Table Of ContentDERIVATION AND ANALYSIS OF A NEW 2D GREEN-NAGHDI
SYSTEM
0
1 SAMERISRAWI
0
2
n
a
J Abstract. We derivehereavariantofthe 2D Green-Naghdi equations that
modelthepropagationoftwo-directional,nonlineardispersivewavesinshallow
6 water. This new model has the same accuracy as the standard 2D Green-
1
Naghdi equations. Its mathematical interest is that itallows a control of the
rotational part of the (vertically averaged) horizontal velocity, which is not
]
P the case forthe usual Green-Naghdi equations. Usingthis property, weshow
that the solution of these new equations can be constructed by a standard
A
Picard iterative scheme so that there is no loss of regularity of the solution
h. with respect to the initial condition. Finally, we prove that the new Green-
t Naghdiequationsconservethealmostirrotationalityoftheverticallyaveraged
a horizontal component ofthevelocity.
m
[
1 1. Introduction
v
0 1.1. General setting. The water-wavesproblem, consists in studying the motion
5 ofthefreesurfaceandtheevolutionofthevelocityfieldofalayeroffluidunderthe
8 following assumptions: the fluid is ideal, incompressible, irrotationnal, and under
2
theonlyinfluenceofgravity. Manyworkshavesetagoodtheoreticalbackgroundfor
.
1 thisproblem. Itslocalwell-posednesshasbeendiscussedamongothersbyNalimov
0 [13], Yosihara [18], Craig [5], Wu [15, 16] and Lannes [8]. Since we are interested
0
hereintheasymptoticbehaviorofthesolutions,itisconvenienttoworkwithanon-
1
dimensionalized version of the equations. In this framework, (see for instance [2]
:
v fordetails),thefreesurfaceisparametrizedbyz =εζ(t,X)(withX =(x,y) R2)
i ∈
X and the bottom by z = 1+βb(X). Here, ε and β are dimensionless parameters
−
r defined as
a
a b
surf bott
ε= , β = ;
h h
0 0
wherea is the typicalamplitude ofthe wavesandb is the typicalamplitude
surf bott
of the bottom deformations, while h is the depth. One can use the incompress-
0
ibility and irrotationality conditions to write the non-dimensionalized water-waves
equations under Bernoulli’s formulation, in terms of a velocity potential ϕ associ-
atedtotheflow,whereϕ(t,.)isdefinedonΩ = (X,z), 1+βb(x)<z <εζ(t,X)
t
{ − }
(i.e. the velocity field is given by U = ϕ) :
X,z
∇
Ce travail a b´en´efici´e d’une aide de l’Agence Nationale de la Recherche portant la r´ef´erence
ANR-08-BLAN-0301-01.
1
2 SAMERISRAWI
µ∂2ϕ+µ∂2ϕ+∂2ϕ=0, at 1+βb<z <εζ,
x y z −
∂zϕ µβ b ϕ=0 at z = 1+βb,
− ∇ ·∇ −
(1) ∂tζ− µ1(µε∇ζ·∇ϕ+∂zϕ)=0, at z =εζ,
1 ε
∂ ϕ+ (ε ϕ2+ (∂ ϕ)2)+ζ =0 at z =εζ,
where = (∂tx,∂y2)T =|∇(∂|1,∂2)µT. zThe shallowness parameter µ appearing in this
∇
set of equations is defined as
h2
µ= 0,
λ2
where, λ is the typical wave-length of the waves. Making assumptions on the size
of ε, β, and µ one is led to derive (simpler) asymptotic models from (1). In the
