A model for Rayleigh-Taylor mixing and interface turn-over

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turn-over

Rafael Granero-Belinchón, Steve Shkoller

To cite this version:

RafaelGranero-Belin hón

UnivLyon,UniversitéClaudeBernardLyon1

CNRSUMR5208,InstitutCamilleJordan

43blvd. du11novembre1918

F-69622Villeurbanne edex, Fran e

email: graneromath.univ-lyon1.fr Steve Shkoller Departmentof Mathemati s University of California Davis, CA 95616 USA email: shkollermath.u davis.edu November 17,2016 Abstra t

We rst develop a new mathemati al model for two-uid interfa e motion, subje ted

to the Rayleigh-Taylor (RT) instabilityintwo-dimensional uidow, whi h inits simplest

form, is given by

h

tt

(α, t) = Ag Λh −

σ

ρ

+

_+ρ

−

Λ

3

h − A∂

_α

(Hh

_t

h

_t

)

, where

Λ = H∂

α

and

H

denotes the Hilbert transform. In this so- alled

h

-model,

A

is the Atwood number,

g

is the a eleration,

σ

is surfa e tension, and

ρ

±

denotes the densities of the two uids.

We derive our

h

-model using asymptoti expansions in the Birkho-Rott integral-kernel formulation forthe evolutionofan interfa e separating two in ompressible and irrotational

uids. The resulting

h

-model equation is shown to be lo ally and globally well-posed in Sobolevspa eswhena ertainstability onditionissatised;thisstability onditionrequires

that the produ t of the Atwood number and the initial velo ity eld be positive. The

asymptoti behavior of these global solutions, as

t → ∞

, is also des ribed. The

h

-model equation is shown to have interesting balan e laws, whi h distinguish the stabledynami s

from theunstable dynami s. Numeri alsimulations ofthe

h

-model showthattheinterfa e an qui kly grow due to nonlinearity, and then stabilizewhen the lighter uid ison top of

the heavier uid and a eleration is dire ted downward. In the unstable ase of a heavier

uid being supported by the lighter uid, we nd good agreement for the growth of the

mixing layerwithexperimentaldatainthero ket rig experiment of Read ofYoungs.

We then derive an interfa e model for RT instability, with a general parameterization

z(α, t) = (z

1

(α, t), z

2

(α, t))

su hthat

z

satises

z

tt

= Λ



A

|∂

α

z|

2

H z

t

· (∂

α

z)

⊥

_H(z

t

· (∂

α

z)

⊥

)

+

[[p]]

ρ

+

_+ρ

−

+ Agz

₂



(∂

α

z)

⊥

|∂

α

z|

2

+ z

t

· (∂

α

z)

⊥



(∂

α

z

t

)

⊥

|∂

α

z|

2

−

(∂

α

z)

⊥

2(∂

α

z·∂

α

z

t

)

|∂

α

z|

4



Contents

1 Introdu tion 2

2 Equations for multi-phase ow 3

3 The integral kernel formulation 4

4 Dynami s of the interfa e and vorti ity amplitude 6

5 The ase that the interfa e

Γ(t)

is a graph 6

6 A simple model equation for RT instability 7

6.1 Theequations inthe linearregime . . . 7

6.2 Thenonlinear regimeand themodelequation . . . 8

7 Well-posedness of the

h

-model equation (27) 10 8 General interfa e parameterization with interfa e turn-over 22 9 Numeri al study 24 9.1 Thealgorithm . . . 24

9.1.1 Numeri alapproximation ofthe

h

-model . . . 25

9.1.2 Numeri alapproximation ofthe

z

-model . . . 26

9.2 Simulation 1:

h

-model,

ρ

+

ρ

−

=

_1.5

1

andAtwood number

A < 0

. . . 26

9.3 Simulation 2:

h

-model,

ρ

+

ρ

−

=

1.23

1027

for Atwoodnumber

A < 0

. . . 28

9.4 Simulation 3:

h

-model, ngering instability,Atwood number

A > 0

. . . 29

9.5 Simulation 4:

h

-model, Stability,Atwood number

A < 0

. . . 30

9.6 Simulation 5: The ro ketrig experiment ofRead and Youngs . . . 30

9.6.1 The

h

-model . . . 31

9.6.2 The

z

-model . . . 33

9.7 Simulation 6: The tilted rig experiment . . . 35

9.7.1 The

h

-model . . . 36

9.7.2 The

z

-model . . . 36 9.8 Simulation 7:

z

-model, Kelvin-Helmholtz instability,Atwood number

A = 0

. 38

1 Introdu tion

Theinstabilityofaheavyuidlayersupportedbyalightoneisgenerallyknownas

Rayleigh-Taylor (RT) instability(see Rayleigh[13 ℄ and Taylor[18 ℄). It an o urundergravity and,

equivalently, under an a eleration of the uid system in the dire tion toward the denser

uid; inparti ular, RT isan interfa e ngering instability,whi h o urs when aperturbed

interfa e,between twouids ofdierent density,is subje tedto anormal pressuregradient.

two uids to mix. See Sharp [17℄, Youngs [19 , 20 ℄, and Kull[12℄ for an overviewof theRT

instability.

TheEulerequationsofinvis idhydrodynami sserveasthebasi mathemati almodelfor

RTinstabilityand mixing between two uids. Thishighly unstablesystemof onservation

laws is both di ult to analyze (as it is ill-posed in the absen e of surfa e tension and

vis osity)and di ultto omputationally simulateat thesmall spatials alesofRTmixing.

As su h, our obje tive is to develop model equations, whi h an be used to predi t theRT

mixing layerand growthrate.

In order to derive our RT interfa e models, we shall assume both in ompressible and

irrotational ow for the two-uid Euler equations. Rather than pro eeding with a dire t

approximation of the Euler equations, we shall instead work with the equivalent

Birkho-Rott singular integral-kernel formulation for the evolution of the material interfa e. We

havefoundthatthisformulationpossessesa ertainrobustnessinregardstoapproximations

founded upon expansionsinvarious parameters.

Inthesimplest ase, inwhi htheinterfa eismodeledasagraph

(α, h(α, t))

ofa signed height fun tion

h(α, t)

,Our approa h yields a se ond-order in-time quadrati ally nonlinear wave equation forthe positionof theinterfa e, the

h

-model, whi h isgiven by

h

tt

(α, t) = Ag Λh −

σ

ρ

+

_{+ ρ}

−

Λ

3

_{h − A∂}

α

(Hh

t

h

t

) ,

where

Λ = H∂

α

and

H

denotes theHilberttransform. Inthis

h

-modelequation,

A

denotes theAtwoodnumber,

g

isthea eleration,

σ ≥ 0

isthesurfa e tension, and

ρ

±

_{> 0}

denotes

thedensitiesof the two uids.

