Well-posedness Of The Free-surface Incompressible Euler

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Well-posedness Of The Free-surface Incompressible Euler

Transcript Of Well-posedness Of The Free-surface Incompressible Euler

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 20, Number 3, July 2007, Pages 829–930 S 0894-0347(07)00556-5 Article electronically published on March 5, 2007
WELL-POSEDNESS OF THE FREE-SURFACE INCOMPRESSIBLE EULER EQUATIONS WITH OR WITHOUT SURFACE TENSION
DANIEL COUTAND AND STEVE SHKOLLER

1. Introduction

1.1. The problem statement and background. For σ ≥ 0 and for arbitrary initial data, we prove local existence and uniqueness of solutions in Sobolev spaces to the free boundary incompressible Euler equations in vacuum:

(1.1a)

∂tu + ∇uu + ∇p = 0

in Q ,

(1.1b)

div u = 0

in Q ,

(1.1c)

p = σ H on ∂Q ,

(1.1d)

(∂t + ∇u)|∂Q ∈ T (∂Q) ,

(1.1e)

u = u0

on Qt=0 ,

(1.1f )

Qt=0 = Ω ,

where Q = 0≤t≤T {t} × Ω(t), Ω(t) ⊂ Rn, n = 2 or 3, ∂Q = 0≤t≤T {t} × ∂Ω(t), ∇uu = uj∂ui/∂xj, and where Einstein’s summation convention is employed. The vector field u is the Eulerian or spatial velocity field defined on the time-dependent domain Ω(t), p denotes the pressure function, H is twice the mean curvature of the boundary of the fluid ∂Ω(t), and σ is the surface tension. Equation (1.1a) is the conservation of momentum, (1.1b) is the conservation of mass, (1.1c) is the wellknown Laplace-Young boundary condition for the pressure function, (1.1d) states that the free boundary moves with the velocity of the fluid, (1.1e) specifies the initial velocity, and (1.1f) fixes the initial domain Ω.
Almost all prior well-posedness results were focused on irrotational fluids (potential flow), wherein the additional constraint curl u = 0 is imposed; with the irrotationality constraint, the Euler equations (1.1) reduce to the well-known waterwaves equations, wherein the motion of the interface is decoupled from the rest of the fluid and is governed by singular boundary integrals that arise from the use of complex variables and the equivalence of incompressibility and irrotationality with the Cauchy-Riemann equations. For 2D fluids (and hence 1D interfaces), the earliest local existence results were obtained by Nalimov [14], Yosihara [22], and Craig [5] for initial data near equilibrium. Beale, Hou, and Lowengrub [4] proved that the linearization of the 2D water-wave problem is well-posed if a Taylor sign condition is added to the problem formulation, thus preventing Rayleigh-Taylor instabilities. Using the Taylor sign condition, Wu [20] proved local existence for

Received by the editors November 9, 2005. 2000 Mathematics Subject Classification. Primary 35Q35, 35R35, 35Q05, 76B03. Key words and phrases. Euler equations, free boundary problems, surface tension.
c 2007 American Mathematical Society Reverts to public domain 28 years from publication 829

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D. COUTAND AND S. SHKOLLER

the 2D water-wave problem for arbitrary (sufficiently smooth) initial data. Later Ambrose [2] and Ambrose and Masmoudi [3] proved local well-posedness of the 2D water-wave problem with surface tension on the boundary replacing the Taylor sign condition.
In 3D, Wu [21] used Clifford analysis to prove local existence of the full waterwave problem with infinite depth, showing that the Taylor sign condition is always satisfied in the irrotational case by virtue of the maximum principle holding for the potential flow. Lannes [11] provided a proof for the finite depth case with varying bottom by implementing a Nash-Moser iteration. The first well-posedness result for the full Euler equations with zero surface tension, σ = 0, is due to Lindblad [13] with the additional “physical condition” that

(1.2)

