Well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability

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problem with discontinuous permeability

Rafael Granero-Belinchón, Steve Shkoller

To cite this version:

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problem with dis ontinuous permeability

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

Werstprovelo al-in-timewell-posednessfortheMuskatproblem,modelinguidowin a two-dimensional inhomogeneous porous media. The permeability of the porous medium is des ribed by a step fun tion, with a jump dis ontinuity a ross the xed-in-time urve

(x

1 , −1 + f (x

1 ))

,whiletheinterfa eseparating theuidfromtheva uumregionisgivenby thetime-dependent urve

(x

1 , h(x

1 , t))

. Ourestimatesarebasedonanewmethodologythat reliesupona arefulstudyofthePDEsystem, ouplingDar y'slawandin ompressibilityof theuid,rather thanthe analysisofthe singularintegral ontourequationfor theinterfa e fun tion

h

. We are able to develop an existen e theory for any initial interfa e given by

h

0 ∈ H

2

andanypermeability urve-of-dis ontinuitythatisgivenby

f ∈ H

2.5

. Inparti ular, our method allows for both urves to have (pointwise) unbounded urvature. In the ase that the permeability dis ontinuity is the set

f = 0

, we prove global existen e and de ay to equilibrium for small initial data. This de ay is obtained using a new energy-energy dissipation inequality that ouples tangential derivatives of thevelo ity in the bulk of the uidwiththe urvatureoftheinterfa e. Tothebestofourknowledge, thisistherstglobal existen eresultfor theMuskatproblem withdis ontinuouspermeability.

Contents

1 Introdu tion 2

2 The Muskat problem in the ALE formulation 10

3 A priori estimates 14

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6 Proof of Theorem 2: Global well-posedness when

f = 0

28

1 Introdu tion

1.1 The Muskat problem

TheMuskatproblem,introdu edin[38 ℄,modelsthedynami sofanevolvingmaterial inter-fa eseparating two uids owing through aporous medium, i.e. a medium onsistingof a solidmatrix withuid-lled pores. Porousmedia owismodelled byDar y'slaw

µ

β

u = −∇p − (0, gρ)

T

_,

(1.1) where

µ

is the vis osity of the uid,

ρ

denotes the density,

u

is the in ompressible uid velo ity,and

p

isthepressure fun tion;additionally,

β > 0

denotes thepermeability ofthe solid matrix, and

g

is the a eleration due to gravity, whi h we shall hen eforth set to

1

. Dar y's law (1.1) is an empiri al relation between momentum and for e (see, for example, [3, 39℄), and repla es onservation of momentum, whi h is used to model theevolution of invis id uidows.

Thepurposeofthispaperistostudytheevolutionofaninterfa emovingthroughporous mediawithadis ontinuouspermeability. Asthepermeabilitytakestwodierentvalues,this ase is known in the literature as theinhomogeneous Muskat problem. Spe i ally, we are interested in the well-posedness and de ay to equilibrium for the inhomogeneous Muskat problem.

Γ

_bot

Ω

−

β

−

Ω

+

(t)

β

+

µ

−

,

ρ

−

Vacuum

Γ

_perm

Γ

(t)

β

+

Figure 1: The solid urve (blue) is the interfa e

Γ(t)

and the dashed urve (red)denotes the interfa e

Γ

perm

,a rosswhi hthepermeabilityisdis ontinuous.

We let

S

1

denote the ir le, so that fun tions

h : S

1 _{→ R}

are identied with

[−π, π]

-periodi fun tionson

R

. AsshowninFigure1,we onsidera porousmedium o upying an opentime-dependent subset

Ω(t) ⊂ S

1 _{× R}

su hthat

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where

Ω

+

_{(t) = {(x}

1 , x

2 ) ∈ S

1 × R, −1 + f (x

1 ) < x

2 < h(x

1 , t)} ,

(1.2a)

Ω

−

_{= {(x}

1 , x

2 ) ∈ S

1 × R, −2 < x

2 < −1 + f (x

1 )} ,

(1.2b)

Γ

perm

= {(x

1 , −1 + f (x

1 )), x

1 ∈ S

1 } ,

(1.2 )

and where the fun tions

f

and

h

satisfy

min

x

1 ∈S

1 f (x

1 ) > −1

and

h(x

1 , 0) > −1 + f (x

1 ).

(1.3)

The xed-in-time permeability interfa e

Γ

perm

denotes the urve, a ross whi h the perme-abilityfun tion

β(x)

isdis ontinuous; spe i ally,thepermeabilityfun tion

β(x)

isdened as

β(x) =

β

+

in

Ω

+

_(t)

β

−

in

Ω

−

,

forgiven onstants

β

±

> 0

. Thedomainfor thisproblemisalsoanunknown;thus,we must tra k theevolution of the time-dependent interfa e or free-boundary

Γ(t)

,whi h isdened astheset

Γ(t) = {(x

1 , h(x

1 , t)), x

1 ∈ S

1 } .

For simpli ity, we shall set the uid density

ρ

and vis osity

µ

to

1

. As the uid is in ompressible, itfollows that

[[u · n

perm

]] = 0

on

Γ

perm

× [0, T ] .

With the domains dened,the Muskat problem onsistsof thefollowing systemof ou-pledequations:

u

±

β

±

+ ∇p

±

= −e

2 ,

in

Ω

±

_{(t) × [0, T ] ,}

(1.4a)

∇ · u

±

= 0,

in

Ω

±

_{(t) × [0, T ] ,}

(1.4b)

[[p]] = 0

on

Γ

perm

× [0, T ],

(1.4 )

[[∇p · n

perm

]] = −

hh 1

β

ii

u

+

_{· n}

perm

on

Γ

perm

× [0, T ],

(1.4d)

p

+

= 0

on

Γ(t) × [0, T ],

(1.4e)

V(Γ(t)) = u

+

· n

on

Γ(t) × [0, T ] ,

(1.4f)

u

−

_{· e}

2 = 0

on

Γ

bot

× [0, T ],

(1.4g )

where

V(Γ(t))

denotes the normal omponent of the velo ity of the time-dependent free-boundary

Γ(t)

,

n

isthe (upward) unitnormal to

Γ(t)

,

n

perm

is the(upward-pointing) unit normalto

Γ

perm

,and

[[f ]] = f

+

_{− f}

−

denotes thejumpof adis ontinuous fun tion

f

a ross

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1.2 A brief history of the analysis of the Muskat problem

Dar y'slaw(1.1 ) isastandardmodelforowinaquifers,oilwells,orgeothermal reservoirs, and it is therefore of pra ti al importan e in geos ien e (see, for example, [9, 28℄ and the referen es therein). Furthermore, the Muskat problem is equivalent to the Hele-Shaw ell problemwithgravity (see[36 ℄)for owbetween twothinly-spa ed parallelplates.

