Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain

J. Differential Equations 250 (2011) 1719–1746

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Journal of Differential Equations www.elsevier.com/locate/jde

Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain Igor Kukavica a , Roger Temam b , Vlad C. Vicol c,∗ , Mohammed Ziane a a b c

Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, CA 90089-2532, United States Institute for Scientific Computing and Applied Mathematics, Rawles Hall, Indiana University, Bloomington, IN 47405-5701, United States Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, United States

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 5 July 2010 Revised 15 July 2010 Available online 19 August 2010

We address the question of well-posedness in spaces of analytic functions for the Cauchy problem for the hydrostatic incompressible Euler equations (inviscid primitive equations) on domains with boundary. By a suitable extension of the Cauchy–Kowalewski theorem we construct a locally in time, unique, real-analytic solution and give an explicit rate of decay of the radius of real-analyticity. © 2010 Elsevier Inc. All rights reserved.

MSC: 35Q35 76N10 76B03 86A05 Keywords: Hydrostatic Euler equations Non-viscous primitive equations Hydrostatic approximation Well-posedness Bounded domain Analyticity

1. Introduction The literature on incompressible homogeneous geophysical flows in the hydrostatic limit is vast, and several models have been proposed both in the viscous and the inviscid cases (cf. J.-L. Lions, Temam, and Wang [13–15], P.-L. Lions [11], Pedloski [19], Temam and Ziane [27], and references therein). In the present article we consider the inviscid hydrostatic model which is classical in geophysical fluid mechanics; see [19] and [11, Section 4.6], where the author raises the question of ex-

*

Corresponding author. E-mail addresses: [email protected] (I. Kukavica), [email protected] (R. Temam), [email protected] (V.C. Vicol), [email protected] (M. Ziane). 0022-0396/$ – see front matter doi:10.1016/j.jde.2010.07.032

© 2010

Elsevier Inc. All rights reserved.

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istence and uniqueness of solutions. These equations are formally derived from the three-dimensional incompressible Euler equations for a fluid between two horizontal plates, in the limit of vanishing distance between the plates [4,5,11]. The problem is to find a velocity field u = ( v 1 , v 2 , w ) = ( v , w ), a scalar pressure P , and a scalar density ρ solving

∂t v + ( v · ∇) v + w ∂z v + ∇ P + f v ⊥ = 0,

(1.1)

div v + ∂z w = 0,

(1.2)

∂z P = −ρ g ,

(1.3)

∂t ρ + ( v · ∇)ρ + w ∂z ρ = 0,

(1.4)

in D × (0, T ), for some T > 0. Here

  D = M × (0, h) = (x1 , x2 , z) = (x, z) ∈ R3 : x ∈ M, 0 < z < h is a three-dimensional cylinder of height h, where M ⊂ R2 is a smooth domain with real-analytic boundary. We denote by div, ∇ , and  the corresponding two-dimensional operators acting on x = (x1 , x2 ), while ∂z = ∂/∂ z. Also, we let v ⊥ = ( v 2 , − v 1 ) be the first two components of u × e 3 , f is the strength of the rotation, and g is the gravitational constant. It follows from (1.9) that the pressure P may be written in terms of the density ρ and the bottom pressure p (x, t ) = P (x, 0, t )

P (x, z, t ) = p (x, t ) − g ψ(x, z, t ),

(1.5)

where we have denoted

z ψ(x, z, t ) =

ρ (x, ζ, t ) dζ,

(1.6)

0

for all (x, z) ∈ D and t  0. The hydrostatic Euler equations (1.1)–(1.4) may then be rewritten as

∂t v + ( v · ∇) v + w ∂z v + ∇ p + f v ⊥ = g ∇ψ,

(1.7)

div v + ∂z w = 0,

(1.8)

∂ z p = 0,

(1.9)

∂t ρ + ( v · ∇)ρ + w ∂z ρ = 0,

(1.10)

where ψ is obtained from ρ via (1.6). The boundary conditions for the top and bottom Γz = M × {0, h} and the side Γx = ∂ M × (0, h) of the cylinder D are

w ( x , z , t ) = 0,

on Γz × (0, T ),

(1.11)

v (x, z, t ) dz · n = 0,

on Γx × (0, T ),

(1.12)

h 0

where n is the outward unit normal to M. Note that there is no evolution equation for w. Instead, the incompressibility condition and (1.11) imply that

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z w ( x, z , t ) = −

div v (x, ζ, t ) dζ,

(1.13)

0

for all 0 < z < h, and 0 < t < T , which combined again with (1.11) shows that the vertical average of div v is zero, i.e.,

h div v (x, z, t ) dz = 0,

(1.14)

0

for all x ∈ M and 0 < t < T . We consider a real-analytic initial datum

v (x, z, 0) = v 0 (x, z),

(1.15)

ρ (x, z, 0) = ρ0 (x, z)

(1.16)

in D , which satisfies the compatibility conditions arising from (1.12) and (1.14), namely

h v 0 (x, z) dz · n = 0,

(1.17)

div v 0 (x, z) dz = 0,

(1.18)

0

for all x ∈ ∂ M, and

h 0

for all x ∈ M. The existence and uniqueness of solutions to the hydrostatic Euler equations is an outstanding open problem (cf. P.-L. Lions [11]). The methods and results for hyperbolic systems cannot be applied to (1.7)–(1.12) in order to find a well-posed set of boundary conditions (cf. Oliger and Sündstrom [23]). The instability results of Grenier [6,5] and Brenier [4] suggest that the problem may be ill-posed in Sobolev spaces, in analogy to the Prandtl equations. Recently, Renardy [20] proved that the linearization of the hydrostatic Euler equations at specific parallel shear flows is ill-posed in the sense of Hadamard. The only local existence result available for the nonlinear problem was obtained in twodimensions by Brenier [3] under the assumptions of convexity of v in the z-variable, constant normal derivative of v on Γz , and of periodicity of ( v , w , p ) in the x-variable. The study of this system in spaces of analytic functions is the point of convergence of a number of essential difficulties which we need to recall before we can properly describe our results in detail. For a given system of partial differential equations, there is the issue of well-posedness in the sense of Hadamard [7]: The objective here is to show that the system of partial differential equations supplemented with suitable boundary (and possibly initial) conditions, possesses a unique solution in certain spaces, and that the solution depends continuously on the data. When we are interested in spaces of analytic functions (in the space variables), two different issues can be considered. The first one is when a well-posedness result has been proven in a suitable function space (e.g. a Sobolev space) for a suitable initial-boundary value problem, and we wish to show that the solution is analytic in space within the domain D , or possibly up to the boundary of D (in which case the solution extends beyond D ). Pertaining to this type of approach are the results of Bardos and Benachour [2] for the Euler equations (see also [9,10,16]), or Masuda [18] for the

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incompressible (viscous) Navier–Stokes equations. The second issue would be to look for a Cauchy– Kowalewski-type of result. In such a case a system of partial differential equations and a number of data are given, analytic, on a manifold V . The equations may permit to compute all the derivatives of the unknown functions, and hence the uniqueness of an analytic solution in the neighborhood of V follows. For the existence, the problem is to derive enough a priori estimates showing that the Taylor series is convergent in a neighborhood of V . Note that for some equations no additional boundary condition, except the data given of V , are necessary. For example one can solve in this way the wave equation utt − u xx = 0, 0 < x < 1, for a short interval of time, without any boundary condition for u at x = 0, 1. Much more elaborate Cauchy–Kowalewski-type results include the results of Sammartino and Cafflish (cf. [24], see also [17] and references therein) for the dissipative Prandtl boundary layer equations. In [24,17], using the abstract Cauchy–Kowalewski theorem (cf. Asano [1]), the authors prove the existence of a real-analytic solution to the Prandtl equations. We note that due to the non-locality created by the pressure it is challenging to verify that the assumptions of the abstract Cauchy–Kowalewski theorem hold for the system (1.1)–(1.4). In the case of the hydrostatic Euler equations (1.1)–(1.4) an additional fundamental set of difficulties arises when we raise the question of well-posedness. Indeed, according to an old result of Oliger and Sündstrom [23], revisited in Temam and Tribbia [26], the boundary value problem for these equations, as well as for a number of other equations from geophysical fluid mechanics, cannot be well-posed in D × [0, T ] for any set of local boundary conditions. The approach in [23] and [26] is the following. Let D = M × (0, h), with M of the form (0, L 1 ) × (0, L 2 ). An expansion of this system is made in a suitable set of cosine and sine functions in the vertical (z) direction. The boundary conditions which are needed for each mode n depends on n, and thus they are of a nonlocal type. In that direction, a linearized system related to (1.1)–(1.4) has been studied by Rousseau, Temam and Tribbia [21,22], and a result of well-posedness for this system has been achieved using the linear semigroup theory. Note that in [21,22] the boundary of M = (0, L 1 ) × (0, L 2 ) is not analytic. For the hydrostatic Euler equations (1.1)–(1.4) initial conditions need to be specified for the variables v = ( v 1 , v 2 ) and ρ , which are called prognostic variables in the language of geophysical fluid mechanics, but P and w are diagnostic variables, giving rise to another set of difficulties. At each instant of time, P and w can be expressed as functions (nonlocal functionals) of v and ρ . Furthermore, as it appears below, P is determined by the solution of an elliptic Neumann problem (cf. (7.24)–(7.25)) which introduces some form of ellipticity in the system (1.1)–(1.4), which is otherwise essentially hyperbolic. The requirements for solvability for this Neumann problem, lead us to introduce a novel side-boundary condition (cf. (1.12)). Since z-independent solutions to the 3D hydrostatic Euler equations (1.1)–(1.4) are solutions to the 2D incompressible Euler equations, the natural boundary condition (1.12) is necessary. After all the preliminaries, we can describe our result as a nonlocal Cauchy–Kowalewski-type of result for the hydrostatic Euler equations (1.1)–(1.4). In the present paper we prove the existence and uniqueness of solutions to the Cauchy problem (1.7)–(1.12) in the two-dimensional case (that is with M and (u , ρ , p ) independent of x2 ), and the three-dimensional cases where M is a half-plane or a periodic box. These results were announced in the note [8]. The three-dimensional case when M is a generic bounded domain with analytic boundary will be treated in a forthcoming paper. Note that this is not a boundary value problem. Indeed we do not require an infinite set of boundary conditions on ∂ D , as seems to be required by the results of [21–23,26]. However, the Neumann problem needed for the determination of P destroys the hyperbolic nature of the equations (and thus also the finite speed of propagation), and for the uniqueness of solutions we cannot argue by unique continuation, but we rather employ methods which pertain to evolution problems. Furthermore, we obtain an explicit rate of decay in time of the radius of analyticity of the solution. To the best of our knowledge this is the first local well-posedness result for the hydrostatic Euler equations in three dimensions, and in the absence of convexity, even in two dimensions. Organization of the paper is as follows. Section 2 contains the functional setting and the statements of the main theorems. In Section 3 we give the a priori estimates for the two-dimensional case. In Section 4 we prove the derivation of the estimates for the velocity. The construction of solutions is given in Section 5, and the uniqueness is proven in Section 6. Section 7 contains the proof for the three-dimensional periodic domain and for the half-space.

