On the analyticity and Gevrey class regularity up to the boundary for the Euler Equations
OON THE ANALYTICITY AND GEVREY CLASS REGULARITY UP TO THEBOUNDARY FOR THE EULER EQUATIONS
IGOR KUKAVICA AND VLAD VICOL
Abstract.
We consider the Euler equations in a three-dimensional Gevrey-class bounded domain. UsingLagrangian coordinates we obtain the Gevrey-class persistence of the solution, up to the boundary, with anexplicit estimate on the rate of decay of the Gevrey-class regularity radius. Introduction
The Euler equations for the velocity vector field u ( x, t ) and the scalar pressure field p ( x, t ) are given by ∂ t u + ( u · ∇ ) u + ∇ p = 0 , in D × (0 , ∞ ) , (E.1) ∇ · u = 0 , in D × (0 , ∞ ) , (E.2) u · n = 0 , on ∂D × (0 , ∞ ) , (E.3)where D is an open bounded Gevrey-class s domain in R , and n is the outward unit normal to ∂D . Weconsider the initial value problem associated to (E.1)–(E.3) with a divergence free Gevrey-class s initialdatum, with s ≥
1, namely u (0) = u , in D. (E.4)The existence of smooth solutions to (E.1)–(E.4) is classical (cf. [BoB, EM, Ka, T, Y]). While in the two-dimensional case smooth initial data yield global solutions, in the three-dimensional case if u ∈ H r ( D ),with r > /
2, the maximal time of existence of the Sobolev solution, T ∗ , might be a priori finite. If T ∗ < ∞ ,the vorticity must accumulate in the sense that (cid:82) T ∗ (cid:107) curl u ( t ) (cid:107) L ∞ ( D ) = ∞ (cf. [BKM, F]). Lastly, upperbounds on (cid:107) u ( · , t ) (cid:107) H r ( D ) are worse than those on (cid:107) u ( · , t ) (cid:107) W , ∞ ( D ) due to the log-Sobolev inequality. Werefer the reader to [BT1, Ch, C2, MB] for the precise formulation of the above statements and for furtherresults cornering the Euler equations.In the present article we address the persistence of Gevrey-class regularity of the solution, i.e., we provethat if u is of Gevrey-class s , then the unique Sobolev solution u ( · , t ) ∈ C ([0 , T ∗ ); H r ( D )) is of Gevrey-class s for all t < T ∗ . Moreover, we are interested in sharp lower bounds on the rate of decay of the radius ofanalyticity and Gevrey-class regularity of the solution. We emphasize that the size of the uniform Gevrey-class radius of the solution provides an estimate for the minimal scale in dissipative flows, that is, thescale below which the Fourier coefficients decay exponentially [HKR, K1]; it moreover gives the rate of thisexponential decay [FT, HKR]. We note that the shear flow example of Bardos and Titi [BT2] (cf. [DM]) maybe used to construct explicit solutions to the three-dimensional Euler equations whose radius of analyticity,or even more generally the Gevrey-class radius, decays for all time (cf. Remark 1.3 below).First we summarize the rich history of this problem.(i) The persistence of C ∞ regularity (cf. Foias, Frisch, and Temam [FFT]) and of real-analyticity(cf. Bardos and Benachour [BB]) holds in both two and three dimensions.(ii) In the two-dimensional analytic case, Bardos, Benachour, and Zerner [BBZ] show that the radiusof analyticity τ ( t ) of the solution u ( · , t ) is bounded from below as τ ( t ) ≥ exp( − C exp( Ct )) /C , forsome sufficiently large constant C depending on the initial data. Their elegant proof is based onanalyzing the complexified equations in vorticity form. Mathematics Subject Classification.
Key words and phrases.