∇p · n < 0 on ∂Q,

where n denotes the exterior unit normal to ∂Ω(t). The condition (1.2) is equivalent to the Taylor sign condition and provided Christodoulou and Lindblad [6] with enough boundary regularity to establish a priori estimates for smooth solutions to (1.1) together with (1.2) and σ = 0. (Ebin [10] provided a counterexample to well-posedness when (1.2) is not satisfied.) Nevertheless, local existence did not follow in [6], as finding approximations of the Euler equations for which existence and uniqueness is known and which retain the transport-type structure of the Euler equations is highly nontrivial, and this geometric transport-type structure is crucial for the a priori estimates. In [12], Lindblad proved well-posedness of the linearized Euler equations, but the estimates were not sufficient for well-posedness of the nonlinear problem. The estimates were improved in [13], wherein Lindblad implemented a Nash-Moser iteration to deal with the manifest loss of regularity in his linearized model and thus established the well-posedness result in the case that (1.2) holds and σ = 0.
Local existence for the case of positive surface tension, σ > 0, remained open. Although the Laplace-Young condition (1.1c) provides improved regularity for the boundary, the required nonlinear estimates are more difficult to close due to the complexity of the mean curvature operator and the need to study time-differentiated problems which do not arise in the σ = 0 case. It appears that the use of the timedifferentiated problem in Lindblad’s paper [13] is due to the use of certain tangential projection operators, but this is not necessary. We note that our energy function is different from that in [13] and provides better control of the Lagrangian coordinate.
After completing this work, we were informed of the paper of Schweizer [16] who studies the Euler equations for σ > 0 in the case that the free-surface is a graph over the two-torus. In that paper, he obtains a priori estimates under a smallness assumption for the initial surface; well-posedness follows under the additional assumption that there is no vorticity on the boundary. We also learned of the paper by Shatah and Zeng [15] who establish a priori estimates for both the σ = 0 and σ > 0 cases without any restrictions on the initial data.

1.2. Main results. We prove two main theorems concerning the well-posedness of (1.1). The first theorem, for the case of positive surface tension σ > 0, is new; for our second theorem, corresponding to the zero surface tension case, we present a new proof that does not require a Nash-Moser procedure and has optimal regularity.

Theorem 1.1 (Well-posedness with surface tension). Suppose that σ > 0, Γ is of class H5.5, and u0 ∈ H4.5(Ω). Then, there exists T > 0 and a solution

WELL-POSEDNESS OF THE FREE-SURFACE EULER EQUATIONS

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(u(t),p(t),Ω(t)) of (1.1) with u ∈ L∞(0, T ; H4.5(Ω(t))), p ∈ L∞(0, T ; H4(Ω(t))), and ∂Ω(t) ∈ H5.5. The solution is unique if u0 ∈ H5.5(Ω) and ∂Ω ∈ H6.5.

Theorem 1.2 (Well-posedness with Taylor sign condition). Suppose σ = 0, ∂Ω is
of class H3, and u0 ∈ H3(Ω) and condition (1.2) holds at t = 0. Then, there exists T > 0 and a unique solution (u(t),p(t),Ω(t)) of (1.1) with u ∈ L∞(0, T ; H3(Ω(t))), p ∈ L∞(0, T ; H3.5(Ω(t))), and ∂Ω(t) ∈ H3.

1.3. Lagrangian representation of the Euler equations. The Eulerian problem (1.1), set on the moving domain Ω(t), is converted to a PDE on the fixed domain Ω, by the use of Lagrangian variables. Let η(·, t) : Ω → Ω(t) be the solution of

∂tη(x, t) = u(η(x, t), t), η(x, 0) = Id and set
v(x, t) := u(η(x, t), t), q(x, t) := p(η(x, t), t), and a(x, t) := [∇η(x, t)]−1 .