shallow-water scaling (µ 1), one can derive (when no smallness assumption is
≪
made on ε and β ) the standard Green-Naghdi equations (see 2.1 below for a
§
derivation and [2] for a rigorous justification). For two-dimensional surfaces and
over uneven bottoms these equations couple the free surface elevation ζ to the
vertically averagedhorizontal component of the velocity,
1 εζ
(2) v(t,X)= ϕ(t,X,z)dz,
1+εζ βb ∇
− Z−1+βb
and can be written as:
∂ ζ+ (hv)=0,
t
∇·
h+µ [h,βb] ∂ v+h ζ+ε h+µ [h,βb] (v )v
(3) T t ∇ T ·∇
(cid:0) +µε 2 [((cid:1)h3(∂ v ∂ v⊥+(cid:0)( v)2)]+ (cid:1)[h,βb](v) =0,
1 2
3∇ · ∇· ℜ
where h=1+εζ β(cid:8)b, v =(V1,V2)T, v⊥ =( V2,V1)⊥, and (cid:9)
− −
1 β
(4) [h,εb]W = (h3 W)+ [ (h2 b W) h2 b W]
T −3∇ ∇· 2 ∇ ∇ · − ∇ ∇·
+β2h b b W,
∇ ∇ ·
while the purely topographic term [h,βb](v) is defined as:
ℜ
β
(5) [h,βb](v) = h2(V2∂2b+2V V ∂ ∂ b+V2∂2b)
ℜ 2∇ 1 1 1 2 1 2 2 2
+βh(cid:0)2 ∂ v ∂ v⊥+( v)2 b (cid:1)
1 2
· ∇· ∇
+β2h(cid:0)V12∂12b+2V1V2∂1∂2b(cid:1)+V22∂22b ∇b.
A rigorous justification of the st(cid:0)andard GN model was given(cid:1)by Li [11] in 1D
andforflatbottoms,andbyB.Alvarez-SamaniegoandD.Lannes[2]inthegeneral
case. This latterreferencerelies onwell-posednessresults forthese equationsgiven
in [3] and based on general well-posedness results for evolution equations using a
Nash-Moser scheme. The result of [3] covers both the case of 1D and 2D surfaces,
andallowsfornonflatbottoms. ThereasonwhyaNash-Moserschemeisusedthere
is because the estimates on the linearized equations exhibit losses of derivatives.
However,in the 1D case, such losses do not occur and it is possible to construct a
solutionwith a standardPicarditerativescheme as in[11, 7] with flatandnonflat
bottoms respectively. But, this is not the case in 2D, since for instance, the term
DERIVATION AND ANALYSIS OF A NEW 2D GREEN-NAGHDI SYSTEM 3
∂ v ∂ v⊥ is not controled by the energy norm naturally associated to (3)
1 2 Ys
· |·|
(see [3]),
(ζ,v)2 = ζ 2 + v 2 +µ v 2 .
| |Ys | |Hs | |Hs |∇· |Hs
This is the motivation for the present derivation of a new variant of the 2D
Green-Naghdi equations (3). This variant has the same accuracy as the standard
2DGreen-Naghdiequations(3)(see 2.2belowforaderivation)andcanbewritten
§
under the form:
∂ ζ+ (hv)=0,
t
∇·
h+µ [h,βb] ⊥curl ∂tv+h ζ
T −∇ ∇
(6) (cid:16) +ε(cid:0)h+µ T[h,βb]−∇(cid:1)⊥(cid:17)curl (v·∇)v
+µ(cid:16)ε 2 [(cid:0)h3(∂ v ∂ v⊥+( (cid:1)v(cid:17))2)]+ [h,βb](v) =0,
where ⊥= ( ∂y,∂xn)3T∇, curlv1=·∂12V2 ∂2∇V1·, while tℜhe linear ooperators [h,εb]
∇ − − T
and [h,βb] are defined in (4) and (5).
ℜ
The reason why the new terms (involving ⊥curl ) do not affect the precision of
∇
the model is because the solutions to (3) are nearly irrotational (in the sense that
curlv is small). This property is of course satisfied also by our new model (6).
