On the motion of angled crested type water waves with surface tension

137 Pages·2017·0.89 MB·English

ON THE MOTION OF ANGLED CRESTED TYPE WATER WAVES WITH
SURFACE TENSION
SIDDHANTAGRAWAL
Abstract. We consider the two-dimensional capillary water wave equation in the presence of
gravity. Weassumethatthefluidisinviscid,incompressibleandirrotationalandtheairdensity
is zero. We construct an energy functional and prove an a priori estimate without assuming the
Taylorsigncondition. Whenthesurfacetensioniszero,theenergyreducestotheenergyobtained
byKinseyandWu[17]forangledcrestwaterwaves. Weshowthatinanappropriateregime,the
zero surface tension limit of our solutions is the one for the gravity water wave equation which
includeswaveswithangledcrest.
1. Introduction
We are concerned with the motion of a fluid in dimension two with a free boundary. The fluid
region Ω(t) and the air is separated by an interface Σ(t), with the fluid being below the air region
andΣ(t)beingaonedimensionalcurve. Theairandthefluidareassumedtohaveconstantdensities
of0and1respectively. Thefluidisalsoassumedtobeinviscid,incompressibleandirrotationaland
we assume that the bottom is at infinite depth. The gravitational field is assumed to be a constant
vector −k pointing in the downward direction. The motion of the fluid is then governed by the
Euler equation
v +(v.∇)v=−k−∇P on Ω(t)
t
(1)
div v=0, curl v=0 on Ω(t)
Along with the boundary conditions
P =−σ∂ θ on Σ(t)
s
(1,v) is tangent to the free surface (t,Σ(t)) (2)
v→0,v →0 as |(x,y)|→∞
t
Here θ = angle the interface makes with the x−axis, ∂ = arc length derivative, σ = coefficient of
s
surface tension ≥0.
The earliest results on local well-posedness for the Cauchy problem are for small data in 2D and
wereobtainedbyNalimov[24],Yoshihara[41,42]andCraig[11]. Inthecaseofzerosurfacetension,
Wu [35, 36] obtained the proof of well-posedness for arbitrary data in Sobolev spaces. Later this
result was extended to the case of bottom with finite depth, non-zero vorticity and in lowering the
Sobolev regularity in [12, 22, 18, 43, 2, 14, 3, 28].
Inthecaseofnon-zerosurfacetension,thelocalwell-posednessoftheequationinSobolevspaces
was established by Beyer and Gunther in [7]. See also the works in [15, 4, 29, 10, 1, 25]. The zero
surface tension limit of the water wave equations in Sobolev spaces was proved by Ambrose and
Masmoudi [5, 6]. See also the works in [26, 30, 23].
Date:January22,2018.
TheauthorwassupportedinpartbyNSFGrantsDMS-1101434,DMS-1361791.
1
2 SIDDHANTAGRAWAL
An important quantity related to the well-posedness of the problem in the zero surface tension
case is the Taylor sign condition. This says that there should exist a constant c>0 such that
∂P
− ≥c>0 on Σ(t)
∂n
In [35] Wu proved that this condition is satisfied for the infinite bottom case if the interface is C1,α
for α >0. This was later shown to be true for flat bottoms and with perturbations to flat bottom
by Lannes [18] and was reproved again in [14]. In all other results mentioned above for zero surface
tension or for zero surface tension limit, the initial data is chosen in such a way that this condition
is satisfied. In the zero surface tension case, for non C1,α curves this condition is only satisfied in
a weak sense with −∂P ≥ 0 [35, 17]. This makes the quasilinear equation degenerative and hence
∂n
standard energy estimates in Sobolev spaces do not work. Kinsey and Wu [17] used a weighted
Sobolev energy with the weight depending nonlinearly on the interface to deal with this case and
proved a priori estimates for interfaces which can have angled crests. Wu [38] obtained existence of
solutions for the Cauchy problem that allows for such waves.
In this paper we extend the results of Kinsey and Wu [17] to the capillary gravity water wave
equation(1)-(2)andshowthatthesolutionsobtainedin[38],includingthosewithangledcrests,are
zero surface tension limits of solutions for the capillary gravity water wave equation (1)-(2) under
appropriate conditions. We first construct an energy functional E (t) and prove that if initially
σ
E (0) < ∞, then E (t) remains finite for a positive time period [0,T], with T > 0 depending only
σ σ
on E (0). E (t) has the property that it is increasing in σ, and its zero surface tension limit E (t)
σ σ 0
is equivalent to the energy functional E (t) of Kinsey and Wu [17] and allows for interfaces with
0
angled crests.1 The energy E (t) does not allow interfaces with angled crest for σ > 0, but allows
σ
the L∞ norm of the curvature of the initial data to grow like σ−13 as σ →0. We then construct an
energy functional E for the difference of the solution of the capillary gravity water wave equation
∆
with surface tension coefficient σ and the solution for the gravity water wave equation, and prove
that for any given data in the class where E (0)+E (0)<∞, we have E (t)→0 as σ →0, on the
σ 0 ∆
time interval where the solution of the gravity water wave equation exists.2
We will follow the approach of Kinsey and Wu [17] and Wu [38], and work with free surface
equations in conformal coordinates, derive quasilinear equations and use weighted Sobolev norms.