There hasabeenagreatdealofmathemati alanalysisofboththeMuskatproblemand theHele-Shaw ell, andweshallonly reviewasmallfra tionoftheresultsthatare,insome sense,most loselyrelatedto our result.

For theMuskatproblemwitha ontinuouspermeability fun tion,existen eofsolutions in the Sobolev spa e

H

3

has been established by Córdoba & Gan edo [18 , 19 ℄, Córdoba, Córdoba&Gan edoin[17 ℄,andCórdoba,Granero-Belin hón&Orive[22 ℄,usingthesingular integral ontour equation for the height fun tion

h

. Cheng,Granero-Belin hón &Shkoller [12 ℄ introdu ed the dire t PDE approa h (modied for use, herein), and established an

H

2

existen e theory (see also Cheng, Coutand & Shkoller [11℄ for a similar approa h to the horizontal Hele-Shaw ell problem). This was followed by an

H

2

existen e theory by Constantin, Gan edo,Shvydkoy&Vi ol[16 ℄usingthesingularintegralapproa h;theyalso obtained a nite-slopeglobal existen e result. Very re ently, lo alexisten e in

H

s

,

s > 3/2

hasbeen obtained by Matio [37℄. Inthepresen e ofsurfa e tension, lo al existen ein

H

6

wasalsoobtained byAmbrose [1,2℄.

In the aseof adis ontinuous permeability fun tion (with a jump a rossthe at urve

(x

1 , −1)

), the lo al-in-time existen e of solutions has been proved by Berselli, Córdoba & Granero-Belin hón [4℄. Inthe ase of two uids withdierent vis osities and densities and permeability fun tion with a jump given by an arbitrary smooth urve

(f

1 (α), f

2 (α))

the lo al-in-time existen eofsolutions hasbeen establishedbyPernás-Castaño [40 ℄ .

For the ase of a ontinuous permeability, there are a variety of results showing global existen e of strong solutions under ertain onditions on the initial data. In parti ular, Cheng,Granero-Belin hón &Shkoller[12 ℄ provedglobal existen eunder restri tionsonthe sizeof

kh

0 k

H

2

,whileCórdoba,Constantin,Gan edo&Strain[15℄andCórdoba,Constantin, Gan edo, Strain & Rodríguez-Piazza [14 ℄ proved global existen eunder restri tionson the sizeof

k b

h

′

0 k

L

1

,where

h

ˆ

denotestheFouriertransform. Theglobalexisten eofweaksolution hasbeenprovedbyCórdoba,Constantin,Gan edo&Strain[15℄andGranero-Belin hón[35 ℄ for initial data satisfying restri tions on

kh

0 k

˙

W

1,∞

and

kh

0 k

W

1,∞

, respe tively. Note that the onditionon

kh

0 k

L

∞

in [35℄isa onsequen e of having abounded porousmedia.

Finitetimesingularitiesofturningtypeareknowntoo ur. Aturningwaveisasolution whi h startsasagraph, butthenturns-overand losesthegraphproperty. Theexisten eof su h waves in the Rayleigh-Taylor stable regime hasbeen established byCastro, Córdoba, Feerman,Gan edo&López-Fernández[8 ℄,Córdoba,Granero-Belin hón&Orive-Illera[22℄, Berselli,Córdoba &Granero-Belin hón [4℄ andGómez-Serrano &Granero-Belin hón [34 ℄.

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interfa essu h that, afterturning, theinterfa e isno longeranalyti and, infa t,

lim sup

t→T

kz(t)k

C

4 = ∞,

for anite time

T > 0

.

Gan edo & Strain [33℄ have shown that the nite-time splash and splat singularities (a self-interse tion of a lo ally smooth interfa e) annot o ur for the two-phase Muskat problem (see also Feerman, Iones u & Lie [32 ℄ and Coutand & Shkoller [27℄). However, in the ase of the one-phase Muskat problem, Castro, Córdoba, Feerman & Gan edo [7℄ provedthatthesplashsingularitiesmayo urwhileCórdoba&Pernás-Castaño [23℄showed thatsplat singularities annoto ur. See also Coutand&Shkoller[26℄ forsplashand splat singularities forthe 3-D Euler equationsand relatedmodels.

Let us also mention that several results for the multiphase Muskat problem have been obtained in the ompletely dierent framework of little Hölder spa es

h

k+α

by Es her & Matio [30 ℄,Es her, Matio &Matio [29℄and Es her, Matio &Walker[31 ℄.

Very re ently,a regularityresultinHölder spa es for theone-phase Hele-Shawproblem hasbeenobtainedbyChang-Lara&Guillén[10℄usingthehodographtransform. Also,Prüss &Simonett[41 ℄studiedthetwo-phaseMuskatprobleminamoregeometri frameworkusing theHanzawa transform. In parti ular,these authors showwell-posedness, hara terize and study thedynami stabilityoftheequilibria.

Finally,usinga onvexintegrationapproa h,Castro,Córdoba&Fara o[5℄veryre ently proved theexisten eofweaksolutionsfor the Muskatprobleminthe ase wherethedenser uidliesabovethelighteruid,so,itisintheRayleigh-Taylorunstableregime. Remarkably, these solutions develop a mixing zone (a strip ontaining parti les from both phases and, onsequently, withuidparti leshaving both densities), growing linearlyintime.

1.3 Methodology

Asnotedabove,mostpriorexisten etheoremshaverelieduponthesingularintegral ontour equation for the height fun tion

h

; in the ase of the innitely deep two-phase Muskat problemwith ontinuous permeability,theevolution equationfor

h

an be written as

h

t

(x

1 ) =

p.v.

Z

R

h

′

(x

1 ) − h

′

(x

1 − y)

y

1 1 +

h(x

1 )−h(x

1 −y)

y

2 dy;

(1.5)

see, for example,[18℄ forthe derivation.

The ontour equation (1.5 ) depends ru ially on the geometry of the domain and the permeability fun tion. In parti ular, when the porous medium has nite depth (equal to

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in[4 ℄ that(1.5 ) takesthe form:

h

t

(x

1 ) =

β

+

_(−[[ρ]])

4π

p.v.

Z

R

(h

′

_(x

₁

_{) − h}

′

_{(y)) sinh(x}

₁

_{− y)}

cosh(x

1 − y) − cos(h(x

1 ) − h(y))

dy

+

β

+

_(−[[ρ]])

4π

p.v.

Z

R

(h

′

(x

1 ) − h

′

(y)) sinh(x

1 − y)

cosh(x

1 − y) + cos(h(x

1 ) + h(y))

dy

+

1 4π

p.v.

Z

R

̟

2 (y)(sinh(x

1 − y) + h

′

(x

1 ) sin(h(x

1 ) + 1))

cosh(x

1 − y) − cos(h(x

1 ) + 1)

dy

+

1 4π

p.v.

