LOCAL EXISTENCE AND UNIQUENESS FOR THE HYDROSTATIC EULER EQUATIONS ON A BOUNDED DOMAIN IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE

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1 LOCAL EXISTENCE AND UNIQUENESS FOR THE HYDROSTATIC EULER EQUATIONS ON A BOUNDED DOMAIN IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE Abstract. We aress te question of well-poseness in spaces of analytic functions for te yrostatic incompressible Euler equations invisci primitive equations on omains wit bounary, wit a novel sie-bounary conition. We construct a locally in time, unique, real-analytic solution an give an explicit rate of ecay of te raius of real-analyticity.. Introuction Te literature on incompressible omogeneous geopysical flows in te yrostatic limit is vast, an several moels ave been propose bot in te viscous an te invisci case cf. J.-L. Lions, Temam, an Wang [2, 3, 4], P.-L. Lions [], Peloski [7], Temam an Ziane [24], an references terein. In te present paper we consier te invisci yrostatic moel wic is classical in geopysical flui mecanics; see [7] an [, Section 4.6], were te autor raises te question of existence an uniqueness of solutions. Tese equations are formally erive from te treeimensional incompressible Euler equations for a flui between two orizontal plates, in te limit of vanising istance between te plates [4, 5, ]. Te problem is to fin a velocity fiel u = v, v 2, w = v, w, a scalar pressure P, an a scalar ensity ρ solving in D, T, for some T >. Here t v + v v + w z v + P + fv =,. iv v + z w =,.2 z P = ρg,.3 t ρ + v ρ + w z ρ =,.4 D = M, = {x, x 2, z = x, z R 3 : x M, < z < } is a 3-imensional cyliner of eigt, were M R 2 is a smoot omain wit real-analytic bounary. We enote by iv,, an te corresponing 2-imensional operators acting on x = x, x 2, wile z = / z. Also, we let v = v 2, v be te first two components of u e 3, f is te strengt of te rotation, an g is te gravitational constant. It follows from.9 tat te pressure P may be written in terms of te ensity ρ an te orizontal pressure px, t = P x,, t P x, z, t = px, t gψx, z, t,.5 Date: April 29, 2. 2 Matematics Subject Classification. 35Q35, 76N, 76B3, 86A5. Key wors an prases. Hyrostatic Euler equations, nonviscous Primitive equations, yrostatic approximation, well-poseness, boune omain, analyticity.

2 2 IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE were we ave enote ψx, z, t = ˆ z ρx, ζ, t ζ,.6 for all x, z D an t. Te yrostatic Euler equations..4 may ten be rewritten as t v + v v + w z v + p + fv = g ψ,.7 iv v + z w =,.8 z p =,.9 t ρ + v ρ + w z ρ =,. were ψ is obtaine from ρ via.6. Te bounary conitions for te top an bottom Γ z = M {, } an te sie Γ x = M, of te cyliner D are wx, z, t =, on Γ z, T,. ˆ vx, z, t z n =, on Γ x, T,.2 were n is te outwar unit normal to M. Note tat tere is no evolution equation for w. Instea, te incompressibility conition an. imply tat wx, z, t = ˆ z iv vx, ζ, t ζ,.3 for all < z <, an < t < T, wic combine again wit. sows tat te vertical average of iv v is zero, i.e., ˆ iv vx, z, t z =,.4 for all x M an < t < T. We consier a real-analytic initial atum vx, z, = v x, z,.5 ρx, z, = ρ x, z.6 in D, wic satisfies te compatibility conitions arising from.2 an.4, namely for all x M, an ˆ ˆ v x, z z n =,.7 iv v x, z z =,.8 for all x M. Te existence an uniqueness of solutions to te yrostatic Euler equations is an outstaning open problem cf. P.-L. Lions []. Te metos an results for yperbolic systems cannot be applie to.7.2 in orer to fin a well-pose set of bounary conitions cf. Oliger an Sünstrom [2]. Te instability results of Grenier [6, 5] an Brenier [4] suggest tat te problem may be ill-pose in Sobolev spaces, in analogy to te Prantl equations. Recently, Renary [8] prove tat te linearization of te yrostatic Euler equations at specific parallel sear flows is ill-pose in te sense of Haamar. Te only local existence result available for te nonlinear problem was obtaine in two-imensions by Brenier [3] uner te assumptions of convexity of v

3 LOCAL EXISTENCE AND UNIQUENESS FOR THE HYDROSTATIC EULER EQUATIONS 3 in te z-variable, constant normal erivative of v on Γ z, an of perioicity of v, w, p in te x- variable. Progress in te linearize case wit a nonlocal bounary conition as been acieve by Rousseau, Temam, an Tribbia [9, 2] using tecniques relate to well-poseness of yperbolic problems. In te present article we introuce te sie-bounary conition.2, an prove te existence an uniqueness of solutions to.7.2 in te two-imensional case tat is wit M an u, ρ, p inepenent of x 2, an te tree-imensional cases were M is a alf-plane or a perioic box. Tese results were announce in te note [7]. Te tree imensional case wen M is a generic boune omain wit analytic bounary will be treate in a fortcoming paper. Te solution we construct is real-analytic up to te bounary an is unique in tis class. Furtermore, we obtain an explicit rate of ecay in time of te raius of analyticity of te solution. To te best of our knowlege tis is te first local well-poseness result for te yrostatic Euler equations in tree imensions, an in te absence of convexity, even in two imensions. For te issipative Prantl bounary layer equations, using te abstract Caucy-Kowalewski teorem, Sammartino an Cafflis cf. [22], see also [6] an references terein prove te existence of a real-analytic solution. We note tat tese metos o not apply in tis case ue to te presence of te lateral bounary. Moreover, for te system.7.2, it is ifficult to verify tat te assumptions of te abstract Caucy-Kowalewski teorem cf. Asano [] ol, an setting up te spaces is callenging, since we ave to account for te ifference in tangential an normal erivatives wen ealing wit te pressure. Moreover, as oppose to te classical incompressible Euler equations cf. Temam [23], te estimates for te yrostatic Euler equations o not close in Sobolev spaces. Aitional ifficulties are create because te bounary conition is not pointwise, i.e., we ave v n z =, as oppose to te bounary conition u n = for te Euler equations. For results concerning te analyticity of solutions to te Euler equations, cf. [2, 8, 9, 5]. Altoug in tis article we eal wit te full nonlinear yrostatic Euler equations, te metos use ere may be applie to te linear case as in [2]. We will aress te question of comparing te corresponing results wit tose of [2] in a future work. Organization of te paper is as follows. Section 2 contains te functional setting an te statements of te main teorems. In Section 3 we give te a priori estimates for te two-imensional case. In Section 4 we prove te erivation of te estimates for te velocity. Te construction of solutions is given in Section 5, an te uniqueness is proven in Section 6. Section 7 contains te proof for te tree-imensional perioic omain an for te alf-space. 2. Main Teorems In te following, α = α, α 2, α z N 3 an =, 2, z N 3 enote multi-inices, were N = {,, 2,...} is te set of all non-negative integers. Te notation α = x α z αz = α x α 2 x 2 z αz, were α = α, α 2 will be use trougout. We enote te omogeneous Sobolev semi-norms m, for m N, by were α v 2 L 2 = α v 2 L 2 + α v 2 2 L 2. v m = α =m α v L 2 D, 2.