The variables v, q and a are functions of the fixed domain Ω and denote the material velocity, pressure, and inverse Jacobian, respectively. Thus, on the fixed domain, (1.1) transforms to

(1.3a) (1.3b)

t
η = Id + v
0
∂tv + a ∇q = 0

in Ω × (0, T ] , in Ω × (0, T ] ,

(1.3c)

Tr(a ∇v) = 0

in Ω × (0, T ] ,

(1.3d)

q aT N/|aT N | = −σ ∆g(η) on Γ × (0, T ] ,

(1.3e)

(η, v) = (Id, u0) on Ω × {t = 0} ,

where N denotes the unit normal to Γ and ∆g is the surface Laplacian with respect

to the induced metric g on Γ, written in local coordinates as

(1.4) ∆g = √g−1∂α[√ggαβ∂β ] , gαβ = [gαβ ]−1 , gαβ = η,α · η,β , and √g =

det g .

Theorem 1.3 (σ > 0). Suppose that σ > 0, ∂Ω is of class H5.5, and u0 ∈ H4.5(Ω)
with div u0 = 0. Then, there exists T > 0 and a solution (v,q) of (1.3) with v ∈ L∞(0, T ; H4.5(Ω)), q ∈ L∞(0, T ; H4(Ω)), and Γ(t) ∈ H5.5. The solution satisfies

sup
t∈[0,T ]

3
|∂Ω(t)|25.5 +
k=0

∂tk v(t)

2

2 4.5−1.5k

+

k=0

∂tk q (t)

2 4−1.5k

≤ M˜ 0

where M˜ 0 denotes a polynomial function of Γ 5.5 and u0 4.5. The solution is unique for u0 ∈ H5.5(Ω) and Γ ∈ H6.5.

Remark 1. Our theorem is stated for a fluid in vacuum, but the analogous theorem holds for a vortex sheet, i.e., for the motion of the interface separating two inviscid immiscible incompressible fluids; the boundary condition (1.1c) is replaced by [p]± = σH, where [p]± denotes the jump in pressure across the interface.

For the zero surface tension case, we have

Theorem 1.4 (σ = 0 and condition (1.2)). Suppose that σ = 0, Γ is of class H3,
u0 ∈ H3(Ω), and condition (1.2) holds at t = 0. Then, there exists T > 0 and a unique solution (v,q) of (1.3) with v ∈ L∞(0, T ; H3(Ω)), q ∈ L∞(0, T ; H3(Ω)), and Γ(t) ∈ H3.

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D. COUTAND AND S. SHKOLLER

Because of the regularity of the solutions, Theorems 1.3 and 1.4 imply Theorems 1.1 and 1.2, respectively.

Remark 2. Note that in 3D, we require less regularity on the initial data than [13].

Remark 3. Since the vorticity satisfies the equation ∂t curl u + £u curl u = 0, where £u denotes the Lie derivative in the direction u, it follows that if curl u0 = 0, then curl u(t) = 0. Thus our result also covers the simplified case of irrotational flow. In particular, Theorem 1.3 shows that the 3D irrotational water-wave problem with surface tension is well-posed. In the zero surface tension case, our result improves the regularity of the data required by Wu [21].

1.4. General methodology and outline of the paper.

1.4.1. Artificial viscosity and the smoothed κ-problem. Our methodology begins with the introduction of a smoothed or approximate problem (4.1), wherein two basic ideas are implemented: first, we smooth the transport velocity using a new tool which we call horizontal convolution by layers; second, we introduce an artificial viscosity term in the Laplace-Young boundary condition (σ > 0) which simultaneously preserves the transport-type structure of the Euler equations, provides a PDE for which we can prove the existence of unique smooth solutions, and for which there exist a priori estimates which are independent of the artificial viscosity parameter κ. With the addition of the artificial viscosity term, the dispersive boundary condition is converted into a parabolic-type boundary condition, and thus finding solutions of the smoothed problem becomes an easier matter. On the other hand, the a priori estimates for the κ problem are more difficult than the formal estimates for the Euler equations.
The horizontal convolution is defined in Section 2. The domain Ω is partitioned into coordinate charts, each the image of the unit cube in R3. A double convolution is performed in the horizontal direction only (this is equivalent to the tangential direction in coordinate patches near the boundary). While there is no smoothing in the vertical direction, our horizontal convolution commutes with the trace operator and avoids the need to introduce an extension operator, the latter destroying the natural transport structure. The development of the horizontal convolution by layers is absolutely crucial in proving the regularity of the weak solutions that we discuss below. Furthermore, it is precisely this tool which enables us to prove Theorem 1.2 without the use of Nash-Moser iteration. To reiterate, this horizontal smoothing operator preserves the essential transport-type structure of the Euler equations.