Z

R

̟

2 (y)(− sinh(x

1 − y) + h

′

(x

1 ) sin(h(x

1 ) − 1))

cosh(x

1 − y) + cos(h(x

1 ) − 1)

dβ,

(1.6) where

̟

2 (x

1 ) =

β

+

_{− β}

−

β

+

_{+ β}

−

β

+

_(−[[ρ]])

2π

p.v.

Z

R

h

′

(y)

sin(1 + h(y))dy

cosh(x

1 − y) − cos(1 + h(y))

−

β

+

_{− β}

−

β

+

_{+ β}

−

β

+

_(−[[ρ]])

2π

p.v.

Z

R

h

′

(y)

sin(−1 + h(y))dy

cosh(x

1 − y) + cos(−1 + h(y))

+

β

+

_−β

−

β

+

_+β

−

2 √

2π

β

+

_(−[[ρ]])

2π

G

β

∗

p.v.

Z

R

h

′

(y) sin(1 + h(y))dy

cosh(x

1 − y) − cos(1 + h(y))

−

β

+

_−β

−

β

+

_+β

−

2 √

2π

β

+

_(−[[ρ]])

2π

G

β

∗

p.v.

Z

R

h

′

_{(y) sin(−1 + h(y))dy}

cosh(x

1 − y) + cos(−1 + h(y))

,

(1.7) with

G

β

(x

1 ) = F

−1







F

cosh(x

sin(2)

1 )+cos(2)

(ζ)

1 +

β+−β−

β++β−

_√

2π

F

_sin(2)

cosh(x

1 )+cos(2)

(ζ)





 ,

a S hwartz fun tion, and where

F

denotes the Fourier transform. Let us emphasize that, duetothenon-lo al hara terof

̟

2

givenby(1.7 ),the ontourequation(1.6)issigni antly more hallenging to analyse than (1.5 ). Note also, from the denition of

G

β

(x

1 )

that the highlynon-lo al onvolution termsin(1.7) arenot expli itlydened.

Be ause of the ompli ations inherent in the singular integral approa h of (1.7 ), we shall instead analyze the system (1.4) dire tly. As (1.4) is set on the time-dependent a priori unknown domain

Ω(t)

, in order to build an existen e theory, we rst pull-ba k this systemof equations onto a xed-in-time spatialdomain. We usea arefully hosen hange-of-variables thattransformsthefree-boundary problem(1.4 ) into asystemof equationsset on asmooth andxed domain,but having time-dependent oe ients.

Topull-ba k our problem, weemploya family ofdieomorphisms

ψ

±

whi h areellipti extensions of the interfa e parametrizations, and thus have optimal

H

s

Sobolev regularity. The time-dependent oe ients (in the pulled-ba k des ription) arise from dierentiation and inversion of these maps

ψ

±

; by studying the transformed Dar y's Law, we obtain a new higher-order energy integral that provides the regularity of the moving interfa e

Γ(t)

. Additionally,we obtainan

L

2

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gainfor solutionsto the heatequation, ex eptthatwe gaina

1/2

-derivative inspa erather than a full derivative. The regularity of the interfa e

Γ(t)

as well as the improved

L

2

-in-time paraboli regularity gain are found from the non-linear stru ture of the pulled-ba k representation ofthe Muskatproblem. Inparti ular, wedonorelyontheexpli itstru ture of thesingular integral ontourequation, and assu h, we arefreeto study general domain geometriesand permeability fun tions.

1.4 The main results

Aswe willshow, theRayleigh-Taylor(RT)stability ondition, given by

−

∂p

∂n

> 0

on

Γ(t)

,is a su ient ondition for well-posedness ofthe Muskat problem (1.4) inSobolev spa es. In parti ular, with

p

0 := p(·, 0)

and

Γ := Γ(0) ,

and letting

N := n(·, 0)

denote theoutward unit normal to

Γ

,we prove that for anyinitial interfa e

Γ

ofarbitrarysize and of lass

H

2

, hosen su hthatthe RTstability ondition

−

∂p

_∂N

0 > 0

on

Γ

(1.8)

issatised,thereexistsauniquesolution

(u

±

_{(x, t), p}

±

_{(x, t), h(x}

1 , t))

totheone-phaseMuskat problemwithdis ontinuous permeabilityfun tion.

Morepre isely,we prove the following Theorem 1 (Lo al well-posedness in

H

2

). Suppose the initial interfa e

Γ

isgiven as the graph

(x

1 , h

0 (x

1 ))

where

h

0 ∈ H

2 _(S

1 ₎

and

R

S

1 h

0 (x

1 )dx

1 = 0

,andthattheRT ondition (1.8) is satised. Let

Γ

perm

be given as the graph

(x

1 , −1 + f (x

1 ))

for a fun tion

f ∈ H

2.5 _(S

1 ₎

. Assumealso that (1.3 ) holds. Then,there exists a time

T (h

0 , f ) > 0

anda unique solution

h ∈ C([0, T (h

0 , f )]; H

2 (S

1 )) ∩ L

2 (0, T (h

0 , f ); H

2.5 (S

1 )) ,

u

±

_{∈ C([0, T (h}

0 , f )]; H

1.5 (Ω

±

(t))) ∩ L

2 (0, T (h

0 , f ); H

2 (Ω

±

(t))) ,

p

±

_{∈ C([0, T (h}

0 , f )]; H

2.5 (Ω

±

(t))) ∩ L

2 (0, T (h

0 , f ); H

3 (Ω

±

(t))) ,

tothe system (1.4 ),satisfying

kh(t)k

2 L

2 _(S

1 ₎

+ 2

Z

t

0 u

+

(s)

β

+

2 L

2 _(Ω

+

_(s))

ds + 2

Z

t

0 u

−

(s)

β

−

2 L

2 _(Ω

−

₎

ds = kh

0 k

2 L

2 _(S

1 ₎

,

and

khk

C([0,T (h

0 ,f )],H

2 (S

1 ))

+ kh

t

k

L

2 (0,T (h

0 ,f );H

1.5 (S

1 ))

+ khk

L

2 (0,T (h

0 ,f );H

2.5 (S

1 ))

+ kpk

C([0,T (h

0 ,f )],H

2.5 (Ω

+

(t)∪Ω

−

))

+ kpk

L

2 (0,T (h

0 ,f );H

3 (Ω

+

(t)∪Ω

−

))

+ kuk

C([0,T (h

0 ,f )],H

1.5 (Ω

+

(t)∪Ω

−

))

+ kuk

L

2 (0,T (h

0 ,f );H

2 (Ω

+

(t)∪Ω

−

))

≤ C(h

0 , f )

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Remark 1. Itis easyto see thatif

h

0 (x

1 ) = f (x

1 ) = 0

,thesolution isgiven by

u

±

(x, t) = 0, h(x

1 , t) = 0, p

±

(x, t) = −x

2 ,

(1.9)

and theRT ondition is satised. There exist innitelymany initial data

h

0

satisfyingthe RT ondition; for example, small perturbations of (1.9) satisfy the RT ondition (1.8 ) via impli it fun tiontheorem arguments.