The presence of surface tension gives rise to several structural and analytical difficulties which are
not present in these works. We explain the difficulties and main results in detail in §3. In addition
to the work of Kinsey and Wu [17] and Wu [38], we also use ideas from the work of Ambrose and
Masmoudi [5] in choosing appropriate variables to work with.
Thepaperisorganizedasfollows: In§2weintroducethenotationandprovesomebasicformulae
including the formula for the Taylor sign condition. In §3 we state our main results, explain the
difficultiesandgiveaheuristicexplanationfortheresults. In§4wederiveourquasilinearequations
bytakingderivativestotheEulerequation. In§5weprovethemainaprioriestimatefortheenergy
E (t). In §6 we prove the equivalence of the energies E (t) and E (t), explain its relation to the
σ σ σ
Sobolev norm of the initial data and then show that the time of existence of the solutions depends
only on E (0). In §7 we prove our convergence result in terms of the energy E . In §8 we give an
σ ∆
example to illustrate all these results. The appendix §9 contains all the commonly used identities
and estimates used throughout this paper.
1More precisely, E0(t) is of lower order as compared with the energy functional in [17] by half weighted spacial
derivative.
2As was shown in [38], for data in E0(0)<∞, the solution of the gravity water wave equation exists on a time
interval[0,T∗)withT∗>0dependingonlyonE0(0).
WATER WAVES 3
Acknowledgment: This work is part of my Ph.D. thesis and I am very grateful to my advisor
Prof. Sijue Wu for proposing the problem and for her guidance during this project. I would also
like to thank Prof. Jeffrey Rauch for many helpful discussions.
2. Notation and Preliminaries
We will try to be as consistent as possible with the notation used in [17]. Most of this section
is essentially taken directly from [17] except for the new definitions and some modifications of the
formulae due to surface tension. The Fourier transform is defined as
1 (cid:90)
fˆ(ξ)= √ eixξf(x)dx
2π
In particular ∂(cid:100)xf(ξ) = −iξfˆ(ξ). We use this definition for the Fourier transform as we will be
concerned with holomorphic functions on the lower half plane instead of the upper half plane. We
will denote by D(R) the space of smooth functions with compact support in R and D(cid:48)(R) will be
the space of distributions. S(R) will denote the Schwartz space of rapidly decreasing functions and
S(cid:48)(R) is the space of tempered distributions. A Fourier multiplier with symbol a(ξ) is the operator
Ta defined formally by the relation T(cid:100)af = a(ξ)fˆ(ξ). The operators |∂α(cid:48)|s for s ∈ R are defined as
the Fourier multipliers with symbol |ξ|s. Note that the 1D Laplacian ∆=∂2 is a Fourier multiplier
x
with symbol −|ξ|2 and hence −∆ = |∂ |2. The Sobolev spaces Hs(R) for s ≥ 0 is the space of
α(cid:48)
functions with (cid:107)f(cid:107)Hs = (cid:107)(1+|ξ|)2sfˆ(cid:107)2 < ∞. The homogenous Sobolev space H˙ 21(R) is the space
of functions modulo constants with (cid:107)f(cid:107)H˙ 12 =(cid:107)|ξ|21fˆ(ξ)(cid:107)2 <∞.