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2. Main theorems In the following, α = (α1 , α2 , αz ) ∈ N3 and β = (β1 , β2 , βz ) ∈ N3 denote multi-indices, where N =  {0, 1, 2, . . .} is the set of all non-negative integers. The notation ∂ α = ∂xα ∂zαz = ∂xα11 ∂xα22 ∂zαz , where α  = (α1 , α2 ) will be used throughout. We denote the homogeneous Sobolev semi-norms | · |m , for m ∈ N, by

| v |m =

  ∂ α v  2 , L (D )

(2.1)

|α |=m

where ∂ α v 2L 2 = ∂ α v 1 2L 2 + ∂ α v 2 2L 2 . Remark 2.1. In the two-dimensional case, when M = (0, 1) × R is independent of x2 , the L 2 norms in the definition of | · |m are considered only with respect to x1 and z, that is

 α 2 ∂ v i  2 = L (D )

1 h 0

 α  ∂ v i (x, z)2 dz dx1 ,

0

for i ∈ {1, 2}. We recall that a function v (x, z) is real-analytic in x and z, with radius of analyticity exists M > 0 such that

τ if there

 α  ∂ v (x, z)  M |α |! , |α |

τ

for all (x, z) ∈ D and α ∈ N3 , where |α | = α1 + α2 + αz . For r  0 and spaces of real-analytic functions

 Xτ =

v∈C

∞

h



h v |Γx dz · n = 0,

(D): 0

τ > 0 fixed, we define the

div v dz = 0,  v  X τ < ∞ ,

(2.2)

0

where

 v  Xτ =

∞ 

| v |m

m =0

(m + 1)r τ m . m!

(2.3)

Similarly, denote





Y τ = v ∈ X τ ,  v Y τ < ∞ ,

(2.4)

where the semi-norm  · Y τ is given by

 v Y τ =

∞  m =1

We write ρ ∈ X τ if notation, let

| v |m

(m + 1)r τ m−1 . (m − 1)!

(2.5)

ρ ∈ C ∞ (D) and ρ  X τ < ∞, and ρ ∈ Y τ if ρ ∈ X τ and ρ Y τ < ∞. For ease of

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  ( v , ρ )

Xτ

=  v  X τ + ρ  X τ ,

and similarly

  ( v , ρ ) =  v Y + ρ Y . τ τ Y τ

Using the Sobolev embedding theorem it is clear from (2.3) that if v ∈ X τ then v is real-analytic τ (and satisfies with radius of analyticity τ . Conversely, if v is real-analytic with radius of analyticity

r +2 (τ  /τ )m < ∞. the boundary conditions), then v ∈ X τ  for any τ  < τ and r  0, since m0 m Moreover, we have the estimate  v  X τ   v  L 2 (D) + τ  v Y τ , and for any ε > 0 we have X τ +ε ⊂ Y τ since  v Y τ  (e τ ln(1 + ε /τ ))−1  v  X τ +ε . The following theorem is our main result for dimension two.

Theorem 2.2. Let the functions u , p , ρ be independent of x2 , and let r  2. Assume that v 0 and ρ0 are real-analytic with radius of analyticity strictly larger than τ0 , and suppose that v 0 satisfies the compatibility conditions (1.17)–(1.18). Then there exists T ∗ = T ∗ (r , g , τ0 , ( v 0 , ρ0 ) X τ0 ) > 0, and a unique real-analytic solution ( v (t ), ρ (t )) of the initial value problem associated with (1.7)–(1.12) with radius of analyticity τ (t ), such that

   v (t ), ρ (t ) 

t X τ (t )

+ Cg





e g (t −s)  v (s), ρ (s) Y

τ (s)

ds

0

  + C ( v 0 , ρ0 )

t X τ0

e

gt









1 + τ −2 (s)  v (s), ρ (s) Y

τ (s)



ds  ( v 0 , ρ0 ) X e gt , τ0

(2.6)

0

for all t ∈ [0, T ∗ ), where C = C (D ) is a fixed positive constant. Moreover, the radius of analyticity of the solution, τ : [0, T ∗ ) → R+ , may be computed explicitly from (3.26) below. In the three-dimensional case, the boundary condition (1.12) allows us to find the pressure implicitly as the solution of an elliptic Neumann problem (cf. (7.24)–(7.25)). The classical higher-regularity estimates (cf. J.-L. Lions and Magenes [12], Temam [25]) may not be used to prove that the pressure has the same radius of analyticity as the velocity, preventing the estimates from closing. To overcome this obstacle we introduce a new analytic norm which combinatorially encodes the transfer of normal to tangential derivatives in the pressure estimate. The following theorem treats the case when M is a half-plane, or the periodic domain. The case when M is a generic real-analytic bounded domain in R2 , and (1.12) holds on ∂ M, requires new ideas and will be treated in a forthcoming paper. Theorem 2.3. Let r  5/2, and let M be either the upper half-plane {x1 > 0} or the periodic box [0, 2π ]2 . Assume that v 0 and ρ0 are real-analytic with radius of analyticity strictly larger than τ0 , and suppose that v 0 satisfies the compatibility conditions (1.17)–(1.18). Then there exists T ∗ = T ∗ (r , f , g , τ0 , ( v 0 , ρ0 ) X τ0 ) > 0, and a unique real-analytic solution ( v (t ), ρ (t )) of the initial value problem associated with (1.7)–(1.12) with radius of analyticity τ (t ), such that

   v (t ), ρ (t ) 

t X τ (t )

+ Cg





e C 1 (t −s)  v (s), ρ (s) Y

τ (s)

ds

0

  + C ( v 0 , ρ0 ) X e C 1 t

t

τ0

0









1 + τ −5/2 (s)  v (s), ρ (s) Y

τ (s)



ds  ( v 0 , ρ0 ) X e C 1 t , τ0

(2.7)

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for all t ∈ [0, T ∗ ), where C = C (D ) and C 1 = C 1 ( f , g ) are fixed positive constants. Moreover, the radius of analyticity of the solution, τ : [0, T ∗ ) → R+ may be computed explicitly from (7.13) or (7.46) below, for the periodic box or the half-plane respectively. The different powers of τ in (2.6) and (2.7) are due to the different exponents in the twodimensional and three-dimensional Agmon’s inequalities. Remark 2.4. We note that the solutions v (t ) constructed in Theorems 2.2 and 2.3 do not depend on the function τ (t ). Moreover, if v (1) (t ) and v (2) (t ) are two such real-analytic solutions, with radii of analyticity τ (1) (t ) and τ (2) (t ) respectively, then v (1) (t ) = v (2) (t ) for all t on the common interval of existence. 3. The two-dimensional case In this section we give the formal a priori estimates needed to prove Theorem 2.2. These estimates will be made rigorous in Sections 5 and 6. Let r  2 be fixed throughout the rest of the section. Assume that ( v , w ), p, and ρ are smooth solutions of (1.7)–(1.12), with v 0 , ρ0 ∈ X τ0 , for some τ0 > 0, which are independent of x2 . The horizontal domain M needs to be x2 independent, translation invariant in x2 , and hence without loss of generality we may consider M = (0, 1) × R. Since all derivatives with respect to x2 vanish we denote ∂x = ∂x1 , and similarly  = ∂x1 x1 . It is convenient to write the system (1.7)–(1.12) in component form

where ψ(x, z) =

z 0

∂t v 1 + v 1 ∂x v 1 + w ∂z v 1 + ∂x p + f v 2 = g ∂x ψ,

(3.1)

∂t v 2 + v 1 ∂x v 2 + w ∂z v 2 − f v 1 = 0,

(3.2)

∂ x v 1 + ∂ z w = 0,

(3.3)

∂ z p = 0,

(3.4)

∂t ρ + v 1 ∂x ρ + w ∂z ρ = 0,

(3.5)

ρ (x, ζ ) dζ . The boundary conditions for w and v 1 are w = 0,

on Γz ,

(3.6)

v 1 dz = 0,

on Γx ,

(3.7)

h 0

where Γx = {0, 1} × R. We note that there is no boundary condition for v 2 . Integrating the incompressibility condition (3.3) in z, and using (3.6) we get ∂x implies that for all x ∈ M we have

h 0

v 1 dz = 0. Combined with (3.7) this

h v 1 dz = 0.