The presence of these new terms allows the definition of a new energy norm that
controls also the rotational part of v. Consequently, we show that it is possible to
use a standard Picard iterative scheme to prove the well-posedness of (6), so that
there is no loss of regularity of the solution with respect to the initial condition.
1.2. Organization of the paper. We first recall the derivation of the standard
2D Green-Naghdi equations in Section 2.1 while in Section 2.2 we derive the new
model (6). We give some preliminary results in Section 3.1; the main theorem
which proves the well-posedness of this new Green-Naghdi system is then stated
in Section 3.2 and proved in Section 3.3. Finally, in Section 3.4 we prove that (6)
conserves the almost irrotationality of v.
1.3. Notation. WedenotebyC(λ ,λ ,...)aconstantdependingontheparameters
1 2
λ , λ , ... and whose dependence on the λ is always assumed to be nondecreasing.
1 2 j
The notation a . b means that a Cb, for some nonegative constant C whose
≤
exact expression is of no importance (in particular, it is independent of the small
parameters involved).
Let p be any constant with 1 p < and denote Lp = Lp(R2) the space of all
≤ ∞
Lebesgue-measurable functions f with the standard norm
f = f(X)pdX 1/p < .
Lp
| | R2| | ∞
Z
(cid:0) (cid:1)
When p = 2, we denote the norm simply by . The inner product of any
L2 2
functions f and f in the Hilbert|sp·a|ce L2(R2) is d|e·n|oted by
1 2
(f ,f )= f (X)f (X)dX.
1 2 1 2
R2
Z
The space L∞ =L∞(R2) consists of all essentially bounded, Lebesgue-measurable
functions f with the norm
f =esssup f(X) < .
L∞
| | | | ∞
4 SAMERISRAWI
We denote by W1,∞ = W1,∞(R2) = f L∞, f (L∞)2 endowed with its
{ ∈ ∇ ∈ }
canonical norm.
Foranyrealconstants,Hs =Hs(R2)denotestheSobolevspaceofalltempereddis-
tributions f with the norm f = Λsf < , where Λ is the pseudo-differential
Hs 2
| | | | ∞
operator Λ=(1 ∆)1/2.
For any function−s u = u(x,t) and v(x,t) defined on R2 [0,T) with T > 0, we
×
denote the inner product, the Lp-norm and especially the L2-norm, as well as the
Sobolev norm, with respect to the spatial variable X, by (u,v) = (u(,t),v(,t)),
· ·
u = u(,t) , u = u(,t) , and u = u(,t) , respectively.
Lp Lp L2 L2 Hs Hs
|Le|t Ck(|R2)· de|note| t|he spa|ce·of|k-times c|on|tinuou|sly· di|fferentiable functions and
C∞(R2) denote the space of infinitely differentiable functions, with compact sup-
0
portinR2;wealsodenotebyC∞(R2)thespaceofinfinitelydifferentiablefunctions
b
that are bounded together with all their derivatives.
For any closed operator T defined on a Banach space X of functions, the commu-
tator[T,f]isdefinedby[T,f]g =T(fg) fT(g)withf,g andfg belongingtothe
−
domain of T.
We denote v⊥ =( V ,V )T and curlv =∂ V ∂ V where v =(V ,V )T.
2 1 1 2 2 1 1 2
− −
2. Derivation of the new Green-Naghdi model
This section is devoted to the derivation of a new Green-Naghdi asymptotic
model for the water-waves equations in the shallow water (µ 1) of the same
≪
accuracy as the standard 2D Green-Naghdi equations (3).
