysupportedbytheNSFgrantDMS-1615239.

References

[1]D.Coutand,J.Hole,S.Shkoller,Well-posednessofthefree-boundarycompressible3-DEulerequationswithsurfacetensionandthe zerosurface tensionlimit,SIAMJ.Math.Anal.45 (6)(2013)3690–3767.

[2]D.Coutand,S.Shkoller,Well-posednessofthefree-surfaceincompressibleEulerequationswithorwithoutsurfacetension,J.Amer.Math.Soc.20 (3) (2007)829–930.

[3]M.M.Disconzi,D.G.Ebin,ThefreeboundaryEulerequationswithlargesurfacetension,J.Differ.Equ.261 (2)(2016)821–889.

[4] M.Disconzi,I.Kukavica,Aprioriestimatesforthefree-boundaryEulerequationswithsurfacetensioninthreedimensions,preprint,2017.

[5] M.Disconzi,I.Kukavica,Aprioriestimatesforthe3dcompressiblefree-boundaryEulerequationswithsurfacetensioninthecaseofaliquid,preprint, 2017.

[6]M.Ignatova,I.Kukavica,Onthelocalexistenceofthefree-surfaceEulerequationwithsurfacetension,Asymptot.Anal.100(2016)63–86.

[7] I.Kukavica,A.Tuffaha,V.Vicol,Onthelocalexistenceforthe3dEulerequationwithafreeinterface,Appl.Math.Optim.76 (3)(2017)535–563, https://doi.org/10.1007/s00245-016-9360-6.

[8]B.Schweizer,Onthethree-dimensionalEulerequationswithafreeboundarysubjecttosurfacetension,Ann.Inst.HenriPoincaré,Anal.NonLinéaire 22 (6)(2005)753–781.

[9]J. Shatah,C.Zeng,GeometryandaprioriestimatesforfreeboundaryproblemsoftheEulerequation,Commun.PureAppl.Math.61 (5)(2008) 698–744.

[10]J.Shatah,C.Zeng,Localwell-posednessforfluidinterfaceproblems,Arch.Ration.Mech.Anal.199 (2)(2011)653–705.