1.4.2. Weak solutions in a variational framework and a fixed point, σ > 0. The

solution to the smoothed κ-problem (4.1) is obtained via a topological fixed-point

procedure, founded upon the analysis of the linear problem (4.2). To solve the lin-

ear problem, we introduce a few new ideas. First, we penalize the pressure function;

in particular, with > 0 the penalization parameter, we introduce the penalized

pressure function q

= 1 Tr(a ∇w).

Second,

we

find

a

new

class

of

[H

3 2

(Ω)]

-weak

solutions of the penalized and linearized smoothed κ-problem in a variational formu-

lation. The penalization allows us to perform difference quotient analysis in order

to prove regularity of our weak solutions; without penalization, difference quotients

of weak solutions do not satisfy the “divergence-free” constraint and as such cannot

be used as test functions. Furthermore, the penalization of the pressure function

WELL-POSEDNESS OF THE FREE-SURFACE EULER EQUATIONS

833

avoids the need to analyze the highest-order time derivative of the pressure, which would otherwise be highly problematic. In the setting of the penalized problem, we crucially rely on the horizontal convolution by layers to establish regularity of our weak penalized solution. Third, we introduce the Lagrange multiplier lemmas, which associate a pressure function to the weak solution of a variational problem for which the test functions satisfy the incompressibility constraint. These lemmas allow us to pass to the limit as the penalization parameter tends to zero, and thus, together with the Tychonoff fixed-point theorem, establish solutions to the smoothed problem (4.1). At this stage, however, the time interval of existence and the bounds for the solution depend on the parameter κ.

1.4.3. Solutions of the κ-problem for σ = 0 via transport. For the σ = 0 problem, we use horizontal convolution to smooth the transport velocity as well as the moving domain. Existence and uniqueness of this smoothed κ problem (17.1) is found using simple transport-type arguments that rely on the pressure gaining regularity just as in the fixed-domain case. Once again, the time interval of existence and the bounds for the solution a priori depend on κ.

1.4.4. A priori estimates and κ-asymptotics. We develop a priori estimates which show that the energy function Eκ(t) in Definition 10.1 associated to our smoothed problem (4.1) is bounded by a constant depending only on the initial data and not on κ. The estimates rely on the Hodge decomposition elliptic estimate (5.1).
In Section 10, we obtain estimates for the divergence and curl of η, v and their space and time derivatives. The main novelty lies in the curl estimate for η. The remaining portion of the energy is obtained by studying boundary regularity via energy estimates.
These nonlinear boundary estimates for the surface tension case σ > 0 are more complicated than the ones for the σ = 0 case with the Taylor sign condition (1.2) since it is necessary to analyze the time-differentiated Euler equations, which is not essential in the σ = 0 case (unless optimal regularity is sought).
We note that the use of the smoothing operator in Definition 2.1, where a double convolution is employed, is necessary in order to find exact (or perfect) derivatives for the highest-order error terms. The idea is that one of the convolution operators is moved onto a function which is a priori not smoothed, and commutation-type lemmas are developed for this purpose.
We obtain the a priori estimate

sup Eκ(t) ≤ M0 + T P ( sup Eκ(t)) ,

t∈[0,T ]

t∈[0,T ]

where M0 depends only on the data and P is a polynomial. The addition of the artificial viscosity term allows us to prove that Eκ(t) is continuous; thus, following the development in [8], there exists a sufficiently small time T , which is independent of κ, such that supt∈[0,T ] Eκ(t) < M˜ 0 for M˜ 0 > M0.
We then find κ-independent nonlinear estimates for the σ = 0 case for the energy
function (20.1).