4 4 IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE Remark 2.. In te two-imensional case, wen M =, R is inepenent of x 2, te L 2 norms in te efinition of m are consiere only wit respect to x an z, tat is for i {, 2}. α v i 2 L 2 D = ˆ ˆ α v i x, z 2 z x, We recall tat a function vx, z is real-analytic in x an z, wit raius of analyticity τ if tere exists M > suc tat α vx, z M α! τ α, for all x, z D an α N 3, were α = α + α 2 + α z. For r an τ > fixe, we efine te spaces of real-analytic functions { ˆ ˆ } X τ = v C D : v Γx z n =, iv v z =, v Xτ <, 2.2 were Similarly, enote were te semi-norm Yτ v Xτ = is given by m= m + r τ m v m. 2.3 Y τ = {v X τ, v Yτ < }, 2.4 v Yτ = m= m + r τ m v m. 2.5 m! We write ρ X τ if ρ C D an ρ Xτ <, an ρ Y τ if ρ X τ an ρ Yτ <. For ease of notation, let an similarly v, ρ Xτ = v Xτ + ρ Xτ, v, ρ Yτ = v Yτ + ρ Yτ. Using te Sobolev embeing teorem it is clear from 2.3 tat if v X τ ten v is real-analytic wit raius of analyticity τ. Conversely, if v is real-analytic wit raius of analyticity τ an satisfies te bounary conitions, ten v X τ for any τ < τ an r, since m mr+2 τ /τ m <. Moreover, we ave te estimate v Xτ v L 2 D + τ v Yτ, an for any ε > we ave X τ+ε Y τ since v Yτ eτ ln + ε/τ v Xτ+ε. Te following teorem is our main result for imension two. Teorem 2.2. Let te functions u, p, ρ be inepenent of x 2, an let r 2. Assume tat v an ρ are real-analytic wit raius of analyticity strictly larger tan τ, an suppose tat v satisfies te compatibility conitions.7.8. Ten tere exists T = T r, g, τ, v, ρ Xτ >, an

5 LOCAL EXISTENCE AND UNIQUENESS FOR THE HYDROSTATIC EULER EQUATIONS 5 a unique real-analytic solution vt, ρt of te initial value problem associate wit.7.2 wit raius of analyticity τt, suc tat vt, ρt Xτt + Cg e gt s vs, ρs Yτs s + C v, ρ Xτ e gt + τ 2 s vs, ρs s v Yτs, ρ Xτ egt, 2.6 for all t [, T, were C = CD is a fixe positive constant. Moreover, te raius of analyticity of te solution, τ : [, T R +, may be compute explicitly from 3.26 below. In te tree-imensional case, te bounary conition.2 allows us to fin te pressure implicitly as te solution of an elliptic Neumann problem cf Te classical igerregularity estimates cf. J.-L. Lions an Magenes [], Temam [23] may not be use to prove tat te pressure as te same raius of analyticity as te velocity, preventing te estimates from closing. To overcome tis obstacle we introuce a new analytic norm wic combinatorially encoes te transfer of normal to tangential erivatives in te pressure estimate. Te following teorem treats te case wen M is a alf-plane, or te perioic omain. Te case wen M is a generic realanalytic boune omain in R 2, an.2 ols on M, requires new ieas an will be treate in a fortcoming paper. Teorem 2.3. Let r 5/2, an let M be eiter te upper alf-plane {x > } or te perioic box [, 2π] 2. Assume tat v an ρ are real-analytic wit raius of analyticity strictly larger tan τ, an suppose tat v satisfies te compatibility conitions.7.8. Ten tere exists T = T r, f, g, τ, v, ρ Xτ >, an a unique real-analytic solution vt, ρt of te initial value problem associate wit.7.2 wit raius of analyticity τt, suc tat vt, ρt Xτt + Cg + C v, ρ Xτ e C t e C t s vs, ρs Yτs s + τ 5/2 s vs, ρs Yτs s v, ρ Xτ ec t, 2.7 for all t [, T, were C = CD an C = C f, g are fixe positive constants. Moreover, te raius of analyticity of te solution, τ : [, T R + may be compute explicitly from 7.3 or 7.46 below, for te perioic box or te alf-plane respectively. Te ifferent powers of τ in 2.6 an 2.7 are ue to te ifferent exponents in te twoimensional an tree-imensional Agmon s inequalities. Remark 2.4. We note tat te solutions vt constructe in Teorem 2.2 an Teorem 2.3 o not epen on te function τt. Moreover, if v t an v 2 t are two suc real-analytic solutions, wit raii of analyticity τ t an τ 2 t respectively, ten v t = v 2 t for all t on te common interval of existence. 3. Te two-imensional case In tis section we give te formal a priori estimates neee to prove Teorem 2.2. Tese estimates will be mae rigorous in Sections 5 an 6. Let r 2 be fixe trougout te rest of te section. Assume tat v, w, p, an ρ are smoot solutions of.7.2, wit v, ρ X τ, for some τ >, wic are inepenent of x 2. Te orizontal omain M nees to be x 2 inepenent, translation invariant in x 2, an ence witout loss of generality we may consier M =, R.

6 6 IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE Since all erivatives wit respect to x 2 vanis we enote x = x, an similarly = x x. It is convenient to write te system of equations.7.2 in component form t v + v x v + w z v + x p + fv 2 = g x ψ, 3. t v 2 + v x v 2 + w z v 2 fv =, 3.2 x v + z w =, 3.3 z p =, 3.4 t ρ + v x ρ + w z ρ =, 3.5 were ψx, z = z ρx, ζ ζ. Te bounary conitions for w an v are w =, on Γ z, 3.6 ˆ v z =, on Γ x, 3.7 were Γ x = {, } R. We note tat tere is no bounary conition for v 2. Integrating te incompressibility conition 3.3 in z, an using 3.6 we get x v z =. Combine wit 3.7 tis implies tat for all x M we ave ˆ v z =. 3.8 In te two-imensional case, te bounary conitions 3.6 an 3.8 give te pressure explicitly as a function of v an ρ, an imply te cancelation property 3. below, wic turns out to be convenient in te a priori estimates below. Lemma 3.. Let v, w, p, ρ be a smoot solution of Ten, after subtracting from p a function of time, te pressure is given at eac instant of time t by px = v 2 x, z z f ˆ x v 2 x, x 2, z z x + g ψx, z z, 3.9 were ψ is obtaine from ρ via.6. In 3.9 we ave suppresse te epenence in t for convenience. Also, we ave te cancelation property for any multi-inex α N 3. x α p, α v =, 3. In 3.9 an in te following we use te notation ffl φx, zz = / φx, zz, for any function φ. In te two imensional case we o not integrate in x 2, so tat in 3. we enote φ, φ 2 = φ x, zφ 2 x, z z x, for any pair of smoot real functions φ an φ 2. Proof of Lemma 3.. Integrating 3. in z an using te bounary conition 3.8, we obtain ˆ ˆ ˆ x vx, 2 z z + px g ψx, z z = f v 2 x, z z. 3. Here we use ˆ w z v z = ˆ v z w z = ˆ v x v z, 3.2 wic ols by w Γz =, an te incompressibility conition 3.3. Te ientity 3.9 ten follows from 3. by integrating in x an subtracting a function of time.

