Abstract We address the local existence of solutions of the 2D and 3D water waveproblems. For the space dimension three, we consider the irrotational datum u0 andprove that the local in time existence holds for initial velocities belonging to H 2.5+δ ,where δ > 0 is arbitrary. For the space dimension two, the data does not need tobe irrotational. We prove the local in time existence when u0 belongs to H 2+δ andcurlu0 to H 1.5+δ , where δ > 0 is arbitrary.
Keywords Water wawes · Euler equation · Free surface
1 Introduction
In this paper, we address the local existence of solutions of the free-surface Eulerequations
ut + u · ∇u + ∇p = 0 in Ω(t) × (0, T ) (1.1)
divu = 0 in Ω(t) × (0, T ). (1.2)
The free boundary Γf = Γf(t) evolves according to the velocity field, while the bound-ary condition for the pressure reads
p(x, t) = εσ (x, t) on ∂Γf × (0, T ) (1.3)
where σ represents the surface tension.
I. KukavicaDepartment of Mathematics, University of Southern California, Los Angeles, CA 90089, USAe-mail: [email protected]
A. Tuffaha (B)Department of Mathematics, The Petroleum Institute, Abu Dhabi, UAEe-mail: [email protected]
The two dimensional problem was initially addressed by Nalimov [19], who es-tablished the existence and uniqueness for the problem with infinite depth provideda certain Sobolev norm is sufficiently small. On the other hand, Shinbrot establishedin [23] the local existence for analytic data. The references [7] and [30, 31] containfurther work—in particular, in [30] Yosihara established the local existence for theirrotational flow with finite depth, again with initial data with a sufficiently smallSobolev norm. Beale, Hou, and Lowengrub proved in [6] the local existence for thelinearized equation with an added Taylor stability condition ∇p · n ≤ 0 on Γf (0).That this condition is indeed necessary was shown by Ebin in [11].
Later, in [27], Wu proved local existence for 2D flows under the Taylor sign con-dition; this condition holds automatically in the irrotational case due to the maximumprinciple for the Laplace equation. The same problem in the 3D case was solved soonafter by Wu in [28]. The case with the surface tension was established by Ambroseand Masmoudi [3, 4]. See also [1, 2, 12, 16–18, 20, 21, 24, 29, 32] for other importantresults on the local existence problem; on the other hand, the viscous problem wasaddressed in [5, 25].
It is an important question what the best Sobolev space is in which the local ex-istence and regularity can be established. In [8, 22], the local well-posedness wasestablished for the initial velocity in the space H 3 under the Taylor condition. In[15], the local existence was proved for the initial velocity in H 2.5+δ , where δ > 0 intwo space dimensions with the Taylor condition.
It can be argued that Hn/2+1+δ is the optimal space which would permit localexistence. Since in the Lagrangian formulation of the problem (cf. (2.4)–(2.10) be-low), the ellipticity coefficients for pressure equation (4.3) contain a (defined as theinverse of the gradient of the Lagrangian displacement), it seems necessary that a
be bounded, i.e., at least in Hn/2+δ . However, since the Euler equations posses nosmoothing mechanism and since a is by one degree rougher than v, we expect, in thebest case, the local existence in Hn/2+1+δ for δ > 0.
For irrotational flows, this is indeed true, and this result was proven by Alazard,Burq, and Zuily in [2] for two and three dimensional cases. The proof was done bywriting the velocity as a gradient of a potential (using irrotationality) and writing thesystem for the surface displacement and the restriction of the velocity on the freesurface (the variables used in [16, 27, 28]).
In the main result of this paper, Theorem 2.2, addressing the 2D case, we lowerthe best known Sobolev exponent in the 2D case for data which are allowed to berotational. Namely, we prove that the local existence holds when the initial velocityu0 belongs to H 2+δ while the initial vorticity lies in H 1.5+δ . Also, in 3D (cf. Theo-rem 2.1), we provide a different proof (one which seems more amenable for the caseof bounded domain) of the local existence for irrotational u0 belonging to H 2.5+δ .The proofs of both theorems are elementary and concise and are inspired by [8–10].For simplicity of presentation, we address the case when the initial boundary free-surface is flat. The general case can be handled similarly by using partition of unity(and the construction by horizontal layers introduced by Coutand and Shkoller in [8]).
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2 The Notation and the Main Result
Let n ∈ {2,3} denote the space dimension. We consider the Euler equation on thedomain
Ω = Rn−1 × (0,1) (2.1)
with periodic boundary conditions along the x1, . . . , xn−1 directions with period 1.The general case is more technical but can then be addressed using a partition ofunity. We assume that the moving boundary is the top
Γ1 = R× {xn = 1} (2.2)
while the rigid bottom is
Γ0 = R× {xn = 0}. (2.3)
Denote by v(x, t) = (v1, . . . , vn) the Lagrangian velocity of the fluid and by q(x, t)
the Lagrangian pressure. The Euler equation in the Lagrangian formulation reads
vit + ak
i ∂kq = 0 in Ω × (0, T ), i = 1, . . . , n (2.4)
aki ∂kv
i = 0 in Ω × (0, T ) (2.5)
where we use the summation convention throughout. The unknown coefficients aij
denote the ij entry of the 2 × 2 matrix
a = (∇η)−1 (2.6)
where η stands for the particle map
ηt (x, t) = v(x, t) (2.7)
η(x,0) = x, x ∈ Ω. (2.8)
The Lagrangian map in turn determines the evolving domain Ω(t) = η(Ω, t).On the top, we assume the no surface tension boundary condition
q = 0 on Γ1 × (0, T ) (2.9)
while on the stationary bottom we use
viNi = 0 on Γ0 × (0, T ); (2.10)
where the vector N = (N1, . . . ,Nn) represents the outward unit normal. In our sim-plified situation, we have N = (0, . . . ,0,−1) on Γ0 and N = (0, . . . ,0,1) on Γ1.Also, denote H = {v ∈ L2(Ω)n : ∂iv
i = 0 in Ω,viNi |Γ0 = 0}.The following is the main result in the three-dimensional case.