(3.8)

0

In the two-dimensional case, the boundary conditions (3.6) and (3.8) give the pressure explicitly as a function of v and ρ , and imply the cancellation property (3.10) below, which turns out to be convenient in the a priori estimates needed to prove Theorem 2.2.

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Lemma 3.1. Let ( v , w , p , ρ ) be a smooth solution of (3.1)–(3.8). Then, after subtracting from p a function of time, the pressure is given at each instant of time t by

h p ( x) = − —

x1h v 21 (x, z) dz

0

−f







h



— v 2 x1 , x2 , z dz dx1 + g — ψ(x, z) dz, 0

0

(3.9)

0

where ψ is obtained from ρ via (1.6). In (3.9) we have suppressed the dependence in t for convenience. Also, we have the cancellation property

 α  ∂ x ∂ p , ∂ α v 1 = 0,

(3.10)

for any multi-index α ∈ N3 .

h

h

In (3.9) and in the following we use the notation —0 φ(x, z) dz = (1/h) 0 φ(x, z) dz, for any function φ . In the two-dimensional case we do not integrate in x2 , so that in (3.10) we denoted

φ1 , φ2 =

1 h 0

0

φ1 (x, z)φ2 (x, z) dz dx1 , for any pair of smooth real functions φ1 and φ2 .

Proof of Lemma 3.1. Integrating (3.1) in z and using the boundary condition (3.8), we obtain

 h ∂x



h v 21 (x, z) dz

+ hp (x) − g

0

h

ψ(x, z) dz = − f 0

v 2 (x, z) dz.

(3.11)

0

Here we used

h

h w ∂z v 1 dz = −

0

h v 1 ∂z w dz =

0

v 1 ∂x v 1 dz,

(3.12)

0

which holds by w |Γz = 0, and the incompressibility condition (3.3). The identity (3.9) then follows from (3.11) by integrating in x1 and subtracting a function of time. In order to prove (3.10), note that if αz = 0 or α2 = 0, then ∂ α p = 0. On the other hand, if α2 = αz = 0 and α1  1, then by (3.3) we have ∂ α v 1 = ∂xα1 v 1 = −∂xα1 −1 ∂z w. Moreover, the boundary α −1 condition (3.6) for w implies ∂x 1 w = 0 on Γz ; therefore integrating by parts in z gives

 α      ∂x ∂ p , ∂ α v 1 = − ∂x ∂ α p , ∂xα1 −1 ∂z w = ∂z ∂xα1 +1 p , ∂xα1 −1 w = 0. Lastly, we need to prove (3.10) in the case α = (0, 0, 0). Integrating by parts in x1 and using (3.3) we have ∂x p , v 1 = p , ∂z w since on Γx = {0, 1} × R by (3.7) we have

h

h p |Γx v 1 |Γx dz = p |Γx

0

v 1 |Γx dz = 0.

(3.13)

0

Integrating by parts in z we obtain p , ∂z w = 0 since w |Γz = 0. Therefore, (3.10) is proven.

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We now turn to a priori estimates needed to prove Theorem 2.2. From (2.1), (2.3), and (2.5) it follows that

  v (t )

d dt

X τ (t )

  = τ˙ (t ) v (t )Y

τ (t )

+

∞  r m   d α ∂ v (t ) 2 (m + 1) τ (t ) . L dt m!

(3.14)

m=0 |α |=m

Given a multi-index α ∈ N3 , we estimate (d/dt )∂ α v (t ) L 2 by applying ∂ α to (3.1)–(3.2) and taking the L 2 (D )-inner product ·,· with ∂ α v. Recall that in ·,· we integrate only with respect to x1 and z. Since ∂ α v ⊥ , ∂ α v = 0, we obtain

       1 d α ∂ v (t )22 + ∂ α ( v · ∇ v + w ∂z v ), ∂ α v + ∂ α ∇ p , ∂ α v = g ∂ α ∇ψ, ∂ α v . L 2 dt

(3.15)

To treat the second term on the left of (3.15), we use the Leibniz rule to write

 α   α   α    ∂ ( v · ∇ v + w ∂z v ), ∂ α v = ∂ β v · ∇∂ α −β v , ∂ α v + ∂ β w ∂z ∂ α −β v , ∂ α v . β β 0βα

0βα

The third term on the left of (3.15) vanishes by Lemma 3.1, and therefore by (3.15) and the Schwarz inequality we have

 d α  ∂ v  2  L dt

0βα

+

   α  β ∂ v · ∇∂ α −β v L 2 β

 α     ∂ β w ∂z ∂ α −β v  2 + g ∇∂ α ψ  2 . L L β

(3.16)

0βα

Substituting estimate (3.16) above into (3.14), and using the independence on the x2 variable, we have the a priori estimate

d dt

 v  X τ  τ˙  v Y τ + U ( v 1 , v 1 ) + U ( v 1 , v 2 ) + V ( w , v 1 ) + V ( w , v 2 ) + g ∂x ψ X τ ,

(3.17)

where for i ∈ {1, 2}, a vector function v ∈ X τ , and a scalar function v˜ ∈ C ∞ (D ), we denoted

U ( v i , v˜ ) =

m  ∞  



m=0 j =0 |α |=m |β|= j , βα

 

α  β

 (m + 1)r τ m , ∂ β v i ∂xi ∂ α −β v˜  L 2 m!

(3.18)

and

V ( w , v˜ ) =

m  ∞  



m=0 j =0 |α |=m |β|= j , βα

z

 

α  β

 (m + 1)r τ m ∂ β w ∂z ∂ α −β v˜  L 2 . m!

(3.19)

In (3.19) we denoted as usual w (x, z) = − 0 div v (x, ζ ) dζ . The following lemma summarizes the estimates on U ( v 1 , v i ) and V ( w , v i ), for i ∈ {1, 2}. It is a corollary of Lemma 4.1, which is proven in Section 4.

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Lemma 3.2. Let v ∈ Y τ , for some τ > 0 and r  2, and let w be given by (1.13). Then we have

 U ( v 1 , v 1 ) + U ( v 1 , v 2 )  C 0 1 + τ −1  v  X τ  v  Y τ ,

(3.20)

 V ( w , v 1 ) + V ( w , v 2 )  C 0 1 + τ −2  v  X τ  v  Y τ ,

(3.21)

and

for some sufficiently large positive constant C 0 = C (D ). To bound the last term on the right of (3.17) we note that

  ∂ α ∂x ψ  2 = |∂x ψ|m = L |α |=m



  z     α  ∂ ∂ x ρ ( x, ζ ) d ζ   

|α |=m, αz =0 0



+

|α |=m,αz 1

L2

 α 1 +1 α z −1  ∂x ∂z ρ (x, z) L 2

 C (h)|ρ |m+1 + |ρ |m , so that by possibly enlarging the constant C 0 from (3.20) and (3.21), we have

g ∂x ψ X τ  C 0 g ρ Y τ + g ρ  X τ .

(3.22)

We fix C 0 = C 0 (D ) as in (3.20)–(3.22) throughout the rest of this section. By (3.17), (3.20), (3.21), and (3.22) we have

d dt

  v (t )

X τ (t )

       v (t )  τ˙ (t ) + 3C 0 1 + τ (t )−2  v (t ) X Y τ (t ) τ (t )     + C 0 g ρ (t )Y + g ρ (t ) X . τ (t )

(3.23)

τ (t )

Similarly, by Lemma 4.1 we obtain from (3.5) an estimate for the growth of ρ (t ) X τ (t ) , namely

d dt

 ρ (t )

X τ (t )

   τ˙ (t )ρ (t )Y

  + U v 1 (t ), ρ (t ) + V w (t ), ρ (t )       τ˙ (t )ρ Y τ (t ) + C 0 1 + τ (t )−2  v (t ) X ρ (t )Y τ (t ) τ (t )      −2     + 2C 0 1 + τ (t ) v (t ) Y ρ (t ) X τ (t ) . τ (t )

(3.24)

τ (t )

By summing the estimates (3.23) and (3.24) we obtain

d dt

 ( v , ρ )

Xτ

         τ˙ + C 0 g + 3C 0 1 + τ −2 ( v , ρ ) X ( v , ρ )Y + g ( v , ρ ) X . τ

Define the decreasing function

τ

τ

(3.25)

τ (t ) by 





τ˙ + 20 C 0 g + 20 C 0 1 + τ −2 ( v 0 , ρ0 ) X τ e gt = 0,

(3.26)

0

and τ (0) = τ0 ; this uniquely determines τ in terms of the initial data. Let T ∗ be the maximal time such that τ (t )  0. It is clear that τ (t ) and T ∗ may be computed from (3.26) in terms of the initial data. By construction, we have at t = 0

I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746







τ˙ (t ) + C 0 g + 3C 0 1 + τ (t )−2  v (t ), ρ (t )  X τ (t ) < 0,

1729

(3.27)

and by (3.25) we then have for a short time

   v (t ), ρ (t ) 

X τ (t )

   ( v 0 , ρ0 ) X e gt .

(3.28)

τ0

It follows that (3.27), and hence (3.28), holds for all t < T ∗ . Moreover, by (3.25), we obtain that the solution is a priori bounded in L ∞ (0, T ∗ ; X τ ) ∩ L 1 (0, T ∗ ; (1 + τ −2 ) Y τ ) in the sense

   v (t ), ρ (t ) 

t X τ (t )

+ 10 C 0 g





e g (t −s)  v (s), ρ (s) Y

τ (s)

ds

0

  + 10 C 0 ( v 0 , ρ0 ) X e gt

t

τ0







1 + τ (s)−2  v (s), ρ (s) Y

τ (s)

ds   v 0  X τ0 e gt ,

(3.29)

0

for all t < T ∗ . The formal construction of the analytic solution ( v , ρ ) satisfying (3.29), with τ given by (3.26) is given in Section 5, which combined with the uniqueness result given in Section 6 completes the proof of Theorem 2.2. 4. The velocity estimate The goal of this section is to prove the next lemma. For convenience of notation, we suppress the time dependence of v, w, v˜ , and τ . By the two-dimensional case d = 2 we mean that all functions and the domain are independent of the variable x2 . In that case we recall that by φ2L 2 (D) we mean

1 h 0

0

|φ(x, z)|2 dz dx1 , for any smooth function φ independent of x2 .