Euler equations, analyticity radius, Gevrey class, Lagrangian coordinates. a r X i v : . [ m a t h . A P ] J u l IGOR KUKAVICA AND VLAD VICOL (iii) In the three-dimensional analytic case, the persistence of analyticity is proven by Bardos and Be-nachour [BB] using an implicit argument. In [B, Be, BG1, BG2, D] using a nonlinear variant ofthe Cauchy-Kowalevski theorem, the authors prove the local in time existence of globally (in space)analytic solutions, with an explicit lower bound on the radius of analyticity which vanishes in finitetime (independent of T ∗ ). See also [LCS, SC] for the dissipative Prandtl boundary layer equations.The proof of [BB] may be modified to yield an explicit rate of decay of the radius of ana-lyticity τ ( t ) which depends exponentially on (cid:107) u ( · , t ) (cid:107) H r . Using different methods, Alinhac andMetivier [AM1, AM2] for the interior, and Le Bail [Lb] for the boundary value problem, obtain theshort time propagation of local analyticity, with lower bounds for τ ( t ) that also decay exponentiallyin (cid:107) u ( · , t ) (cid:107) H r . Note that the lower bounds for τ ( t ) obtained in [AM1, AM2, BB, Lb] do not recoverthe lower bounds of [BBZ] in the two-dimensional case, since the presently known upper bounds onhigh Sobolev norms of the solution increase as C exp( C exp( Ct )), for some C >
0. Moreover, themethods used in [AM1, AM2, BB, BG1, BG2, D, Lb] explicitly use the special properties of complexholomorphic functions, and hence may not be applied to the non-analytic Gevrey-class case.(iv) For the non-analytic Gevrey-class case, on a periodic domain, in both two and three dimensions,the persistence of Gevrey-class regularity follows from the elegant proof of Levermore and Oliver[LO]. Their proof builds on the Fourier-based method introduced by Foias and Temam [FT] forthe Navier-Stokes equations. The lower bound for the radius of Gevrey-class regularity obtainedin [LO] also decays exponentially in (cid:107) u ( · , t ) (cid:107) H r . This bound was improved by the authors of thepresent paper in [KV1], by proving that the radius of Gevrey-class regularity decays algebraically in a high Sobolev norm of the solution, and exponentially in (cid:82) t (cid:107)∇ u ( · , s ) (cid:107) L ∞ ds . Therefore, theFourier-based method may be employed (cf. [KV1]) to recover the bounds of [BBZ]. For furtherresults on analyticity cf. [Bi, BGK1, BGK2, CTV, FTi, GK1, GK2, K1, K2, KTVZ, OT].(v) The only result in the non-analytic Gevrey-class case, on domains with boundary, was obtained bythe authors in [KV2] for D a half-space. As opposed to the periodic case, here the main difficultyarises from the equation for the pressure. The classical methods of [LM, MN] are not sufficient toprove that the pressure has the same radius of Gevrey-class regularity as the velocity. In [KV2]we overcome this by defining suitable norms that combinatorially encode the transfer of normal totangential derivative in the elliptic estimate for the pressure.The proof of [KV2] does not apply directly to the case when D is a general bounded domainof Gevrey-class s . The main obstruction is that if s >
1, under composition with a Gevrey-class(or even analytic) boundary straightening map, the Gevrey-class regularity radius of the velocitymay deteriorate (cf. [CS, KP] and Remark 2.2 below). As a consequence, we need to localize theequation using particle trajectories and define suitable Lagrangian Gevrey-class norms. This givesrise to additional difficulties because the pressure is the solution of an elliptic Neumann problem(cf. [T]), and hence is non-local.The following is our main theorem.
Theorem 1.1.
Let u be divergence-free and of Gevrey-class s on D , a Gevrey-class s , open bounded domainin R , where s ≥ , and let r ≥ . Then the unique solution u ( · , t ) ∈ C ([0 , T ∗ ); H r ( D )) to the initial valueproblem (E.1) – (E.4) is of Gevrey-class s for all t < T ∗ , where T ∗ ∈ (0 , ∞ ] is the maximal time of existencein H r ( D ) . Moreover, the radius τ ( t ) of Gevrey-class regularity of the solution u ( · , t ) satisfies τ ( t ) ≥ Cτ exp (cid:32) − C (cid:18)(cid:90) t (cid:107) u ( s ) (cid:107) W , ∞ ( D ) ds (cid:19) (cid:33) exp (cid:16) − C t − Ct (cid:107) u (cid:107) H r ( D ) (cid:17) , (1.1) for all t < T ∗ , where C is a sufficiently large constant depending only on the domain D , τ is the radius ofGevrey-class regularity of u , and C has additional dependence on the Gevrey-class norm of u . Remark 1.2.