2.1. Derivation of the standard Green-Naghdi equations (3). We recall
here the main steps of [10] for the derivation of the standard 2D Green-Naghdi
equations (3). In order to reduce the model (1) into a model of two equations we
introduce the trace of the velocity potential at the free surface, defined as
ψ =ϕ ,
|z=εζ
and the Dirichlet-Neumann operator [εζ,βb] as
µ
G ·
[εζ,βb]ψ = µε ζ ϕ +∂ ϕ ,
Gµ − ∇ ·∇ |z=εζ z |z=εζ
with ϕ solving the boundary value problem
µ∂2ϕ+µ∂2ϕ+∂2ϕ=0,
x y z
(7) ∂ ϕ =0,
n |z=−1+βb
ϕ =ψ.
|z=εζ
As remarked in [19, 6], the equations (1) are equivalent to a set of two equations
on the free surface parametrizationζ and the trace of the velocity potential at the
surface ψ =ϕ involving the Dirichlet-Neumann operator. Namely
|z=εζ
1
∂ ζ+ [εζ,βb]ψ =0,
t µ
µG
(8) ∂tψ+ζ+ 2ε|∇ψ|2−εµ(µ1Gµ[ε2ζ(,1βb+]ψε2+µ∇(ζεζ2))·∇ψ)2 =0.
|∇ |
It is a straightforwardconsequence of Green’s identity that
1
[εζ,βb]ψ = (hv),
µ
µG −∇·
DERIVATION AND ANALYSIS OF A NEW 2D GREEN-NAGHDI SYSTEM 5
with h = 1+εζ βb and v = 1 εζ ϕ(t,X,z)dz. Therefore, the first
− 1+εζ−βb −1+βb∇
equation of (8) exactly coincides with the first equation of (3). In order to derive
R
the evolution equation on v, the key point is to obtain an asymptotic expansion of
ψ withrespecttoµandintermspfv andζ. Asin[10],welookforanasymptotic
∇
expansion of ϕ under the form
N
(9) ϕ = µjϕ .
app j
j=0
X
Plugging this expression into the boundary value problem (7) one can cancel the
residual up to the order O(µN+1) provided that
(10) j =0,...,N, ∂2ϕ = ∂2ϕ ∂2ϕ
∀ z j − x j−1− y j−1
(with the convention that ϕ =0), together with the boundary conditions
−1
ϕ =δ ψ,
(11) j =0,...,N, j|z=εζ 0,j
∀ ( ∂zϕj =β∇b·∇ϕj−1|z=−1+βb
(where δ =1 if j =0 and 0 otherwise).
0,j
By solving the ODE (10) with the boundary conditions (11), one finds (see [10])
(12) ϕ = ψ,
0
1
(13) ϕ = (z εζ) (z+εζ) 1+βb ∆ψ+β(z εζ) b. ψ.
1
− − 2 − − ∇ ∇
According to formulae (12(cid:0)), (13), the horizontal(cid:1)component of the velocity in the
fluid domain is given by
V(z)= ϕ (z)+ ϕ (z)+O(µ2).
0 1
∇ ∇
The averagedvelocity is thus given by
1 εζ
v = ( ϕ (z)+ ϕ (z))dz+O(µ2),
0 1
h ∇ ∇
Z−1+βb
or equivalently
1
v = ψ µ [h,βb] ψ+O(µ2),
∇ − hT ∇
and thus
1
(14) ψ =v+µ [h,βb] ψ+O(µ2),
∇ hT ∇
where [h,βb] is as in (4). As in [10], taking the gradient of the second equation
T
of (8), replacing ψ by its expression (14) and 1 [εζ,βb]ψ by (hv) in the
∇ µGµ −∇·
resulting equation, gives the standardGreen-Naghdi equations (after dropping the
O(µ2) terms),
∂ ζ+ (hv)=0,
t
∇·
h+µ [h,βb] ∂ v+h ζ+ε h+µ [h,βb] (v )v
T t ∇ T ·∇
(cid:0) +µε 2 [h(cid:1)3(∂ v ∂ v⊥+((cid:0) v)2)]+ [h(cid:1),βb](v) =0,
1 2
3∇ · ∇· ℜ
where h =1+εζ β(cid:8)b, v = (V1,V2)T, v⊥ = ( V2,V1)T, and the(cid:9)linear operators
− −
[h,εb] and [h,βb] being defined in (4) and (5).