As we will des ribe below, our

h

-model equation for RT instability is both lo ally and globally well-posedinSobolev spa es when a stability ondition issatised, whi h requires

theprodu tof theAtwood numberand the initial velo ityeldto be positive. Under su h

a stability ondition, we also derivetheasymptoti behavior ofsolutions to the

h

-modelas

t → ∞

. A number of interesting energy laws are also derived, whi h distinguish between stableand unstableinterfa e dynami s

Numeri alsimulationsareperformed, whi hshowthatthe

h

-modelis apableof produ -ingremarkablegrowthoftheinterfa e,followedby(possiblyos illatory)de aytoareststate

inthe asethatthelighteruidissupportedbytheheavieruid. Inthehighlyunstable ase,

where a heavy uid is supported by the lighter uid, we perform the lassi al ro ket rig

experiment of Read [14 ℄ and Youngs [20℄ for the ase of unstable RT mixing-layer growth,

andndverygoodagreement forthegrowthrateswiththepredi tedquadrati growthrate

of Youngs [20℄,andwithboth Dire tNumeri al Simulations andexperimental data.

Finally, in order to allow for the interfa e to turn-over (rather than simply remaining

a graph) we develop a more general

z

-model for the interfa e parameterization

z(α, t) =

Ratherthan onstrainingtheinterfa e amplitudeto growattheRTinstability,the

z

-model an allow for interfa e turn-over. We perform numeri al simulations to demonstrate the

improvement aordedbythismoregeneralmodel,andtheevenbettera ura yinmat hing

thepredi tedgrowthoftheRTmixinglayerforthero ketrig experiment. Wealsoperform

theso- alled tilted rig experiment,inwhi h thetankholding theuidistitledbya small

angle away from verti al. Again, our simulations demonstrate good qualitative agreement

with DNS. To on lude, we also perform a numeri al simulation for a Kelvin-Helmholtz

problem, for whi htheAtwood numberisset tozero to prevent anRTinstability; starting

from a steep mode-

1

wave, we show roll-over of the interfa e with a very lo alized energy spe trum.

2 Equations for multi-phase ow

The Euler equations are a system of onservation laws, modeling the dynami s of

multi-phase invis id uid ow. For in ompressible two-dimensional motion, the onservation of

momentum foran homogeneous, invis id uid an be written as

ρ u

t

+ (u · ∇)u

+ ∇p = −gρe

2

,

(1)

where

u = (u

1

, u

2

)

denotes the velo ity ve tor eld of the uid,

p

is the s alar pressure fun tion,

ρ

is the density,

g

is the a eleration,

∇ = (∂

x

1

, ∂

x

2

)

is the gradient ve tor, and

e

2

= (0, 1)

.

For in ompressibleow, the onservationof massisgiven by

∇ · u = 0 .

We shallfurther assumethat theuidis irrotational sothat

ω := curl u = 0,

(2)

where

ω

isthe vorti ityof theuid,and

curl u = ∂

1

u

2

_{− ∂}

2

u

1

.

We assumethat thereare two uidswith densities

ρ

+

and

ρ

−

,separated bya material

interfa e

Γ(t)

,where

t

denotes an instant oftime intheinterval

[0, T ]

. AsshowninFigure 1 ,theuidwithdensity

ρ

+

liesabove

Γ(t)

,whiletheuidwithdensity

ρ

−

liesbelow

Γ(t)

.

Γ(t)

u

+

, ρ

+

u

−

, ρ

−

Figure 1: Theblue urveisanillustrationoftheinterfa e

Γ(t)

,separatingtwouidsatatime

t

∈ [0, T ]

. Theuidontopof

Γ(t)

hasdensity

ρ

+

,whiletheuidonthebottomhasdensity

ρ

−

.

For ea h instant of time

t ∈ [0, T ]

, the interfa e

Γ(t)

is parameterized by a fun tion

Wedenote the jumpofa eldvariable

f (x, t)

a ross

Γ(t)

by

[[f ]] = f

+

− f

−

.

We let

n

and

τ

denotetheunitnormaland tangentve torsto

Γ(t)

,respe tively;for theRT instability,we have thefollowing jump onditions:

[[u · n]] = 0 ,

[[u · τ ]] 6= 0 ,

on

Γ(t) .

(3) Note, that

∂

α

z(α, t)

isa tangent ve tor to

Γ(t)

at thepoint

z(α, t)

,sothat

[[u · ∂

α

z]] 6= 0.

Consequently, the velo ity is not ontinuous at the interfa e. Due to the fa t that the

shapeof the interfa e isdeterminedonly bythe normal omponent of theuidvelo ity,the

motion of the interfa e has a tangential reparameterization symmetry,so that we an add

anytangential velo ity to the motion of the interfa e, withthe hope that this willsimplify

theanalysis. Assu h,theevolutionequation for the parameterizationof

Γ(t)

iswritten as

z

t

(α, t) = u(z(α, t), t) + c(α, t)∂

α

z(α, t),

(4) where the fun tion

c(α, t)

will be spe ied below.

3 The integral kernel formulation

AnumberofmodalmodelshavebeenproposedfortheevolutionoftheRTmixinglayer;see,

for example, Rollin & Andrews [16℄ and Gon harov [8℄, and the referen es therein. These

modal models are based on a modal de omposition and approximation of the evolution

equations for the parameterization of theinterfa e and thevelo ity potential; su h models

onsistof a large oupled systemof nonlinearODEs for anite setof Fourier modes.

Rather than developing a modal model for RT mixing and approximating the partial

dierential equations themselves, we shall take another approa h to the development of

an RTmodel, whi h is founded upon the Birkho-Rott integral-kernel formulation for the

evolution ofan interfa e,separating two in ompressibleand irrotational uids.

Inordertointrodu ethe integral-kernelformulation,whi h onsistsofsingularintegrals,

webeginbydeningtheprin ipal value integral ofa given fun tion

f

as P.V.

Z

R

f (β)dβ = lim

ǫ→0

+

Z

(−1/ǫ,−ǫ)∪(ǫ,1/ǫ)

f (β)dβ .

The well known Biot-Savart kernel

K

BS

is an integral representation for

∇

⊥

_∆

−1

where

∇

⊥

= (−∂

x

2

, ∂

x

1

)

,and

∆ = ∂

2

x

1

+ ∂

2

x

2

denotes the Lapla e operator intheplane. In other words,ifthetwo uidsllthe plane,then

∆

−1

isgiven bytheNewtonian potential andthe

Biot-Savart kernelis givenby

Similarly,ifthe uidowis periodi inthe horizontalvariable, then

K

BS

(x, y) =

1

4π



− sinh(x

2

− y

2

)

cosh(x

2

− y

2

) − cos(x

1

− y

1

)

,

sin(x

1

− y

1

)

cosh(x

2

− y

2

) − cos(x

1

− y

1

)



.

(6)

Due to the hara teristi s of the irrotational ow, the vorti ity is a measure whi h is

supported onthe interfa e

Γ(t)

,written as

ω = ̟δ

_Γ(t)

,

where

δ

Γ(t)

is the Dira delta fun tion supported on theinterfa e

Γ(t)

. Morepre isely,the vorti ity

ω

isadistributiondened asfollows: forall smoothtest fun tions

ϕ

with ompa t support,

ω(ϕ) =

Z

R

̟(β, t)ϕ(z(β, t))dβ .

Thefun tion

̟

istheamplitudeofthe vorti ityalong

Γ(t)

. Noti ethat

̟

isminusthejump of thevelo ityinthetangential dire tion:

̟ = −[[u · ∂

α

z]].