Outline. Sections 2–15 are devoted to the case of positive surface tension σ > 0. Sections 16–27 concern the problem with zero surface tension σ = 0 together with the Taylor sign condition (1.2) imposed.

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D. COUTAND AND S. SHKOLLER

1.5. Notation. Throughout the paper, we shall use the Einstein convention with

respect to repeated indices or exponents. We specify here our notation for certain

vector and matrix operations.

We write the Euclidean inner-product between two vectors x and y as x · y,

so that x · y = xi yi.

The

transpose

of

a

matrix

A

will

be

denoted

by

AT ,

i.e.,

(

A

T

)

i j

=

Aji .

We write the product of a matrix A and a vector b as A b, i.e., (A b)i = Aijbj.

The product of two matrices A and S will be denoted by A·S, i.e., (A·S)ij =

Aik Sjk.

The trace of the product of two matrices A and S will be denoted by A : S,

i.e., (A : S)ij = Aik Sik.

For Ω, a domain of class Hs (s ≥ 2), there exists a well-defined extension operator

that we shall make use of later.

Lemma 1.1. There exists E(Ω), a linear and continuous operator from Hr(Ω) into Hr(R3) (0 ≤ r ≤ s), such that for any v ∈ Hr(Ω) (0 ≤ r ≤ s), E(Ω)(v) = v in Ω.

We will use the notation Hs(Ω) to denote either Hs(Ω; R) (for a pressure func-

tion, for instance) or Hs(Ω; R3) (for a velocity vector field) and we denote the

standard norm of Hs(Ω) (s ≥ 0) by · s. The Hs(Ω) inner-product will be de-

noted (·, ·)s.

We shall use the following notation for derivatives: ∂t or (·)t denotes the partial

time derivative, ∂ denotes the tangential derivative on Γ (or in a small enough

neighborhood of Γ), and ∇ denotes the three-dimensional gradient.

Letting (x1, x2) denote a local coordinate system on Γ, for α = 1, 2, we let either

∂α

or

(·),α

denote



∂ xα

.

We

define

∂α := g0αβ ∂β , |∂kφ|2 = ∂α1 ∂α2 · · · ∂αk ∂α1 ∂α2 · · · ∂αk

for integers k ≥ 0, where g0 = gt=0 is the (induced) metric on Γ. In particular,

|∂0φ| = |φ|, |∂1φ|2 = |∂φ|2 = ∂αφ∂αφ and ∂kφ will mean any kth tangential

derivative of φ.



The area element on Γ in local coordinates is dS0 = g0dx1 ∧ dx2 a√nd the pull-

back of the area element dS on Γ(t) = η(Γ) is given by η∗(dS) = gdS0. Let

{Ui}Ki=1 denote an open covering of Γ, and let {ξi}Ki=1 denote the partition of unity subordinate to this cover. The L2(Γ) norm is

|φ|0 := φ L2(Γ) =

1 2
φ2dS0 ,
Γ

and the Hk(Γ) norm for integers k ≥ 1 is

|φ|k := φ Hk(Γ) =

kK

2

|ξl∂iφ|20 .

i=1 l=1

Similarly, for the Hilbert space inner-products, we use

[φ, ψ]0 := [φ, ψ]L2(Γ) = φ ψ dS0,
Γ

kK

[φ, ψ]k := [φ, ψ]Hk(Γ) = [φ, ψ]0 +

[ξl∂iφ, ξl∂iψ]0 .