T ℜ
6 SAMERISRAWI
2.2. Derivation of the new Green-Naghdi system (6). Itisquite commonin
the literature to work with variants of the Green-Naghdi equations (6) that differ
onlyuptotermsoforderO(µ2)inordertoimprovethedispersivepropertiesofthe
model (see for instance [14, 4]) or to change its mathematical properties [12]. Our
approach here is in the same spirit since our goal is to derive a new model with
better energy estimates.
In order to obtain the new Green-Naghdi system (6), let us remark that, from the
expression of v one has
µ
ψ =v+ [h,βb] ψ+O(µ2);
∇ hT ∇
and since v = ψ+O(µ), one gets the following Lemma:
∇
Lemma 1. Let v be the vertically averaged horizontal component of the velocity
given above, one obtains
curl∂ v = O(µ),
t
(15) curl(v )v = O(µ).
·∇
Remark 1. For the sake of simplicity, we denote by O(µ) any family of functions
(f ) such that 1f remains bounded in L∞([0,T],Hr(R2)), for some
µ 0<µ<1 µ µ 0<µ<1 ε
r large enough.
(cid:0) (cid:1)
Proof. Byapplyingtheoperator(curl∂ )()totheidentityv = ψ+O(µ)onegets
t
· ∇
curl∂ v =O(µ).
t
In order to prove the second identity of (15), replace v = ψ+O(µ) in (v )v
∇ ·∇
and apply the operator curl() to deduce
·
curl(v )v =O(µ).
·∇
(cid:3)
Using Lemma 1, the quantities µ ⊥curl∂ v and µ ⊥curlε(v )v are of size
t
∇ ∇ ·∇
O(µ2), which is the precision of the GN equations (3). We can thus include these
new terms in the second equation of (3) to get
∂ ζ+ (hv)=0,
t
∇·
h+µ [h,βb] ⊥curl ∂ v+h ζ
t
T −∇ ∇
(cid:16) +ε(cid:0)h+µ T[h,βb]−∇(cid:1)⊥(cid:17)curl (v·∇)v
+µ(cid:16)ε 2 [(cid:0)h3(∂ v ∂ v⊥+( (cid:1)(cid:17)v)2)]+ [h,βb](v) =O(µ2),
(the linear operatonrs3∇[h,εb]1an·d2 [h,βb∇] b·eing defiℜned in (4)oand (5)).
T ℜ
Remark 2. We added the quantity µ ⊥curl∂ v = O(µ2) to the standard GN
t
∇
equations (3) to obtain an energy norm :
Xs
|·|
(ζ,v)2 = ζ 2 + v 2 +µ v 2 +µcurlv 2 .
| |Xs | |Hs | |Hs |∇· |Hs | |Hs
The last term is absent from the energy associated to the standard GN equa-
Ys
|·|
tions (3) (see [3]). We will see in the next section that it allows a control of the
term ∂ v ∂ v⊥.
1 2
·
DERIVATION AND ANALYSIS OF A NEW 2D GREEN-NAGHDI SYSTEM 7
Remark 3. The bilinear operator [h,βb](v) only involves second order deriva-
ℜ
tives of v while third order derivatives of v have been factorized by h+µ [h,εb]
T −
⊥curl . Thefact that theoperator [h,βb](v)does notinvolve thirdorder deriva-
∇ ℜ (cid:0)
tives is of great interest for the well-posedness of the new Green-Naghdi model.
(cid:1)
3. Mathematical analysis of the new Green-Naghdi model
3.1. Preliminaryresults. Forthesakeofsimplicity,wetakehereandthroughout
the rest of this paper (β =ε) and we write
T=h+µ [h,εb] ⊥curl .