7 LOCAL EXISTENCE AND UNIQUENESS FOR THE HYDROSTATIC EULER EQUATIONS 7 In orer to prove 3., note tat if α z or α 2, ten α p =. On te oter an, if α 2 = α z = an α, ten by 3.3 we ave α v = α x v = α x z w. Moreover, te bounary conition 3.6 for w implies α x w = on Γ z ; terefore integrating by parts in z gives x α p, α v = x α p, α x z w = z α + x p, α x w =. Lastly, we nee to prove 3. in te case α =,,. Integrating by parts in x an using 3.3 we ave x p, v = p, z w since on Γ x = {, } R by 3.7 we ave ˆ ˆ p Γx v Γx z = p Γx v Γx z =. 3.3 Integrating by parts in z we obtain p, z w = since w Γz =. Terefore, 3. is proven. We now turn to a priori estimates neee to prove Teorem 2.2. From 2., 2.3, an 2.5 it follows tat t vt m + r τt m X τt = τt vt Yτt + t α vt L m= α =m Given a multi-inex α N 3, we estimate /t α vt L 2 by applying α to an taking te L 2 D-inner prouct, wit α v. Recall tat in, we integrate only wit respect to x an z. Since α v, α v =, we obtain 2 t α vt 2 L + α v v + w 2 z v, α v + α p, α v = g α ψ, α v. 3.5 To treat te secon term on te left of 3.5, we use te Leibniz rule to write α v v + w z v, α v = α v α v, α v + α w z α v, α v. α α Te tir term on te left of 3.5 vanises by Lemma 3., an terefore by 3.5 an te Scwarz inequality we ave t α v L 2 α v α v L 2 + α w z α v L 2 + g α ψ L 2. α α 3.6 Substituting estimate 3.6 above into 3.4, an using te inepenence on te x 2 variable, we ave te a priori estimate t v X τ τ v Yτ + Uv, v + Uv, v 2 + Vw, v + Vw, v 2 + g x ψ Xτ, 3.7 were for i {, 2}, a vector function v X τ, an a scalar function ṽ C D, we enote m α Uv i, ṽ = v i xi α m + r τ m ṽ L 2, 3.8 an Vw, ṽ = m= j= α =m =j, α m m= j= α =m =j, α α w z α ṽ L 2 m + r τ m. 3.9

8 8 IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE In 3.9 we enote as usual wx, z = z iv vx, ζ ζ. Te following lemma summarizes te estimates on Uv, v i an Vw, v i, for i {, 2}. It is a corollary of Lemma 4., wic is proven in Section 4. Lemma 3.2. Let v Y τ, for some τ > an r 2, an let w be given by.3. Ten we ave an Uv, v + Uv, v 2 C + τ v Xτ v Yτ, 3.2 Vw, v + Vw, v 2 C + τ 2 v Xτ v Yτ, 3.2 for some sufficiently large positive constant C = CD. To boun te last term on te rigt of 3.7 we note tat x ψ m = ˆ α z x ψ L 2 = α x ρx, ζ ζ + L 2 α =m C ρ m+ + ρ m, α =m,α z= α =m,α z so tat by possibly enlarging te constant C from 3.2 an 3.2, we ave α + x z αz ρx, z L 2 g x ψ Xτ C g ρ Yτ + g ρ Xτ We fix C = C D as in trougout te rest of tis section. By 3.7, 3.2, 3.2, an 3.22 we ave t vt X τt τt + 3C + τt 2 vt Xτt vt Yτt + C g ρt Yτt + g ρt Xτt. Similarly, by Lemma 4. we obtain from 3.5 an estimate for te growt of ρt Xτt, namely 3.23 t ρt X τt τt ρt Yτt + Uv t, ρt + Vwt, ρt τt ρ Yτt + C + τt 2 vt Xτt ρt Yτt + 2C + τt 2 vt Yτt ρt Xτt By summing te estimates 3.23 an 3.24 we obtain t v, ρ X τ τ + C g + 3C + τ 2 v, ρ Xτ v, ρ Yτ + g v, ρ Xτ Define te ecreasing function τt by τ + 2 C g + 2 C + τ 2 v, ρ Xτ e gt =, 3.26 an τ = τ ; tis uniquely etermines τ in terms of te initial ata. Let T be te maximal time suc tat τt. It is clear tat τt an T may be compute from 3.26 in terms of te initial ata. By construction, we ave at t = an by 3.25 we ten ave for a sort time τt + C g + 3C + τt 2 vt, ρt Xτt <, 3.27 vt, ρt Xτt v, ρ Xτ e gt. 3.28

9 LOCAL EXISTENCE AND UNIQUENESS FOR THE HYDROSTATIC EULER EQUATIONS 9 It follows tat 3.27, an ence 3.28, ols for all t < T. Moreover, by 3.25, we obtain tat te solution is a priori boune in L, T ; X τ L, T ; + τ 2 Y τ in te sense vt, ρt Xτt + C g e gt s vs, ρs Yτs s + C v, ρ Xτ e gt + τs 2 vs, ρs s v Yτs Xτ egt, 3.29 for all t < T. Te formal construction of te analytic solution v, ρ satisfying 3.29, wit τ given by 3.26 is given in Section 5, wic combine wit te uniqueness result given in Section 6 completes te proof of Teorem Te velocity estimate Te goal of tis section is to prove te next lemma. For convenience of notation, we suppress te time epenence of v, w, ṽ, an τ. By te two-imensional case = 2 we mean tat all functions an te omain are inepenent of te variable x 2. In tat case we recall tat by φ 2 L 2 D we mean φx, z 2 z x, for any smoot function φ inepenent of x 2. Lemma 4.. Let v Y τ for some τ >, an ṽ C D L 2 D wit ṽ Yτ <. Fix r /2+ an {2, 3}. Let w be etermine from v via.3 i.e., wx, z = z iv vx, ζ ζ. If Uv i, ṽ, for i {, 2}, an Vw, ṽ are as in 3.8 an 3.9 respectively, ten we ave te estimates an Uv i, ṽ C + τ θ v Xτ ṽ Yτ, 4. Vw, ṽ C + τ θ v Xτ ṽ Yτ + C τ + τ θ v Yτ ṽ Xτ, 4.2 for some positive constant C = Cr, D, were θ = if = 2, an θ = 3/2 if = 3. By setting ṽ = v, an ten ṽ = v 2, Lemma 3.2 is a corollary of Lemma 4. for te case = 2. For te rest of tis section we fix = 2. In te tree-imensional case te proof is ientical except for two moifications. Te integration is one also in x 2, an ence te exponents in Agmon s inequality are ifferent. We omit furter etails. Here, L p = L p D, for all p. Proof of Lemma 4.. In orer to prove te lemma, we nee to estimate te terms v i xi α ṽ L 2 an w z α ṽ L 2, for all α, N 3. For tis purpose it is convenient to istinguis between two cases: α an α < α. Wen α, by te Höler inequality, an by te two-imensional Agmon inequality we ave v i xi α ṽ L 2 C v i L xi α ṽ L 2 C v i /2 L 2 + zz v i /2 L 2 xi α ṽ L 2 + C v i L 2 xi α ṽ L 2, 4.3 for some constant C = CD >. Recall tat is te orizontal Laplacian x x wen = 2, an x x + x2 x 2 wen = 3. Similarly, we estimate w z α ṽ L 2 C w L z α ṽ L 2 C w /2 L 2 + zz w /2 L 2 z α ṽ L 2 + C w L 2 z α ṽ L