Theorem 2 (Global well-posedness and de ayto equilibrium in

H

2

). Suppose the initial interfa e

Γ

isgiven as the graph

(x

1 , h

0 (x

1 ))

where

h

0 ∈ H

2 _(S

1 ₎

and

R

S

1 h

0 (x

1 )dx

1 = 0

. Let

Γ

perm

be given asthe graph

(x

1 , −1)

. Then, there existsa onstant

C

su h that if

|h

0 |

2 < C ,

the RT ondition (1.8) is satised and there exists a unique solution

h ∈ C([0, ∞); H

2 (S

1 _{)) ∩ L}

2 _{(0, ∞; H}

2.5 (S

1 )) ,

u

±

_{∈ C([0, ∞); H}

1.5 (Ω

±

_{(t))) ∩ L}

2 _{(0, ∞; H}

2 (Ω

±

(t))) ,

p

±

_{∈ C([0, ∞); H}

2.5 (Ω

±

_{(t))) ∩ L}

2 _{(0, ∞; H}

3 (Ω

±

(t))) ,

tothe system (1.4 ),satisfying

kh(t)k

H

2 _(S

1 ₎

≤ kh

₀

k

_H

2 _(S

1 ₎

e

−γt/2

for a onstant

γ(h

0 , β

±

)

, whi h dependson

h

0

and

β

±

.

Remark 2. Notethatthe questionofwhetherthefreeboundary

Γ(t)

anrea hthe urve

Γ

perm

innitetime, ina asituation thatresembles thesplash/splat singularity,remains an open problem. In fa t, su h behavior an be seen as a singular phenomena (for instan e, some ofthe (non-singular) termsin(1.6 ) and(1.7 ) be ome singular integraloperators). As Theorem2impliesthat

Γ(t)

annotrea hthe urve

Γ

perm

innitetimeif

h

0

issmallenough, this resultrules out the possibilityof interfa e ollisioninnite timefor smallinitial data. Remark 3. Wenotethatthedryzone (theregionwithoutuid)liesabovethe urve

Γ(t)

, and so, as long as (1.3) holds, the dry zone lies above

Γ

perm

. The question of whether a dry zone an forminside

Ω

−

remains an open problem. In other words, assumethat there exists a solution

h(x

1 , t)

up to time

T

and assume also that

Γ(t)

interse ts

Γ

perm

at the point

(x

0 , t

′

) ∈ S

1 _{× (0, T )}

,i.e.

h(x

0 , t

′

) = −1 + f (x

0 ).

Then, it isnot lear ifthe urve

Γ(t)

may ross the urve

Γ

perm

,i.e.

h(x

1 , t) < −1 + f (x

1 ), ∀ (x

1 , t) ∈ (x

0 − ǫ, x

0 + ǫ) × (t

′

, t

′

+ δ), ,

for ertain

ǫ, δ > 0

. Notealso that,ifthis happens,thenthe region

{(x

1 , x

2 ), x

1 ∈ (x

0 − ǫ, x

0 + ǫ), h(x

1 , t) < x

2 < −1 + f (x

1 )} ⊂ Ω

−

(10)

Remark 4. The exponential de ay of the solution

h(t)

is a onsequen e of an energy-energydissipation inequality establishingarelationshipbetweentheinterfa eregularityand theregularityof the semi-ALE velo ity(see Se tions5and 6):

kh

′′

(t)k

L

2 _(S

1 ₎

≤ Ckw

′′

k

_L

2 _(S

1 _{×(−2,−1)∪S}

1 _×(−1,0))

.

Remark 5. Note that the linearized evolution equation for a small perturbation of the at interfa e an be written as

h

t

= −Λ

_D

+

h,

where

Λ

D

+

is theDiri hlet-to-Neumann map asso iated withtheellipti system(5.1 ):

Λ

_D

+

h(x

₁

) = δψ,

₂

(x

₁

, 0).

An integration byparts shows that

Z

S

Z

0 −1

δψ

+

∆δψ

+

dx

2 dx

1 =

Z

S

δψ

+

(x

1 , 0)δψ

+

,

2 (x

1 , 0)dx

1 −

Z

S

δψ

+

(x

1 , −1)δψ

+

,

2 (x

1 , −1)dx

1 −

Z

S

Z

0 −1

|∇δψ

+

_|

2 _dx

2 dx

1 ,

sothat,

Z

Γ

Λ

_D

+

hhdx

₁

=

Z

D

+

|∇δψ

+

|

2 dx.

We also have the following Poin aré-Wirtinger inequality

Z

D

+

|δψ

+

_(x)|

2 _{dx =}

Z

D

+

Z

1

0 δψ

+

,

2 (x

1 , sx

2 + (1 − s)(−1))(x

2 + 1)ds

2 dx

≤

Z

S

Z

0 −1

Z

1

0 δψ

+

_,

2 (x

1 , sx

2 + (1 − s)(−1))

2 (x

2 + 1)

2 dsdx

2 dx

1 =

Z

0 −1

(x

2 + 1)

Z

S

Z

x

2 −1

δψ

+

_,

2 (y)

2 dy

1 dx

2 ≤

Z

0 −1

(x

2 + 1)

Z

D

+

δψ

+

_,

2 (y)

2 dydx

2 ≤

1 ₂

Z

D

+

δψ

+

_,

2 (y)

2 dy.

Thus,using the tra etheorem, we on lude that

Z

Γ

Λ

_D

+

hhdx

₁

≥ 0.5kδψ

+

k

2 _1,+

≥ ν|h|

2 _0.5

≥ ν|h|

2 ₀

,

(11)

1.5 Notation used throughout the paper For amatrix

A

,we write

A

i

j

forthe omponentof

A

lo atedinrow

i

and olumn

j

. Weuse theEinstein summation onvention, whereinrepeated indi esaresummed from

1

to

2

. We denotethe

j

th anoni albasisve tor in

R

2

by

e

j

. For

s ≥ 0

,we set

kuk

s,+

:= ku

+

k

H

s

_(D

+

₎

, kuk

_s,−

:= ku

−

k

_H

s

_(D

−

₎

, kuk

_s,±

:= ku

+

k

_s,+

+ ku

−

k

_s,−

and

|h|

s

:= khk

H

s

_(Γ)

.

For fun tions

h

dened on

Γ

perm

,weshallalso denotethe

H

s

norm by

|h|

s

:= khk

H

s

_(Γ

_perm

₎

, wheneverthe ontext is lear.

Wewrite

f

′

=

∂ f

∂ x

1 , f,

k

=

∂f

∂x

k

,

and

f

t

=

∂f

∂t

.

For a dieomorphism

ψ

,welet

A = (∇ψ)

−1

,and dene

curl

ψ

v = A

j

1 v

2 ,

j

−A

j

2 v

1 ,

j

,

(1.10)

div

ψ

v = A

i

j

v

j

,

i

.