Giventhe vorti itymeasure

̟

,we an re onstru t thevelo ity eldeverywhere; spe i- ally,we have, thankstotheBiot-Savartlaw, that

u(x, t) =

P.V.

Z

R

̟(β)K

BS

(x, z(β, t))dβ.

(7)

Sin e by (2) , the velo ity is irrotational, there exist a velo ity potential fun tion

φ :

R

2

→ R

su hthat

u

±

_{= ∇φ}

±

. The potential

φ

satisesBernoulli's equation

ρφ

t

+

ρ

2

|∇φ|

2

_{+ p + ρgx}

2

= f (t) ,

(8)

and

φ

isrelatedto

̟

bythe singular integral equation

φ(x, t) =

1

2π

P.V.

Z

R

̟(β) arctan



x

2

− z

2

(β, t)

x

1

− z

1

(β, t)



dβ.

(9)

As the kernel

K

BS

(x, z(β, t))

is singular for

x = z(α, t)

, it follows that thetangential om-ponent of

u

is dis ontinuous a rossthe interfa e

Γ(t)

. By similar reasoning, the potential fun tion

φ

is also dis ontinuous a ross

Γ(t)

. Moreover, the vorti ity

̟

along theinterfa e

Γ(t)

is relatedto thejumpofthevelo ity potential by

̟ = −∂

α

[[φ]] .

(10)

Now,usingtheparameterization

z(α, t)

fortheinterfa e

Γ(t)

,wedenetheBirkho-Rott prin ipal-value integral, denoting

Inparti ular, (11 )is expli itlygiven by

K

BR

(α, t) =

1

2π

P.V.

Z

̟(β)



−

z

2

(α, t) − z

2

(β, t)

|z(α, t) − z(β, t)|

2

,

z

1

(α, t) − z

1

(β, t)

|z(α, t) − z(β, t)|

2



dβ.

(12)

TheBirkho-Rottfun tion

K

BR

(α, t)

denotes theaveragevelo ityatagivenpoint

α ∈ Γ(t)

, sothat

K

BR

=

u

+

+ u

−

2

on theinterfa e

z(α, t).

4 Dynami s of the interfa e and vorti ity amplitude

We nowformulate thedynami s of theinterfa e. Sin e

z

t

= u + c∂

α

z

,we seethat

z

t

(α, t) =

1

2π

P.V.

Z

̟(β)

(z(α, t) − z(β, t))

⊥

|z(α, t) − z(β, t)|

2

dβ + c(α, t)∂

α

z(α, t) .

(13) A lengthy omputation thenshows [6℄that

̟

t

= −∂

α



A

4π

2









Z

̟(β)

(z(α, t) − z(β, t))

⊥

|z(α, t) − z(β, t)|

2

dβ









2

−

A

4

̟(α, t)

2

|∂

α

z(α, t)|

2

+

A

π

Z

̟(β)

(z(α, t) − z(β, t))

⊥

|z(α, t) − z(β, t)|

2

· ∂

α

z(α, t)c(α, t)dβ

− c(α, t)̟(α, t) −

2[[p]]

ρ

+

_{+ ρ}

−

− 2Agz

2



+

A

π

∂

t

 Z

̟(β)

(z(α, t) − z(β, t))

⊥

|z(α, t) − z(β, t)|

2

· ∂

α

z(α, t)dβ



.

(14)

5 The ase that the interfa e

Γ(t)

is a graph

We nowassumethattheinterfa e

Γ(t)

isthegraph of thesignedheight fun tion

h(α, t)

,so thattheparameterization

z(α, t)

isgiven by

(z

1

(α, t), z

2

(α, t)) = (α, h(α, t)) .

(15) If at time

t = 0

,

Γ(0)

is given as the graph

(α, h(α, 0))

, then we an ensure that

Γ(t)

stays a graph for future time

t > 0

by making an expli it hoi e of the fun tion

c(α, t)

in (4 ). To thisend,we dene

c(α, t) =

1

2π

P.V.

Z

̟(β)

h(α, t) − h(β, t)

theevolutionequation for

Γ(t)

an bewritten intermsof theheight fun tion

h(α, t)

as

h

t

(α, t) =

1

2π

P.V.

Z

_{̟(β, t)}

α − β

1

1 +



h(α,t)−h(β,t)

_α−β



2

dβ + c(α, t)∂

α

h(α, t) ,

(18)

and theevolutionequation forvorti ity

̟

on

Γ(t)

takes theform

̟

t

(α, t) =

A

4

∂

α



̟

2

(α, t)

1 + (∂

α

h(α, t))

2



+ ∂

α

(c(α, t)̟(α, t)) +

2

ρ

+

_{+ ρ}

−

∂

α

[[p]]

+ 2A∂

α

K

BR

(α, t) · (1, ∂

α

h)c(α, t) + 2Ag∂

α

h + 2A∂

t

K

BR

(α, t) · (1, ∂

α

h) .

(19)

6 A simple model equation for RT instability

In this se tion, we shall derive a model of RT instability in whi h the interfa e

Γ(t)

is a graphofthe height fun tion

h(α, t)

. Inorderto pro eedwithour derivation,weshallmake further approximations to the oupled systemof equations(18 ) and (19 ).

6.1 The equations in the linear regime

Inthelinear regime,equations (18 ) and(19) redu eto thefollowing oupled system:

h

t

(α, t) =

1

2

H̟ ,

(20a)

̟

t

(α, t) = +2Ag∂

α

h +

2σ

ρ

+

_{+ ρ}

−

∂

3

α

h ,

(20b)

where

H

denotes the Hilbert transform,dened as

H̟(α) =

1

π

P.V.

Z

̟(β)

α − β

dβ .

(21)

6.2 The nonlinear regime and the model equation

Having found the linear dynami s,weturn our attention to thethenonlinear regime. Our

obje tive isto derive a modelequation whi h ontains the quadrati nonlinearities. To do

so,weshall introdu e some furthernotation.

Wedene the operator

Λ

by

Λ̟ = H∂

α

̟ ,

and thespa e-integratedvorti ity as

h̟i(t) =

Z

̟(β, t)dβ .

We shallusethe notation

hf i(t)

to denote

R

f (β, t)dβ

for anyintegrable fun tion

f (β, t)

. Wemake useof the following power seriesexpansionsfor

|ζ| < 1

:

1

1 + ζ

2

= 1 − ζ

and

ζ

1 + ζ

2

= ζ − ζ

3

_{+ ζ}

5

_{− ζ}

7

_{+ · · · ,}

Using these identities togetherwiththeapproximation

h(α) − h(β)

α − β

≈ ∂

α

h(α) +

1

2

∂

2

α

h(α)(β − α),

thenonlo al termsinequations (18)and (19 ) an beapproximated as

c(α, t)∂

α

h(α, t) ≈

∂

α

h(α, t)

2π

P.V.

Z

_̟(β)

α − β

h(α, t) − h(β, t)

α − β

dβ ,

K

BR

(α, t) ≈

1

2π

P.V.

Z

̟(β, t)

α − β

−

h(α, t) − h(β, t)

α − β

, 1 −



h(α, t) − h(β, t)

α − β



2

!

,

∂

α

K

BR

(α, t) · (1, ∂

α

h)c(α, t) ≈ −

1

2

∂

α



1

2π

P.V.