Let the interface Σ(t) : z = z(α,t) ∈ C be given by a Lagrangian parametrization α satisfying
z (α,t) (cid:54)= 0 for all α ∈ R. Hence z (α,t) = v(z(α,t),t) is the velocity of the fluid on the interface
α t
and ztt(α,t)=(vt+(v.∇)v)(z(α,t),t) is the acceleration. As |zzαα|(α,t)=eiθ(α,t) and |z1α|∂α is the
arc length derivative in Lagrangian coordinates, the pressure can be rewritten as
1 z
P(z(α,t),t)=iσ ∂ α (α,t)
z α|z |
α α
Note that z1α∂α|zzαα| is purely imaginary. Hence the Euler equation becomes
∂P ∂P
z (α,t)+i=−nˆ (z(α,t))−tˆ (z(α,t))
tt ∂nˆ ∂tˆ
where
z
tˆ= α =eiθ =unit tangent vector
|z |
α
z
nˆ =i α =ieiθ =unit outward normal vector
|z |
α
Define
1 ∂P
a(α,t)=− (z(α,t)) ∈R
|z | ∂nˆ
α
Hence we get
z 1
z +i=iaz − α ∂ (P(z(α,t),t))
tt α |z ||z | α
α α
1 1 z
Therefore z −i=−iaz −iσ ∂ ∂ α (3)
tt α z αz α|z |
α α α
4 SIDDHANTAGRAWAL
Let Φ(.,t):Ω(t)→P be a Riemann map satisfying lim Φ (z,t)=1 and define
− z→∞ z
h(α,t)=Φ(z(α,t),t) (4)
hence h:R→R is a homeomorphism. Let h−1(α(cid:48),t) be its inverse i.e.
h(h−1(α(cid:48),t),t)=α(cid:48)
Note that for each fixed time t, a Riemann map Φ(·,t) : Ω(t) → P with the condition
−
lim Φ (z,t) = 1 is not determined uniquely. We have one degree of freedom namely that
z→∞ z
oftranslation. Tofixthis, weimposetheconditionthath (α,t)→0as|α|→∞forallt≥0. With
t
this, the only freedom left is that of the choice of translation of the Riemann map at t = 0, which
does not play any role in the analysis. From now on, we will fix our Lagrangian parametrization at
t=0 by imposing
h(α,0)=α for all α∈R
Hence the Lagrangian parametrization is the same as conformal parametrization at t=0.
From now on compositions of functions will always be in the spatial variables. We write f =
f(·,t),g =g(·,t),f ◦g(·,t):=f(g(·,t),t). Define the operator U as given by U f =f ◦g. Observe
g g
that U U = U . Let [A,B] := AB−BA be the commutator of the operators A and B. If A is
f g g◦f
an operator and f is a function, then (A+f) will represent the addition of the operators A and the
multiplication operator T where T (g)=fg. For functions f,g,h we define the function
f f
1 (cid:90) (cid:18)f(α(cid:48))−f(β(cid:48))(cid:19)(cid:18)g(α(cid:48))−g(β(cid:48))(cid:19)
[f,g;h](α(cid:48))= h(β(cid:48))dβ(cid:48)
iπ α(cid:48)−β(cid:48) α(cid:48)−β(cid:48)
Define the variables
z
Z(α(cid:48),t)=z◦h−1(α(cid:48),t) Z (α(cid:48),t)=∂ Z(α(cid:48),t) Hence ( α)◦h−1 =Z
,α(cid:48) α(cid:48) h ,α(cid:48)
zα
Z (α(cid:48),t)=z ◦h−1(α(cid:48),t) Z (α(cid:48),t)=∂ Z (α(cid:48),t) Hence ( tα)◦h−1 =Z
t t t,α(cid:48) α(cid:48) t h t,α(cid:48)
zα
Z (α(cid:48),t)=z ◦h−1(α(cid:48),t) Z (α(cid:48),t)=∂ Z (α(cid:48),t) Hence ( ttα)◦h−1 =Z
tt tt tt,α(cid:48) α(cid:48) tt h tt,α(cid:48)
α
Note that as Z(α(cid:48),t)=z(h−1(α(cid:48),t),t) we see that ∂ Z (cid:54)=Z . Similarly ∂ Z (cid:54)=Z . The substitute
t t t t tt
for the time derivative is the material derivative. Define the operators
D =material derivative=∂ +b∂ where b=h ◦h−1
t t α(cid:48) t
1 1 1
D = ∂ D = ∂ |D |= ∂
α(cid:48) Z,α(cid:48) α(cid:48) α(cid:48) Z,α(cid:48) α(cid:48) α(cid:48) |Z,α(cid:48)| α(cid:48)
H=Hilbert transform = Fourier multiplier with symbol sgn(ξ)
1 (cid:90) 1
Hf(α(cid:48))= p.v. f(β(cid:48))dβ(cid:48)
iπ α(cid:48)−β(cid:48)
I+H
P =Holomorphic projection=
H 2
I−H
P =Antiholomorphic projection=
A 2
√
|∂ |= −∆= Fourier multiplier with symbol |ξ| = iH∂
α(cid:48) α(cid:48)
|∂ |1/2 = Fourier multiplier with symbol |ξ|1/2
α(cid:48)
WATER WAVES 5
Now we have D Z = Z and D Z = Z and more generally D (f(·,t)◦h−1) = (∂ f(·,t))◦h−1 or
t t t t tt t t
equivalently ∂ (F(·,t)◦h)=(D F(·,t))◦h. This means that D =U−1∂ U i.e. D is the material
t t t h t h t
derivativeinRiemmaniancoordinates. Inthiswork,thematerialderivativeD ismoreheavilyused
t
as compared to the time derivative ∂ .