(1.11)

2 The Muskat problem in the ALE formulation 2.1 Constru ting the family of dieomorphisms

ψ

(·, t)

2.1.1 The idea for the onstru tion

OuranalysisoftheMuskatproblem(1.4 )isfoundedonatime-dependent hange-of-variables whi h onverts thefreeboundaryproblemto one set onsmooth referen edomains

D

±

D

+

= S

1 _{× (−1, 0) , D}

−

= S

1 _{× (−2, −1) ,}

(2.1)

The boundaries ofthedomains

D

±

aredened as

Γ

bot

= {(x

1 , −2), x

1 ∈ S

1 } , Γ

perm

= {(x

1 , −1), x

1 ∈ S

1 } ,

and

Γ = {(x

1 , 0), x

1 ∈ S

1 _{} .}

(2.2) We let

N = e

2

denotethe unitnormal ve toron

Γ

(outwards),

Γ

perm

and

Γ

bot

.

Asour analysis ru iallyrelies onobtaining aparaboli regularity gain,we need a refer-en e domain

D

+

with

C

∞

boundary. In parti ular, the initial domain

Ω

+

₍₀₎

annot serve asareferen e domain.

Weadapttheideas from[12 ℄to onstru tthetime-dependent familyofdieomorphisms with optimal Sobolev regularity,

ψ(x, t)

,that we shalluse to pull-ba k (1.4) onto thexed domain

D

±

. Before detailing this onstru tion, let us sket h the pro edure. First, we onstru ta dieomorphismwith optimalSobolev regularityat

t = 0

:

ψ

+

_{(0) : D}

+

_{→ Ω}

+

(0), ψ

−

_{: D}

−

_{→ Ω}

−

.

(12)

•

For

0 < δ ≪ 1

a su iently small parameter (to be xed later), we dene auxiliary domains,

D

±,δ

₍₀₎

. Theseauxiliarydomainsare onstru tedviamolli ationof

h(x

1 , 0)

and

f (x

1 )

and,thus,they areinnitelysmooth. We dene thegraph dieomorphism

φ

±

₁

_{: D}

±

_{→ D}

±,δ

.

Thesedieomorphisms areof lass

C

∞

be auseofthesmoothnessof thedomains

D

±

and

D

±,δ

.

•

Weneed anotherdieomorphismsfromthe auxiliarydomain

D

±,δ

to

Ω

±

₍₀₎

. Weneed these dieomorphisms to gain1/2 derivatives with respe t totheregularity of

Ω

±

(0)

. Inorderthatthisoptimalregularityisobtained,we makeuseofthedenitionof

D

±,δ

and the propertiesof ourmolliers. We dene

φ

±

₂

_{: D}

±,δ

_{→ Ω}

±

(0)

asthesolution toLapla e problems withappropriate boundary onditions. Usingthe boundary dataand the inverse fun tion theorem, these mappings

φ

±

2

are

H

2.5 ₋

lass dieomorphisms.

•

Finally,wedene

ψ

+

(0) = φ

+

₂

_{◦ φ}

+

₁

, ψ

−

= φ

−

₂

_{◦ φ}

−

₁

.

As ompositionof dieomorphisms,

ψ

±

₍₀₎

is adieomorphism.

On ethe initialdieomorphismwithoptimalSobolevregularityis onstru ted,we solve Poissonproblems(tobedetailedbelow)for

ψ

±

(x, t)

. Anappli ation oftheinverse fun tion theorem together with standard ellipti estimates will show that these mappings

ψ

±

_{(x, t)}

area familyof dieomorphisms withthedesiredsmoothness.

2.1.2 Constru ting the initial regularizing dieomorphism

ψ(·, 0)

Given a fun tion

h ∈ C(0, T ; H

2 ₎

withinitial data

h(x

1 , 0) = h

0 (x

1 )

,we x

0 < δ ≪ 1

and dene our auxiliary domainsand boundaries

D

+,δ

(0) = {(x

1 , x

2 ), x

1 ∈ S

1 , −1 + J

δ

f (x

1 ) < x

2 < J

δ

h

0 (x

1 )},

D

−,δ

= {(x

1 , x

2 ), x

1 ∈ S

1 , −2 < x

2 < −1 + J

δ

f (x

1 )},

Γ

δ

_{(0) = {(x}

1 , J

δ

h

0 (x

1 )), x

1 ∈ S

1 }, Γ

δ

perm

= {(x

1 , −1 + J

δ

f (x

1 )), x

1 ∈ S

1 }.

Aswesaid previously,we dene thegraph dieomorphism

φ

+

₁

(x

1 , x

2 ) = (x

1 , (x

2 + 1)J

δ

h

0 (x

1 ) − (−1 + J

δ

f (x

1 ))x

2 ) ,

φ

−

₁

(x

1 , x

2 ) = (x

1 , x

2 + J

δ

f (x

1 )(x

2 + 2)) ,

where

J

δ

denotes the onvolution withastandard Friedri h's mollier. This fun tion

(13)

isa

C

∞

dieomorphism.

Next, we have to dene the regularizing dieomorphisms

φ

±

₂

_{: D}

±,δ

_{(0) → Ω}

±

(0).

We dene these mappingsasthe solution to thefollowing ellipti problems:

∆φ

+

₂

= 0

in

D

+,δ

_{(0) ,}

(2.3a)

φ

+

₂

= (x

1 , x

2 ) + [h

0 (x

1 ) − J

δ

h

0 (x

1 )]e

2

on

Γ

δ

_{(0) ,}

(2.3b)

φ

+

₂

= (x

1 , x

2 ) + [f (x

1 ) − J

δ

f (x

1 )]e

2

on

Γ

δ

perm

,

(2.3 )

∆φ

−

₂

= 0

in

D

−,δ

_,

(2.4a)

φ

−

₂

= (x

1 , x

2 ) + [f (x

1 ) − J

δ

f (x

1 )]e

2

on

Γ

δ

perm

(2.4b)

φ

−

₂

= (x

1 , x

2 )

on

Γ

bot

.

(2.4 )

Using standard ellipti regularitytheory,wehave that

kφ

2 − ek

H

2.5 _(D

±

,δ

₎

≤ C(|h

₀

− J

_δ

h

₀

|

₂

+ |f − J

_δ

f |

₂

),

where

e = (x

1 , x

2 )

denotes the identity mapping. Using the Sobolev embedding theorem, and taking

δ > 0

su iently small,we have that

kφ

2 − ek

C

1 _(D

±

,δ

₎

≪ 1,

so,due to the inverse fun tiontheorem, we obtain that

φ

±

2

is an

H

2.5

- lass dieomorphism. Asin[12 ℄, we dene

ψ

+

(0) = φ

+

₂

_{◦ φ}

+

₁

_{: D}

+

_{→ Ω}

+

(0), ψ

−

= φ

−

₂

_{◦ φ}

−

₁

_{: D}

−

_{→ Ω}

−

.