Z

_̟(β)

α − β

h(α, t) − h(β, t)

α − β

dβ



2

+

1

2

Λ̟∂

α

h

1

2π

P.V.

Z

_̟(β)

α − β

h(α, t) − h(β, t)

α − β

dβ

≈ −

1

2

∂

α



∂

α

h

2

H̟ −

∂

2

α

h

4π

h̟i



2

+

1

2

Λ̟∂

α

h



∂

α

h

2

H̟ −

∂

2

α

h

4π

h̟i



,

and

∂

t

K

BR

(α, t) · (1, ∂

α

h) ≈

1

2π

P.V.

Z

̟

t

(β)

α − β



−

h(α, t) − h(β, t)

α − β

+ ∂

α

h(α)



dβ

−

1

2

H̟∂

t

∂

α

h +

1

4π

h̟i∂

t

∂

2

α

h

≈

1

4π

∂

2

α

h

d

dt

h̟i −

1

2

H̟∂

t

∂

α

h +

1

4π

h̟i∂

t

∂

2

α

h.

If we negle ttermsof order

O(h

2

₎

inequations(18 ) and (19 ),wend that

h

t

(α, t) =

1

2

H̟ ,

(22a)

̟

t

(α, t) = 2Ag∂

α

h +

2σ

ρ

+

_{+ ρ}

−

∂

3

α

h

+

A

2

̟∂

α

̟ + ∂

α

̟

2

∂

α

h H̟



− ∂

α

 ̟

4π

∂

2

α

h



h̟i

+

A

4π

∂

α

Λ̟h̟i −

A

2

H̟Λ̟ +

A

2π

∂

2

α

h(α)

d

dt

h̟i .

(22b) Integratingin spa e,weobtain thatthevorti ityaverage

h̟i

veries

d

thus,

h̟i(t) = h̟i(0) = h̟

0

i.

Finally, we an usethe Tri omi relationfor theHilberttransform,

2H(f Hf ) = (Hf )

2

− f

2

,

(23) to obtain

1

2

∂

α

((H̟)

2

_{− ̟}

2

_{) = H̟Λ̟ − ̟∂}

α

̟ = ∂

α

H(̟H̟).

Then, the system(22 a,b)is equivalent to

h

t

(α, t) =

1

2

H̟ ,

(24a)

̟

t

(α, t) = 2Ag∂

α

h +

2σ

ρ

+

_{+ ρ}

−

∂

3

α

h + ∂

α

̟

2

∂

α

h H̟



− ∂

α

 ̟

4π

∂

2

α

h



h̟

0

i +

A

4π

∂

α

Λ̟h̟

0

i −

A

2

Λ(̟H̟) .

(24b) The oupledrst-order system(24 ) an be writtenasone se ond-orderequation forthe

evolution ofthe height fun tion:

h

tt

(α, t) = AgΛh −

σ

ρ

+

_{+ ρ}

−

Λ

3

_{h − Λ (Hh}

t

∂

α

h h

t

)

− A∂

α

(Hh

t

h

t

) + Λ



Hh

t

4π

∂

2

α

h



h̟

0

i +

A

4π

∂

α

Λh

t

h̟

0

i .

(25) Themodelequation(25 ) ontainsbothquadrati and ubi nonlinearities,butwe ansimply

further.

Keepingonlythequadrati nonlinearities,wendthatthegraphoftheinterfa e

(x, h(x, t))

evolvesa ording to

h

tt

(α, t) = AgΛh −

σ

ρ

+

_{+ ρ}

−

Λ

3

_h

− A∂

α

(Hh

t

h

t

) + Λ



Hh

t

4π

∂

2

α

h



h̟

0

i +

A

4π

∂

α

Λh

t

h̟

0

i .

(26) By assuming thatourinitial vorti ityhaszero average, wearriveat thenonlinear

equa-tion

h

tt

(α, t) = AgΛh −

σ

ρ

+

_{+ ρ}

−

Λ

3

_{h − A∂}

α

(Hh

t

h

t

) .

(27)

Equation (27) is supplemented with initial onditions. In parti ular, we spe ify theinitial

interfa e positionand velo ity,respe tively,by

h(·, 0) = h

0

and

h

t

(·, t) = h

1

.

As we explain in the next se tion, this model equation has a natural stability ondition,

requiring the produ t of the Atwood number

A

and and the initial velo ity eld

h

1

to be positive.

These ond-order-in-time nonlinearwaveequation(27)isanewmodelfor themotion of

an interfa e under the inuen e of RTinstability. We shallrefer to either thesystem (24)

Remark 1. The quadrati nonlinearity

u 7→ ∂

α

(u Hu)

has been previously studied by many authors (see for instan e [1 , 2 , 3 , 4 , 5 ℄). In those papers, the model problem was

paraboli andthus yieldedamaximumprin iple,fromwhi hmany oftheanalyti al results

followed. Ourstudy ofequation(27)involvesahyperboli -typeproblemwithnomaximum

prin iple. Of parti ular signi an e of the stru ture of (27) is the re overy of an RT-like

stability ondition for the model;this will bedis ussed ingreatdetail next.

7 Well-posedness of the

h

-model equation (27)

Wenowprovethatour

h

-modelequation(27)iswell-posedinSobolevspa es,whena ertain stability ondition issatised bythedata.

In thisse tion we onsiderthe periodi Hilbert transform

H

,dened as

Hf (α) =

1

2π

P.V.

Z

T

f (β)

tan



α−β

₂

 dβ ,

(28)

and theperiodi Zygmundoperator

Λf (α) = H∂

α

f (α) =

1

4π

P.V.

Z

T

f (α) − f (β)

sin

2



α−β

₂

 dβ .

(29)

Re allalso that

Λ = (−∆)

1

2

.

The spa e

L

2

_(T)

onsists of the Lebesgue measurable fun tions on the ir le

T

whi h are square-integrable withnorm

khk

2

0

=

R

T

|h|

2

_dx

. For

s ≥ 0

, we dene the homogeneous Sobolevspa e

˙

H

s

(T) =

(

h ∈ L

2

(T) :

X

k∈Z

|k|

2s

|ˆ

h(k)|

2

< ∞

)

,

with normdened as

khk

2

_s

=

Z

T

|Λ

s

h|

2

dx .

(30) For

s ≥ 0

,fun tions

H

˙

s

_(T)

areidentiedwith

2π

-periodi fun tions

[−π, π]

withnitenorm

k · k

s

. Fromequation (27),

hh

t

(t)i = hh

1

i.

Thus,

d

dt

hh(t)i =

Z

T

h

t

(t)dx = hh

1

i,

sothat

hh(t)i = hh

0

i + hh

1

it.

These identities, together with the Poin aré inequality, show that (30 ) is an equivalent

H

s

(T)

-norm. (Re all that we are using the notation

hf i(t)

to denote

R

Theorem 1 (Lo al well-posedness for the

h

-model (27 )). Let

σ ≥ 0

,

ρ

+

_{, ρ}

−

_{> 0}

,

g 6= 0

be xed onstants and let

(h

0

, h

1

)

denote initialpositionand velo ity pair for equation (27 ). Suppose that

(h

0

, h

1

) ∈ H

2.5+

sgn

(σ)

(T) × H

2

(T)

and let

λ := min

α∈T

A h

1

(α)

(31) If

λ > 0 ,

(32)

thenthereexistsatime

0 < T

∗

_(h

0

, h

1

) ≤ ∞

andaunique lassi alsolutionof (27 )satisfying

h ∈ C

0

([0, T

∗

]; H

2.5+

sgn

(σ)

), h

t

∈ C

0

([0, T

∗

]; H

2

) ∩ L

2

(0, T

∗

; H

2.5

).