T −∇
We always assume that the nonzero(cid:0)depth condition (cid:1)
(16) h >0, inf h h , h=1+ε(ζ b)
min min
∃ X∈R2 ≥ −
is valid initially, which is a necessary condition for the new GN type system (6) to
be physically valid. We shall demonstrate that the operator T plays an important
roleintheenergyestimateandthe localwell-posednessofthe GNtypesystem(6).
Therefore, we give here some of its properties.
ThefollowinglemmagivesanimportantinvertibilityresultonTandsomeprop-
erties of the inverse operator T−1.
Lsaetmisfimeda. T2.heLne,tthbe∈opCerb∞at(oRr2T),hta0s>a b1ouannddedζin∈veHrset0+o1n(R(L2)2(Rbe2)s)u2c,hantdhat (16) is
(i) For all 0 s t +1, one has
0
≤ ≤
1
|T−1f|Hs +√µ|∇·T−1f|Hs +√µ|curlT−1f|Hs ≤C(h ,|h−1|Ht0+1)|f|Hs,
min
and
1
√µ|T−1∇g|Hs +√µ|T−1∇⊥g|Hs ≤C(h ,|h−1|Ht0+1)|g|Hs.
min
(ii) If s t +1 and ζ Hs(R2) then
0
≥ ∈
T−1 (Hs)2→(Hs)2 +√µ T−1 (Hs)2→(Hs)2 +√µ T−1 ⊥ (Hs)2→(Hs)2 cs,
k k k ∇k k ∇ k ≤
and
√µ T−1 (Hs)2→(Hs)2 +√µ curlT−1 (Hs)2→(Hs)2 cs,
k∇· k k k ≤
where c is a constant depending on 1 , h 1 and independent of (µ,ε)
s hmin | − |Hs
(0,1)2.
∈
Remark 4. Here and throughout the rest of this paper, and for the sake of sim-
plicity, we do not try to give some optimal regularity assumption on the bottom
parametrization b. This could easily be done, but is of no interest for our present
purpose. Consequently, we omit to write the dependence on b of the different quan-
tities that appear in the proof.
Proof. It can be remarked that the operator T is L2 self-adjoint; since, one has
h+µ [h,εb] ⊥curl v,v =(hv,v)
T −∇
(cid:16) (cid:0)h √3 (cid:1) (cid:17)h √3 µε2
+µ h v ε b v , v ε b v + (h b v, b v)
√3∇· − 2 ∇ · √3∇· − 2 ∇ · 4 ∇ · ∇ ·
(cid:16) (cid:16) (cid:17) (cid:17)
+µ(curlv,curlv),
8 SAMERISRAWI
and using the fact that infR2h hmin, one deduces that
≥
(Tv,v) E(εb,v),
≥
with
h √3 µε2h
E(εb,v):=h v 2+µh v ε b v 2+ min b v 2+µcurlv 2,
min| |2 min|√3∇· − 2 ∇ · |2 4 |∇ · |2 | |2
proceeding exactly as in the proof of the Lemma 1 of [7] it follows that T has an
inverse bounded on (L2(R2))2.
For the rest of the proof, One can proceeding as in the proof of Lemma 4.7 of [3],
to get the result. (cid:3)
3.2. Linear analysis of (6). In order to rewrite the new GN type equations (6)
in a condensed form, let us write U =(ζ,vT)T, v =(V ,V )T and
1 2
Q=h3( ∂ v ∂ v⊥ ( v)2).