i=1 l=1

WELL-POSEDNESS OF THE FREE-SURFACE EULER EQUATIONS

835

Fractional-order spaces are defined via interpolation using the trace spaces of Lions (see, for example, [1]).
The dual of a Banach space X is denoted by X , and the corresponding norm in X will be denoted · X . For L ∈ Hs(Ω) and v ∈ Hs(Ω), the duality pairing between L and v is denoted by L, v s.
Throughout the paper, we shall use C to denote a generic constant, which may
possibly depend on the coefficient σ or on the initial geometry given by Ω (such as a Sobolev constant or an elliptic constant), and we use P (·) to denote a generic polynomial function of (·). For the sake of notational convenience, we will often write u(t) for u(t, ·).

2. Convolution by horizontal layers and the smoothed transport velocity

Let Ω ⊂ Rn denote an open subset of class H6, and let {Ui}Ki=1 denote an open covering of Γ := ∂Ω, such that for each i ∈ {1, 2, ..., K},
θi : (0, 1)2 × (−1, 1) → Ui is an H6 diffeomorphism ,
Ui ∩ Ω = θi((0, 1)3) and Ui ∩ Γ = θi((0, 1)2 × {0}) ,
θi(x1, x2, x3) = (x1, x2, ψi(x1, x2) + x3) and det ∇θi = 1 in (0, 1)3 .
Next, for L > K, let {Ui}Li=K+1 denote a family of open sets contained in Ω such that {Ui}Li=1 is an open cover of Ω. Let {αi}Li=1 denote the partition of unity subordinate to this covering.
Thus, each coordinate patch is locally represented by the unit cube (0, 1)3 and for the first K patches (near the boundary), the tangential (or horizontal) direction is represented by (0, 1)2 × {0}.

Definition 2.1 (Horizontal convolution). Let 0 ≤ ρ ∈ D((0, 1)2) denote an even

Friederich mollifier, normalized so that

ρ = 1, with corresponding dilated

(0,1)2

function

1x ρ 1δ (x) = δ2 ρ δ , δ > 0.

For w ∈ H1((0, 1)3) such that supp(w) ⊂ [δ, 1 − δ]2 × (0, 1), set

ρ 1 h w(xH , x3) = ρ 1 (xH − yH )w(yh, x3)dyH , yH = (y1, y2) .

δ

R2 δ

We then have the tangential integration by parts formula

while

ρ 1 h w,α (xH , x3) = ρ 1 ,α (xH − yH )w(yh, x3)dyH , α = 1, 2 ,

δ

R2 δ

ρ 1 h w,3 (xH , x3) = ρ 1 (xH − yH )w,3 (yh, x3)dyH .

δ

R2 δ

It should be clear that h smooths w in the horizontal directions, but not in the vertical direction. Fubini’s theorem ensures that

(2.1)

ρ 1 h w s,(0,1)3 ≤ Cs w s,(0,1)3 for any s ≥ 0 , δ

and we shall often make implicit use of this inequality.

836

D. COUTAND AND S. SHKOLLER

Remark 4. The horizontal convolution hw does not smooth w in the vertical direction; however, it does commute with the trace operator, so that

ρ 1δ h w (0,1)2×{0} = ρ 1δ h w|(0,1)2×{0} ,
which is essential for our methodology. Also, note that h smooths without the introduction of an extension operator, required by standard convolution operators on bounded domains; the extension to the full space would indeed be problematic for the transport structure of the divergence and curl of solutions to the Euler-type PDEs that we introduce.

Definition

2.2

(Smoothing

the

velocity

field).

For

v

∈ L2(Ω)

and

any

κ



(0,

κ0 2

)

with

K
κ0 = min dist supp(αi ◦ θi) , [(0, 1)2 × {0}]c ∩ ∂[0, 1]3 ,
i=1

set

K√



L −1

vκ =

αi ρ 1 h [ρ 1 h (( αiv) ◦ θi)] ◦ θi +

αiv .