10 IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE As in te above estimates, for multi-inices suc tat α < α, we ave an v i xi α ṽ L 2 C v i L 2 xi α ṽ L C v i L 2 xi α ṽ /2 L 2 xi + zz α ṽ /2 L 2 + C v i L 2 xi α ṽ L 2, 4.5 w z α ṽ L 2 C w L 2 z α ṽ L C w L 2 z α ṽ /2 L 2 z + zz α ṽ /2 + C w L 2 L 2 z α ṽ L Trougout tis section we use te inequality α α, wic ols for all α, N 3 wit α. Moreover, since α = α for α, te ientity a b α =, 4.7 α =m =j, α =j a γ =m j ols for all sequences {a } an {b γ }, an for j m; tis ientity is useful wen estimating U = Uv i, ṽ an V = Vw, ṽ. Wit 4.3 an 4.5 in min, we split U = U low + U ig, accoring to j [m/2] an [m/2] + j m respectively. Using 4.3, we ave U low C m= [m/2] j= + C α =m =j, α m= By 4.7 it follows tat α α =m =j, α C m j =j [m/2] j= α α =m =j, α v i /2 L 2 + zz v i /2 L 2 xi α ṽ L 2 b γ α v i L 2 xi α ṽ L 2 v i /2 L 2 + zz v i /2 L 2 xi α ṽ L 2 v i L 2 /2 + zz v i L 2 =j /2 m + r τ m m + r τ m. 4.8 γ =m j γ xi ṽ L We observe tat γ =m j x i γ ṽ L 2 C ṽ m j+ an =j + zz v i L 2 C v j+2. Hence, from 4.8 it follows by te iscrete Höler inequality an 4.9 tat U low is boune from above by C m= [m/2] j= C v i /2 j v i /2 j+2 ṽ m j+ m= [m/2] j= + C m m + r τ m + C j m= [m/2] j= v i j ṽ m j+ m j m + r τ m j + v r τ j 2 j + 3 r τ j+2 2 m j + 2 r τ m j i j v i j+2 ṽ m j+ m= [m/2] j= j! j + 2! j + v r τ j m j + 2 i j ṽ r τ m j m j+ j! m j! m j! τ, 4.

11 LOCAL EXISTENCE AND UNIQUENESS FOR THE HYDROSTATIC EULER EQUATIONS were we ave use te inequality m m + r m j! j! /2 j + 2! /2 m + r j + /2 j + 2 /2 j m j + 2 r j + r/2 = j + 3 r/2 m j + 2 r j + r/2 C, 4. j + 3 r/2 wic ols for all m, j [m/2], r, an a sufficiently large constant C, epening only on r. By 4., te iscrete Höler an Young inequalities imply U low C + τ v Xτ ṽ Yτ. By symmetry, from 4.5 an 4.7, we obtain an estimate for te ig values of j. Namely U ig C m m= j=[m/2]+ + C v i j xi ṽ /2 m j x i + zz ṽ /2 m j m m= j=[m/2]+ v i j xi ṽ m j m j m m + r τ m j m + r τ m C + τ v Xτ ṽ Yτ, 4.2 for some positive constant C epening only on r an D. Tis completes te proof of 4.. Te estimate on V is in te same spirit, but we nee to account for te erivative loss tat occurs wen estimating w in terms of v. First, note tat te efinition.3 of w an tat of te semi-norms j imply w j =j, 3 x 3 z iv v L 2 + =j, 3 = ˆ z iv vx, ζζ v j + C v j+, L 2 for some constant C = C >. We recall tat iv is te ivergence operator acting on x. Similarly we ave tat w j v j+2 + C v j+3, an also zz w j w j+2. Next, we split V = V low + V ig, accoring to j [m/2] an [m/2] < j m. By 4.4 an 4.7 we ave V low C m= [m/2] j= + C v /2 j + v /2 j+ v /2 m m + r j+2 + τ m v /2 j+3 ṽ m j+ j m= [m/2] j= m m + r τ m v j + v j+ ṽ m j+, 4.3 j for some constant C = Cr, D >. Using te fact tat m m + r m j! j +! /2 j + 3! /2 j m j + 2 r j + 2 r/2 C, j + 4 r/2 ols for all m an j [m/2], for some sufficiently large positive constant C epening only on r 2, we obtain V low C + τ 2 v Xτ ṽ Yτ.

12 2 IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE Lastly, to estimate V ig, we note tat by 4.6 an 4.7, we ave m V ig C w j z ṽ /2 m j z + zz ṽ /2 m m + r τ m m j j C m= j=[m/2]+ + C m= j=[m/2]+ + C m m= j=[m/2]+ m w j z ṽ m j m j m + r τ m v j + v j+ ṽ /2 m j+ ṽ /2 m j+3 m m= j=[m/2]+ m m + r τ m j m m + r τ m v j + v j+ ṽ m j j By symmetry wit te V low estimate, te lower orer terms te ones containing v j on te rigt sie of 4.4 are estimate by C + τ v Xτ ṽ Yτ. On te oter an, te terms containing v j+ are similarly boune by Cτ + τ 2 v Yτ ṽ Xτ concluing te proof of 4.2, an ence of te lemma. 5. Construction of te solution Te formal construction of te solutions is via te Picar iteration. Let v = v, ρ = ρ be given, wit v satisfying te compatibility conitions.7 an.8. For n N, let an Te ensity iterate is given by ρ n+ t = ρ w n x, z, t = ψ n x, z, t = ˆ z ˆ z an motivate by Lemma 3., we efine te pressure p n+ ˆ 2 x x, t = x, z, t z f + g v n iv v n x, ζ, t ζ, 5. ρ n x, ζ, t ζ. 5.2 v n s + w n s z ρ n s s, 5.3 v n 2 x, x 2, z, t z x ψ n+ x, z, t z. 5.4 Lastly, te velocity iterate is constructe as v n+ t = v v n s + w n s z v n s s p n+ s g ψ n+ s + fv n s s, 5.5 for all n N. Taking te time erivative of te first component of 5.5, integrating in z, an using te fact tat x p n is obtaine from 5.4, we obtain tat t vn+ z =. Since te