1 2
− · − ∇·
We decompose now the O(εµ) term of the second equation of (6) under the form
2
Q+R[h,εb](v)=R [U]v+r (U),
1 2
−3∇
with for all f =(F ,F )T
1 2
2 ε
R [U]f = Q[U]f + (h2(V F ∂2b+2V F ∂ ∂ b+V F ∂2b))
1 −3∇ 2∇ 1 1 1 1 2 1 2 2 2 2
εh2( ∂ v ∂ f⊥ ( v)( f)) b;
1 2
− − · − ∇· ∇· ∇
r (U) = ε2h(V2∂2b+2V V ∂ ∂ b+V2∂2b) b,
2 1 1 1 2 1 2 2 2 ∇
where
Q[U]f =h3( ∂ v ∂ f⊥ ( v)( f)),
1 2
− · − ∇· ∇·
(in particular, Q = Q[U]v). The new Green-Naghdi equations (6) can thus be
written after applying T−1 to both sides of the second equation in (6) as
(17) ∂ U +A[U]U +B(U)=0,
t
with U =(ζ,V ,V )T, v =(V ,V )T and where
1 2 1 2
εv h
(18) A[U]= T−1(·h∇) ε(v )+ε∇µ·T−1R [U]
1
(cid:18) ∇ ·∇ (cid:19)
and
ε b v
(19) B(U)= εµT∇−1r· (U) .
2
(cid:18) (cid:19)
This subsection is devoted to the proof of energy estimates for the following initial
value problem around some reference state U =(ζ,V ,V )T:
1 2
∂ U +A[U]U +B(U)=0;
(20) t
U =U .
(cid:26) |t=0 0
We define first the Xs spaces, which are the energy spaces for this problem.
DERIVATION AND ANALYSIS OF A NEW 2D GREEN-NAGHDI SYSTEM 9
Definition1. For alls 0andT >0,wedenotebyXs thevectorspaceHs(R2)
(Hs+1(R2))2 endowed w≥ith the norm ×
U =(ζ,v) Xs, U 2 := ζ 2 + v 2 +µ v 2 +µcurlv 2 ,
∀ ∈ | |Xs | |Hs | |(Hs)2 |∇· |Hs | |Hs
while Xs stands for C([0,T];Xs) endowed with its canonical norm.
T ε
We define the matrix S as
1 0
(21) S = ,
0 T
(cid:18) (cid:19)
withh=1+ε(ζ b)andT=h+µ( [h,εb] ⊥( )). Anaturalenergyforthe
− T −∇ ∇∧
IVP (20) is given by
(22) Es(U)2 =(ΛsU,SΛsU).
The link between Es(U) and the Xs-norm is investigated in the following Lemma.
Lemma 3. Let b C∞(R2), s 0 and ζ W1,∞(R2). Under the condition (16),
∈ b ≥ ∈
Es(U) is uniformly equivalent to the -norm with respect to (µ,ε) (0,1)2:
Xs
|·| ∈
Es(U) C h U ,
W1,∞ Xs
≤ | | | |
and (cid:0) (cid:1)
1
U C Es(U).
Xs
| | ≤ h
min
(cid:0) (cid:1)
Proof. Notice first that
Es(U)2 = Λsζ 2+(Λsv,TΛsv),
| |2
one gets the first estimate using the explicit expression of T, integration by parts
and Cauchy-Schwarzinequality.
The other inequality can be proved by using that infx∈R2h hmin > 0 and pro-
≥
ceeding as in the proof of Lemma 2. (cid:3)
We prove now the energy estimates in the following proposition. It is worth
insisting on the fact that these estimates are uniform with respect to ε,µ (0,1);
∈
sincethecontrolofthes+1orderderivativesbytheXs-normdisappearsasµ 0,
→
the uniformity with respect to µ requires particular care (see the control of B in
46
the proof for instance), but is very important for the application since µ 1.
≪
Proposition 1. Let b C∞(R2), t > 1, s t +1. Let also U = (ζ,V ,V )T
∈ b 0 ≥ 0 1 2
Xs be such that ∂ U Xs−1 and satisfying the condition (16) on [0,T]. Then
∈ T t ∈ T ε
for all U Xs there exists a unique solution U = (ζ,V ,V )T Xs to (20) and
0 ∈ 1 2 ∈ T
for all 0 t T
≤ ≤ ε
t
Es(U(t)) eελTtEs(U )+ε eελT(t−t′)C(Es(U)(t′))dt′,
0
≤
Z0
for some λ =λ (sup Es(U(t)),sup ∂ h(t) ) .