In the proof of Theorem 1.1 we also address the local (in space) propagation of Gevrey-class regularity of the solution (cf. Theorem 3.4 below), in the interior of the smooth domain, or in theneighborhood of a point where ∂D is locally of Gevrey-class s . This extends the results of [AM1, Lb] to thenon-analytic Gevrey-classes. NALYTICITY AND GEVREY CLASS REGULARITY FOR THE EULER EQUATIONS IN A BOUNDED DOMAIN 3
Remark 1.3.
Note that there exist explicit examples of solutions to (E.1)–(E.4) whose radius of Gevrey-class s regularity, where s ≥
1, decays for all time, and vanishes as t → ∞ . Namely, consider the three-dimensionalshear flow example (cf. Bardos and Titi [BT2], DiPerna and Majda [DM]) given by u ( x, t ) = ( f ( x ) , , g ( x − tf ( x ))) , (1.2)which is divergence free and satisfies (E.1) for smooth functions f and g .In the analytic category s = 1 we may let f ( x ) = sin( x ) and g ( x ) = 1 / ( τ + cos ( x )), where for simplicity D is the periodic box [0 , π ] . Substituting these particular functions f and g into (1.2) we obtain that u ( · ,
0) has radius of analyticity τ , while u ( x, t ) = (sin( x ) , , / ( τ + cos ( x − t sin( x )) has uniform radiusof analyticity that decreases with the rate 1 /t . For a similar example in the non-analytic Gevrey-classes, s >
1, let D = R and define g ( x ) = exp (cid:0) −| x | − / ( s − (cid:1) cf. [Le]. Note that g ( x ) is of Gevrey-class s , but notanalytic. Organization of the paper.
In Section 2 we introduce the notation used to define the Lagrangian Gevrey-class norms. Section 3 consists of a priori estimates needed to prove the short time propagation of localanalyticity (cf. Theorem 3.4). Lemmas 3.2 and 3.3 are proven in Sections 4 and 5 respectively. Lastly, inSection 6 we show how the local in space and time results may be patched together to obtain the globalpersistence of Gevrey-class regularity (cf. Theorem 6.1).2.
Notation and preliminary remarks
The existence of a unique H r solution, where r > /
2, on a maximal time interval [0 , T ∗ ), where T ∗ ∈ (0 , ∞ ], implies the existence and uniqueness of the particle trajectories (cf. [C1, MB]), that is solutions to ddt X ( t ) = u ( X ( t ) , t ) (2.1) X (0) = a, (2.2)where a ∈ ¯ D . For simplicity we denote by φ t ( a ) the solution of (2.1)–(2.2). It is well known that for all t < T ∗ the maps φ t : D (cid:55)→ D , and φ t | ∂D : ∂D (cid:55)→ ∂D are diffeomorphisms. Local change of coordinates.