13 LOCAL EXISTENCE AND UNIQUENESS FOR THE HYDROSTATIC EULER EQUATIONS 3 first component of te initial ata, v, as zero vertical average, we obtain all x M an n. Terefore te compatibility conitions conition iv vn vn x, z z = for z = an te bounary vn Γ x z = are conserve for all n N. Recall tat we enote by iv te ifferential operator acting only on x. Assume tat v, ρ X τ +ε, for some < ɛ < τ. In particular, we ave v, ρ Y τ. We efine τt by τ = τ an τt + 2 C g + 2 C + τ 2 t v, ρ Xτ e gt =, 5.6 were te constant C = C r, D is fixe in Lemma 4.. First we sow tat te sequence of approximations v n is boune in L, T ; X τ L, T ; + τ 2 Y τ for some sufficiently small T >, epening solely on te initial ata. Lemma 5.. Let v, ρ X τ +ε an τt be efine by 5.6. {v n, ρ n } n, constructe via , satisfies ˆ T Te approximating sequence sup v n t, ρ n t Xτt + C g e gt t v n t, ρ n t Yτt t t [,T ] ˆ T + 2 C v, ρ Xτ e gt + τ 2 t v n t, ρ n t Yτt t 3e gt v, ρ Xτ, 5.7 for all n, were T = T v, ρ > is sufficiently small. Proof of Lemma 5.. Wen n =, since τ is ecreasing an since ɛ < τ, for all t we ave v, ρ Yτ C v, ρ Xτ +ε /ε, for some constant C. Sufficient conitions for te boun 5.7 to ol in te case n = are tat T is cosen suc tat an C Ce gt v, ρ Xτ +ε ε v, ρ Xτ e gt C C v, ρ Xτ +ε ˆ T + τ 2 t t ε. 5.9 Te conition 5.8 ols if T T, were T ɛ, C, C, g, v, ρ Xτ, v, ρ Xτ +ε > is etermine explicitly. Also, by te construction of τ cf. 5.6 we ave 2 C +τ 2 < τ/ v, ρ Xτ, so tat te conition 5.9 is satisfie if we coose T so tat τ τt ε v, ρ Xτ, 5. C v, ρ Xτ +ε wic is satisfie if T T 2, were T 2 ε, τ, C, C, g, v, ρ Xτ, v, ρ Xτ +ε > may be compute explicitly from 5.6 an 5.. Tus 5.7 ols for n = if T min{t, T 2 }. We procee by inuction. By 5.3, 5.5, an Lemma 4., similarly to estimate 3.25, we obtain t vn+, ρ n+ Xτ τ + C g v n+, ρ n+ Yτ + g v n+, ρ n+ Xτ + 3C + τ 2 v n, ρ n Xτ v n, ρ n Yτ τ + C g v n+, ρ n+ Yτ + g v n+, ρ n+ Xτ + 9C + τ 2 e gt v, ρ Xτ v n, ρ n Yτ, 5.

14 4 IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE by te inuction assumption. In te above we also use te fact tat by Lemma 3. we ave α p n, α v n+ =. Using tat τ was cosen to satisfy 5.6, te above estimate an Grönwall s inequality give v n+ t, ρ n+ t Xτt + C g + 2 C v, ρ Xτ e gt v, ρ Xτ e gt + 9e gt C v, ρ Xτ e gt e gt s v n+ s, ρ n+ s Yτs s + τ 2 s v n+ s, ρ n+ s Yτs s + τ 2 s v n s, ρ n s Yτs s. 5.2 Te proof of Lemma 5. is complete by taking te supremum over t [, T ] of te above inequality, by te inuction assumption, an by aitionally letting T be small enoug so tat 27e gt 4. We conclue te construction of te solution by sowing tat te map v n v n+ is a contraction in L, T ; X τ L, T ; + τ 2 Y τ, for some sufficiently small T epening on te initial ata. Lemma 5.2. Let ṽ n = v n+ v n, an ρ n = ρ n+ ρ n for all n. Let v, ρ X τ +ε, τt be efine by 5.6, an T be as in Lemma 5.. If for all n we let a n = sup ṽ n t, ρ n t Xτt + C g t [,T ] ˆ T + 2 C v, ρ Xτ e gt ˆ T ten we ave 2 a n 9 a n for all n. e gt t ṽ n t, ρ n t Yτt t + τ 2 t ṽ n t, ρ n t Yτt t, 5.3 Proof of Lemma 5.2. Denote w n = w n+ w n, ψ n = ψ n+ ψ n, an p n = p n+ p n. Ten we ave tat te ifference of two iterations satisfies te equations an t ṽ n + v n + w n z ṽn + ṽ n + w n z v n + p n g ψ n + fṽ n =, 5.4 t ρ n + v n + w n z ρ n + ṽ n + w n z ρ n =, 5.5 wit initial conitions ṽ n = an ρ n =, for all n. Since te approximate solutions u n satisfy te bounary conitions..2, similarly to te proof of 3., it can be sown tat α p n, α ṽ n = for all α N 3. Hence, from 5.4, 5.5, an Lemma 4., we obtain t ṽn, ρ n Xτ τ + C g ṽ n, ρ n Yτ + g ṽ n, ρ n Xτ + 3 C + τ 2 v n, ρ n Xτ + v n, ρ n Xτ ṽ n, ρ n Yτ + 3 C + τ 2 v n, ρ n Yτ + v n, ρ n Yτ ṽ n, ρ n Xτ. 5.6

15 LOCAL EXISTENCE AND UNIQUENESS FOR THE HYDROSTATIC EULER EQUATIONS 5 Using te efinition of τ cf. 5.6, te estimate in Lemma 5., Grönwall s inequality, an taking te supremum for t [, T ], we obtain ˆ T a n 8C e gt v, ρ Xτ + τ 2 e gt t ṽ n, ρ n Yτ t ˆ T + sup ṽ n, ρ n Xτ 3C + τ 2 e gt t v n, ρ n Yτ + v n, ρ n Yτ t. t [,T ] 5.7 If T is taken suc tat 8 e gt 9, ten te above estimate an te efinition of a n cf. 5.3 imply tat a n 9 2 a n. 5.8 Tis conclues te proof of te lemma, sowing tat te map v n, ρ n v n, ρ n is a strict contraction. Te existence of a solution to.7.2 in te class L, T ; X τ L, T ; + τ 2 Y τ, wit τt given by 5.6 follows from te classical fixe point teorem. 6. Uniqueness Fix v, ρ X τ +ε, a real-analytic function on D wit raius of analyticity strictly larger tan τ, for some positive ε < τ. Let τt be efine by τ = τ an τ + 2C g + 2C + τ 2 v, ρ Xτ e gt =, were C = CD, r > is te fixe constant efine in Lemma 3.2. Let T be te maximal time suc tat τt. Assume tat tere exist two solutions v, ρ an v 2, ρ 2 to.7.2 evolving from initial ata v, ρ, suc tat for i =, 2, we ave v i t, ρ i Xτt + C g e gt s v i s, ρ i s Yτs s + C v, ρ Xτ e gt + τ 2 s v i s, ρ i s Yτs s <, for all t < T. Similarly to 3.28 we ave tat v i t Xτt v, ρ Xτ e gt for i {, 2} an for all t < T. Let w i an p i be te vertical velocity an te pressure associate to v i, an let ψ i x, z = z ρi x, ζ ζ, for i {, 2}. We enote te ifference of te solutions v v 2 = v, ρ ρ 2 = ρ, an similarly efine w, ψ an p. Ten v, w, p, ρ satisfy te equations 6. t v + v + w z v + v + w z v 2 + p + fv = g ψ, 6.2 iv v + z w =, 6.3 z p =, 6.4 t ρ + v + w z ρ + v + w z ρ 2 =, 6.5