(2.5)

Then, this mapping isalso an

H

2.5

- lassdieomorphism.

2.1.3 Constru tingthe time-dependent family of regularizing dieomorphisms

ψ(·, t)

Wedenethetime-dependentfamilyofdieomorphisms

ψ(t) = ψ(·, t)

assolutionstoPoisson equationswithfor ingdependingon

ψ(0)

. Themainpointofthis onstru tionisthatdueto the ontinuityintime oftheinterfa e

h

andstandardellipti estimates,thetime-dependent familyofdieomorphisms

ψ(t) = ψ(·, t)

isgoingtoremain losetotheinitialdieomorphism

ψ(0)

.

In parti ular,we onsider thefollowing ellipti system:

∆ψ

+

(t) = ∆ψ

+

(0)

in

D

+

× [0, T ] ,

(2.6a)

ψ

+

(t) = (x

1 , x

2 ) + h(x

1 , t)e

2

on

Γ × [0, T ] ,

(2.6b)

(14)

and

ψ

−

(t) = ψ

−

. Be auseofthefor ingterm present in(2.6 a), wehavethat

ψ

+

_{(t) − ψ}

+

₍₀₎

solves

∆(ψ

+

_{(t) − ψ}

+

(0)) = 0

in

D

+

_{× [0, T ] ,}

ψ

+

_{(t) − ψ}

+

(0) = (h(x

1 , t) − h(x

1 , 0))e

2

on

Γ × [0, T ] ,

ψ

+

_{(t) − ψ}

+

(0) = 0

on

Γ

perm

× [0, T ] .

Due toellipti estimates,we havethebound

kψ(t) − ψ(0)k

2.25,±

≤ C|h(t) − h

0 |

1.75 .

(2.8)

By takingsu iently small time

t

and re alling that

h ∈ C(0, T ; H

2 ₎

,we have that

kψ(t)k

2.25,±

≤ C|h(t) − h

0 |

1.75 + C(|h

0 |

1.75 + |f |

1.75 + 1) ≤ 2C(|h

0 |

1.75 + |f |

1.75 + 1).

Writing

J(t) =

det

(∇ψ(t)) = ψ

1 _,

1 ψ

2 ,

2 −ψ

2 ,

1 ψ

1 ,

2 ,

wehave the bound

kJ(t) − J(0)k

1.25,±

≤ C|h(t) − h

0 |

1.75 .

(2.9)

Consequently,using

h ∈ C(0, T ; H

2 ₎

,for su iently smalltime

t

,we have that

min

x∈D

±

J(0)

2 < J(t) < 2 max

x∈D

±

J(0),

and,thanksto (2.8) ,wehave that

kψ(t) − ψ(0)k

C

1 ≤ C|h(t) − h

₀

|

_1.75

≪ 1.

Due to the inverse fun tion theorem and using thefa t that

ψ(0)

is a dieomorphism, we seethat

ψ

±

_{(t) : D}

±

_{→ Ω}

±

(t)

isa dieomorphism. Fromthe ellipti estimate

kψ(t)k

2.5,±

≤ C(|h(t)|

2 + |f |

2 + 1),

wehave that

ψ(t)

is an

H

2.5

- lass dieomorphism.

2.1.4 The matrix

A(·, t)

We write

A = (∇ψ)

−1

. Thus,

A

i

_r

ψ

r

,

j

= δ

j

i

and we obtaintheuseful identities

(A

t

)

i

k

= −A

i

r

(ψ

t

)

r

,

j

A

j

_k

,

A

′′

= −2A

′

∇ψ

′

A − A∇ψ

′′

A.

(2.10)

We will also make useof thePiola'sidentity:

(JA

k

(15)

2.2 The Muskat problem in the referen e domains

D

±

With

ψ(t) = ψ(·, t)

dened in Se tion 2.1, we dene our new variables in the referen e domains

D

±

:

v = u ◦ ψ, q = p ◦ ψ

. Welet

˜

τ = ψ

′

, ˜

n = (ψ

′

)

⊥

_{, g = |ψ}

′

_|

2

denote the (non-normalized) tangent and normal ve tors and the indu ed metri , respe -tively,on

Γ(t)

. Wealsodene theunittangentve tor

τ = ˜

τ /√g

andtheunitnormalve tor

n = ˜

n/√g

. In the same way, we dene

τ

˜

perm

, ˜

n

perm

, g

perm

, τ

perm

, n

perm

as the analogous quantities on

Γ

perm

. Re allthat

JA

k

_i

N

k

= ˜

n

i

on

Γ, JA

k

i

N

k

= ˜

n

i

perm

on

Γ

perm

.

Hen e,theALErepresentationoftheone-phaseinhomogeneousMuskatproblemisgiven by

(v

±

₎

i

β

±

+ (A

±

)

k

i

(q

±

+ ψ

±

· e

2 ),

k

= 0

in

D

±

_{× [0, T ] ,}

(2.11a)

(A

±

)

k

_i

(v

±

)

i

,

k

= 0

in

D

±

_{× [0, T ] ,}

(2.11b)

h

t

(t) = (v

+

)

i

J

+

(A

+

)

j

_i

N

j

on

Γ × [0, T ] ,

(2.11 )

q

+

= 0

on

Γ × [0, T ] ,

(2.11d)

[[q]] = 0

on

Γ

perm

× [0, T ] ,

(2.11e)

[[q,

k

A

k

i

JA

j

i

N

j

]] = −

hh 1

β

ii

v

i

JA

j

_i

N

j

on

Γ

perm

× [0, T ]

(2.11f)

v

−

₂

= 0

on

Γ

bot

× [0, T ] .

(2.11g ) 3 A priori estimates

In this se tion we establish the a priori estimates for theone-phase Muskat problem with dis ontinuouspermeability (1.4).

Wedene the higher-order energy fun tion

E(t) = max

0≤s≤t

|h(s)|

2

2 +

Z

t

0 kv(s)k

2 2,±

+ |h(s)|

2

2.5 ds.

Remark 6. Anotherpossible denitionfor ahigher-order energy fun tionis (see[12℄)

E(t) = max

0≤s≤t

|h(s)|

2

2 +

Z

t

0 kv(s)k

2 2,±

.

Infa t, aswill be shown,

(16)

As in[12 ℄, our goalisto obtain thepolynomial inequality

E(t) ≤ M

0 + Q(E(t))t

α

,

for ertain

α > 0

,ageneri polynomial

Q

,anda onstant

M

0

dependingon

h

0

and

f

. When

E(t)

is ontinuous, theprevious inequality impliestheexisten eof

T

∗

_(h

₀

_{, f )}

su hthat

E(t) ≤ 2M

0 .