Appl Math Optim
Theorem 2.1 Let n = 3, let Ω be as above, and let δ ∈ (0,0.5). Assume that v(·,0) =v0 ∈ H 2.5+δ(Ω) ∩ H satisfies
curlv0 = 0 (2.11)
and that the associated initial pressure q(·,0) satisfies the Rayleigh-Taylor condition
∂q
∂N(x,0) ≤ − 1
C0< 0 (2.12)
for all x ∈ Γ1. Then there exists a solution (v, q, η) to the system (2.4)–(2.5) and(2.7) with the boundary conditions (2.9)–(2.10) and the initial conditions (2.8) andv(0) = v0 such that
η ∈ L∞([0, T ];H 3+δ(Ω)) ∩ C
([0, T ];H 2.5+δ(Ω))
v ∈ L∞([0, T ];H 2.5+δ(Ω)) ∩ C
([0, T ];H 2+δ(Ω))
vt ∈ L∞([0, T ];H 2+δ(Ω))
q ∈ L∞([0, T ];H 3+δ(Ω))
qt ∈ L∞([0, T ];H 2.5+δ(Ω))
(2.13)
for some time T > 0 depending on the initial data.
In the two-dimensional case, we do not need to assume that the flow is irrotational.
Theorem 2.2 Let n = 2, let Ω be as above, and let δ ∈ (0,0.5). Assume that v(·,0) =v0 ∈ H 2+δ(Ω) ∩ H satisfies
curlv0 ∈ H 1.5+δ(Ω) (2.14)
and that the associated initial pressure q(·,0) satisfies the Rayleigh-Taylor condition(2.12) for all x ∈ Γ1. Then, there exists a solution (v, q, η) to the system (2.4)–(2.5)and (2.7) with the boundary conditions (2.9)–(2.10) and the initial conditions (2.8)and v(0) = v0 such that
η ∈ L∞([0, T ];H 2.5+δ(Ω)) ∩ C
([0, T ];H 2+δ(Ω))
v ∈ L∞([0, T ];H 2+δ(Ω)) ∩ C
([0, T ];H 1.5+δ(Ω))
vt ∈ L∞([0, T ];H 1.5+δ(Ω))
q ∈ L∞([0, T ];H 2.5+δ(Ω))
qt ∈ L∞([0, T ];H 2+δ(Ω))
(2.15)
for some time T > 0 depending on the initial data.
We first present the proof of Theorem 2.1 by means of a priori estimates, which inturn can be made rigorous by the horizontal mollification procedure due to Coutand
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and Shkoller [8, 9]. Then we indicate the necessary changes to obtain the Theo-rem 2.2.
3 A Preliminary Lemma
The first lemma provides preliminary estimates on the coefficient matrix a and theparticle map η.
Lemma 3.1 Assume that ‖∇v‖L∞([0,T ];H 1.5+δ(Ω)) ≤ M . If
T ≤ 1
CM(3.1)
where C is a sufficiently large constant, the following statements hold:
(i) ‖∇η(·, t)‖H 1.5+δ(Ω) ≤ C for t ∈ [0, T ],(ii) det(∇η(x, t)) = 1 for (x, t) ∈ Ω × [0, T ],
(iii) ‖a(·, t)‖H 1.5+δ(Ω) ≤ C (and thus also ‖a(·, t)‖L∞(Ω) ≤ C) for t ∈ [0, T ],(iv) ‖at (·, t)‖Lp(Ω) ≤ C‖∇v(·, t)‖Lp(Ω) for p ∈ [1,∞] and t ∈ [0, T ],(v) ‖at (·, t)‖Hr(Ω) ≤ C‖∇v(·, t)‖Hr(Ω) for r ∈ [0,1.5 + δ) and t ∈ [0, T ],
(vi) ‖att (·, t)‖Hσ (Ω) ≤ C‖∇v(·, t)‖H 1.5+δ(Ω)‖∇v(·, t)‖Hσ (Ω) +C‖∇vt (·, t)‖Hσ (Ω),for t ∈ [0, T ] and all 0 < σ ≤ 1.5 + δ, and
(vii) for every ε ∈ (0,1] there exists a constant C > 0 such that for all t ∈ [0, T ′],where T ′ = min{ε/CM,T } > 0, we have
∥∥ajl − δjl
∥∥H 1.5+δ(Ω)
≤ ε (3.2)
for j, l = 1,2 and
∥∥ajl ak
l − δjk
∥∥H 1.5+δ(Ω)
≤ ε (3.3)
for j, k = 1,2.