16 6 IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE in D, T, wit te corresponing bounary an initial value conitions wx, z, t =, on Γ z, T, 6.6 ˆ vx, z, tz n =, on Γ x, T, 6.7 vx, z, =, in D, 6.8 ρx, z, =, in D. 6.9 Similarly to te a priori estimates of Section 3, by an Lemma 4., we obtain tat t v, ρ X τ τ + C g + 3C + τ 2 v, ρ Xτ + v 2, ρ 2 Xτ v, ρ Yτ + g v, ρ Xτ + 3C + τ 2 v, ρ Yτ + v 2, ρ 2 Yτ v, ρ Xτ, 6. were C > is te constant from Lemma 3.2. But by te construction of τ we ave τ + 2C g + 2C + τ 2 v, ρ Xτ e gt =, an by also using v, ρ Xτ + v 2, ρ 2 Xτ 2 v, ρ Xτ e gt, we obtain t v, ρ X τ + C g v, ρ Yτ + C v, ρ Xτ e gt + τ 2 v, ρ Yτ g v, ρ Xτ + 3C + τ 2 v Yτ + v 2 Yτ v, ρ Xτ. 6. It is straigtforwar to ceck tat 6., 6.8, 6.9, 6., an Grönwall s inequality imply tat v, ρ Xτ = for all t [, T, an tereby proving te uniqueness of te solutions. 7. Te tree-imensional case In tis section we sketc te proof of Teorem 2.3. As oppose to te two-imensional case, ere Lemma 3. oes not ol, an ence we nee to estimate te analytic norm of te pressure. We only empasize te necessary canges from te two-imensional case. In Section 7. we give te proof of te pressure estimate in te case of perioic bounary conitions in te x-variable. In tis case p may be written explicitly as a function of v an ρ cf. 7.6 below, tereby simplifying te analysis. Wen M is an analytic omain wit bounary, te pressure is given implicitly as a solution of an elliptic Neumann problem cf. Temam [23] for te Euler equations. We explore te transfer of normal to tangential erivatives in te iger-orer estimates for te pressure an introuce a new suitable analytic norm to combinatorially encoe tis transfer. Tis gives us te necessary estimate cf. Lemma 7. to prove tat te pressure as te same raius of analyticity as te velocity. In Section 7.2 we give te proof of te pressure estimate in te case wen M is a alf-space. 7.. Te perioic case: M = [, 2π] 2. Here we give te a priori estimates for te case wen te bounary conition.2 is replace by te perioic bounary conition in te x-variable. Assume tat u, p, ρ is a smoot solution to.7., M-perioic in te x-variable. Since te pressure is efine up to a function of time, we may assume tat M p x =. Let v, ρ X τ for some τ >, an fixe r 5/2. Similarly to estimate 3.7 we ave t vt X τt τt vt Yτt + Uv, v + Vw, v + P + g ψ Xτ, 7.

17 LOCAL EXISTENCE AND UNIQUENESS FOR THE HYDROSTATIC EULER EQUATIONS 7 were Uv, v = 2 i,j= Uv i, v j an Vw, v = 2 j= Vw, v j, wit Uv i, v j an Vw, v j being efine by 3.8 an 3.9 respectively. In 7. above, we ave enote te upper boun on te pressure term by P = m= α =m α m + r τ m p L 2 D = /2 m= α =m,α 3 = α m + r τ m p L 2 M. 7.2 Here we use te fact tat p is z-inepenent, an te fact tat ue to te bounary conition.2 we ave p, v = p, iv v = p, z w =. We note tat in te tree-imensional case te cancelation property 3. oes not ol, an terefore te pressure term oes not vanis in te estimate 7.. To estimate P, we use te fact tat te pressure may be compute explicitly from te velocity. First, note tat iv v z =, an terefore, by integrating.7 in te z-variable, an ten applying te ivergence operator in te x-variable, we obtain ˆ ˆ p = k v j j v k + w z v k z + f v 2 2 v z g ψ z. 7.3 In 7.3 we ave use te summation convention over repeate inices j, k 2, an enote by j te partial erivative / x j, for all j 2. Integrating by parts in te z variable, it follows from.8 an. tat ˆ an terefore, by 7.3 we ave p = k j w z v k z = ˆ v j v k z + f v k z w z = ˆ v 2 2 v z g v k j v j z, 7.4 ψ z. 7.5 Te perioic bounary conitions in te x-variable allow for a simple solution to 7.5, namely p = R j R k v j v k z + f /2 R v 2 R 2 v z + g ψ z, 7.6 were R j is te j t Riesz transform, classically efine by its Fourier symbol iξ j / ξ. Te bouneness of te Riesz transforms on L 2 M, te Höler inequality, an te Leibniz rule give te boun P C /2 i α m + v j v k z r τ m m= α =m,α 3 = i,j,k 2 L 2 M + /2 f α v z + g α ψ z m + r τ m L 2 M C C m= α =m,α 3 = m= α =m,α 3 = i,j,k 2 m= α =m,α 3 = α α L 2 M α m + r τ m v j i v k L 2 D + f v Xτ + g ψ Xτ i,j,k 2 v j α m + r τ m i v k L 2 D + f v Xτ + g ψ Xτ. 7.7

18 8 IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE Te first term on te rigt of 7.7 is estimate similarly to Uv, v via Lemma 4., te case = 3, an we obtain te a priori pressure estimate P C + τ 5/2 v Xτ v Yτ + f v Xτ + g ψ Xτ. 7.8 By possibly enlarging C, as sown in 3.22 we also ave ψ Xτ C ρ Yτ + ρ Xτ, an terefore P C + τ 5/2 v Xτ v Yτ + f v Xτ + g ρ Xτ + C g ρ Yτ. 7.9 Combining te a priori estimate 7., te bouns on Uv, v an Vw, v obtaine from Lemma 4., an te pressure estimate 7.9, in analogy to 3.23, we obtain te boun t v X τ τ + 4C + τ 5/2 v Xτ v Yτ + 2C g ρ Yτ + f v Xτ + 2g ρ Xτ. 7. Since te evolution of te ensity ρ cf.. oes not involve te pressure term, using Lemma 4., in analogy to 3.24 we ave t ρ X τ τ ρ Yτ + 2C + τ 5/2 v Xτ ρ Yτ + 4C + τ 5/2 v Yτ ρ Xτ, 7. an terefore, by combining 7. an 7. we obtain t v, ρ X τ τ + 2C g + 4C + τ 5/2 v, ρ Xτ v, ρ Yτ + C v, ρ Xτ, 7.2 were C = max{f, 2g}. Wit τt efine by τ = τ an τ + 2C g + 2C + τ 5/2 v, ρ Xτ e C t =, 7.3 te rest of te proof of Teorem 2.3, namely te estimate 2.7, follows in analogy to te twoimensional case cf. Section 3. Te uniqueness of x-perioic solutions in te space L, T ; X τ L, T ; + τ 5/2 Y τ follows as in Section 6, wit te only moification being te power of τ in te estimates is now 5/2 instea of 2. Te construction of te x-perioic solution is similar to Section 5, wit one aitional moification: instea of p n being efine by 7.6, we efine te n t iterate of te pressure via p n+ = R j R k v n j v n k z + f /2 To avoi reunancy we omit furter etails. R v n 2 R 2 v n z + g ψ n+ z A omain wit bounary: M is te upper alf-plane. Let M = {x R 2 : x > }, so tat Γ x = M, = {x, z R 3 : x =, < z < }. Terefore te sie bounary conition.2 is v, x 2, z z =. In orer to close te estimates for te pressure cf. Lemma 7., in te case of te alf-pane it is necessary to use a moifie Sobolev semi-norm see also Kukavica an Vicol [9], instea of te classical m from 2.. We let [v] m = α =m 2 α α v L 2 D, 7.5 an efine te corresponing analytic X τ norm m + r τ m v Xτ = [v] m, 7.6 m=