t
The Hilbert transform defined above satisfies the following property (see [33])
Lemma 2.1. Let 1 < p < ∞ and let F(z) be a holomorphic function in the lower half plane with
F(z)→0 as z →∞. Then the following are equivalent
(1) sup(cid:107)F(·+iy)(cid:107) <∞
p
y<0
(2) F(z)hasaboundaryvaluef,non-tangentiallyalmosteverywherewithf ∈Lp andH(f)=f.
Inparticularthissaysifvdecaysappropriatelyatinfinity,thenitsboundaryvalueZ willsatisfy
t
HZ =Z . We can now define the main variables used in this paper
t t
A=(ah )◦h−1
α
A ∂P
A =A|Z |2 Hence 1,σ =− ◦h−1
1,σ ,α(cid:48) |Z | ∂nˆ
,α(cid:48)
A =1−Im[Z ,H]Z
1 t t,α(cid:48)
Z Z
g =θ◦h−1 Hence ,α(cid:48) =eig and |D |g =(∂ θ)◦h−1 =−iD ,α(cid:48)
|Z | α(cid:48) s α(cid:48)|Z |
,α(cid:48) ,α(cid:48)
Z
Θ=(I+H)|D |g =−i(I+H)D ,α(cid:48)
α(cid:48) α(cid:48)|Z |
,α(cid:48)
Z
ω =eig = ,α(cid:48) Hence |D |ω =iωReΘ
|Z | α(cid:48)
,α(cid:48)
Observe that ReΘ = κ◦h−1 where κ is the curvature. With this notation, by precomposing (3)
with h−1 we get
A Z
Z −i=−i 1,σ −iσD D ,α(cid:48) (5)
tt Z α(cid:48) α(cid:48)|Z |
,α(cid:48) ,α(cid:48)
Let us now derive the formulae of A and b.
1,σ
2.1. Formula for A
1,σ
Let F =v and hence F is holomorphic in Ω(t) and z =F(z(α,t),t). Hence
t
z =F (z(α,t),t)+F (z(α,t),t)z (α,t) z =F (z(α,t),t)z (α,t)
tt t z t tα z α
z
Hence z =F ◦z+z tα
tt t t z
α
Z
Precomposing with h−1 we obtain Z = F ◦Z +Z t,α(cid:48). Now Multiply by Z and use (5) to
tt t t Z ,α(cid:48)
,α(cid:48)
get
Z
A =iZ F ◦Z+Z +iZ Z −σ∂ D ,α(cid:48)
1,σ ,α(cid:48) t ,α(cid:48) t t,α(cid:48) α(cid:48) α(cid:48)|Z |
,α(cid:48)
Apply (I−H) and use the fact that H(Z −1)=Z −1 and H1=0 to obtain
,α(cid:48) ,α(cid:48)
Z
(I−H)A =1+i[Z ,H]Z −σ(I−H)∂ D ,α(cid:48)
1,σ t t,α(cid:48) α(cid:48) α(cid:48)|Z |
,α(cid:48)
6 SIDDHANTAGRAWAL
Now take the real part
Z
A =1−Im[Z ,H]Z +σ∂ HD ,α(cid:48)
1,σ t t,α(cid:48) α(cid:48) α(cid:48)|Z |
,α(cid:48)
Hence
A1,σ =A1+σ∂α(cid:48)HDα(cid:48)|ZZ,α(cid:48)| and in particular A1,σ(cid:12)(cid:12)σ=0 =A1 (6)
,α(cid:48)
Using this relation above we get another formula for A
1
Z
A =iZ F ◦Z+Z +iZ Z −σ∂ (I+H)D ,α(cid:48) (7)
1 ,α(cid:48) t ,α(cid:48) t t,α(cid:48) α(cid:48) α(cid:48)|Z |
,α(cid:48)
Note that the only non-holomorphic quantity in the above formula is iZ Z . Also note that as
t t,α(cid:48)
A = 1−Im[Z ,H]Z by the calculation in [17, 39] we have that A ≥ 1. From (6) the Taylor
1 t t,α(cid:48) 1
sign condition term can be written as
−∂P ◦h−1 = A1,σ = 1 (cid:0)A +σ|∂ |(κ◦h−1)(cid:1) (8)
∂nˆ |Z | |Z | 1 α(cid:48)
,α(cid:48) ,α(cid:48)
where κ◦h−1 =−iD Z,α(cid:48) is the curvature in conformal coordinates. Hence we see that for large
α(cid:48)|Z,α(cid:48)|
values of σ, the Taylor sign condition may fail. For σ = 0, this formula was first derived by Wu
[35] to prove the Taylor sign condition for C1,α interfaces with α > 0 and was crucially used in
Kinsey-Wu [17] to prove apriori estimates for angled crest interfaces. We will also use this formula
in an essential way in this paper.