T T 0≤t≤T/ε 0≤t≤T/ε| t |L∞
Proof. Existence and uniqueness of a solution to the IVP (20) is achievedby using
classicalmethods as in appendix A of [7] for the standard 1D Green-Naghdiequa-
tions and we thus focus our attention on the proof of the energy estimate. For any
λ R, we compute
∈
eελt∂ (e−ελtEs(U)2)= ελEs(U)2+∂ (Es(U)2).
t t
−
10 SAMERISRAWI
Since
Es(U)2 =(ΛsU,SΛsU),
and U =(ζ,V ,V )T, v =(V ,V )T, we have
1 2 1 2
(23) ∂ (Es(U)2)=2(Λsζ,Λsζ )+2(Λsv,TΛsv )+(Λsv,[∂ ,T]Λsv).
t t t t
One gets using the equations (20) and integrating by parts,
1 ελ
eελt∂ (e−ελtEs(U)2)= Es(U)2 (SA[U]ΛsU,ΛsU)
t
2 − 2 −
1
(24) Λs,A[U] U,SΛsU (ΛsB(U),SΛsU)+ (Λsv,[∂ ,T]Λsv).
t
− − 2
We now turn(cid:0)t(cid:2)o bound f(cid:3)rom abov(cid:1)e the different components of the r.h.s of (24).
Estimate of (SA[U]ΛsU,ΛsU). Remarking that
•
εv h
SA[U]= h·∇ T(εv )∇+·εµR [U] ,
1
(cid:18) ∇ ·∇ (cid:19)
we get
(SA[U]ΛsU,ΛsU) = (εv Λsζ,Λsζ)+(h Λsv,Λsζ)
·∇ ∇·
+(h Λsζ,Λsv)+ (T(εv )+εµR [U])Λsv,Λsv
1
∇ ·∇
=:A +A +A +A .
1 2 3 (cid:0) 4 (cid:1)
We now focus on the control of (A ) .
j 1≤j≤4
Control of A . Integrating by parts, one obtains
1
−
1
A =(εv Λsζ,Λsζ)= (ε vΛsζ,Λsζ),
1
·∇ −2 ∇·
and one can conclude by Cauchy-Schwarzinequality that
A εC( v )Es(U)2.
1 L∞
| |≤ |∇· |
Control of A +A . First remark that
2 3
−
A +A = ( h Λsv,Λsζ) h Es(U)2;
2 3 (L∞)2
| | | ∇ · |≤|∇ |
we get,
A +A εC( h Es(U)2.
2 3 (L∞)2
| |≤ |∇ |
Control of A . One computes,
4
−
A = ε T((v )Λsv),Λsv +(εµR [U]Λsv,Λsv)
4 1
·∇
=:A +A .
(cid:0) 41 42 (cid:1)
Note first that
εµ
A = ε(h(v )Λsv,Λsv)+ (h3 (v )Λsv),Λs v)
41
·∇ 3 ∇· ·∇ ∇·
ε2µ ε2µ
(h2 b (v )Λsv,Λs v) (h2 b (v )Λsv,Λsv)
− 2 ∇ · ·∇ ∇· − 2 ∇ ∇· ·∇
+ε3µ(h b b (v )Λsv,Λsv)+εµ(curl(v )Λsv),Λscurlv);
∇ ∇ · ·∇ ·∇
remark also that
1
(curl(v )Λsv),Λscurlv) = (curlΛsv,∂ V curlΛsv)+( Λsv,∂ V Λsv)
1 1 2 2
·∇ −2 ∇∧ ∇∧
+(c(cid:16)urlΛsv,∂ v ΛsV ) (curlΛsv,∂ v ΛsV ), (cid:17)
1 2 2 1
·∇ − ·∇