(3.1)

We assume that we have a smooth solution dened for

t ∈ [0, T ]

. We take

0 < T ≤ 1

small enough su h that the following onditions hold: for a xed onstant

0 < ǫ ≪ 1

(possiblydependingon

h

0

and

f

) andfor

t ∈ [0, T ]

,

kψ(t) − ψ(0)k

L

∞

+ kA(t) − A(0)k

_L

∞

+ kJ(t) − J(0)k

_L

∞

≤ ǫ ;

(3.2a)

kh(t) − h

0 k

L

∞

+ k∇q(t) − ∇q(0)k

_L

∞

≤ ǫ ;

(3.2b)

E(t) ≤ 3M

0 ;

(3.2 )

min

0≤t≤T

x

min

1 ∈S

1 q,

2 (t) ≥ min

x

1 ∈S

1 q,

2 (0)/4.

(3.2d)

Wewillshowthat onditionsevenstri terthan(3.2 ,d)a tuallyholds. Letusemphasize that, due tothe RT ondition, we have that

min

x

1 ∈S

1 q,

2 (0) > 0.

Again, we let

C = C(h

0 , f, δ)

denote a onstant thatmay hange fromline to line. We let

P(x)

denoteapolynomialwith oe ientsthatmaydependon

h

0 (·) := h(·, 0), f, δ

. This polynomialmay hange fromline to line.

3.1 Estimates for some lower-order norms

In thefollowing, we olle t some estimates of lower-order norms. Theproofs aresimilar to those in[12 ℄,so,we omit them.

Lemma3 (Estimatesforsomelower-ordernormsof

h

,[12 ℄,Se tion8.4.1). Givena smooth solutionto the Muskat problem (2.11a-g),

Z

t

0 |h

t

|

2

1 ds ≤ C E(t).

(3.3a)

|h(t) − h

0 |

1 ≤ C

p

E(t)t

1/2

.

(3.3b)

Lemma 4 (Estimates for some lower-order norms of the ALE mapping

ψ

, [12 ℄, Se tion 8.4.2). Givena smooth solutionto the Muskatproblem(2.11a-g),

kψ(t)k

2.5,±

≤ C(1 + |h(t)|

2 ), kψ(t)k

3,±

≤ C(1 + |h(t)|

2.5 )

(3.4a)

kψ(t) − ψ(0)k

2.25,±

+ kA(t) − A(0)k

1.25,±

+ kJ(t) − J(0)k

1.25,±

≤

4 √

tC

p

E(t).

(3.4b) Noti e that(3.4b) implies astri ter versionof (3.2 a). Asa onsequen e we obtain that

|h(t)|

1.75 , kψ(t)k

2.25,±

, kJ(t)k

1.25,±

and

kA(t)k

1.25,±

are bounded by

C(h

0 , f )

uniformly for all

t ∈ [0, T ]

. Furthermore,wealso havethat

0 <

1

2 x∈D

min

+

_∪D

−

J(0) ≤ J(t) ≤ 1.5

max

x∈D

+

_∪D

−

J(0).

(17)

3.2 Basi

L

2

energy law

Lemma5(Estimatesforsomelower-ordernormsof

v

). ForasmoothsolutiontotheMuskat problem (2.11a-g),

|h(t)|

2

0 + 2

Z

t

0 s

J

β

v

2 0,±

ds = |h

0 |

2

0 .

(3.6)

Proof. We test the equation (2.11 a) against

Jv

and integrate. Using Piola's identity, inte-grating by partsand using thedivergen e free ondition (2.11b),we obtain that

s

J

β

v

2 0,±

+

Z

Γ

v

i

JA

k

_i

_{(q + ψ · e}

2 )N

k

dx

1 −

Z

Γ

perm

[[v

i

JA

k

_i

_{(q + ψ · e}

2 )N

k

]]dx

1 −

Z

Γ

bot

v

i

JA

k

_i

_{(q + ψ · e}

2 )N

k

dx

1 = 0.

Then, using the jumpand boundary onditions on

Γ

perm

and

Γ

bot

,

s

J

β

v

2 0,±

+

1

2 d

dt

|h|

2

0 = 0.

3.3 Estimates for

h

∈ L

2 _{(0, T ; H}

2.5 _(Γ))

and

h

t

∈ L

2 _{(0, T ; H}

1.5 _(Γ))

From (2.11 a)

(v

i

+ β

+

δ

i

2 )τ

i

= 0

on

Γ,

and

v

′

i

τ

i

= −β

+

JA

2 i

A

2 i

q,

2 h

′′

g

3/2

on

Γ .

(3.7)

Lemma6(Paraboli smoothing,[12℄,Se tion8.4.6). GivenasmoothsolutiontotheMuskat problem (2.11a-g),

h ∈ C([0, T ], H

2 (S

1 )).

In parti ular,

Z

t

0 |h

t

(s)|

2

1.5 ds +

Z

t

0 |h(s)|

2

2.5 ds ≤ C

max

0≤s≤t

|h(s)|

2

2 +

Z

t

0 kv(s)k

2 2,±

ds

.

(3.8)

(18)

3.4 Pressure estimates Using (2.11 a)and(2.11 b),

q

±

solves

−(β

±

J

±

(A

±

)

j

_i

(A

±

)

k

_i

q

±

,

k

),

j

= 0

in

D

±

_,

(3.9)

q

+

= 0

on

Γ ,

(3.10)

[[q]] = 0

on

Γ

perm

,

(3.11)

[[βq,

k

A

k

i

JA

r

i

N

r

]] = −[[β]]δ

2 i

JA

r

i

N

r

on

Γ

perm

,

(3.12)

β

−

q

−

,

k

(A

−

)

k

i

J

−

(A

−

)

r

i

N

r

= −β

−

on

Γ

bot

.

(3.13)

Wehave that

A(0)A(0)

T

issymmetri and positivedenite:

[A(0)A(0)

T

]

i

_j

ξ

i

ξ

j

≥ L|ξ|

2 ;

onsequently, dueto (3.4 b),

kA

0 A

T

0 − A(t)A

T

(t)k

L

∞

≤ C

√

t

p

E(t) ,

and we seethatfor

t

su iently small,

L

2 |ξ|

2 _{≤ [A(·, t)A}

T

_{(·, t)]}

i

j

ξ

i

ξ

j

≤ 2L|ξ|

2 .

Thus,

A(t)A

T

_(t)

form a uniformly ellipti operator for

t

on

[0, T ]

, and ellipti estimates (following the same approa h asin[12 ℄ and[13℄) lead to

kqk

2.5,±

≤ C

p

E(t) , kv(t)k

1.5,±

≤ C

p

E(t).

(3.14)

Furthermore,using thesame argument asin[12 , Se tion 8.4.5℄, we obtainthat

kq(t) − q(0)k

2.25,±

≤ t

1/8

P(E(t)),

(3.15)

and,

kq,

2 (t) − q,

2 (0)k

L

∞

_(Γ)

≤ C|q,

₂

(t) − q,

₂

(0)|

_0.75

≤ t

1/8

P(E(t)).