Proof of Lemma 3.1 (i) Abbreviating η(t) = η(·, t), we have
∇η(t) = I +∫ t
0∇v(s) ds (3.4)
whence ‖∇η‖H 1.5+δ ≤ I + tM .(ii) Holds since u is divergence-free.(iii) Using
at = −a : ∇v : a (3.5)
where the symbol : denotes the matrix multiplication, we have a(t) = I +∫ t
0 at (s) ds = I − ∫ t
0 (a : ∇v : a)(s) ds whence
∥∥a(t)∥∥
H 1.5+δ ≤ C + M
∫ t
0
∥∥a(s)∥∥2
H 1.5+δ ds (3.6)
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and the claim follows by solving the Gronwall inequality.(iv) and (v) follow from (3.5) and (iii).(vi) Differentiating (3.5) leads to
att = 2a : ∇v : a : ∇v : a − a : ∇vt : a (3.7)
and (vi) follows by using the multiplicative Sobolev inequality.(vii) The inequality (3.2) is a consequence of a(t) − I = ∫ t
0 at (s) ds and (v). Onthe other hand, (3.3) follows from
(a
jl ak
l
)(t) − δjk =
∫ t
0
(∂ta
jl ak
l + ajl ∂ta
kl
)(s) ds (3.8)
combined with (iii) and (v). �
4 The Pressure Estimates
In the following lemma, we establish elliptic estimates satisfied by the pressure. Fromhere on, the dependence of norms on the domain Ω is omitted. Thus we write ‖u‖L2
and ‖u‖Hs for ‖u‖L2(Ω) and ‖u‖Hs(Ω) respectively.
Lemma 4.1 Assume that (v, q, a) satisfies the Euler equation (2.4)–(2.5) in Ω ×[0, T ) and that we have ‖∇v‖L∞([0,T ];H 1.5+δ(Ω)) ≤ M . Assume that a satisfies theestimates in Lemma 3.1 for a sufficiently small constant ε > 0. Then the pressure q
satisfies
∥∥q(t)∥∥
H 3+δ ≤ P + P
∫ t
0
∥∥qt (s)∥∥
H 2+δ ds, t ∈ [0, T ] (4.1)
where P is a polynomial in ‖v‖H 2.5+δ , ‖η‖H 3+δ , and ‖v0‖H 2.5+δ , and
∥∥qt (t)∥∥
H 2.5+δ ≤ P + P
∫ t
0
∥∥qt (s)∥∥
H 2+δ ds, t ∈ [0, T ] (4.2)
where P is a polynomial in ‖v‖H 2.5+δ , ‖vt‖H 2+δ , ‖q‖H 3+δ , ‖η‖H 3+δ , and ‖v0‖H 2.5+δ .
Proof of Lemma 4.1 Applying aji ∂j to the Euler equation (2.4), summing over i, j =
1,2,3, we get
aji ∂j
(aki ∂kq
) = −aji ∂j v
it = ∂ta
ji ∂j v
i (4.3)
where we used the time differentiated divergence condition (2.5). We may rewritethis as
∂kkq = ∂taji ∂j v
i + ∂j
((δjk − a
ji ak
i
)∂kq
)(4.4)
where we used the Piola identity
∂j aji = 0, i = 1,2,3. (4.5)
Appl Math Optim
This equation is supplemented with the boundary conditions (2.9) on Γ1 and
aki ∂kqNi = 0 on Γ0 × (0, T ). (4.6)
The last condition may be rewritten as
∂iqNi = (δik − ak
i
)∂kqNi on Γ0 × (0, T ). (4.7)
Applying the elliptic regularity to Eq. (4.4) with the boundary conditions (2.9) and(4.7), we get
‖q‖H 3+δ ≤ C∥∥∂ta
ji ∂j v
i∥∥
H 1+δ + C∑
j
∥∥(δjk − a
ji ak
i
)∂kq
∥∥H 2+δ
+ C∥∥(
δik − aki
)∂kqNi
∥∥H 1.5+δ(Γ )
≤ C‖at‖H 1+δ‖∇v‖H 1.5+δ + C∥∥I − a : aT
∥∥L∞‖∇q‖H 2+δ
+ C∥∥I − a : aT
∥∥
H 2+δ‖∇q‖L∞ + C‖I − a‖L∞‖∇q‖H 2+δ
+ C‖I − a‖H 2+δ‖∇q‖H 1.5+δ . (4.8)
Now, with T as in Lemma 3.1 and with C in (3.1) sufficiently large, we have‖I − a‖L∞,‖I − a : aT ‖L∞ ≤ ε with
‖a‖H 2+δ ≤ C‖η‖2H 3+δ . (4.9)
Therefore,
‖q‖H 3+δ ≤ C‖v‖H 2+δ‖v‖H 2.5+δ + Cε‖q‖H 3+δ