Proof. Step 1. Approximate problem for

h

ǫ

,

ǫ > 0

. For

ǫ > 0

, we introdu e a sequen e of approximations to equation (27). We let

P

ǫ

denote theproje tion operator in

L

2

_(T)

,given by

P

ǫ

f (α) =

X

|k|≤1/ǫ

ˆ

f (k)e

ikα

,

where

f (k)

ˆ

denotes the

k

th Fourier mode of

f

. Then, we let

h

ǫ

be asolution to

h

ǫ

_tt

(α, t) = AgP

ǫ

ΛP

ǫ

h

ǫ

−

σ

ρ

+

_{+ ρ}

−

P

ǫ

Λ

3

_P

ǫ

h

ǫ

− AP

ǫ

∂

α

(P

ǫ

h

ǫ

t

P

ǫ

Hh

ǫ

t

) ,

(33)

with initial datagiven by

(P

ǫ

h

0

, P

ǫ

h

1

)

. The proje tion operator

P

ǫ

ommutes with

∂

α

and with

H

(and hen ewith

Λ

).

We let the parameter

ǫ

range inthe interval

(0, ǫ

0

]

for a onstant

ǫ

0

≪ 1

to be hosen later. Then, ODE theory provides a unique short-time solution

h

ǫ

to equation (33 ) on a

timeinterval

[0, T

ǫ

]

;thesolution

h

ǫ

issmooth and anbe taken in

C

2

_{([0, T}

ǫ

]; H

s

(T))

for all

s ≥ 0

.

A ording to the equation (33 ),

h

ǫ

tt

= P

ǫ

h

ǫ

tt

; sin e

(h

ǫ

_{(0), h}

ǫ

t

(0)) = (P

ǫ

h

0

, P

ǫ

h

1

)

, the fundamental theoremof al ulusshows that

h

ǫ

= P

ǫ

h

ǫ

and

h

ǫ

t

= P

ǫ

h

ǫ

t

.

Assu h,(33 ) an be written as

h

ǫ

_tt

(α, t) = AgΛP

_ǫ

2

h

ǫ

−

σ

ρ

+

_{+ ρ}

−

Λ

3

_P

2

ǫ

h

ǫ

− AP

ǫ

∂

α

(h

ǫ

t

Hh

ǫ

t

) .

(33 ')

Step 2. The higher-order energy norm. We dene thehigher-order energy norm

E(t)

by

E(t) = sup

0≤s≤t



σ

ρ

+

_{+ ρ}

−

kh

ǫ

_(s)k

2

3.5

+ kh

ǫ

t

(s)k

2

2

+

λ

2

Z

t

0

kh

ǫ

_t

(s)k

2

_2.5

ds



.

(34)

Notethat forea h

ǫ ∈ (0, ǫ

0

]

,

t 7→ E(t)

is ontinuouson

[0, T

ǫ

_]

.

Step 3. Lower bound for

h

ǫ

t

. We shallassumethat by hoosing

T

ǫ

and

ǫ

0

su iently small,

A h

ǫ

_t

(α, t) ≥

λ

Below, we shall verify that this assumption holds on a time interval

[0, T ]

, where

T

is independent of

ǫ

.

Step 4.

ǫ

-independent energy estimates. We now establish a time of existen e and bounds for

h

ǫ

whi h areindependent of

ǫ

. Wemultiply equation (27)by

Λ

4

_h

ǫ

t

,integrate over

T

,and nd that

1

2



d

dt

kh

ǫ

t

k

2

2

+

σ

ρ

+

_{+ ρ}

−

d

dt

kh

ǫ

_k

2

3.5



≤ |g|kh

ǫ

k

2.5

kh

ǫ

t

k

2.5

+ I,

(36) where

I = −A

Z

T

∂

α

(h

ǫ

t

Hh

ǫ

t

) ∂

4

α

h

ǫ

t

dα.

(37) We rst writethe integral

I

as

I = −A

Z

T

∂

_α

3

(h

ǫ

_t

Hh

ǫ

_t

)∂

_α

2

h

ǫ

_t

dα

= −A

Z

T

Λ∂

_α

2

h

ǫ

_t

h

ǫ

_t

∂

_α

2

h

ǫ

_t

dα

|

{z

}

I

1

− A

Z

T

Hh

ǫ

_t

∂

_α

3

h

ǫ

_t

∂

2

_α

h

ǫ

_t

dα

|

{z

}

I

2

− 3A

Z

T



Λ∂

α

h

ǫ

t

∂

α

h

ǫ

t

+ Λh

ǫ

t

∂

α

2



∂

_α

2

h

ǫ

_t

dα

|

{z

}

I

3

.

The integral

I

2

hasanexa t derivative,soupon integration-by-parts,

I

2

=

A

2

Z

T

Λh

ǫ

_t

∂

2

_α

h

ǫ

_t

∂

_α

2

h

_t

ǫ

dα ≤ kh

ǫ

_t

k

2

₂

kΛh

ǫ

_t

k

L

∞

≤ Ckh

ǫ

t

k

2

2

kh

ǫ

t

k

1.75

,

the last inequality following from the Sobolev embedding theorem. The integral

I

3

learly hasthesame bound.

Equipped with the assumption (35) and the estimate for

I

2

, the integral

I

1

provides extraregularityfor

h

The last equality follows fromthe fa tthat

Z

T

Z

T

∂

2

_α

h

ǫ

_t

(β)∂

_α

2

h

ǫ

_t

(α) (h

ǫ

_t

(α) − h

ǫ

_t

(β))

sin

2



α−β

₂



dαdβ = 0 .

Withour assumedlowerbound for

h

ǫ

t

in(35), wend that

−I

1

≥

λ

2

1

8π

Z

T

Z

T

∂

_α

2

h

ǫ

_t

(α) − ∂

2

_α

h

ǫ

_t

(β)

2

sin

2



α−β

₂



dαdβ .

Using the omputation

Z

T

f Λf dα =

Z

T





1

4π

Z

T

f (α) − f (β)

sin

2



α−β

₂

 dβ



 f(α)dα

= −

Z

T

1

4π

Z

T

f (α) − f (β)

sin

2



α−β

₂

 f(β)dβdα

=

1

8π

Z

T

Z

T

(f (α) − f (β))

2

sin

2



α−β

₂

 dβdα ,

itfollows that

I

1

≤ −

λ

2

kh

ǫ

t

k

2

2.5

,and hen e

I ≤ −

λ

2

kh

ǫ

t

k

2

2.5

+ Ckh

ǫ

t

k

2

2

kh

ǫ

t

k

1.75

.