Lemma 4.1. Let v ∈ Y τ for some τ > 0, and v˜ ∈ C ∞ (D ) ∩ L 2 (D ) with  v˜ Y τ < ∞. Fix r  d/2 + 1 and z d ∈ {2, 3}. Let w be determined from v via (1.13) i.e., w (x, z) = − 0 div v (x, ζ ) dζ . If U ( v i , v˜ ), for i ∈ {1, 2}, and V ( w , v˜ ) are as in (3.18) and (3.19) respectively, then we have the estimates

 U ( v i , v˜ )  C 0 1 + τ −θ (d)  v  X τ  v˜ Y τ ,

(4.1)

  V ( w , v˜ )  C 0 1 + τ −1−θ (d)  v  X τ  v˜ Y τ + C 0 τ −1 + τ −1−θ (d)  v Y τ  v˜  X τ ,

(4.2)

and

for some positive constant C 0 = C (r , D ), where θ(d) = 1 if d = 2, and θ(d) = 3/2 if d = 3. By setting v˜ = v 1 , and then v˜ = v 2 , Lemma 3.2 is a corollary of Lemma 4.1 for the case d = 2. For the rest of this section we fix d = 2. In the three-dimensional case the proof is identical except for two modifications. The integration is done also in x2 , and hence the exponents in Agmon’s inequality are different. We omit further details. Here,  ·  L p =  ·  L p (D) , for all 1  p  ∞. Proof of Lemma 4.1. In order to prove the lemma, we need to estimate the terms ∂ β v i ∂xi ∂ α −β v˜  L 2 and ∂ β w ∂z ∂ α −β v˜  L 2 , for all α , β ∈ N3 . For this purpose it is convenient to distinguish between two cases: 0  |β|  |α − β| and |α − β| < |β|  |α |. When 0  |β|  |α − β|, by the Hölder inequality, and by the two-dimensional Agmon inequality we have

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 β  ∂ v i ∂x ∂ α −β v˜  i

L2

     C ∂ β v i  L ∞ ∂xi ∂ α −β v˜  L 2  1/2  1/2    C ∂ β v i  L 2 ( + ∂zz )∂ β v i  L 2 ∂xi ∂ α −β v˜  L 2     + C ∂ β v i  2 ∂x ∂ α −β v˜  2 , L

L

i

(4.3)

for some constant C = C (D ) > 0. Recall that  is the horizontal Laplacian ∂x1 x1 when d = 2, and ∂x1 x1 + ∂x2 x2 when d = 3. Similarly, we estimate

 β      ∂ w ∂z ∂ α −β v˜  2  C ∂ β w  ∞ ∂z ∂ α −β v˜  2 L L L  β 1/2  1/2    C ∂ w  L 2 ( + ∂zz )∂ β w  L 2 ∂z ∂ α −β v˜  L 2     + C ∂ β w  L 2 ∂z ∂ α −β v˜ L 2 .

(4.4)

As in the above estimates, for multi-indices such that |α − β| < |β|  |α |, we have

 β  ∂ v i ∂x ∂ α −β v˜  i

L2

     C ∂ β v i L 2 ∂xi ∂ α −β v˜ L ∞    1/2  1/2  C ∂ β v i L 2 ∂xi ∂ α −β v˜ L 2 ∂xi ( + ∂zz )∂ α −β v˜  L 2     + C ∂ β v i L 2 ∂xi ∂ α −β v˜ L 2 ,

(4.5)

     C ∂ β w  L 2 ∂z ∂ α −β v˜  L ∞    1/2  1/2  C ∂ β w  L 2 ∂z ∂ α −β v˜  L 2 ∂z ( + ∂zz )∂ α −β v˜ L 2     + C ∂ β w  L 2 ∂z ∂ α −β v˜  L 2 .

(4.6)

and

 β  ∂ w ∂z ∂ α −β v˜ 

L2

α

|α |

Throughout this section we use the inequality β  |β| , which holds for all Moreover, since |α − β| = |α | − |β| for β  α , the identity





aβ bα −β =

|α |=m |β|= j , βα

 |β|= j

aβ

 

α , β ∈ N3 with β  α .

 bγ

(4.7)

|γ |=m− j

holds for all sequences {aβ } and {bγ }, and for j  m; this identity is useful when estimating U = U ( v i , v˜ ) and V = V ( w , v˜ ). With (4.3) and (4.5) in mind, we split U = Ulow + Uhigh , according to 0  j  [m/2] and [m/2] + 1  j  m respectively. Using (4.3), we have

Ulow  C

m/2] ∞ [  



m=0 j =0 |α |=m |β|= j , βα

+C

m/2] ∞ [  

 

1/2  1/2   (m + 1)r τ m ∂ β v i  L 2 ( + ∂zz )∂ β v i  L 2 ∂xi ∂ α −β v˜  L 2 m!

α  β



m=0 j =0 |α |=m |β|= j ,βα

  α  β   α −β  (m + 1)r τ m . v˜ L 2 ∂ v i L 2 ∂ xi ∂ β m!

(4.8)

I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

1731

By (4.7) it follows that





|α |=m |β|= j , βα

C

  1/2   α  β 1/2  ∂ v i L 2 ( + ∂zz )∂ β v i  L 2 ∂xi ∂ α −β v˜  L 2 β

          β  1/2    1/2  γ  m ∂ v i  2 ( + ∂zz )∂ β v i  2 ∂ ∂x v˜  2 . i L L L j

|β|= j

|β|= j

(4.9)

|γ |=m− j





γ ˜  2  C | v˜ | β We observe that m− j +1 and L |γ |=m− j ∂xi ∂ v |β|= j ( + ∂zz )∂ v i  L 2  C | v | j +2 . Hence, from (4.8) it follows by the discrete Hölder inequality and (4.9) that Ulow is bounded from above by

C

m/2] ∞ [  m =0 j =0

1/2 1/2 | v i | j | v i | j +2 | v˜ |m− j +1

 

m (m + 1)r τ m m!

j

+C

m/2] ∞ [ 

  | v i | j | v˜ |m− j +1

m (m + 1)r τ m

m =0 j =0

j

m!

1  1   m/2] ∞ [  ( j + 1)r τ j 2 ( j + 3)r τ j +2 2 (m − j + 2)r τ m− j −1 C |v i | j | v i | j +2 | v˜ |m− j +1 τ j! ( j + 2)! (m − j )! m =0 j =0

+C

m/2] ∞ [ 

|v i | j

m =0 j =0

( j + 1)r τ j j!

  (m − j + 2)r τ m− j | v˜ |m− j +1 , (m − j )!

(4.10)

where we have used the inequality

 

m (m + 1)r j

m!

(m − j )! (m + 1)r ( j + 1)1/2 ( j + 2)1/2 j !1/2 ( j + 2)!1/2 =  C , (4.11) r r / 2 r / 2 (m − j + 2) ( j + 1) ( j + 3) (m − j + 2)r ( j + 1)r /2 ( j + 3)r /2

which holds for all m  0, 0  j  [m/2], r  1, and a sufficiently large constant C , depending only on r. By (4.10), the discrete Hölder and Young inequalities imply

 Ulow  C 1 + τ −1  v  X τ  v˜ Y τ . By symmetry, from (4.5) and (4.7), we obtain an estimate for the high values of j. Namely

Uhigh  C

∞ 

m 

m=1 j =[m/2]+1

+C

∞ 

1/2 1/2  | v i | j |∂xi v˜ |m− j ∂xi ( + ∂zz ) v˜ m− j

m 

 

m (m + 1)r τ m m!

j

  | v i | j |∂xi v˜ |m− j

m (m + 1)r τ m

m=1 j =[m/2]+1

j

m!

  C 1 + τ −1  v  X τ  v˜ Y τ ,

(4.12)

for some positive constant C depending only on r and D . This completes the proof of (4.1). The estimate on V is in the same spirit, but we need to account for the derivative loss that occurs when estimating w in terms of v. First, note that the definition (1.13) of w and that of the semi-norms | · | j imply

|w| j 

 |β|= j , β3 1

 β  β3 −1  ∂x ∂z div v  L 2 +



 z      β  ∂ div v (x, ζ ) dζ   

|β|= j , β3 =0 0

 | v | j + C | v | j +1 , L2

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I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

for some constant C = C (h) > 0. We recall that div is the divergence operator acting on x. Similarly we have that | w | j  | v | j +2 + C | v | j +3 , and also |∂zz w | j  | v | j +2 . Next, we split V = Vlow + Vhigh , according to 0  j  [m/2] and [m/2] < j  m. By (4.4) and (4.7) we have

Vlow  C

m/2] ∞ [  

1/2

|v| j

m =0 j =0

+C

m/2] ∞ [   m =0 j =0

  m (m + 1)r τ m 1/2  1/2 1/2 + | v | j +1 | v | j +2 + | v | j +3 | v˜ |m− j +1 m! j

  m (m + 1)r τ m | v | j + | v | j +1 | v˜ |m− j +1 , m! j

(4.13)

for some constant C = C (r , D ) > 0. Using the fact that

 

m (m + 1)r

(m − j )! ( j + 1)!1/2 ( j + 3)!1/2 C (m − j + 2)r ( j + 2)r /2 ( j + 4)r /2

m!

j

holds for all m  0 and 0  j  [m/2], for some sufficiently large positive constant C depending only on r  2, we obtain

 Vlow  C 1 + τ −2  v  X τ  v˜ Y τ . Lastly, to estimate Vhigh , we note that by (4.6) and (4.7), we have

Vhigh  C

∞ 

m 

m=1 j =[m/2]+1

+C

∞ 

1/2 1/2  | w | j |∂z v˜ |m− j ∂z ( + ∂zz ) v˜ m− j

∞ 

m 

| w | j |∂z v˜ |m− j 

m=1 j =[m/2]+1

+C

∞ 

m 

m (m + 1)r τ m j

m!