+ C(1 + ‖a‖H 2+δ + ‖a‖2
H 2+δ
)‖q‖H 2.5+δ
≤ C‖v‖2H 2.5+δ + Cε‖q‖H 3+δ + C
(1 + ‖η‖4
H 3+δ
)‖q‖1/2H 2+δ‖q‖1/2
H 3+δ . (4.10)
Choosing ε > 0 sufficiently small, we obtain
‖q‖H 3+δ ≤ C‖v‖2H 2.5+δ + C
(1 + ‖η‖8
H 3+δ
)‖q‖H 2+δ
≤ C‖v‖2H 2.5+δ + C
(1 + ‖η‖8
H 3+δ
)∫ t
0
∥∥qt (s)∥∥
H 2+δ ds
+ C(1 + ‖η‖8
H 3+δ
)∥∥qt (0)∥∥
H 2+δ (4.11)
where we also used T ≤ 1. The estimate (4.1) is thus established.Now, differentiating Eq. (4.4) in time, we get
∂kkqt = ∂tt aji ∂j v
i + ∂taji ∂j v
it − ∂j
(∂ta
ji ak
i ∂kq) − ∂j
(a
ji ∂ta
ki ∂kq
)
+ ∂j
((δjk − a
ji ak
i
)∂kqt
)(4.12)
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with the boundary conditions
qt = 0 on Γ1 (4.13)
and
∂iqtNi = −∂ta
ki ∂kqNi + (
δik − aki
)∂kqtN
i on Γ0. (4.14)
Using ellipticity, we get
‖qt‖H 2.5+δ ≤ C∥∥∂tt a
ji ∂j v
i∥∥
H 0.5+δ + C∥∥∂ta
ji ∂j v
it
∥∥H 0.5+δ + C
∑
j
∥∥∂taji ak
i ∂kq∥∥
H 1.5+δ
+ C∑
j
∥∥aji ∂ta
ki ∂kq
∥∥H 1.5+δ + C
∑
j
∥∥(δjk − a
ji ak
i
)∂kqt
∥∥H 1.5+δ
+ C∥∥∂ta
ki ∂kqNi
∥∥H 1+δ(Γ0)
+ C∥∥(
δik − aki
)∂kqtN
i∥∥
H 1+δ(Γ0). (4.15)
By the multiplicative Sobolev inequalities, we get
where P and Q are polynomials in indicated arguments.
Henceforth, unless the arguments are explicitly written, denote by P a genericpositive polynomial depending on ‖v‖H 2.5+δ , ‖vt‖H 2+δ , ‖q‖H 3+δ , ‖qt‖H 2.5+δ , and‖η‖H 3+δ .
Proof of Lemma 6.1 Applying the differential operator S to (2.4), we obtain
Svit + S
(aki ∂kq
) = 0. (6.3)
We then multiply by Svi , integrate, and sum over i = 1,2,3 in order to obtain
1
2
d
dt‖Sv‖2
L2 = −∫
Ω
S(aki ∂kq
)Svi dx
= −∫
Ω
Saki ∂kqSvi dx −
∫
Ω
aki ∂kSqSvi dx
−∫
Ω
(S(aki ∂kq
) − Saki ∂kq − ak
i ∂kSq)Svi dx
= I1 + I2 + I3. (6.4)
We now estimate each of the terms I1, I2, and I3.
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First, we write
S =2∑
m=1
Sm∂m + S0 (6.5)
where Sm = −(−2)0.25+δ/2∂m for m = 1,2 and S0 = (I − 2)
0.25+δ/2. Using
∂maki = −ak
l ∂s∂mηlasi , m = 1,2 (6.6)
which follows by differentiating a : ∇η = I , we may rewrite the term I1 =− ∫
ΩSak
i ∂kqSvi dx as
I1 = −2∑
m=1
∫
Ω
Sm∂maki ∂kqSvi dx −
∫
Ω
S0aki ∂kqSvi dx
=2∑
m=1
∫
Ω
Sm
(akl ∂s∂mηlas
i
)∂kqSvi dx −
∫
Ω
S0aki ∂kqSvi dx
=2∑
m=1
∫
Ω
akl Sm∂s∂mηlas
i ∂kqSvi dx
+2∑
m=1
∫
Ω
(Sm
(akl ∂s∂mηlas
i
) − akl Sm∂s∂mηlas
i
)∂kqSvi dx
−∫
Ω
S0aki ∂kqSvi dx
= I11 + I12 + I13. (6.7)
We now integrate by parts in the variable xs in order to obtain
I11 =∫
Ω
akl S∂sη
lasi ∂kqSvi dx −
∫
Ω
akl S0∂sη
lasi ∂kqSvi dx
= −∫
Ω
∂sakl Sηlas
i ∂kqSvi dx −∫
Ω
akl Sηlas
i ∂s∂kqSvi dx
−∫
Ω
akl Sηlas
i ∂kq∂sSvi dx +∫
Γ1
akl Sηlas
i ∂kqSviNs dσ (x)
+∫
Γ0
akl Sηlas
i ∂kqSviNs dσ (x) −∫
Ω
akl S0∂sη
lasi ∂kqSvi dx
= I111 + I112 + I113 + I114 + I116 (6.8)
where we used (4.5). By Hölder’s inequality, we have
I111 ≤ C∥∥∂sa
kl
∥∥L6
∥∥Sηl∥∥
L3
∥∥asi
∥∥L∞‖∂kq‖L∞
∥∥Svi∥∥
L2 . (6.9)