Thus,

1

2



d

dt

kh

ǫ

t

k

2

2

+

σ

ρ

+

_{+ ρ}

−

d

dt

kh

ǫ

_k

2

3.5



≤ |g|kh

ǫ

k

2.5

kh

ǫ

t

k

2.5

−

λ

2

kh

ǫ

t

k

2

2.5

+ Ckh

ǫ

t

k

2

2

kh

ǫ

t

k

1.75

.

(38)

The fundamental theoremof al ulusshows that

By Gronwall's inequality,

E(t) ≤ C



kh

0

k

2

2.5

+ kh

1

k

2

2



.

(40)

Step 5. Verifyingthelower-bound on

h

ǫ

t

. Asthebound(40 )for

E(t)

isindependent of

ǫ

and

σ

,there existsa time interval

[0, T ]

,with

T

independent of

ǫ

and

σ

,su hthat

E(t) ≤ C



kh

0

k

2

2.5

+ kh

1

k

2

2



on

[0, T ]

with

T

independent of

ǫ, σ .

(41) Using theequation(33'), we seethat

h

ǫ

tt

isbounded in

L

2

_{(0, T ; L}

2

_(T))

. Thus,

kh

t

(T ) − P

ǫ

h

1

k

_L

2

≤

Z

T

0

kh

tt

(s)k

L

2

ds ≤ C T

Sin e

kh

t

(T ) − P

ǫ

h

1

k

2

≤ C

, we an interpolate between the last two inequalities and use theSobolev embedding theoremto on lude that

h

t

(T

2

) − P

ǫ

h

1

L

∞

≤ CT

1/8

_.

By hoosing

T

and

ǫ

0

su iently small,and using (31) , we verify(35 ). Step 6. Existen e of solutions. From(41 ), for all

1 < p < ∞

,

h

ǫ

⇀ h

in

W

1,p

_{(0, T ; H}

2

_{(T)) ,}

h

ǫ

_t

⇀ h

in

L

2

_{(0, T ; H}

2.5

_{(T)) ∩ H}

1

_{(0, T ; L}

2

_{(T)) ∩ L}

p

_{(0, T ; H}

2

_{(T)) .}

By using Relli h's theorem, we an pass to the limit as

ǫ → 0

in (33 '). Sin e

kf k

L

∞

=

lim

p→∞

kf k

L

p

,we ndthat thelimit

h

isasolution of (27)on

[0, T ]

and

h ∈ L

∞

(0, T ; H

2.5

(T)) , h

t

∈ L

2

(0, T ; H

2.5

(T)) ∩ L

∞

(0, T ; H

2

(R)) .

It is easy to prove that

t 7→ h(·, t)

and

t 7→ h

t

(·, t)

are ontinuousinto

H

2.5

_(T)

and

H

2

_(T)

,

respe tively, with respe t to the weak topologies on

H

2.5

_(T)

and

H

2

_(T)

; furthermore, we

an againndadierential inequality thatis almostidenti alto (39 )(butthistimeforthe

solution

h

ratherthanthe sequen e

h

ǫ

). It followsthat

t 7→ kh(t)k

2.5

and

t 7→ kh

t

(t)k

2.5

are ontinuous, hen e

h ∈ C

0

(0, T ; H

2.5

(T)) , h

t

∈ L

2

(0, T ; H

2.5

(T)) ∩ C

0

(0, T ; H

2

(R)) .

When

σ > 0

,bythesame argument,we have thebetter regularity

h ∈ C

0

(0, T ; H

3.5

(T)) .

Step 7. Uniqueness of solutions. Uniqueness of solutions follows from a standard

L

2

-type

energy estimate, andweomit the details.



Remark 2. Ifthestability ondition

A h

1

> 0

isnotsatised,theevolutionequation(27) may not be well-posed in Sobolev spa es. For analyti initial data, however, the equation

doesnotrequireastability onditionforashort-timeexisten etheorem. Weshallinvestigate

Having establishedexisten eanduniquenessof lassi alsolutions to theRTmodel(27),

wenext establisha numberofinteresting energy lawssatised byits solutions.

Proposition 2 (Energy laws for solutions to the

h

-model (27 )). Given onstants

σ ≥

0, ρ

+

, ρ

−

> 0

, and

g 6= 0

, suppose that

h

is a solution to (27 ) with initial data

(h

0

, h

1

)

. Then, with

T

∗

given by Theorem 1 and for all

0 ≤ t ≤ T

∗

,

h

veries the following energy laws:

kh

t

k

2

0

+

σ

ρ

+

_{+ ρ}

−

khk

2

1.5

− Agkhk

2

0.5

+

Z

t

0

D

1

(s)ds

= kh

1

k

2

0

+

σ

ρ

+

_{+ ρ}

−

kh

0

k

2

1.5

− Agkh

0

k

2

0.5

,

(42a)

kh

t

k

2

0.5

+

σ

ρ

+

_{+ ρ}

−

khk

2

2

− Agkhk

2

1

+

Z

t

0

D

2

(s)ds

= kh

1

k

2

0.5

+

σ

ρ

+

_{+ ρ}

−

kh

0

k

2

2

− Agkh

0

k

2

1

,

(42b)

kh

t

− hh

1

ik

2

−0.5

+

σ

ρ

+

_{+ ρ}

−

khk

2

1

− Agkhk

2

0

+

Z

t

0

D

3

(s)ds + 2Ag

Z

t

0

2hh

1

ihh(s)ids

= kh

1

− hh

1

ik

2

−0.5

+

σ

ρ

+

_{+ ρ}

−

kh

0

k

2

1

− Agkh

0

k

2

0

,

(42 ) where the terms

D

1

,

D

2

, and

D

3

are given by

D

1

= 2

Z

T

P.V.

Z

T

A (h

t

(α) + h

t

(β)) (h

t

(α) − h

t

(β))

2

8π sin

2



α−β

₂



dαdβ ,

D

2

= A

Z

T

h

t



(Λh

t

)

2

+ (∂

α

h

t

)

2



dα ,

D

3

= 2A

Z

T

h

t

|Hh

t

|

2

dα .

Proof. Theproof redu es to a areful manipulation ofthefollowing integrals:

I

1

= −A

Z

T

∂

α

(h

t

Hh

t

)h

t

dα ,

I

2

= −A

Z

T

∂

α

(h

t

Hh

t

)Λh

t

dα ,

I

3

= −A

Z

T

∂

α

(h

t

Hh

t

)Λ

−1

(h

t

− hh

1

i)dα ,

where

I

j

= −D

j

/2

,for

j = 1, 2, 3

. First,

Next,

I

2

= −A

Z

T

∂

α

h

t

Hh

t

∂

α

Hh

t

hdα − A

Z

T

h

t

|Λh

t

|

2

dα.

Using theskew-adjointness ofthe Hilbert transformand Tri omi relation (23 ),

−A

Z

T

Hh

t

∂

α

h

t

H∂

α

h

t

dα = A

Z

T

h

t

H (∂

α

h

t

H∂

α

h

t

) dα =

A

2

Z

T

h

t



(Λh

t

)

2

− (∂

α

h

t

)

2



dα,

sothat

I

2

= −

A

2

Z

T

h

t



(Λh

t

)

2

+ (∂

α

h

t

)

2



dα.