 

m 

m=1 j =[m/2]+1

C

 

m (m + 1)r τ m j

m!

  1/2 m (m + 1)r τ m 1/2 | v | j + | v | j +1 | v˜ |m− j +1 | v˜ |m− j +3 m! j 

m=1 j =[m/2]+1

  m (m + 1)r τ m | v | j + | v | j +1 | v˜ |m− j +1 . m! j

(4.14)

By symmetry with the Vlow estimate, the lower-order terms (the ones containing | v | j ) on the right side of (4.14) are estimated by C (1 + τ −1 ) v  X τ  v˜ Y τ . On the other hand, the terms containing | v | j +1 are similarly bounded by C (τ −1 + τ −2 ) v Y τ  v˜  X τ concluding the proof of (4.2), and hence of the lemma. 2 5. Construction of the solution The formal construction of the solutions is via the Picard iteration. Let v (0) = v 0 , given, with v 0 satisfying the compatibility conditions (1.17) and (1.18). For n ∈ N, let

ρ (0) = ρ0 be

z w (n) (x, z, t ) = −

div v (n) (x, ζ, t ) dζ, 0

(5.1)

I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

1733

and

z ψ

(n)

ρ (n) (x, ζ, t ) dζ.

( x, z , t ) =

(5.2)

0

The density iterate is given by

ρ (n+1) (t ) = ρ0 −

t



v (n) (s) · ∇ + w (n) (s)∂z



ρ (n) (s) ds,

(5.3)

0

and motivated by Lemma 3.1, we define the pressure

p

(n+1)

h



(n) 2

( x, t ) = − — v 1

x1h (x, z, t ) dz − f

(n)  

— v2

0

0



x1 , x2 , z, t dz dx1

0

h

+ g — ψ (n+1) (x, z, t ) dz.

(5.4)

0

Lastly, the velocity iterate is constructed as

v

(n+1)

t (t ) = v 0 −





v (n) (s) · ∇ + w (n) (s)∂z v (n) (s) ds

0

t −



∇ p (n+1) (s) − g ∇ψ (n+1) (s) + f v (n)⊥ (s) ds,

(5.5)

0

for all n ∈ N. Taking the time derivative of the first component of (5.5), integrating in z, and using the fact that ∂x1 p (n) is obtained from (5.4), we obtain that ∂t

h

the initial data, v 01 , has zero vertical average, we obtain

h

(n)

(n+1)

v1

h

0

0

dz = 0. Since the first component of

(n)

v 1 (x, z) dz = 0 for all x ∈ M and n  0.

h

(n)

Therefore the compatibility conditions 0 div v 1 dz = 0 and the boundary condition 0 v 1 |Γx dz = 0 are conserved for all n ∈ N. Recall that we denote by div the differential operator acting only on x. Assume that ( v 0 , ρ0 ) ∈ X τ0 +ε , for some 0 <  < τ0 . In particular, we have ( v 0 , ρ0 ) ∈ Y τ0 . We define τ (t ) by τ (0) = τ0 and







τ˙ (t ) + 20 C 0 g + 20 C 0 1 + τ −2 (t ) ( v 0 , ρ0 ) X τ e gt = 0,

(5.6)

0

where the constant C 0 = C 0 (r , D ) is fixed in Lemma 4.1. First we show that the sequence of approximations v (n) is bounded in L ∞ (0, T ; X τ ) ∩ L 1 (0, T ; (1 + τ −2 )Y τ ) for some sufficiently small T > 0, depending solely on the initial data. Lemma 5.1. Let ( v 0 , ρ0 ) ∈ X τ0 +ε and τ (t ) be defined by (5.6). The approximating sequence {( v (n) , ρ (n) )}n0 , constructed via (5.1)–(5.5), satisfies

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I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

  sup  v (n) (t ), ρ (n) (t )  X

t ∈[0, T ]

T τ (t )

+ 10 C 0 g





e g ( T −t )  v (n) (t ), ρ (n) (t ) Y

τ (t )

dt

0

  + 20 C 0 ( v 0 , ρ0 ) X e g T

T



τ0





1 + τ −2 (t )  v (n) (t ), ρ (n) (t ) Y

τ (t )

dt

0

   3e g T ( v 0 , ρ0 ) X ,

(5.7)

τ0

for all n  0, where T = T ( v 0 , ρ0 ) > 0 is sufficiently small. Proof. When n = 0, since τ is decreasing and since  < τ0 , for all t  0 we have ( v 0 , ρ0 )Y τ0  C ( v 0 , ρ0 ) X τ0 +ε /ε , for some constant C . Sufficient conditions for the bound (5.7) to hold in the case n = 0 are that T is chosen such that







10 C 0 C e g T − 1 ( v 0 , ρ0 ) X

τ0 + ε

   ε ( v 0 , ρ0 ) X e g T

(5.8)

τ0

and

  20 C 0 C ( v 0 , ρ0 )

T X τ0 + ε





1 + τ −2 (t ) dt  ε .

(5.9)

0

The condition (5.8) holds if T  T 1 , where T 1 ( , C 0 , C , g , ( v 0 , ρ0 ) X τ0 , ( v 0 , ρ0 ) X τ0 +ε ) > 0 is deter-

mined explicitly. Also, by the construction of τ (cf. (5.6)) we have 20 C 0 (1 + τ −2 ) < −τ˙ /( v 0 , ρ0 ) X τ0 , so that the condition (5.9) is satisfied if we choose T so that

τ0 − τ ( T ) 

ε( v 0 , ρ0 ) X τ0 , C ( v 0 , ρ0 ) X τ0 +ε

(5.10)

which is satisfied if T  T 2 , where T 2 (ε , τ0 , C 0 , C , g , ( v 0 , ρ0 ) X τ0 , ( v 0 , ρ0 ) X τ0 +ε ) > 0 may be computed explicitly from (5.6) and (5.10). Thus (5.7) holds for n = 0 if T  min{ T 1 , T 2 }. We proceed by induction. By (5.3), (5.5), and Lemma 4.1, similarly to estimate (3.25), we obtain

      d  v (n+1) , ρ (n+1)   (τ˙ + C 0 g ) v (n+1) , ρ (n+1)  + g  v (n+1) , ρ (n+1)  X Y Xτ τ τ dt      + 3C 0 1 + τ −2  v (n) , ρ (n)  X  v (n) , ρ (n) Y τ τ  (n+1) (n+1)   (n+1) (n+1)   + g v   (τ˙ + C 0 g ) v ,ρ ,ρ Yτ Xτ       + 9C 0 1 + τ −2 e g T ( v 0 , ρ0 ) X  v (n) , ρ (n) Y , τ0

τ

(5.11)

by the induction assumption. In the above we also used the fact that by Lemma 3.1 we have ∂ α ∇ p (n) , ∂ α v (n+1) = 0. Using that τ was chosen to satisfy (5.6), the above estimate and Grönwall’s inequality give

I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

 (n+1)   v (t ), ρ (n+1) (t ) 

t + 10 C 0 g

X τ (t )

1735





e g (t −s)  v (n+1) (s), ρ (n+1) (s) Y

τ (s)

ds

0

  + 20 C 0 ( v 0 , ρ0 )

t X τ0

e



gt





1 + τ −2 (s)  v (n+1) (s), ρ (n+1) (s) Y

τ (s)

ds

0

   ( v 0 , ρ0 ) X e gt τ0

+ 9e

gT

  C 0 ( v 0 , ρ0 )

t X τ0

e

gt







1 + τ −2 (s)  v (n) (s), ρ (n) (s) Y

τ (s)

ds.

(5.12)

0

The proof of Lemma 5.1 is completed by taking the supremum over t ∈ [0, T ] of the above inequality, by the induction assumption, and by additionally letting T be small enough so that 27e g T  40. 2 We conclude the construction of the solution by showing that the map v (n) → v (n+1) is a contraction in L ∞ (0, T ; X τ ) ∩ L 1 (0, T ; (1 + τ −2 )Y τ ), for some sufficiently small T depending on the initial data. Lemma 5.2. Let v˜ = v (n+1) − v (n) , and ρ˜ (n) = ρ (n+1) − ρ (n) for all n  0. Let ( v 0 , ρ0 ) ∈ X τ0 +ε , τ (t ) be defined by (5.6), and T be as in Lemma 5.1. If for all n  0 we let (n)

 (n)  an = sup  v˜ (t ), ρ˜ (n) (t )  X t ∈[0, T ]

T τ (t )

+ 10 C 0 g



e g ( T −t )  v˜

(n)

 (t ), ρ˜ (n) (t ) Y

τ (t )

dt

0

  + 20 C 0 ( v 0 , ρ0 )

T X τ0

e

gT







(n) 1 + τ −2 (t )  v˜ (t ), ρ˜ (n) (t ) Y

τ (t )

dt ,

(5.13)

0

then we have 20 an  19 an−1 for all n  1.