κ

κ

i=1

i=K +1

It follows from (2.1) that there exists a constant C > 0 which is independent of κ such that for any v ∈ Hs(Ω) for s ≥ 0,

(2.2)

vκ s ≤ C v s and |vκ|s−1/2 ≤ C|v|s−1/2 .

The smoothed particle displacement field is given by

(2.3)

t
ηκ = Id + vκ .
0

For each x ∈ Ui, let x˜ = θi−1(x). The difference of the velocity field and its smoothed counterpart along the boundary Γ then takes the form

(2.4) vκ(x) − v(x)
K
=
i=1

ζi(x)ρ 1 (y˜)ρ 1 (z˜) [(ζiv)(θi(x˜ − (y˜ + z˜))) − (ζiv)(θi(x˜))] dz˜ dy˜ ,

B(0,κ)2

κ

κ

where ζi(x) = αi(θi(x˜)). Combining (1.1a), (2.3), and (2.4),

(2.5) ηκ(x) − η(x)

K
=
i=1

ζi(x)ρ 1 (y˜)ρ 1 (z˜) [(ζiη)(θi(x˜ − (y˜ + z˜))) − (ζiη)(θi(x˜))] dz˜ dy˜ .

B(0,κ)2

κ

κ

For any u ∈ H1.5(Γ) and for y ∈ B(x, κ), where B(x, κ) denotes the disk of radius κ centered at x, the mean value theorem shows that

|u(y) − u(x)| ≤ C|r−1|Lq(B(x,κ))|∂u|Lp(B(x,κ)), r = radial coordinate,

so that in particular, with p = 4 and q = 43 ,





|u(y) − u(x)| ≤ C κ|∂u|L4 ≤ C κ|u|1.5 ,

WELL-POSEDNESS OF THE FREE-SURFACE EULER EQUATIONS

837

the last inequality following from the Sobolev embedding theorem.

U ∈ H1.5(Γ), (2.6)

√ |Uκ(x) − U (x)|L∞ ≤ C κ|U |1.5 .

Hence, for

Note that the co√nstant C depends on maxi∈{1,...,K} |θi|5.5. Letting ζi = αi and R = (0, 1)2, we also have that for any φ ∈ L2(Γ),

K

K

vκ φ =

ρ 1 h ρ 1 h ζiv(x) ζiφ(x) =

ρ 1 h ζiv(x) ρ 1 h ζiφ(x)

Γ

i=1 R κ

κ

i=1 R κ

κ

(2.7)

K

=

[ρ 1 h (ζiv ◦ θi)] ◦ θi−1 [ρ 1 h (ζiφ ◦ θi)] ◦ θi−1 .

Γ i=1 κ

κ

Finally, we need the following

Lemma 2.1 (Commutation-type lemma). Let g ∈ L2(Γ) satisfy dist(supp(g), ∂R) < κ0 and let f ∈ Hs(Γ) for s > 1. Then independently of κ ∈ (0, κ0), there exists
a constant C > 0 such that

We also have

ρ 1 h [f g] − f ρ 1 h g ≤ C κ|f |s+1,R |g|0,R .

κ

κ

0,R

ρ 1 h [f g] − f ρ 1 h g

≤ C κ f s+ 3 ,[0,1]3 g 0,[0,1]3

κ

κ

0,[0,1]3

2

whenever g ∈ L2(Ω), f ∈ Hs(Ω) and κ < min(dist(supp f g, {1} × [0, 1]2), dist(supp f g, {0} × [0, 1]2)). 2

Proof. Let = ρ 1 h [f g] − f ρ 1 h g. Then

κ

κ

| (x)| =

ρ 1 (x − y)[f (y) − f (x)]g(y)dy

B(x,κ) κ

so that

≤ C κ|f |s+1,R

ρ 1 (x − y)|g(y)|dy ,

B(x,κ) κ

| |0,R ≤ C κ|f |s+1,R ρ 1 h |g| ≤ C κ|f |s+1,R |g|0,R .