2.2. Formula for b
Recall that h(α,t)=Φ(z(α,t),t) and so by taking derivatives we get
h =Φ ◦z+(Φ ◦z)z h =(Φ ◦z)z
t t z t α z α
h
Hence h =Φ ◦z+ αz
t t z t
α
Z
Precomposing with h−1 we obtain h ◦h−1 =Φ ◦Z+ t . Apply (I−H) and take real part, to
t t Z
,α(cid:48)
get
(cid:18) (cid:19)
Z
b=Re(I−H) t (9)
Z
,α(cid:48)
By taking a derivative and rearranging we obtain the relation
(cid:16) 1 (cid:17)
b =(Φ ◦Z) +D Z +Z ∂ (10)
α(cid:48) t α(cid:48) α(cid:48) t t α(cid:48)Z
,α(cid:48)
This formula is the same as the one in [17] and surface tension does not affect the formula.
WATER WAVES 7
2.3. Fundamental Equation
Substituting the formula for A in equation (5), we get
1,σ
A Z Z
Z −i=−i 1 −iσD HD ,α(cid:48) −iσD D ,α(cid:48)
tt Z α(cid:48) α(cid:48)|Z | α(cid:48) α(cid:48)|Z |
,α(cid:48) ,α(cid:48) ,α(cid:48)
Now combine the second and third term and use Θ = −i(I+H)D Z,α(cid:48) to get the fundamental
α(cid:48)|Z,α(cid:48)|
equation
A
Z −i=−i 1 +σD Θ (11)
tt Z α(cid:48)
,α(cid:48)
Note that as A does not depend on σ, the effect of surface tension is that it adds a holomorphic
1
quantity to the conjugate of the acceleration. We also see that
A ∂P (cid:12)
1 =− ◦h−1(cid:12) ≥0
|Z,α(cid:48)| ∂nˆ (cid:12)σ=0
and hence it represents the Taylor sign condition in the absence of surface tension. As the equation
is written in terms of A and not A , our energy E will always be positive irrespective of the
1 1,σ σ
value of surface tension.
3. Discussion and Main results
3.1. Discussion.
Wenowgiveabriefheuristicexplanationintothenatureofourresults. Oneofourmainresultsis
Theorem3.1whichisanaprioriestimatefortheenergyE . Thisresultextendstheaprioriestimate
σ
of Kinsey-Wu [17] to include the effect of surface tension σ ≥ 0. The key ideas and difficulties in
proving this theorem are as follows:
1) Taylor sign condition:
In previous works on zero surface tension limit, the Taylor sign condition is assumed which
restrictsthemaximumvalueofsurfacetensionalloweddependingontheinitialdata. Wemanage
toremovethisrestrictionbyworkingwithanewquasilinearequation. In§4wederivequasilinear
equations in conformal coordinates of the form
(cid:40) A (cid:18) 1 (cid:19) (cid:18) 1 (cid:19)3 (cid:41)
D2+i 1 ∂ H−iσ ∂ H f =l.o.t (12)
t |Z | |Z | α(cid:48) |Z | α(cid:48)
,α(cid:48) ,α(cid:48) ,α(cid:48)
where H= Hilbert transform iH∂ =|∂ |
α(cid:48) α(cid:48)
1
D = material derivative ∂ = arc length derivative on the interface
t |Z | α(cid:48)
,α(cid:48)
Here f is a suitably chosen variable and the right hand side consists of lower order terms.
Observe that the coefficient of the second term on the left-hand side is −∇P ·nˆ| which is
σ=0
alwaysnon-negativeanddoesnotdependonσ. Thisallowsustoproveenergyestimateswithout
assuming the Taylor sign condition.