˜ (n) = w (n+1) − w (n) , ψ˜ (n) = ψ (n+1) − ψ (n) , and p˜ (n) = p (n+1) − p (n) . Then we have that Proof. Denote w the difference of two iterations satisfies the equations

  ˜ (n−1) ∂z v (n−1) ∂t v˜ (n) + v (n) · ∇ + w (n) ∂z v˜ (n−1) + v˜ (n−1) · ∇ + w + ∇ p˜ (n) − g ∇ ψ˜ (n) + f v˜ (n−1)⊥ = 0,

(5.14)

and

  ˜ (n−1) ∂z ρ (n−1) = 0, ∂t ρ˜ (n) + v (n) · ∇ + w (n) ∂z ρ˜ (n−1) + v˜ (n−1) · ∇ + w (n)

(5.15)

with initial conditions v˜ (0) = 0 and ρ˜ (n) (0) = 0, for all n  0. Since the approximate solutions u (n) satisfy the boundary conditions (1.11)–(1.12), similarly to the proof of (3.10), it can be shown that ∂ α ∇ p˜ (n−1) , ∂ α v˜ (n) = 0 for all α ∈ N3 . Hence, from (5.14), (5.15), and Lemma 4.1, we obtain

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I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

    d  (n) (n)   v˜ , ρ˜   (τ˙ + C 0 g ) v˜ (n) , ρ˜ (n)  + g  v˜ (n) , ρ˜ (n)  X Y Xτ τ τ dt    + 3 C 0 1 + τ −2  v (n) , ρ (n)  X τ     +  v (n−1) , ρ (n−1)   v˜ (n−1) , ρ˜ (n−1)  Xτ



−2

 (n) (n)   v ,ρ 

Yτ

+ 3 C0 1 + τ Yτ  (n−1) (n−1)   (n−1) (n−1)    v˜  . + v ,ρ , ρ˜ Y X τ

(5.16)

τ

Using the definition of τ˙ (cf. (5.6)), the estimate in Lemma 5.1, Grönwall’s inequality, and taking the supremum for t ∈ [0, T ], we obtain

an  18C 0 e

  ( v 0 , ρ0 ) X

gT 

T



τ0







(n−1) 1 + τ −2 e g ( T −t )  v˜ , ρ˜ (n−1) Y dt τ

0

+



T  (n−1) (n−1)       sup  v˜ , ρ˜ 3C 0 1 + τ −2 e g ( T −t )  v (n) , ρ (n) Y X τ

t ∈[0, T ]

τ

0

  +  v (n−1) , ρ (n−1) Y dt .

(5.17)

τ

If T is taken such that 18 e g T  19, then the above estimate and the definition of an (cf. 5.13) imply that

an 

19 20

(5.18)

an−1 .

This concludes the proof of the lemma, showing that the map ( v (n) , ρ (n) ) → ( v (n+1) , ρ (n+1) ) is a strict contraction. The existence of a solution to (1.7)–(1.12) in the class L ∞ (0, T ; X τ )∩ L 1 (0, T ; (1 + τ −2 )Y τ ), with τ (t ) given by (5.6) follows from the classical fixed point theorem. 2 6. Uniqueness Fix the initial data ( v 0 , ρ0 ) ∈ X τ0 +ε , real-analytic on D with radius of analyticity strictly larger than τ0 , for some positive ε < τ0 . Let τ (t ) be defined by τ (0) = τ0 and τ˙ + 20C 0 g + 20C 0 (1 + τ −2 )( v 0 , ρ0 ) X τ0 e gt = 0, where C 0 = C (D, r ) > 0 is the fixed constant defined in Lemma 3.2. Let T ∗ be the maximal time such that τ (t )  0. Assume that there exist two solutions ( v (1) , ρ (1) ) and ( v (2) , ρ (2) ) to (1.7)–(1.12) evolving from initial data ( v 0 , ρ0 ), such that for i = 1, 2, we have

 (i )   v (t ), ρ (i ) 

t X τ (t )

+ 10C 0 g





e g (t −s)  v (i ) (s), ρ (i ) (s) Y

τ (s)

ds

0

  + 10C 0 ( v 0 , ρ0 ) X e g T

t

τ0







1 + τ −2 (s)  v (i ) (s), ρ (i ) (s) Y

τ (s)

ds < ∞,

(6.1)

0

for all 0  t < T ∗ . Similarly to (3.28) we have that  v (i ) (t ) X τ (t )  ( v 0 , ρ0 ) X τ0 e gt for i ∈ {1, 2} and

for all 0  t < T ∗ . Let w (i ) and p (i ) be the vertical velocity and the pressure associated to v (i ) , and

I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

let ψ (i ) (x, z) =

1737

z

ρ (i) (x, ζ ) dζ , for i ∈ {1, 2}. We denote the difference of the solutions v (1) − v (2) = v, 0 ρ (1) − ρ (2) = ρ , and similarly define w, ψ and p. Then ( v , w , p , ρ ) satisfy the equations  ∂t v + v (1) · ∇ + w (1) ∂z v + ( v · ∇ + w ∂z ) v (2) + ∇ p + f v ⊥ = g ∇ψ,

(6.2)

div v + ∂z w = 0,

(6.3)



∂t ρ + v

(1 )

∂ z p = 0, · ∇ + w ∂z ρ + ( v · ∇ + w ∂z )ρ (2) = 0,

(6.4)

(1 )

(6.5)

in D × (0, T ), with the corresponding boundary and initial value conditions

w ( x , z , t ) = 0,

on Γz × (0, T ),

(6.6)

v (x, z, t ) dz · n = 0,

on Γx × (0, T ),

(6.7)

in D,

(6.8)

ρ (x, z, 0) = 0, in D.

(6.9)

h 0

v ( x , z , 0) = 0,

Similarly to the a priori estimates of Section 3, by (6.2)–(6.9) and Lemma 4.1, we obtain that

d dt

         ( v , ρ ) ( v , ρ )  τ˙ + C 0 g + 3C 0 1 + τ −2  v (1) , ρ (1)  +  v (2) , ρ (2)  X X X Y τ

τ

τ

     + g ( v , ρ ) X + 3C 0 1 + τ −2  v (1) , ρ (1) Y τ τ  (2) (2)    +  v , ρ  ( v , ρ ) , Yτ

τ

(6.10)

Xτ

where C 0 > 0 is the constant from Lemma 3.2. But by the construction of τ we have τ˙ + 20C 0 g + 20C 0 (1 + τ −2 )( v 0 , ρ0 ) X τ0 e gt = 0, and by also using ( v (1) , ρ (1) ) X τ + ( v (2) , ρ (2) ) X τ  2( v 0 , ρ0 ) X τ0 e gt , we obtain

d dt

 ( v , ρ )

Xτ

       + 10C 0 g ( v , ρ )Y + 10C 0 ( v 0 , ρ0 ) X e gt 1 + τ −2 ( v , ρ )Y τ

τ0

          g ( v , ρ ) X + 3C 0 1 + τ −2  v (1) Y +  v (2) Y ( v , ρ ) X . τ

τ

τ

τ

τ

(6.11)

It is straightforward to check that (6.1), (6.8), (6.9), (6.11), and Grönwall’s inequality imply that ( v , ρ ) X τ = 0 for all t ∈ [0, T ∗ ), and thereby proving the uniqueness of the solutions. 7. The three-dimensional case In this section we sketch the proof of Theorem 2.3. As opposed to the two-dimensional case, Lemma 3.1 does not hold, and hence we need to estimate the analytic norm of the pressure. We emphasize the necessary changes from the two-dimensional case. In Section 7.1 we give the proof of the pressure estimate in the case of periodic boundary ditions in the x-variable. In this case p may be written explicitly as a function of v and ρ (cf. below), thereby simplifying the analysis.

here only con(7.6)

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When M is an analytic domain with boundary, the pressure is given implicitly as a solution of an elliptic Neumann problem (cf. Temam [25] for the Euler equations). We explore the transfer of normal to tangential derivatives in the higher-order estimates for the pressure and introduce a new suitable analytic norm to combinatorially encode this transfer. This gives us the necessary estimate (cf. Lemma 7.1) to prove that the pressure has the same radius of analyticity as the velocity. In Section 7.2 we give the proof of the pressure estimate in the case when M is a half-space. 7.1. The periodic case: M = [0, 2π ]2 Here we give the a priori estimates for the case when the boundary condition (1.12) is replaced by the periodic boundary condition in the x-variable. Assume that (u , p , ρ ) is a smooth solution to (1.7)– in the x-variable. Since the pressure is defined up to a function of time, we may (1.11), M-periodic

assume that M p dx = 0. Let v 0 , ρ0 ∈ X τ0 for some τ0 > 0, and fixed r  5/2. Similarly to estimate (3.17) we have

d dt

  v (t )

X τ (t )

   τ˙ (t ) v (t )Y

τ (t )

+ U ( v , v ) + V ( w , v ) + P + g ∇ψ X τ ,

2

(7.1)

2

where U ( v , v ) = i , j =1 U ( v i , v j ) and V ( w , v ) = j =1 V ( w , v j ), with U ( v i , v j ) and V ( w , v j ) being defined by (3.18) and (3.19) respectively. In (7.1) above, we have denoted the upper bound on the pressure term by ∞    α  ∇∂ p 

P=

m=1 |α |=m

= h1/2

∞ 



L 2 (D )

(m + 1)r τ m m!

 α  ∇∂ p 

m=1 |α |=m, α3 =0

L 2 ( M)

(m + 1)r τ m . m!

(7.2)

Here we used the fact that p is z-independent, and the fact that due to the boundary condition (1.12) we have ∇ p , v = p , div v = − p , ∂z w = 0. We note that in the three-dimensional case the cancellation property (3.10) does not hold, and therefore the pressure term does not vanish in the estimate (7.1). To estimate P , we use the fact that the pressure may be computed explicitly from the

h

velocity. First, note that 0 div v dz = 0, and therefore, by integrating (1.7) in the z-variable, and then applying the divergence operator in the x-variable, we obtain

h

h

− p = ∂k — ( v j ∂ j v k + w ∂z v k ) dz + f 0

h (∂1 v 2 − ∂2 v 1 ) dz − g 

0

ψ dz.