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Using the Gagliardo-Nirenberg inequality and (4.9), the term I111 is bounded by P ,which stands for a generic positive polynomial in the arguments ‖v‖H 2.5+δ , ‖vt‖H 2+δ ,‖q‖H 3+δ , ‖qt‖H 2.5+δ , and ‖η‖H 3+δ as in Sect. 4. The term I112 is treated similarlyusing Hölder’s inequality as
I112 ≤ C∥∥ak
l
∥∥L∞
∥∥Sηl∥∥
L3
∥∥asi
∥∥L∞‖∂s∂kq‖L6
∥∥Svi∥∥
L2 (6.10)
which is also bounded by P . In order to estimate I113 = − ∫Ω
akl Sηlas
i ∂kqS∂svi dx,
we may rewrite it as
I113 = −∫
Ω
akl SηlSas
i ∂kq∂svi dx
+∫
Ω
akl Sηl∂kq
(S(asi ∂sv
i) − Sas
i ∂svi − as
i S∂svi)dx (6.11)
since asi ∂sv
i = 0. The first term on the far right side of (6.11) is rewritten as
− ∫Ω
∂1/2
(akl Sηl∂kq∂sv
i)∂2+δ
asi dx and is bounded by P using (4.9) and the com-
mutator estimate (5.4). The second term on the right side of (6.11) is bounded by P
using the commutator inequality (5.3). Now, since ∂mq = 0 on Γ1 for m = 1,2, thefirst boundary term I114 equals
I114 =∫
Γ1
a3l Sηla3
i ∂3qSvi dσ (x) =∫
Γ1
a3l Sηla3
i ∂3qS∂tηi dσ (x)
= 1
2
d
dt
∫
Γ1
a3l Sηla3
i Sηi∂3q dσ(x) −∫
Γ1
∂ta3l Sηla3
i Sηi∂3q dσ(x)
− 1
2
∫
Γ1
a3l Sηla3
i Sηi∂3qt dσ (x). (6.12)
Invoking the Taylor condition (2.12) which guarantees
∂3q ≤ − 1
C0< 0 on Γ1 (6.13)
we get∫ t
0I114 ds ≤ − 1
2C0
∫
Γ1
a3l Sηla3
i Sηi dσ (x)
∣∣∣∣t
−∫
Γ1
a3l Sηla3
i Sηi∂3q dσ(x)
∣∣∣∣0
−∫ t
0
∫
Γ1
∂ta3l Sηla3
i Sηi∂3q dσ(x)ds
− 1
2
∫ t
0
∫
Γ1
a3l Sηla3
i Sηi∂3t q dσ (x) ds (6.14)
which leads to∫ t
0I114 ds ≤ − 1
2C0
∥∥a3l (t)Sηl(t)
∥∥2L2(Γ1)
+∫ t
0P ds + Q (6.15)
Appl Math Optim
where Q denotes a generic positive polynomial depending on ‖v0‖H 2.5+δ . As for thesecond boundary term, we have I115 = ∫
Γ0akl Sηla3
i ∂kqSvi dσ (x). This term vanishes
since a3m = 0 on Γ0 for m = 1,2 due to ∂mη3 = 0 on Γ0 for m = 1,2 whence a3
i Svi =a3
3Sv3 = 0 on Γ0 by (2.10). Thus the treatment of I11 is complete. For I116, we useI116 ≤ C‖ak
l ‖L6‖S0∂sηl‖L3‖as
i ‖L∞‖∂kq‖L∞‖Svi‖L2 which is bounded by P .For I12, we use (5.1) in order to estimate
I12 ≤2∑
m=1
∥∥Sm
(akl ∂msη
lasi
) − akl Sm∂msη
lasi
∥∥L2‖∂kq‖L∞
∥∥Svi∥∥
L2
≤ C(∥∥ak
l asi
∥∥W 1,6
∥∥∂msηl∥∥
W 0.5+δ,3 + ∥∥akl a
si
∥∥W 1.5+δ,3
∥∥∂msηl∥∥
L6
)‖∂kq‖L∞∥∥Svi
∥∥L2
≤ P. (6.16)
Finally, for I13, we write
I13 ≤ ∥∥S0aki
∥∥H 1‖∂kq‖H 1/2
∥∥Svi∥∥
L2 ≤ P. (6.17)
Summing up all the resulting inequalities, we obtain∫ t
0I1 ds ≤ − 1
2C0
∥∥a3l (t)Sηl(t)
∥∥2L2(Γ1)
+∫ t
0P ds + Q. (6.18)
For I2 = ∫Ω
aki ∂kSqSvi dx, we integrate by parts in xk in order to obtain
I2 =∫
Ω
aki Sq∂kSvi dx −
∫
Γ0
aki SqSviNk dσ(x) (6.19)
since the boundary term over Γ1 vanishes due to (2.9). The second term on the rightside of (6.19) also vanishes for the same reason as I115 above; namely, on Γ0 we haveaki SviNk = a3
i Svi = a33Sv3 = 0 due to a3
1 = a32 = 0 on Γ0. Therefore,
I2 = −∫
Ω
Saki Sq∂kv
i dx −∫
Ω
(S(aki ∂kv
i) − Sak
i ∂kvi − ak
i S∂kvi)Sq dx
= −∫
Ω
∂2+δ
aki ∂
1/2(Sq∂kv
i)dx −
∫
Ω
(S(aki ∂kv
i) − Sak
i ∂kvi − ak
i S∂kvi)Sq dx.