2) The energy estimate:
Let us do a heuristic energy estimate to understand the difficulties. If the interface is C1,α
then we have 0 < c ≤ 1 ≤ c < ∞ and hence for smooth enough interfaces the main
1 |Z,α(cid:48)| 2
operator in (12) behaves like ∂2+|∂ |+σ|∂ |3 for which standard energy estimates in Sobolev
t α(cid:48) α(cid:48)
8 SIDDHANTAGRAWAL
spaceswork. Iftheinterfacehasanangledcrestofangleνπ atα=0,thenZ(α)∼αν andhence
1 ∼|α|1−ν near α=0 and hence the quasilinear equation near α=0 behaves like
|Zα|
(cid:110) (cid:111)
∂2+|α|2−2ν|∂ |+σ|α|3−3ν|∂ |3 f
t α α
=|α|1−2νf +σ|α|2−3ν|∂ |2f +σ|α|1−3ν|∂ |f +σ|α|−3νf +other l.o.t
α α
We have included a few simplified versions of the lower order terms to demonstrate the issues
in proving an energy estimate. We obtain our energies by multiplying the above equation with
either ∂ f or |∂ |∂ f and then integrating. If we multiply the equation by ∂ f and integrate, we
t α t t
obtaincontrolof∂tf ∈L2,|α|1−νf ∈H˙ 12 andσ12|α|23−32ν|∂α|f ∈H˙ 12. Forsimplicityalsoassume
thatf ∈L2 whichiscompatiblewith∂ f ∈L2. Toclosetheenergyestimate,weneedtocontrol
t
theL2 normoftherighthandside. Hencetocontrolthefirstterm|α|1−2νf ∈L2 ,weobtainthe
restrictionν ≤ 1 whichisoneofthemainreasonsoftherestrictionsontheanglesinKinseyand
2
Wu [17]. Note that we cannot control the term σ|α|2−3ν|∂ |2f ∈ L2 as we only have control of
α
3/2 derivatives on f. For smooth enough interfaces, it was observed by Ambrose-Masmoudi [5]
thatbycarefullychoosingf (byusingvariablesderivedfromθ),thistermdoesnotappearinthe
quasilinear equation and we follow the same approach. We do not use the modified tangential
velocityasin[5]andinsteadusethemorenaturalmaterialderivativeD alongwiththevariable
t
θ to obtain our highest order quasilinear equation.
Now we need to control σ|α|1−3ν|∂ |f ∈ L2 and σ|α|−3νf ∈ L2. As we only have f ∈ L2,
α
thereisnowaywecancontrolthetermσ|α|−3νf ∈L2 andthisisthereasonwhywedonotallow
angled crest data if σ >0. Hence if we work with the smooth interface Z(cid:15) =Z∗P where P is
(cid:15) (cid:15)
thePoissonkernel,thenthishastheeffectofchanging|α|(cid:55)→|−i(cid:15)+α|nearα=0. Hencetoclose
the energy estimate, we obtain the restriction σ(cid:15)−3ν (cid:46) 1. For the interface Z(cid:15), the curvature
κ ∼ (cid:15)−ν and hence this can be written as σκ3 (cid:46) 1. A similar argument for σ|α|1−3ν|∂ |f ∈ L2
α
also yields the same restriction. Note that these restrictions do not depend on the choice of f,
but is purely a consequence of the structure of the quasilinear equation and attempting to prove
an L2 based energy estimate.
A key difficulty in implementing this heuristic energy estimate is to find a suitable f and
obtain a corresponding quasilinear equation. We need the following properties for f
(a) f needs to have appropriate weights so that the energy allows angled crests solutions when
σ =0
(b) The quasilinear equation for f should not have terms like σ|α|2−3ν|∂ |2f in the errors, to
α
be able to close the energy estimate.
These are rather severe restrictions on f and in particular if f has such properties, then no
weighted derivate of the form w∂ f will satisfy the same properties (Kinsey-Wu [17] used the
α
weightedderivative 1 ∂ andAmbrose-Masmoudi[5]usedthearclengthderivative 1 ∂
|Z,α(cid:48)|2 α(cid:48) |Z,α(cid:48)| α(cid:48)
to derive the higher order energies). Fortunately the material derivative D f will satisfy both
t
the properties and hence we use the material derivate quite heavily in this paper. The highest
order energy in E corresponds to an f which is related to the material derivative of the angle.