Finally,

I

3

= −A

Z

T

Λ

−1

∂

α

(h

t

Hh

t

)(h

t

− hh

1

i)dα

= A

Z

T

H(h

t

Hh

t

)(h

t

− hh

1

i)dα

= −A

Z

T

h

t

|Hh

t

|

2

dα

where wehave usedthe fa tthat

Λ

−1

_∂

α

= −H

,sothat

H∂

α

= Λ

.



Corollary 3. If

ρ

+

< ρ

−

sothattheAtwoodnumber

A < 0

,andifgravitya tsdownward so that

g > 0

, then whenever the initial velo ity

h

1

satises the stability ondition

h

1

< 0

, then the energy law (42 b) shows that

kh

t

(t)k

2

0.5

+

σ

ρ

+

_{+ ρ}

−

kh(t)k

2

2

+ |Ag|kh(t)k

2

1

de ays in time for all

t ∈ [0, T

∗

_{] ,}

where

T

∗

is given by Theorem 1.

Remark 3. The energy law (42a) provides de ay for lower-order norms when

A h

1

> 0

, whiletheenergylaw(42 )maya tually auseagrowth-in-timewhi hbehaveslike

t

2

. There

may indeedbe other higher-order energy laws to the modelequation (27 ) thathave yet to

beestablished.

Wenowdene the average value ofa fun tion

h(α)

on

[−π, π]

:

¯

h :=

hhi

2π

=

1

2π

Z

T

h(α)dα .

Theorem 4 (Global well-posedness and asymptoti behavior for the

h

-model (27 )). Let

σ ≥ 0

,

ρ

+

_{, ρ}

−

_{> 0}

,

g 6= 0

be xed onstants su h that

Ag < 0

, and let

(h

0

, h

1

)

denote the initial position andvelo ity, respe tively, for the

h

-model (27 ). Setting

andwith

λ

dened by (31 ), suppose that

(h

0

, h

1

) ∈ H

2.5+

sgn

(σ)

(T) × H

2

_(T)

isgivensu h that

λ > 0 ,

(43) and

kh

2

k

2

0.5

+ kh

1

k

2

1

+ σkh

1

k

2

2

<



−¯

h

1

5



2

.

(44)

Then there exists a unique lassi al solution of (27 ) satisfying

h ∈ C

0

([0, T ]; H

2.5+

sgn

(σ)

), h

t

∈ C

0

([0, T ]; H

2

) ∩ L

2

(0, T ; H

2.5

) ∀ 0 ≤ T < ∞.

Furthermore, as

t → ∞

, the solution

h(·, t)

onverges to the homogeneous solution

h

∞

₌

¯

h

0

+ ¯

h

1

t

; spe i ally,

lim sup

t→∞

kh(t) − h

∞

_k

1+

sgn

(σ)

+ kh

t

(t) − ¯

h

1

k

1+

sgn

(σ)

= 0.

Proof. Without loss of generality, we onsider

ρ

+

_{= 0, ρ}

−

_{= 1}

and

g = 1

so that

A = −1

. Ouranalysis willrely onthe fa tthat

hh

1

i < 0 ,

sin e we haveassumed the initial velo ity satises

h

1

< 0

. It is onvenient to introdu e a newvariable

f (α, t)

given by

f = h − ¯

h

0

− ¯

h

1

t

sothat

f

t

= h

t

− ¯

h

1

.

It follows that

f

satisesthe evolution equation

f

tt

(α, t) + Λf + σΛ

3

f − ¯

h

1

Λf

t

= ∂

α

(Hf

t

f

t

) ,

(45)

with initial data

f

0

= h

0

− ¯

h

0

, f

1

= h

1

− ¯

h

1

.

The lo al existen e for

h

and

h

t

(and onsequently, for

f

and

f

t

) follows from Theorem 1 . Thus, in order to prove that solutions exist for all time, it remains only to establish

estimateswhi hareuniformintime,andthedesiredasymptoti behaviorwillbeestablished

by showing thatboth

f

and

f

t

onvergeto zero.

With

λ

dened by(31 ), thestability ondition(43)for

h

t

redu es to

sup

0≤t

kf

t

(t)k

L

∞

< −¯

h

₁

.

(46)

Furthermore, we have that

Wetest equation (45 )against

Λf

t

and obtainthat

1

2

d

dt

kf

t

k

2

0.5

+ kf k

2

1

+ σkf k

2

2

− ¯

h

1

kf

t

k

2

1

=

Z

T

∂

α

(Hf

t

f

t

)Λf

t

dα

≤ kf

t

k

2

1

(kf

t

k

L

∞

+ kHf

t

k

L

∞

).

Weshallmake useof therenedCarlson inequality[10℄:

kwk

2

_L

∞

≤ kwk

₀

kwk

₁

−

1

π

kwk

2

0

∀w ∈ ˙

H

1

(T)

with

hwi = 0 .

(47) Using (47 )together withthe Poin aré inequality shows that

kwk

L

∞

< kwk

₁

.

As a onsequen e,we obtain that

1

2

d

dt

kf

t

k

2

0.5

+ kf k

2

1

+ σkf k

2

2

− ¯

h

1

kf

t

k

2

1

< 2kf

t

k

3

1

.

(48)

Taking atime derivative ofequation (45), we ndthat

f

t

satises

f

ttt

(α, t) + Λf

t

+ σΛ

3

f

t

− ¯

h

1

Λf

tt

= ∂

α

(Hf

tt

f

t

+ Hf

t

f

tt

),

(49)

with initial datagiven by

f

1

(α) = f

t

(α, 0), f

2

= f

tt

(α, 0) = −Λf

0

− σΛ

3

f

0

+ ¯

h

1

Λf

1

+ ∂

α

(Hf

1

f

1

) .

Testing(49 )against

Λf

tt

,and using(47), we obtain that

1

2

d

dt

kf

tt

k

2

0.5

+ kf

t

k

2

1

+ σkf

t

k

2

2

− ¯

h

1

kf

tt

k

2

1

≤ kf

tt

k

2

1

(kf

t

k

L

∞

+ kHf

t

k

L

∞

)

+ kf

tt

k

1

kf

t

k

1

(kf

tt

k

L

∞

+ kHf

_tt

k

_L

∞

)

< 4kf

tt

k

2

1

kf

t

k

1

.

(50)

Next, we dene the following energy and dissipationfun tions, respe tively,as

E(t) = kf

tt

k

2

0.5

+ kf

t

k

2

1

+ σkf

t

k

2

2

, D(t) = −2¯

h

1

kf

tt

k

2

1

.

Then (50) an bewritten as

d

dt

E(t) + D(t) ≤

4

p

E(t)

−¯

h

1

D(t),

and we ndde ayof theenergy

provided thattheinitial datasatises

E(0) <



−¯

h

1

4



2

.

Consequently,due to (44 ), theinitial data

f

0

and

f

1

satisfy

E(0) = kf

2

k

2

0.5

+ kf

1

k

2

1

+ σkf

1

k

2

2

<



−¯

h

1

5



2

,

(51)

and we have thatequations (48) and(50) redu eto

d

dt

kf

t

k

2

0.5

+ kf k

2

1

+ σkf k

2

2

− ¯

h

1

kf

t

k

2

1

< 0,

d

dt

kf

tt

k

2

0.5

+ kf

t

k

2

1

+ σkf

t

k

2

2

−

2¯

h

1

5

kf

tt

k

2

1

< 0.