(6.20)
Both terms are bounded by P —for the first term, we use (4.9) and (5.4), while forthe second we apply (5.3) in order to write
∫
Ω
(S(aki ∂kv
i) − Sak
i ∂kvi − ak
i S∂kvi)Sq dx
≤ ∥∥S(aki ∂kv
i) − Sak
i ∂kvi − ak
i S∂kvi∥∥
L3/2‖Sq‖L3
≤ C(∥∥ak
i
∥∥W 1.5+δ,3
∥∥∂kvi∥∥
W 1,3 + ∥∥aki
∥∥W 1,6
∥∥∂kvi∥∥
W 1.5+δ,2
)‖Sq‖L3
≤ (C‖a‖H 2+δ‖v‖H 2.5 + ‖a‖H 2‖v‖H 2.5+δ
)‖Sq‖L3 . (6.21)
Finally, the term I3 is treated easily using (5.3). �
Appl Math Optim
7 A Divergence-Curl Estimates for η and v
For any given matrix function a(x), introduce the variable curl operator Ba acting onthe vector function f = (f 1, f 2, f 3) according to
Baf =⎡
⎢⎣
ak2∂kf
3 − ak3∂kf
2
ak3∂kf
1 − ak1∂kf
3
ak1∂kf
2 − ak2∂kf
1
⎤
⎥⎦ . (7.1)
For a as in (2.6), this corresponds to curl f in Eulerian coordinates. Similarly, weintroduce the variable divergence operator Aa defined by
Aaf = aki ∂kv
i . (7.2)
Observe that if a = I , then BI and AI agree with the usual curl and divergenceoperators.
Proof of Lemma 7.1 Since curlv0 = 0, it follows from the Kelvin principle that theEulerian velocity remains irrotational, i.e., curlu = 0. In Lagrangian coordinates thisis expressed as
Bav = 0. (7.4)
We next establish an estimate for curlη by first using the inequality
‖ curlη‖H 2+δ ≤ C‖ curl∇η‖H 1+δ + C‖ curlη‖H 1+δ
≤ C‖Ba∇η‖H 1+δ + C∥∥(BI − Ba)∇η
∥∥
H 1+δ + C‖ curlη‖H 1+δ (7.5)
where we decomposed curl∇η = Ba∇η + (BI −Ba)∇η. Using the fundamental the-orem of calculus, we have
(Ba∇η)(t) =∫ t
0(Bat ∇η + Ba∇v)ds =
∫ t
0(Bat ∇η − B∇av) ds (7.6)
where we used Ba∇v = −B∇av, which follows from (7.4). Returning to the inequal-ity (7.5) and using (7.6), we get
Appl Math Optim
∥∥curlη(t)∥∥
H 2+δ ≤ C∥∥(Ba∇η)(t)
∥∥H 1+δ + C
∥∥(Ba − BI )∇η(t)∥∥
H 1+δ
+ C∥∥curlη(t)
∥∥H 1+δ
≤ C
∫ t
0‖Bat ∇η‖H 1+δ ds + C
∫ t
0‖B∇av‖H 1+δ ds
+ C‖a − I‖H 1.5+δ‖η‖H 3+δ + C‖η‖H 2+δ
≤ C
∫ t
0‖at‖H 1.5+δ‖η‖H 3+δ ds
+ C
∫ t
0‖∇a‖H 1+δ‖∇v‖H 1.5+δ ds + Cε‖η‖H 3+δ
+ C
∫ t
0‖v‖H 2+δ ds + ‖η0‖H 2+δ (7.7)
and thus, using (4.9) and η0(x) = x,
∥∥curlη(t)∥∥
H 2+δ ≤ C
∫ t
0‖v‖H 2.5+δ‖η‖H 3+δ ds
+ C
∫ t
0‖η‖2
H 3+δ‖v‖H 2.5+δ ds + Cε‖η‖H 3+δ
+ C
∫ t
0‖v‖H 2+δ ds + C. (7.8)
An identical argument can be used to derive the same estimate for divη. Here we usediv∇η = Aa∇η + (AI − Aa)∇η, while instead of (7.6) we have
(Aa∇η)(t) =∫ t
0(Aat ∇η + Aa∇v)ds =
∫ t
0(Aat ∇η − A∇av) ds (7.9)
since Aa∇v + A∇av = 0. This identity in turn follows from Aav = 0. This leads to
∥∥divη(t)
∥∥
H 2+δ ≤ C
∫ t
0‖v‖H 2.5+δ‖η‖H 3+δ ds
+ C
∫ t
0‖η‖2
H 3+δ‖v‖H 2.5+δ ds + Cε‖η‖H 3+δ
+ C
∫ t
0‖v‖H 2+δ ds + C. (7.10)
Adding the inequalities (7.8) and (7.10), we get∥∥curlη(t)
Now, from [8, 9], we appeal to the following result, which gives regularity of a func-tion in terms of regularity of its divergence, curl, and the values on the boundary.Namely, for s > 1.5 we have
for any vector function f ∈ Hs(Ω), where ∇2 denotes the tangential derivative of f .Applying this result to the flow map η(x, t) for s = 3 + δ we obtain
∥∥η(t)∥∥
H 3+δ ≤ C∥∥η(t)
∥∥L2 + C
∥∥curlη(t)∥∥
H 2+δ + C∥∥divη(t)
∥∥H 2+δ + C
∥∥Sη3∥∥
L2(Γ1).