σ
3) Lower order terms:
It is very tempting to think that the quasilinear equation is of the form
 
(cid:32) (cid:33) (cid:32) (cid:33)3
 1 (cid:16) (cid:17)3 1 
Dt2+iA1 |Z,α(cid:48)|2∂α(cid:48) H−i σ13|Z,α(cid:48)| |Z,α(cid:48)|2∂α(cid:48) Hf =l.o.t
WATER WAVES 9
1
Wecanthenthinkof ∂ asthemainweightedderivate(aswasdoneinKinsey-Wu[17])
|Z |2 α(cid:48)
,α(cid:48)
and assume σ13|Z,α(cid:48)| ∈ L∞. Indeed adding the term (cid:13)(cid:13)σ13Z,α(cid:48)(cid:13)(cid:13)∞ in the energy Eσ considerably
simplifies the proof of the apriori estimate Theorem 3.1 and the proof of the convergence result
Theorem 3.8. However this has the drawback of yielding a worse result in terms of the scaling,
σ
as instead of the restriction ≤ 1 in Proposition 3.10, we would need the more restrictive
σ (cid:15)32 σ
condition of ≤1. The restriction ≤1 is more natural as this corresponds to letting ν ↑ 1
(cid:15)3 (cid:15)32 2
in σκ3 ∼σ(cid:15)−3ν (cid:46)1 which we saw in the heuristic energy estimate. Our energy E has no lower
σ
order terms with respect to the scaling which keeps gravity constant (See Theorem 3.1)
4) Analytical difficulties:
In addition to the structural difficulties due to surface tension explained above, we also face
numerous analytical difficulties. Even in the special case of σ = 0, the energy E | is lower
σ σ=0
order as compared to the energy in Kinsey-Wu [17] by half weighted spacial derivative and we
crucially do not have D Z ∈ L∞. This makes our estimates even in the case of σ = 0 much
α(cid:48) tt
more subtle. In addition we now have a lot of nonlinear terms due to surface tension which we
need to control. To overcome these issues, we define weighted function spaces adapted to our
problem and prove estimates for these spaces (see Lemma 5.1). We use these function spaces
along with estimates from harmonic analysis to control the nonlinear terms.
3.2. Results.
We now describe our results. Define
Eσ,0 =(cid:13)(cid:13)(cid:13)(cid:13)σ12|Z,α(cid:48)|12∂α(cid:48)Z1,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)2∞+(cid:13)(cid:13)(cid:13)(cid:13)σ16|Z,α(cid:48)|12∂α(cid:48)Z1,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)62+(cid:13)(cid:13)(cid:13)(cid:13)∂α(cid:48)Z1,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)22+(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)|Zσ,α21(cid:48)|12∂α2(cid:48)Z1,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)22
(cid:13) (cid:13)2
Eσ,1 =(cid:13)(cid:13)(Ztt−i)Z,α(cid:48)(cid:13)(cid:13)2H˙ 12 +(cid:13)(cid:13)(cid:13)(cid:112)A1Zt,α(cid:48)(cid:13)(cid:13)(cid:13)22+(cid:13)(cid:13)(cid:13)(cid:13)|Zσ,α12(cid:48)|12∂α(cid:48)Zt,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)2
Eσ,2 =(cid:13)(cid:13)DtZt,α(cid:48)(cid:13)(cid:13)22+(cid:13)(cid:13)(cid:13)(cid:13)(cid:112)A1|ZZt,,αα(cid:48)(cid:48)|(cid:13)(cid:13)(cid:13)(cid:13)2H˙ 12 +(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)|Zσ,α12(cid:48)|32∂α(cid:48)Zt,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2H˙ 12
Eσ,3 =(cid:107)DtΘ(cid:107)22+(cid:13)(cid:13)(cid:13)(cid:13)(cid:112)A1|ZΘ,α(cid:48)|(cid:13)(cid:13)(cid:13)(cid:13)2H˙ 12 +(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)|Zσ,α21(cid:48)|32∂α(cid:48)Θ(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2H˙ 21
(cid:13) (cid:13)2
Eσ,4 =(cid:13)(cid:13)DtDα(cid:48)Zt(cid:13)(cid:13)2H˙ 21 +(cid:13)(cid:13)(cid:13)(cid:112)A1|Dα(cid:48)|Dα(cid:48)Zt(cid:13)(cid:13)(cid:13)22+(cid:13)(cid:13)(cid:13)(cid:13)|Zσ,α21(cid:48)|12∂α(cid:48)|Dα(cid:48)|Dα(cid:48)Zt(cid:13)(cid:13)(cid:13)(cid:13)2
E =E +E +E +E +E
σ σ,0 σ,1 σ,2 σ,3 σ,4
Observe that the variables used above are all very natural. Z and Z are the velocity and
t tt
acceleration on the boundary respectively, Θ is twice the holomorphic projection of the curvature
andD Z isrelatedtothematerialderivativeoftheanglebytherelationIm(cid:0)D Z (cid:1)=−(∂ θ)◦h−1
α(cid:48) t α(cid:48) t t
from (14). The weight 1 is related to the Taylor sign condition from (8).
|Z,α(cid:48)|
TheenergiesE for1≤i≤4areobtainedfromthequasilinearequationsderivedin§4whereas
σ,i
theenergyE isaddedasalowerorderterm. Forσ =0,theenergiesE andE areequivalent,
σ,0 σ,3 σ,4
but for σ >0 the energy E is higher order than E . We now state our main result.