19 LOCAL EXISTENCE AND UNIQUENESS FOR THE HYDROSTATIC EULER EQUATIONS 9 an respectively te Y τ semi-norm v Yτ = m + r τ m [v] m. 7.7 m! m= As in te perioic case, we ave te a priori estimate t vt X τt τt vt Yτt + Uv, v + Vw, v + P + g ψ Xτ, 7.8 were Uv, v an Vw, v are efine similarly to 3.8 an 3.9, namely by an by Uv, ṽ = Vw, ṽ = m 2 α m= j= α =m =j, α m an te pressure term is given by 2 α m= j= α =m =j, α P = /2 m= α =m,α 3 = α v α ṽ L 2 α w z α ṽ L 2 2 α α p L 2 M m + r τ m, 7.9 m + r τ m, 7.2 m + r τ m. 7.2 Recall tat te term corresponing to m = in 7.2 is missing since te sie bounary conition.2 implies tat p, v =. It is straigtforwar to ceck tat te proof of Lemma 4. also applies to te above efine operators U an V, an ence we ave te tree-imensional bouns Uv, v + Vw, v 2C + τ 5/2 v Xτ v Yτ To estimate P, we note tat by 7.3 te vertical average of te full pressure satisfies p = k px = p n = P x, z z = px g v j j v k + v k j v j z + f ψx, z z 7.23 v 2 2 v z = F 7.24 for all x M, were te summation convention over j, k 2 is use. By applying ffl z to.7, an taking te inner prouct wit n =,, te outwar unit normal vector to M, we obtain tat 7.24 is supplemente wit te bounary conition v j j v k + w z v k z n k f vk z n k = v j v j + v j j v z + f v 2 z = G, 7.25 were j {, 2}. We note tat as oppose to te Euler equations on a alf-space cf. [9] te nonlocal bounary conition on te velocity implies tat te bounary conition for p is nonomogeneous i.e., p/ n may be nonzero, creating aitional ifficulties. After subtracting a function of time from te full-pressure we ave D P x, z xz =, an terefore tere exists a unique smoot solution to te bounary value problem

20 2 IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE Lemma 7.. Te smoot solution p = p g ffl ψ z to te elliptic Neumann problem , satisfies were C is a universal constant, inepenent of m. Proof of Lemma 7.. In orer to boun [ p] m C [F ] m + C [G] m, 7.26 [ p] m = α =m, α 3 = 2 α α p L 2 M, 7.27 we estimate tangential an normal erivatives separately. To estimate tangential erivatives of te pressure, we note tat for any α 2, te function α 2 2 p is a solution of te elliptic Neumann problem α 2 2 p = α 2 2 F 7.28 α 2 2 p n = α 2 2 G, 7.29 an ence te classical H 2 regularity teorem, an te trace teorem give tat tere exists C > suc tat To estimate normal erivatives, we note tat an by inuction one may sow tat α 2 2 p Ḣ 2 M C α 2 2 F L 2 M + C α 2 2 G H M. 7.3 p = p + 22 p = F + 22 p p = F + 22 F + 22 p = F + 22 F p, 7.32 k+ p = k+ 22 p + k l= l 2l 2k 2l 2 F Terefore, if α = α, α 2, N 3 is suc tat α 2, an α = 2k + 2 is even, ten by 7.33 an 7.3 we ave α p L 2 M 2k+2+α 2 2 p L 2 M + k l= 2l 2k 2l+α 2 2 F L 2 M 7.34 α C α 2 2 G H M + C j α j 2 2 F L 2 M Similarly, if α 3, an α = 2k + 3 is o, ten similar arguments sow tat j= α α p L 2 M C α 2 2 G H M + C j α j 2 2 F L 2 M Lastly, wen α, an α, ten α p L 2 M C α 2 F L 2 M + α 2 G H M j=

21 LOCAL EXISTENCE AND UNIQUENESS FOR THE HYDROSTATIC EULER EQUATIONS 2 Summarizing estimates we obtain α =m,α 3 = 2 α α p L 2 M C m α = + C m α =2 2 α m 2 G H M + 2C m 2 F L 2 M α 2 j= 2 j j α j 2 2 F L 2 M 2 α j, 7.38 for all m. Here we see wy te introuction of te normalizing factors /2 α was necessary. Witout tem te constant C in te above estimates woul epen linearly on m. However since k /2k <, by possibly enlarging C wic is inepenent of m we ave α =m,α 3 = 2 α α p L 2 M C m 2 G H M + C α =m, α 3 = 2 α α F L 2 M, 7.39 concluing te proof of te lemma. Remark 7.2. If C woul epen on m, an woul grow unbounely as m, ten te aitional loss of one full erivative coming from estimating w in terms of v, prevents te estimates from closing. Te normalizing weigts /2 α may be viewe as a suitable combinatorial encoing of te transfer of normal to tangential erivatives in Lemma 7.3. Let v, ρ X τ, an p be te unique smoot solution of te elliptic Neumann-problem Ten te term P as efine in 7.2, wit p = p + g ffl ψ, is boune by P C + τ 5/2 v Xτ v Yτ + g ρ Xτ + C g ρ Yτ + C f v Xτ, 7.4 for some positive universal constat C. Proof of Lemma 7.3. By te triangle inequality an te efinition of p cf. 7.23, we ave tat α =m, α 3 = 2 α α p L 2 M α =m, α 3 = 2 α α p L 2 M + g α =m, α 3 = 2 α α ψ z L 2 M. From 7.39 an te above estimate it follows tat te pressure term is boune by P C /2 2 α m= α =m, α 3 = + g /2 m= α =m, α 3 = α F L 2 M + α G H M m + r τ m 2 α α ψ z L 2 M. 7.4