(7.13)
In particular, we used η3 = 0 on the stationary boundary Γ0. Using (7.11) and (7.13)and choosing ε > 0 sufficiently small (and thus adjusting T accordingly), we get
where we chose ε > 0 sufficiently small so that the second term on the far right sideof (7.15) can be absorbed.
We also apply (7.12) to v with s = 2.5 + ε, and utilize curlv = Bav + (BI −Ba)v = (BI − Ba)v and divv = Aav + (AI − Aa)v = (AI − Aa)v in order to obtain
In the final section, we provide a priori estimates leading to the proof of the maintheorem. These a priori estimates can be used to give a proof of existence of solutionsby using the horizontal mollification procedure due to Coutand and Shkoller [8, 9].
Proof of Theorem 2.1 For simplicity, we allow all constants to depend on ‖v0‖H 2.5+δ .In the proof, we denote by P a generic positive polynomial depending on the indi-cated argument.
First, adding the squared estimates (7.16) and (7.22), taking into account (6.2), weget
where P1, P2, and P3 are positive polynomials. Finally, a Gronwall type argu-ment leads to desired a priori bounds, as we now show. Without loss of gener-ality, we may assume that P1, P2, and P3 are increasing in their arguments. Let[0, T ] be the maximal interval such that X(t) ≤ 2C0, Y(t) ≤ 2P2(2C0), and Z(t) ≤2P3(2C0,2P2(2C0)). Then, unless the interval is infinite (in which case we have aglobal bound on X, Y , and Z), we have either (i) X(T ) = 2C0, (ii) Y(T ) = 2P2(2C0),or (iii) Z(T ) = 2P3(2C0,2P2(2C0)). In the case (i), we have, substituting t = T
in (8.9)
2C0 ≤ C0 + T P1(2C0,2P2(2C0),2P3
(2C0,2P2(2C0)
))(8.12)
which gives T ≥ C0/P1(2C0,2P2(2C0),2P3(2C0,2P2(2C0))). In the case (ii), wededuce, again substituting t = T in (8.10)
2P2(2C0) ≤ P2(2C0) + T P2(2C0)(2P3
(2C0,2P2(2C0)
))2 (8.13)
leading to T ≥ P2(2C0)/P2(2C0)(2P3(2C0,2P2(2C0)))2. Finally, in the case (iii),
we obtain
2P3(2C0,2P2(2C0)
)
≤ P3(2C0,2P2(2C0)
) + T P3(2C0,2P2(2C0)
)(2P3
(2C0,2P2(2C0)
))2 (8.14)
Appl Math Optim
from where T ≥ P3(2C0,2P2(2C0))/P3(2C0,2P2(2C0))(2P3(2C0,2P2(2C0)))2. In
either of the cases (8.12), (8.12), or (8.14), we get an explicit lower bound
T0 = min
{C0
P1(2C0,2P2(2C0),2P3(2C0,2P2(2C0))),
P2(2C0)
P2(2C0)(2P3(2C0,2P2(2C0)))2,
P3(2C0,2P2(2C0))
P3(2C0,2P2(2C0))(2P3(2C0,2P2(2C0)))2
}(8.15)
for T , and the desired a priori bounds 2C0, 2P2(2C0), and 2P3(2C0,2P2(2C0)) forX(t), Y(t), and Z(t) respectively are established on [0, T0]. �
9 The Modifications for the Case n = 2
First, we state the analog of Lemma 3.1 in the two dimensional case.
Lemma 9.1 Assume that ‖∇v‖L∞([0,T ];H 1+δ(Ω)) ≤ M . If T ≤ 1/CM , where C is asufficiently large constant, the following statements hold:
(i) ‖∇η(·, t)‖H 1+δ(Ω) ≤ C for t ∈ [0, T ],(ii) det(∇η(x, t)) = 1 for (x, t) ∈ Ω × [0, T ],
(iii) ‖a(·, t)‖H 1+δ(Ω) ≤ C (and thus also ‖a(·, t)‖L∞(Ω) ≤ C) for t ∈ [0, T ],(iv) ‖at (·, t)‖Lp(Ω) ≤ C‖∇v(·, t)‖Lp(Ω) for p ∈ [1,∞] and t ∈ [0, T ],(v) ‖at (·, t)‖Hr(Ω) ≤ C‖∇v(·, t)‖Hr(Ω) for r ∈ [0,1 + δ) and t ∈ [0, T ],
(vi) ‖att (·, t)‖Hσ (Ω) ≤ C‖∇v(·, t)‖H 1+δ(Ω)‖∇v(·, t)‖Hσ (Ω) + C‖∇vt (·, t)‖Hσ (Ω),for t ∈ [0, T ] and all 0 < σ ≤ 1 + δ, and
(vii) for every ε ∈ (0,1] there exists a constant C > 0 such that for all t ∈ [0, T ′],where T ′ = min{ε/CM,T } > 0, we have
∥∥ajl − δjl
∥∥H 1+δ(Ω)
≤ ε (9.1)
for j, l = 1,2 and
∥∥ajl ak
l − δjk
∥∥H 1+δ(Ω)
≤ ε (9.2)
for j, k = 1,2.