σ,4 σ,3
10 SIDDHANTAGRAWAL
Theorem 3.1. Let T >0 and let (Z,Z ) be a smooth solution 3 to the capillary gravity water wave
t
equation with surface tension σ ≥ 0 in the time interval [0,T) with E (t) < ∞ for all t ∈ [0,T).
σ
Then there exists a polynomial P with universal non-negative coefficients such that for all t∈[0,T)
we have
dE (t)
σ ≤P(E (t))
dt σ
Remark 3.2. Wementionaminortechnicalpointinthestatementofthetheorem. TheenergyE
σ,0
contains a term which is the L∞ norm of a function and hence may not in general be differentiable
in time even for smooth solutions. Hence for this term we replace the time derivative by the upper
Dini derivative limsup (cid:107)f(t+s)(cid:107)∞−(cid:107)f(t)(cid:107)∞
s→0+ s
Properties:
(1) Energyispositveforallσ : Foranysmoothdatadecayingsufficientlyatinfinity,theenergy
is positive and finite for all σ ≥ 0. In particular the energy is positive even if the Taylor
sign condition fails which may happen for large values of σ as may be observed from the
formula of A . This is in contrast to the work in [5], where the authors assume σ small to
1,σ
obtain a positive energy.
(2) Energy estimate is completely independent of σ: In the above theorem the polynomial P
has universal coefficients and hence is independent of σ
(3) Energy allows angled crest solutions for σ = 0: If we put σ = 0 in the energy E , then it
σ
allows solutions with angled crest with angle less than 90◦. These are exactly the solutions
allowed by the energy obtained by Kinsey and Wu in [17] and our energy is equivalent to
theirs. (More precisely the above energy for σ =0 is lower order as compared to the energy
in [17] by half spacial derivative)
(4) Energy does not allow angled crest solution for σ > 0: In the proof of this theorem we
show the estimate (cid:13)(cid:13)∂sθ(cid:13)(cid:13)∞ ≤σ−13P(Eσ) and hence for σ >0 the curvature is finite, which
automatically excludes angled crest solutions. Note however that for small values of σ, the
energy allows data with quite large curvature of the order of σ−31. (See Proposition 3.10
where for any given (cid:15) > 0 arbitrarily small, we construct examples where E = O(1) and
σ
the curvature of the initial data gro√ws like σ−13+(cid:15) as σ →0) √
(5) Scaling: Let Zλ(α(cid:48),t)=λ−1Z(λα(cid:48), λt) and Ztλ =λ−21Zt(λα(cid:48), λt). Then as can be seen
directlyfromtheequation,(Z ,Z )isasolutiontothewaterwaveequationwiththesame
λ tλ
gravity −k but surface tension σλ−2. The energy E has the following property: If (E )
σ σ λ
represents the energy corresponding to the data (Z ,Z ), then we have (E ) ≤ E for
λ tλ σ λ σ
λ ≤ 1 and (E ) ≥ E for λ ≥ 1. This says that there are no lower order terms in the
σ λ σ
energy with respect to this scaling.
We now give a more convenient expression for the energy and its relation to the Sobolev norm
of the data. Define
Eσ,1 =(cid:13)(cid:13)(cid:13)(cid:13)∂α(cid:48)Z1,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)22+(cid:13)(cid:13)(cid:13)(cid:13)Z1,α(cid:48)∂α(cid:48)Z1,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)2H˙12 +(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)σ61Z,12α(cid:48)∂α(cid:48)Z1,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)62+(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)σ12Z,21α(cid:48)∂α(cid:48)Z1,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2∞+(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Zσ,12α12(cid:48)∂α2(cid:48)Z1,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)22
+(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Zσ,32α12(cid:48)∂α2(cid:48)Z1,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2H˙ 12 +(cid:107)σ∂α(cid:48)Θ(cid:107)2H˙ 12 +(cid:13)(cid:13)(cid:13)(cid:13)Zσ,α(cid:48)∂α3(cid:48)Z1,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)22+(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Zσ,2α(cid:48)∂α3(cid:48)Z1,α(cid:48)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2H˙ 12
3Itisenoughtoassumethatthesolutionsatisfies(Z,α(cid:48)−1,Zt)∈C([0,T],Hs+12(R)×Hs(R))fors≥10