(7.3)

0

In (7.3) we have used the summation convention over repeated indices 1  j , k  2, and denoted by ∂ j the partial derivative ∂/∂ x j , for all 1  j  2. Integrating by parts in the z variable, it follows from (1.8) and (1.11) that

h

h w ∂z v k dz = −

0

h v k ∂z w dz =

0

v k ∂ j v j dz, 0

(7.4)

I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

1739

and therefore, by (7.3) we have

h h h − p = ∂k ∂ j — v j v k dz + f — (∂1 v 2 − ∂2 v 1 ) dz − g  — ψ dz. 0

0

(7.5)

0

The periodic boundary conditions in the x-variable allow for a simple solution to (7.5), namely

h

h

h

p = R k R j — v j v k dz + f (−)−1/2 — ( R 1 v 2 − R 2 v 1 ) dz + g — ψ dz, 0

0

(7.6)

0

where R j is the jth Riesz transform, classically defined by its Fourier symbol i ξ j /|ξ |. The boundedness of the Riesz transforms on L 2 (M), the Hölder inequality, and the Leibniz rule give the bound

P  Ch1/2

∞ 





  h      α — v j v k dz  ∂i ∂  

m=1 |α |=m, α3 =0 1i , j ,k2

+h

1/2

∞ 

   h    α  f ∂ — v dz  



m=1 |α |=m, α3 =0

C

∞ 



0

0



 α  ∂ ( v j ∂i v k )

m=1 |α |=m, α3 =0 1i , j ,k2

C

∞ 



 α 

m=1 |α |=m, α3 =0 βα

L 2 ( M)

β



L 2 ( M)

(m + 1)r τ m m!

 h     α  + g ∂ — ∇ψ dz  

L 2 (D )

0

 L 2 ( M)

(m + 1)r τ m m!

(m + 1)r τ m + f  v  X τ + g ∇ψ X τ m!

r m  β  ∂ v j ∂ α −β ∂i v k  2 (m + 1) τ L (D ) m!

1i , j ,k2

+ f  v  X τ + g ∇ψ X τ .

(7.7)

The first term on the right of (7.7) is estimated similarly to U ( v , v ) via Lemma 4.1, the case d = 3, and we obtain the a priori pressure estimate

 P  C 0 1 + τ −5/2  v  X τ  v Y τ + f  v  X τ + g ∇ψ X τ .

(7.8)

By possibly enlarging C 0 , as shown in (3.22) we also have ∇ψ X τ  C 0 ρ Y τ + ρ  X τ , and therefore

 P  C 0 1 + τ −5/2  v  X τ  v Y τ + f  v  X τ + g ρ  X τ + C 0 g ρ Y τ .

(7.9)

Combining the a priori estimate (7.1), the bounds on U ( v , v ) and V ( w , v ) obtained from Lemma 4.1, and the pressure estimate (7.9), in analogy to (3.23), we obtain the bound

d dt

   v  X τ  τ˙ + 4C 0 1 + τ −5/2  v  X τ  v Y τ + 2C 0 g ρ Y τ + f  v  X τ + 2g ρ  X τ . (7.10)

Since the evolution of the density in analogy to (3.24) we have

d dt

ρ (cf. (1.10)) does not involve the pressure term, using Lemma 4.1,

  ρ  X τ  τ˙ ρ Y τ + 2C 0 1 + τ −5/2  v  X τ ρ Y τ + 4C 0 1 + τ −5/2  v Y τ ρ  X τ ,

(7.11)

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I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

and therefore, by combining (7.10) and (7.11) we obtain

d dt

 ( v , ρ )

Xτ

         τ˙ + 2C 0 g + 4C 0 1 + τ −5/2 ( v , ρ ) X ( v , ρ )Y + C 1 ( v , ρ ) X , τ

where C 1 = max{ f , 2g }. With

τ

τ

(7.12)

τ (t ) defined by τ (0) = τ0 and 





τ˙ + 20C 0 g + 20C 0 1 + τ −5/2 ( v 0 , ρ0 ) X τ e C 1 t = 0,

(7.13)

0

the rest of the proof of Theorem 2.3, namely the estimate (2.7), follows in analogy to the twodimensional case (cf. Section 3). The uniqueness of x-periodic solutions in the space L ∞ (0, T ∗ ; X τ ) ∩ L 1 (0, T ∗ ; (1 + τ −5/2 )Y τ ) follows as in Section 6, with the only modification being the power of τ in the estimates is now −5/2 instead of −2. The construction of the x-periodic solution is similar to Section 5, with one additional modification: Instead of p (n) being defined by (7.6), we define the nth iterate of the pressure via

p

(n+1)

h

−1/2

(n) (n)

h

= R j R k — v j v k dz + f (−) 0



(n)

(n)

— R1 v2 − R2 v1 0

h

dz + g — ψ (n+1) dz.

(7.14)

0

To avoid redundancy we omit further details. 7.2. A domain with boundary: M is the upper half-plane Let M = {x ∈ R2 : x1 > 0}, so that Γx = ∂ M × (0, h) = {(x, z) ∈ R3 : x1 = 0, 0 < z < h}. Therefore

h

the side boundary condition (1.12) is 0 v 1 (0, x2 , z) dz = 0. In order to close the estimates for the pressure (cf. Lemma 7.1), in the case of the half-pane it is necessary to use a modified Sobolev seminorm (see also Kukavica and Vicol [10]), instead of the classical | · |m from (2.1). We let



[ v ]m =

|α |=m

1  α  ∂ v  2 , L (D )

(7.15)

(m + 1)r τ m , m!

(7.16)

(m + 1)r τ m−1 . (m − 1)!

(7.17)

2α 1

and define the corresponding analytic X τ norm

 v  Xτ =

∞ 

[ v ]m

m =0

and respectively the Y τ semi-norm ∞ 

 v Y τ =

[ v ]m

m =1

As in the periodic case, we have the a priori estimate

d dt

  v (t )

X τ (t )

   τ˙ (t ) v (t )Y

τ (t )

+ U ( v , v ) + V ( w , v ) + P + g ∇ψ X τ ,

where U ( v , v ) and V ( w , v ) are defined similarly to (3.18) and (3.19), namely by

(7.18)

I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

m  ∞  

U ( v , v˜ ) =

m=0 j =0 |α |=m

 



1 2α 1

α  β

|β|= j , βα

1741

 (m + 1)r τ m , ∂ β v · ∇∂ α −β v˜  L 2 m!

(7.19)

 (m + 1)r τ m ∂ β w ∂z ∂ α −β v˜  L 2 , m!

(7.20)

and by

V ( w , v˜ ) =

m  ∞   m=0 j =0 |α |=m

2α 1

 



1

α  β

|β|= j , βα

and the pressure term is given by ∞ 

P = h1/2



m=1 |α |=m, α3 =0

1  

2α 1

 (m + 1)r τ m ∇∂ α p  L 2 (M) . m!

(7.21)

Recall that the term corresponding to m = 0 in (7.21) is missing since the side boundary condition (1.12) implies that ∇ p , v = 0. It is straightforward to check that the proof of Lemma 4.1 also applies to the above defined operators U and V , and hence we have the three-dimensional bounds

 U ( v , v ) + V ( w , v )  2C 0 1 + τ −5/2  v  X τ  v Y τ .

(7.22)

To estimate P , we note that by (7.3) the vertical average of the full pressure

h

h

p˜ (x) = — P (x, z) dz = p (x) − g — ψ(x, z) dz 0

(7.23)

0

satisfies

h

h

− p˜ = ∂k — ( v j ∂ j v k + v k ∂ j v j ) dz + f — (∂1 v 2 − ∂2 v 1 ) dz = F 0

(7.24)

0

h

for all x ∈ M, where the summation convention over 1  j , k  2 is used. By applying —0 dz to (1.7), and taking the inner product with n = (−1, 0), the outward unit normal vector to M, we obtain that (7.24) is supplemented with the boundary condition

h h ∂ p˜ = − — ( v j ∂ j v k + w ∂z v k ) dz · nk − f — v k⊥ dz · nk ∂n 0

h

0

h

= — ( v 1 ∂ j v j + v j ∂ j v 1 ) dz + f — v 2 dz = G , 0

(7.25)

0

where j ∈ {1, 2}. We note that as opposed to the Euler equations on a half-space (cf. [10]) the nonlocal boundary condition on the velocity implies that the boundary condition for p is non-homogeneous (i.e., ∂ p /∂ n may be nonzero),

creating additional difficulties. After subtracting a function of time from the full-pressure we have D P (x, z) dx dz = 0, and therefore there exists a unique smooth solution to the boundary value problem (7.24)–(7.25).

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I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

h

Lemma 7.1. The smooth solution p˜ = p − g —0 ψ dz to the elliptic Neumann problem (7.24)–(7.25), satisfies

[∇ p˜ ]m  C 1 [ F ]m−1 + C 1 [G ]m ,

(7.26)

where C 1 is a universal constant, independent of m. Proof. In order to bound

[∇ p˜ ]m =



1  

|α |=m, α3 =0

2α 1

 ∇∂ α p˜  L 2 (M) ,

(7.27)

we estimate tangential and normal derivatives separately. To estimate tangential derivatives of the α pressure, we note that for any α2  0, the function ∂2 2 p˜ is a solution of the elliptic Neumann problem

 − ∂2α2 p˜ = ∂2α2 F ,

(7.28)

α2

∂(∂2 p˜ ) = ∂2α2 G , ∂n

(7.29)

and hence the classical H 2 regularity theorem, and the trace theorem give that there exists C 1 > 0 such that

 α2  ∂ p˜  ˙ 2

H ( M)

2

     C 1 ∂2α2 F  L 2 (M) + C 1 ∂2α2 G  H 1 (M) .