The proof is analogous to the three dimensional case.It is easy to check that Lemma 4.1 does not use the irrotationality of the flow. Thus
we have the following replacement for Lemma 4.1.
Lemma 9.2 Suppose that (v, q, a) satisfies the Euler equation (2.4)–(2.5) in Ω ×[0, T ) and that we have ‖∇v‖L∞([0,T ];H 1+δ(Ω)) ≤ M . Assume that a satisfies the es-timates in Lemma 9.1 for a sufficiently small constant ε > 0. Then the pressure q
Appl Math Optim
satisfies
∥∥q(t)∥∥
H 2.5+δ ≤ P + P
∫ t
0
∥∥qt (s)∥∥
H 1.5+δ ds, t ∈ [0, T ] (9.3)
where P is a positive polynomial depending on ‖v‖H 2+δ , ‖η‖H 2.5+δ , and ‖v0‖H 2+δ ,and
∥∥qt (t)∥∥
H 2+δ ≤ P + P
∫ t
0
∥∥qt (s)∥∥
H 1.5+δ ds, t ∈ [0, T ] (9.4)
where P is a positive polynomial depending on ‖v‖H 2+δ , ‖vt‖H 1.5+δ , ‖q‖H 2.5+δ ,‖η‖H 2.5+δ , and ‖v0‖H 2+δ .
Similarly, the tangential estimates directly adapt to the two-dimensional case aswell. Here, we define
Proof of Theorem 2.2 Instead of Ba used in (7.1), we now define
Baf = ak1∂kf
2 − ak2∂kf
1
which with a as in (2.6) corresponds to curlf = ∂1f2 − ∂2f1 in Eulerian coordinates.The definition of the divergence in (7.2) remains the same.
Now, since the flow is rotational, we no longer have (7.4). Instead, due to thevorticity preservation along the particle trajectories in space dimension 2,
ω(η(x, t), t
) = ω0(x)
(in Eulerian coordinates). Therefore, we obtain
Bav(x, t) = ω0(x) (9.7)
in the Lagrangian coordinates. Differentiating (9.7) gives
B∇av(x, t) + Ba∇v(x, t) = ∇ω0(x) (9.8)
Appl Math Optim
from where we obtain
(Ba∇η)(t) =∫ t
0(Bat ∇η + Ba∇v)ds =
∫ t
0(Bat ∇η − B∇av) ds + t∇ω0. (9.9)
Now, we have
‖ curlη‖H 1.5+δ ≤ C‖Ba∇η‖H 0.5+δ + C∥∥(BI − Ba)∇η
∥∥H 0.5+δ + C‖ curlη‖H 0.5+δ
(9.10)
and thus, combined with (9.9),∥∥curlη(t)
∥∥H 1.5+δ
≤ C
∫ t
0‖Bat ∇η‖H 0.5+δ ds + C
∫ t
0‖B∇av‖H 0.5+δ ds + C‖∇ω0‖H 0.5+δ
+ C‖a − I‖H 1+δ‖η‖H 2.5+δ + C‖η‖H 1.5+δ
≤ C
∫ t
0‖at‖H 1+δ‖η‖H 2.5+δ ds + C
∫ t
0‖∇a‖H 0.5+δ‖∇v‖H 1+δ ds + C‖∇ω0‖H 0.5+δ
+ Cε‖η‖H 2.5+δ + C
∫ t
0‖v‖H 1.5+δ ds + C. (9.11)
Hence,∥∥curlη(t)
∥∥H 1.5+δ
≤ C
∫ t
0‖v‖H 2+δ‖η‖H 2.5+δ ds + C
∫ t
0‖η‖2
H 2.5+δ‖v‖H 2+δ ds + Cε‖η‖H 2.5+δ
+ C
∫ t
0‖v‖H 1.5+δ ds + C‖∇ω0‖H 0.5+δ + C. (9.12)
The derivation analogous to the one in Sect. 7 leads to∥∥divη(t)
∥∥H 1.5+δ
≤ C
∫ t
0‖v‖H 2+δ‖η‖H 2.5+δ ds + C
∫ t
0‖η‖2
H 2.5+δ‖v‖H 2+δ ds + Cε‖η‖H 2.5+δ
+ C
∫ t
0‖v‖H 1.5+δ ds + C. (9.13)
Now, we add the inequalities (9.12) and (9.13) and continue as in the proof ofLemma 7.1 to write
The proof of Theorem 2.2 is then concluded by repeating the arguments fromSect. 8. �
Acknowledgements I.K. was supported in part by the NSF grant DMS-1311943, while A.T. was sup-ported in part by the Petroleum Institute Research Grant Ref. Number 11014.
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