22 22 IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE Using Höler s inequality an te efinitions of F an G, we obtain P C C m= α =m, α 3 = + g ψ Xτ + C f m m + r τ m 2 α α v j j v k + v k j v j L 2 D m= α =m, α 3 = 2 α m= j= α =m, α 3 = =j, α α m + r τ m 2 α α v L 2 D v α v L 2 D m + r τ m + g ρ Xτ + C g ρ Yτ + C f v Xτ Te first term on te rigt of te above term is boune as Uv, v using te tree-imensional case of Lemma 4., concluing te proof of te lemma. We now conclue te proof of Teorem 2.3 in te case wen M is a alf-space. Combining 7.4 wit 7.8 an 7.22, we obtain te analytic a priori estimate t v X τ τ + C 2 + τ 5/2 v Xτ v Yτ + g ρ Xτ + C 2 g ρ Yτ + C 2 f v Xτ, 7.43 for some positive constant C 2 = C 2 C, C. Since te equation for te evolution of te ensity ρ oes not contain a pressure term, similarly to 7. we ave t ρ X τ τ ρ Yτ + C 2 + τ 5/2 v Xτ ρ Yτ + C 2 + τ 5/2 v Yτ ρ Xτ, 7.44 an terefore, by combining te above wit 7.43 we obtain t v, ρ X τ τ + C 2 g + C 2 + τ 5/2 v, ρ Xτ v, ρ Yτ + C 3 v, ρ Xτ, 7.45 for some fixe positive constant C 2 >, were C 3 = g + C 2 f. Lastly, we let τt be te solution of te orinary ifferential equation τ + 2C 2 g + 2C 2 + τ 5/2 v, ρ Xτ e C 3t =, 7.46 wit initial ata τ. Arguments similar to tose for te perioic case an to tose for te twoimensional case, give te existence an uniqueness of solutions satisfying vt, ρt Xτt + C 2 g + C 2 v, ρ Xτ e C 3t e C 3t s vs, ρs Yτs s + τ 5/2 s vs, ρs Yτs s v, ρ Xτ ec 3t, 7.47 were C 3 = C 3 C 2, f, g is a fixe constant, for all t [, T, were T can be estimate from te ata. We point out tat tese a priori estimates can be mae rigorous using verbatim arguments to tose in Sections 5 an 6. Te only ifference is tat in te construction of te solutions, te n t iterate p n is efine ere by p n+ x = p n x + g ψ n+ x, z z, 7.48

23 LOCAL EXISTENCE AND UNIQUENESS FOR THE HYDROSTATIC EULER EQUATIONS 23 were p n is te unique smoot mean-free solution of te elliptic Neumann-problem p n = k v n j j v n k + v n k j v n j z + f v n 2 2 v n z 7.49 p n n = v n jv n j + v n j j v n z + f v n 2 z. 7.5 We recall tat te velocity iterate v n+ is efine via cf. 5.5, an ence after taking te erivative in time, te average value in z, an integrating by parts in z, it satisfies v n+ z + v n + iv v n v n z + p n + f v n z =. 7.5 t By taking te ot prouct of 7.5 wit te outwar unit normal n to Γ x, an using 7.5, we obtain tat ˆ v n+, x 2, z, t z = ˆ v n+, x 2, z, z = ˆ v, x 2, z z Terefore, te bounary conition vn, x 2, z, t z = cf..2 is satisfie by all iterates if it is satisfie by te initial ata. Similarly, by taking te two-imensional ivergence of 7.5, an using 7.49, we obtain tat te compatibility conition iv vn x, z, t z = cf..4 is satisfie by all iterates if it is satisfie by te initial ata. We omit furter etails. Acknowlegements Te work of I.K. an V.V. was supporte in part by te NSF grant DMS-64886, te work of R.T. was partially supporte by te National Science Founation uner te grant NSF-DMS an by te Researc Fun of Iniana University, wile M.Z. was supporte in part by te NSF grant DMS References [] K. Asano, A note on te abstract Caucy-Kowalewski teorem. Proc. Japan Aca. Ser. A , 2-5. [2] C. Baros an S. Benacour, Domaine analycité es solutions e l équation Euler ans un ouvert e R n. Ann. Scuola Norm. Sup. Pisa Cl. Sci , [3] Y. Brenier, Homogeneous yrostatic flows wit convex velocity profiles. Nonlinearity 2 999, [4] Y. Brenier, Remarks on te erivation of te yrostatic Euler equations. Bull. Sci. Mat , [5] E. Grenier, On te erivation of omogeneous yrostatic equations. M2AN Mat. Moel. Numer. Anal , no. 5, [6] E. Grenier, On te Nonlinear Instability of Euler an Prantl Equations. Comm. Pure Appl. Mat. 53 2, no. 9, [7] I. Kukavica, R. Temam, V. Vicol, an M. Ziane, Local Existence an Uniqueness for te Hyrostatic Euler Equations on a Boune Domain. C. R. Aca. Sci. Paris, to appear. [8] I. Kukavica an V. Vicol, On te raius of analyticity of solutions to te tree-imensional Euler equations. Proc. Amer. Mat. Soc , [9] I. Kukavica an V. Vicol, Te Domain of Analyticity of Solutions to te Tree-Dimensional Euler Equations in a Half Space. Discrete Contin. Dyn. Syst., to appear. [] P.-L. Lions, Matematical topics in flui mecanics. Vol.. Incompressible moels. Oxfor Lecture Series in Matematics an its Applications, 3. Oxfor University Press, New York, 996. [] J.-L. Lions an E. Magenes, Problemès aux limites non omogènes et applications. Vol. 3, S.A. Duno, Paris, 968 [2] J.-L. Lions, R. Temam, an S. Wang, New formulations of te primitive equations of atmospere an applications. Nonlinearity 5 992, no. 2,

24 24 IGOR KUKAVICA, ROGER TEMAM, VLAD C. VICOL, AND MOHAMMED ZIANE [3] J.-L. Lions, R. Temam, an S. Wang, On te equations of te large-scale ocean. Nonlinearity 5 992, no. 5, [4] J.-L. Lions, R. Temam, an S. Wang, Moèles et analyse matématiques u système océan/atmospère. I. Structure es sous-systèmes. C. R. Aca. Sci. Paris Sér. I Mat , no., 3 9. [5] C.D. Levermore an M. Oliver, Analyticity of solutions for a generalize Euler equation. J. Differential Equations , no. 2, [6] M.C. Lombaro, M. Cannone, an M. Sammartino, Well-poseness of te bounary layer equations, SIAM J. Mat. Anal , no. 4, [7] J. Pelosky, Geopysical flui ynamcs. New York, Springer-Verlag, 987. [8] M. Renary, Ill-poseness of te Hyrostatic Euler an Navier-Stokes Equations. Arc. Rational Mec. Anal , [9] A. Rousseau, R. Temam, an J. Tribbia, Bounary conitions for te 2D linearize PEs of te ocean in te absence of viscosity. Discrete an Continuous Dynamical Systems, Series A 3 25, No. 5, [2] A. Rousseau, R. Temam, an J. Tribbia, Te 3D Primitive Equations in te absence of viscosity: Bounary conitions an well-poseness in te linearize case. J. Mat. Pures Appl , [2] J. Oliger an A. Sunström, Teoretical an practical aspects of some initial bounary value problems in flui ynamics. SIAM J. Appl. Mat , no. 3, [22] M. Sammartino an R.E. Caflisc, Zero Viscosity Limit for Analytic Solutions of te Navier-Stokes Equation on a Half-Space I. Existence for Euler an Prantl equations. Commun. Mat. Pys , [23] R. Temam, On te Euler equations of incompressible perfect fluis. J. Functional Analysis 2 975, no., [24] R. Temam an M. Ziane, Some matematical problems in geopysical flui ynamics. Hanbook of matematical flui ynamics. Vol. III, , Nort-Hollan, Amsteram, 24. Department of Matematics, University of Soutern California, 362 S. Vermont Avenue, Los Angeles, CA aress: kukavica@usc.eu Institute for Scientific Computing an Applie Matematics, Rawles Hall, Iniana University, Bloomington, IN aress: temam@iniana.eu Department of Matematics, University of Soutern California, 362 S. Vermont Avenue, Los Angeles, CA aress: vicol@usc.eu Department of Matematics, University of Soutern California, 362 S. Vermont Avenue, Los Angeles, CA aress: ziane@usc.eu