Thus,we nd theestimates

kf

t

(t)k

2

0.5

+ kf (t)k

2

1

+ σkf (t)k

2

2

− ¯

h

1

Z

t

0

kf

t

(s)k

2

1

ds ≤ kf

1

k

2

0.5

+ kf

0

k

2

1

+ σkf

0

k

2

2

,

kf

tt

(t)k

2

0.5

+ kf

t

(t)k

2

1

+ σkf

t

(t)k

2

2

−

2¯

h

1

5

Z

t

0

kf

tt

(s)k

2

1

ds ≤ kf

2

k

2

0.5

+ kf

1

k

2

1

+ σkf

1

k

2

2

.

and theasymptoti behavior

lim sup

t→∞

kf (t)k

1+

sgn

(σ)

+ kf

t

(t)k

1+

sgn

(σ)

+ kf

tt

(t)k

0.5

= 0.

(52)

Finally,using (47 )and (51), we on ludethat

kf

t

(t)k

2

L

∞

< kf

_t

(t)k

2

1

<



−¯

h

1

5



2

,

(53)

andthestability ondition(46 )issatisedeveninastri tersense. Finally,wetestequation

(45 )against

Λ

4

_f

t

. We nd that

1

2

d

dt

kf

t

k

2

2

+ kf k

2

2.5

+ σkf k

2

3.5

− ¯

h

1

kf

t

k

2

2.5

=

Z

T

∂

α

(f

t

Hf

t

)∂

α

4

f

t

dα.

Using integration byparts,Sobolev embedding and interpolation, we havethat

where wehave usedthe lassi al ommutator estimate [11℄

k[Λ

0.5

, u]vk

_L

2

≤ C(k∂

_α

uk

_L

4

kΛ

−0.5

vk

_L

4

+ kΛ

0.5

uk

_L

4

kvk

_L

4

).

Using Young'sinequality and(53 ), we nd that

d

dt

kf

t

k

2

2

+ kf k

2

2.5

+ σkf k

2

3.5

+ δkf

t

k

2

2.5

≤ C

1

(h

0

, h

1

)kf

t

k

2

2

,

(54)

for a ertain expli it

0 < δ(h

0

, h

1

)

. UsingGronwall's inequality,we on lude that

kf

t

(t)k

2

2

+ kf (t)k

2

2.5

+ σkf (t)k

2

3.5

≤ C

2

(h

0

, h

1

)e

C

1

(h

0

,h

1

)t

.

(55) Finally, fromthe regularityof

f

and

f

t

,we an on ludetheregularity of

h

and

h

t

.



Remark 4. Letus emphasize that the ondition(44) does not require small initial data;

rather,werequirethe datatobesu iently lose toanarbitrarilylargehomogeneousstate.

For example,inthe ase thatsurfa e tension

σ = 0

,we an onsider

h

0

= A + Be

αi

and

h

1

= −1000 +

e

αi

6

for onstants

A

and

B

. A simple omputation using the expli it form of

h

0

and

h

1

shows thatthe ondition (44 )issatised when

B ≤ 110

andany

A

.

Remark 5. With

h

2

= Λh

0

+ σΛ

3

_h

0

− ∂

α

(Hh

1

h

1

)

,ourasymptoti behaviorrequiresthat

kh

2

k

2

0.5

+ kh

1

k

2

1

+ σkh

1

k

2

2

<



−¯

h

1

5



2

. Let us set surfa e tension

σ = 0

, in whi h ase our initial position

h

0

and initial velo ity

h

1

mustbe hosensothat

kΛh

0

− ∂

α

(Hh

1

h

1

)k

2

0.5

+ kh

1

k

2

1

<



−¯

h

1

5



2

.

Thus,a onstraintispla edonthe

H

1.5

-normoftheinitialpositionfun tion

h

0

,eventhough thenonlinearity a tsonly on thevelo ity eld

h

1

. As an beseen intheproof of Theorem 4,theneedto obtainasign onthederivative oftheenergy fun tionleads tothis onstraint,

butthereisaninteresting pointwiseargument whi halsoexplainstheneedto onstrainthe

size of

Λh

0

.

In order to be able to ontinue our solution for all time, it is ne essaryto ensure that

the initial stability ondition

h

1

< 0

is propagated, and that we thus maintain

h

t

(·, t) < 0

for all

t ≥ 0

inwhi h ase we mustalso have that

kΛh(·, t) − ∂

α

(Hh

t

(·, t) h

t

(·, t))k

2

0.5

+ kh

t

(·, t)k

2

1

<



−¯

h

1

5



2

∀t ≥ 0 .

Togiveanother(andperhapsmoretransparent)explanationofhowthis anbea hieved,

weintrodu e the time-dependent point

x

t

whi h satises

h

t

(x

t

, t) = max

and we dene thetime-dependent fun tion

y(t)

by

y(t) = h

t

(x

t

, t) − ¯

h

1

sothat

y(t) ≥ 0 .

The Hilberttransform,a ting on periodi

2π

-periodi fun tions,is dened by

Λh(α) = H∂

α

h(α) =

1

2π

P.V.

Z

T

∂

α

h(α − y)

tan(y/2)

dy.

We time-dierentiate this formulaand use integration-by-parts (following the omputation

in[9 ℄):

Λh

t

(x

t

) =

1

2π

P.V.

Z

T

∂

y

(h

t

(α) − h

t

(α − y))

tan(y/2)

dy

=

1

4π

P.V.

Z

T

h

t

(α) − h

t

(α − y)

sin

2

_(y/2)

dy

=

1

4π

P.V.

Z

T

h

t

(x

t

) − h

t

(y)

sin

2

((x

t

− y)/2)

dy

≥

1

4π

(2πh

t

(x

t

) − hh

1

i)

= h

t

(x

t

) − ¯

h

1

/2.

Thus,we have that

d

dt

h

t

(x

t

, t) = Λh(x

t

, t) + Λh

t

(x

t

, t)h

t

(x

t

, t) ≤ Λh(x

t

, t) +

h

t

(x

t

, t) − ¯

h

1

2

h

t

(x

t

, t) .

(56) From (56 ), itthenfollowsthat

dy

dt

≤ Λh(x

t

, t) + y(t)

2

₊

¯

h

1

2

y(t) .

Now, the onditionthat

h

t

(·, t) < 0

isequivalent toshowing that

0 ≤ y(t) < −¯

h

1

∀t ≥ 0 .

(57)

From theODE (56), we see that inorder for (57 ) to hold, we mustpla e a size restri tion

on

Λh(x

t

, t)

. This size restri tion, in turn, requires

L

∞

- ontrol of

Λh(x

t

, t)

; in parti ular, we must have that

16Λh(x

t

, t) ≤ ¯

h

2

1

. It is, therefore, somewhat remarkable that our proof of Theorem4 requiresonly ontrol on

kΛh

0

k

0.5

ratherthan

kΛh

0

k

L

∞

.

Remark 6. Finally, we note that theasymptoti ondition (52 ) remains true for

higher-orderSobolev norms ifthe ondition(53 ) isrepla edwithfurther onstraints ontheinitial