(7.30)

To estimate normal derivatives, we note that

−∂11 p˜ = − p˜ + ∂22 p˜ = F + ∂22 p˜ ,

(7.31)

2 (−∂11 )2 p˜ = −∂11 F + ∂22 ( F + ∂22 p˜ ) = −∂11 F + ∂22 F + ∂22 p˜ ,

(7.32)

and by induction one may show that

k +1 (−∂11 )k+1 p˜ = ∂22 p˜ +

k  (−1)l ∂12l ∂22k−2l F .

(7.33)

l =0

Therefore, if α = (α1 , α2 , 0) ∈ N3 is such that (7.30) we have

α1  2, and α1 = 2k + 2 is even, then by (7.33) and

k   α   2l 2k−2l+α2   2k+2+α2  ∇∂ p˜  2 2 ∂ ∂ ∂ ˜  ∇ p + ∇ F  L 2 ( M) 1 2 2 L ( M) L ( M)

(7.34)

l =0

 |α |−1   C 1 ∂ G 2

Similarly, if

H 1 ( M)

+ C1

α 1 −2

 j |α |− j −2  ∂ ∂ ∇F 1 2

j =0

L 2 ( M)

.

(7.35)

α1  3, and α1 = 2k + 3 is odd, then similar arguments show that  α  ∇∂ p˜ 

L 2 ( M)

α 1 −2  |α |−1   j |α |− j −2  ∂ ∂  C 1 ∂2 G  H 1 ( M) + C 1 ∇ F  L 2 ( M) . 1 2 j =1

(7.36)

I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

Lastly, when

1743

α1  1, and |α |  1, then  α   |α |−1   |α |−1  ∇∂ p˜  2  C 1 ∂2 F  L 2 ( M) +  ∂ 2 G  H 1 ( M) . L ( M)

(7.37)

Summarizing estimates (7.30)–(7.37) we obtain



m     1  m −1  ∇∂ α p˜  2 ∂  C G  H 1 (M) + 2C 1 ∂2m−1 F  L 2 (M) 1 2 L ( M ) α α 2 1 2 1

1 

|α |=m, α3 =0

α 1 =0

+ C1

 m α 1 −2   1  j |α |− j −2 1 ∂ ∂ 2 ∇ F , (7.38) L ( M) 2 α 1 − j 2j 1 2

α 1 =2 j =0

for all m  1. Here we see why the introduction of the normalizing factors 1/2α1 was necessary. Without them the constant C 1 in the above estimates would depend linearly on m. However since

k k 1/2 < ∞, by possibly enlarging C 1 (which is independent of m) we have

 |α |=m, α3 =0

2α 1

   ∇∂ α p˜  2  C 1 ∂2m−1 G  H 1 (M) + C 1 L ( M)



1 

concluding the proof of the lemma.

|α |=m−1, α3 =0

1  α  ∂ F  2 , L ( M) 2α 1

(7.39)

2

Remark 7.2. If C 1 would depend on m, and would grow unboundedly as m → ∞, then the additional loss of one full derivative coming from estimating w in terms of v, prevents the estimates from closing. The normalizing weights 1/2α1 may be viewed as a suitable combinatorial encoding of the transfer of normal to tangential derivatives in (7.33). Lemma 7.3. Let ( v , ρ ) ∈ X τ , and p˜ be the unique smooth solution of the elliptic Neumann problem (7.24)–

h

(7.25). Then the term P as defined in (7.21), with p = p˜ + g —0 ψ , is bounded by

 P  C 1 1 + τ −5/2  v  X τ  v Y τ + g ρ  X τ + C 1 g ρ Y τ + C 1 f  v  X τ ,

(7.40)

for some positive universal constat C 1 . Proof. By the triangle inequality and the definition of p˜ cf. (7.23), we have that

 |α |=m, α3 =0

1  2α 1

 ∇∂ α p  2  L ( M)



1 

|α |=m, α3 =0

+g

2α 1

 ∇∂ α p˜ 

 |α |=m, α3 =0



1  

L 2 ( M)

h

 

 ∇∂ α — ψ dz

2α 1 



0

. L 2 ( M)

From (7.39) and the above estimate it follows that the pressure term is bounded by

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I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

P  C 1 h1/2

∞ 



 α  (m + 1)r τ m 1  α  ∂ F  2 ∂ G  1 + L ( M) H ( M) 2α 1 m!

m=1 |α |=m−1, α3 =0

+ gh

1/2

∞ 



1  



 

h

 ∇∂ α — ψ dz

m=1 |α |=m, α3 =0

2α 1 



0

(7.41)

. L 2 ( M)

Using Hölder’s inequality and the definitions of F and G, we obtain

P  C1

∞ 



m=0 |α |=m, α3 =0

r m  1  ∂ α ( v j ∂ j v k + v k ∂ j v j ) 2 (m + 1) τ L ( D ) α 1 2 m!

+ g ∇ψ X τ + C 1 f

∞ 



m=0 |α |=m, α3 =0

 C1

m ∞  



m=0 j =0 |α |=m, α3 =0



1 2α 1

|β|= j , βα

r m 1  α  ∂ v  2 (m + 1) τ L ( D ) α 2 1 m!

 

 (m + 1)r τ m ∂ β v · ∇∂ α −β v  L 2 (D) m!

α  β

+ g ρ  X τ + C 1 g ρ Y τ + C 1 f  v  X τ .

(7.42)

The first term on the right of the above term is bounded as U ( v , v ) using the three-dimensional case of Lemma 4.1, concluding the proof of the lemma. 2 We now conclude the proof of Theorem 2.3 in the case when M is a half-space. Combining (7.40) with (7.18) and (7.22), we obtain the analytic a priori estimate

d dt

   v  X τ  τ˙ + C 2 1 + τ −5/2  v  X τ  v Y τ + g ρ  X τ + C 2 g ρ Y τ + C 2 f  v  X τ ,

for some positive constant C 2 = C 2 (C 0 , C 1 ). Since the equation for the evolution of the density not contain a pressure term, similarly to (7.11) we have

d dt

  ρ  X τ  τ˙ ρ Y τ + C 2 1 + τ −5/2  v  X τ ρ Y τ + C 2 1 + τ −5/2  v Y τ ρ  X τ ,

(7.43)

ρ does

(7.44)

and therefore, by combining the above with (7.43) we obtain

d dt

 ( v , ρ )

Xτ

         τ˙ + C 2 g + C 2 1 + τ −5/2 ( v , ρ ) X ( v , ρ )Y + C 3 ( v , ρ ) X , τ

τ

for some fixed positive constant C 2 > 0, where C 3 = g + C 2 f . Lastly, we let the ordinary differential equation







τ˙ + 2C 2 g + 2C 2 1 + τ −5/2 ( v 0 , ρ0 ) X τ e C 3 t = 0,

τ

(7.45)

τ (t ) be the solution of

(7.46)

0

with initial data τ0 . Arguments similar to those for the periodic case and to those for the twodimensional case, give the existence and uniqueness of solutions satisfying

I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

   v (t ), ρ (t ) 

t X τ (t )

+ C2 g





e C 3 (t −s)  v (s), ρ (s) Y

τ (s)

1745

ds

0

  + C 2 ( v 0 , ρ0 ) X e C 3 t

t



τ0







1 + τ −5/2 (s)  v (s), ρ (s) Y

τ (s)



ds  ( v 0 , ρ0 ) X e C 3 t , τ0

(7.47)

0

where C 3 = C 3 (C 2 , f , g ) is a fixed constant, for all t ∈ [0, T ∗ ), where T ∗ can be estimated from the data. We point out that these a priori estimates can be made rigorous using verbatim arguments to those in Sections 5 and 6. The only difference is that in the construction of the solutions, the nth iterate p (n) is defined here by

p

(n+1)

h

˜ (n)

( x) = p

(x) + g — ψ (n+1) (x, z) dz,

(7.48)

0

where p˜ (n) is the unique smooth mean-free solution of the elliptic Neumann problem

h ˜ (n)

− p



(n)

(n)

(n)

(n)

= ∂k — v j ∂ j v k + v k ∂ j v j

h



(n)

(n)

dz + f — ∂1 v 2 − ∂2 v 1

0

dz,

(7.49)

0

h

h

 (n) (n) ∂ p˜ (n) (n) (n) (n) = — v 1 ∂ j v j + v j ∂ j v 1 dz + f — v 2 dz. ∂n 0

(7.50)

0

We recall that the velocity iterate v (n+1) is defined via cf. (5.5), and hence after taking the derivative in time, the average value in z, and integrating by parts in z, it satisfies

h ∂t — v

(n+1)

0

h



dz + — v

(n)

· ∇ + div v

(n)



v

(n)

h ⊥ (n) ˜ dz + ∇ p + f — v (n) dz = 0.

0

(7.51)

0

By taking the dot product of (7.51) with the outward unit normal n to Γx , and using (7.50), we obtain that

h

(n+1)

v1

h (0, x2 , z, t ) dz =

0

(n+1)

v1

h (0, x2 , z, 0) dz =

0

h

v 01 (0, x2 , z) dz.

(7.52)

0

(n)

Therefore, the boundary condition 0 v 1 (0, x2 , z, t ) dz = 0 (cf. (1.12)) is satisfied by all iterates if it is satisfied by the initial data. Similarly, by taking the two-dimensional divergence of (7.51), and using

h

(7.49), we obtain that the compatibility condition 0 div v (n) (x, z, t ) dz = 0 (cf. (1.14)) is satisfied by all iterates if it is satisfied by the initial data. We omit further details.

1746

I. Kukavica et al. / J. Differential Equations 250 (2011) 1719–1746

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