Asa onsequen e of thelatterinequality, theRayleigh-Taylor sign ondition holds in

[0, T ]

for small enough

T

. Furthermore, usingtheSobolev embeddingtheorem,

k∇q(t) − ∇q(0)k

L

∞

≤ t

1/8

P(E(t)),

and astronger versionofthe bootstrapassumption (3.2 d)also holds.

3.5 The energy estimates

In this se tion we will perform the basi energy estimates. Integrals of lower-order terms will bedenoted by

R(t)

,meaning that

Z

t

0 R(s)ds ≤ M

0

(19)

We take two horizontal derivatives of (2.11a), test against

Jv

′′

, and integrate by parts to ndthat

Z

t

0 Z

D

+

_∪D

−

J

β

|v

′′

_|

2 _dxds+

Z

t

0 Z

D

+

_∪D

−

J

h

A

k

_i

(q + ψ

2 )

′′

,

k

+(A

k

i

)

′′

(q + ψ

2 ),

k

i

v

′′

_i

dxds+

Z

t

0 R(s)ds = 0.

Dueto the divergen e free ondition(2.11 b)we obtain that

A

k

_i

(v

i

)

′′

,

k

= −(A

′′

)

k

i

v

i

,

k

+R(t).

(3.16)

Thus,integratingbyparts andusing (3.16 )andtheidentities

JA

k

i

N

k

= ˜

n

i

, JA

k

i

N

k

= ˜

n

i

perm

and

JA

k

i

N

k

= N

i

on

Γ

,

Γ

perm

and

Γ

bot

,respe tively,we nd that

I

1 =

Z

t

0 Z

D

+

_∪D

−

JA

k

_i

(q + ψ

2 )

′′

,

k

v

′′

i

dxds

=

Z

t

0 Z

Γ

˜

n

i

(q + ψ

2 )

′′

v

′′

_i

dx

1 ds −

Z

t

0 Z

Γ

perm

[[v

_i

′′

(q + ψ

2 )

′′

n

˜

i

_perm

]]dx

1 ds

−

Z

t

0 Z

Γ

bot

N

i

(q + ψ

2 )

′′

v

_i

′′

dx

1 ds −

Z

t

0 Z

D

+

_∪D

−

JA

k

_i

(q + ψ

2 )

′′

(v

i

),

′′

_k

dxds

=

Z

t

0 Z

D

+

_∪D

−

J(A

′′

)

k

_i

(q + ψ

2 )

′′

(v

i

),

k

dxds +

Z

t

0 Z

Γ

˜

n

i

h

′′

v

′′

_i

dx

1 ds

−

Z

t

0 Z

Γ

perm

(q

+

+ f )

′′

[[v

′′

_i

n

˜

i

_perm

]]dx

1 ds +

Z

t

0 R(s)ds.

The2-Dintegralisnowalower-orderterm that an beestimatedwitha

L

2 _{− L}

4 _{− L}

∞

Hölder argument togetherwiththeSobolevembeddingtheorem. Thus,we areleftwiththe integrals on the boundaries

Γ

and

Γ

perm

. Due to the in ompressibility ondition, we have that

[[v

i

n

˜

i

perm

]] = 0

,sothat

I

1 =

Z

t

0 Z

Γ

h

′′

h

′′

_t

dx

1 ds −

Z

t

0 Z

Γ

(

√

gn

i

)

′′

h

′′

v

i

dx

1 ds

+

Z

t

0 Z

Γ

perm

(q

+

+ f )

′′

[[v

1 ]]f

′′′

dx

1 ds +

Z

t

0 R(s)ds

=

1

2 |h

′′

_|

2

0 −

1

2 |h

′′

0 |

2

0 −

1

2 Z

t

0 Z

Γ

(h

′′

)

2 v

₁

′

dx

1 ds +

Z

t

0 |q

′′

_[[v

1 ]]|

0.5 |f |

2.5 ds

−

1

2 Z

Γ

perm

(f

′′

)

2 [[v

′

₁

]]dx

1 ds +

Z

t

0 R(s)ds

≥

1 ₂

|h

′′

|

2

0 −

1

2 |h

′′

0 |

2

0 −

√

tP(E(t)),

where wehave usedHölder inequality, the tra etheorem, (3.14 )and theinequality

(20)

The remaining high order term an be handled asfollows: using (2.10 )and integrating by parts,

I

2 =

Z

t

0 Z

D

+

_∪D

−

J(A

k

_i

)

′′

(q + ψ

2 ),

k

v

i

′′

dxds

= −

Z

t

0 Z

D

+

_∪D

−

JA

k

_r

ψ

r

,

11j

A

_i

j

(q + ψ

2 ),

k

v

′′

i

dxds +

Z

t

0 R(s)ds

=

Z

t

0 Z

D

+

_∪D

−

ψ

r

,

11 JA

j

i

(A

k

r

(q + ψ

2 ),

k

v

′′

i

),

j

dxds −

Z

t

0 Z

Γ

ψ

r

,

11 JA

j

i

(A

k

r

(q + ψ

2 ),

k

v

′′

i

)N

j

dx

1 ds

+

Z

t

0 Z

Γ

perm

[[ψ

r

,

11 JA

j

i

(A

k

r

(q + ψ

2 ),

k

v

i

′′

)]]N

j

dx

1 ds

−

Z

t

0 Z

Γ

bot

ψ

r

,

11 JA

j

i

(A

k

r

(q + ψ

2 ),

k

v

′′

i

)N

j

dx

1 ds +

Z

t

0 R(s)ds

=

Z

t

0 Z

D

+

_∪D

−

ψ

r

,

11 JA

j

i

(A

k

r

(q + ψ

2 ),

k

v

′′

i

),

j

dxds −

Z

t

0 Z

Γ

h

′′

n

˜

i

A

k

2 (q + ψ

2 ),

k

v

i

′′

dx

1 ds

+

Z

t

0 Z

Γ

perm

f

′′

[[˜

n

i

_perm

A

k

₂

(q + ψ

2 ),

k

v

i

′′

]]dx

1 ds +

Z

t

0 R(s)ds.

The integralinthe bulk ofthe uid an beestimated using(3.16) sothat

Z

t

0 Z

D

+

_∪D

−

ψ

r

,

11 JA

j

i

(A

k

r

(q + ψ

2 ),

k

v

i

′′

),

j

dxds ≥ −

√

tP(E(t)).

Theintegralover

Γ

perm

anbeestimatedusingthe

H

0.5 _−H

−0.5

dualityand(3.17 )asfollows:

(21)

Thus,

Z

t

0 s

J

β

v

′′

2 0,±

ds +

1

2 s

−q,

2 (0)

J(0)

h

′′

_(t)

2

0 ≤

1

2 s

−q,

2 (0)

J(0)

h

′′

0

2

0 +

√

_tP(E(t)).

(3.18)

3.6 Ellipti estimates via the Hodge de omposition Inthis se tion, we usethefollowing