κ

0,R

The inequality on [0, 1]3 follows the identical argument with an additional inte-

gration over the vertical coordinate. The hypothesis on the support of f g makes

the integral well-defined.

Remark 5. Higher-order commutation-type lemmas will be developed for the case of zero surface tension in Section 21.

3. Closed convex set used for the fixed point for σ > 0
In order to construct solutions for our approximate model (4.1), we use a topological fixed-point argument which necessitates the use of high-regularity Sobolev spaces. In particular, we shall assume that the initial velocity u0 is in H13.5(Ω) and that Ω is of class C∞; after establishing our result for the smoothed initial domain and velocity, we will show that both Ω and u0 can be taken with the optimal regularity stated in Theorem 1.3.

838

D. COUTAND AND S. SHKOLLER

For T > 0, we define the following closed convex set of the Hilbert space L2(0, T ; H13.5(Ω)):
CT = {v ∈ L2(0, T ; H13.5(Ω))| sup v 13.5 ≤ 2 u0 13.5 + 1}.
[0,T ]

It is clear that CT is nonempty, since it contains the constant (in time) function u0, and is a convex, bounded and closed subset of the separable Hilbert space L2(0, T ; H13.5(Ω)).
Let v ∈ CT be given, and define η by (1.3a), the Bochner integral being taken in the separable Hilbert space H13.5(Ω).
Henceforth, we assume that T > 0 is given such that independently of the choice
of v ∈ CT , we have the injectivity of η(t) on Ω, the existence of a normal vector to η(Ω, t) at any point of η(Γ, t), and the invertibility of ∇η(t) for any point of Ω and for any t ∈ [0, T ]. Such a condition can be achieved by selecting T small enough so that

(3.1)

∇η − Id L∞(0,T ;H13.5(Ω)) ≤ 0 ,

for 0 > 0 taken sufficiently small. Condition (3.1) holds if T ∇u0 H2 ≤ 0. Thus,

(3.2)

a = [∇η]−1

is well-defined.

Then choosing T > 0 even smaller, if necessary, there exists κ0 > 0 such that for

any

κ



(0,

κ0 2

),

we

have

the

injectivity

of

ηκ(t)

on



for

any

t



[0,

T

];

furthermore,

∇ηκ satisfies the condition (3.1) with ηκ replacing η. We let nκ(ηκ(x)) denote the

exterior unit normal to ηκ(Ω) at ηκ(x) with x ∈ Γ.

Our notational convention will be as follows: if we choose v¯ ∈ CT , then η¯ is the

flow map coming from (1.3a), and a¯ is the associated pull-back, a¯ = [∇η¯]−1. Thus,

a bar over the velocity field will imply a bar over the Lagrangian variable and the

associated pull-back.

For a given vκ, our notation is as follows:

t

ηκ(t) = Id + vκ and ηκ(0) = Id ,
0
aκ = Cof ∇ηκ , Jκ = det ∇ηκ , gκαβ = ∂αηκ · ∂βηκ .

We take T (which a priori depends on κ) even smaller if necessary to ensure that

for t ∈ [0, T ],

(3.3a)

−1
g(t)



2√g0−1

,

(3.3b)

−1
gκ(t)



2√g0−1

,

(3.3c)

1

3

≤ Jκ(t) ≤ .

2

2

Lemma 3.1. For v ∈ CT and for any s ≥ 0, we have independently of the choice of v ∈ CT that

sup |vκ|s ≤ Cκ,s P ( u0 13.5) .
[0,T ]

Proof. By the standard properties of the convolution a.e. in [0, T ]:

(3.4)

C

C

|vκ|s ≤ [ κs−13 + 1]|v|13 ≤ [ κs−13 + 1][2 u0 13.5 + 1],
EstimatesRegularitySolutionsSolutionExistence