Fix x ∈ ∂D . In a sufficiently small neighborhood of x , the boundary ∂D is the graph of a Gevrey-class s function γ , i.e., for 0 < r (cid:28) D r ,x = D ∩ B r ( x ) = { x ∈ B r ( x ) : x > γ ( x , x ) } . Moreover, since the Euler equations are invariant under rigid body rotationsof R , modulo composition with a rigid body rotation about x , we may assume that (cid:107) ∂ γ (cid:107) L ∞ ( ¯ D (cid:48) r ,x ) + (cid:107) ∂ γ (cid:107) L ∞ ( ¯ D (cid:48) r ,x ) ≤ ε (cid:28)
1, for r sufficiently small, where ε is a fixed, sufficiently small universal constant,to be chosen later. Here we have denoted D (cid:48) r ,x = { x (cid:48) : x ∈ D r ,x } , where we write x (cid:48) = ( x , x ) for x = ( x , x , x ). Define a boundary straightening map θ : R → R by θ ( x , x , x ) = ( x , x , x − γ ( x , x )) = ( y , y , y ) . (2.3)Note that det( ∂θ/∂x ) = 1. By the construction of θ we have (cid:101) D r ,x = θ ( D r ,x ) = { y ∈ θ ( B r ( x )) : y > } .Let Ω = D ∩ B r / ( x ) ⊂ D r ,x be a neighborhood of x . Also let (cid:101) Ω = θ (Ω) and (cid:101) Ω t = θ (Ω t ).There exists T = T ( r , u ) such that for all 0 = T ≤ t ≤ T we have Ω t = φ t (Ω) ⊂ D r ,x . The value of T may be estimated from below by using the representation formula for solutions of (2.1)–(2.2). We have | φ t ( a ) − a | ≤ (cid:90) t | u ( φ s ( a ) , s ) | ds ≤ (cid:90) t (cid:107) u ( · , s ) (cid:107) L ∞ ( D r ,x ) ds ≤ K ( t ) , (2.4)where we set K ( t ) = (cid:90) t (cid:107) u ( s ) (cid:107) W , ∞ ( D ) ds. (2.5)Therefore, it is sufficient to chose T such that K ( T ) ≤ dist( ¯Ω , ∂B r ( x )) = r / IGOR KUKAVICA AND VLAD VICOL ΩΩ δ DD D r ,x D r ,x Ω t Ω δ,t ~ D r x ~ ~ Ω Ω δ ~ ~~ D r x Ω t Ω δ,t θθ t θ t θ -1 On the closure of (cid:101) D r ,x we let (cid:37) be the Euclidean distance to the curved part of the boundary of (cid:101) Ω, thatis, (cid:37) ( y ) = 0 if y ∈ (cid:101) Ω c and (cid:37) ( y ) = dist( y, ∂ (cid:101) Ω \ { y = 0 } ) if y ∈ (cid:101) Ω. As in [AM1, AM2, Lb], for all 0 < δ ≤ δ ,where δ > (cid:101) Ω δ = { y ∈ (cid:101) Ω : (cid:37) ( y ) > δ } . (2.6)By the triangle inequality and the definition of (cid:37) it follows that | y (1) − y (2) | ≥ r for all y (1) ∈ (cid:101) Ω δ + r and y (2) ∈ Ω cδ . Also let Ω δ = θ − ( (cid:101) Ω δ ), Ω t,δ = φ t (Ω δ ) and (cid:101) Ω t,δ = θ (Ω t,δ ). Here δ = δ ( γ ) ≤ δ ∈ [0 , δ ), the set Ω δ is a Gevrey-class s domain, i.e., it lies on one side of aGevrey-class surface.If y (1) ∈ (cid:101) Ω δ + r,t and y (2) ∈ (cid:101) Ω δ,t , where δ, r + δ ∈ (0 , δ ), it follows by the mean value theorem that r ≤ | θ ◦ φ − t ◦ θ − ( y (1) ) − θ ◦ φ − t ◦ θ − ( y (2) ) | ≤ C | y (1) − y (2) |(cid:107)∇ φ − t (cid:107) L ∞ ( D ) ≤ C | y (1) − y (2) | (1 + K ( t )) , (2.7)where C is a constant depending on θ . In (2.7) we have used that ∇ φ − t is the inverse matrix of ∇ φ t (whosedeterminant is 1 since div u = 0), and the fact that the 2 × K .Therefore, by (2.7), we have | y (1) − y (2) | ≥ r/ ( C + CK ( t )). Hence there exists a smooth cut-off function η such that η ≡ (cid:101) Ω δ + r,t and η ≡ (cid:101) Ω cδ,t , with |∇ η | ≤ C + CK ( t ) r , (2.8)for some positive constant C = C ( D ). We denote (cid:101) u ( y, t ) = u ( x, t ) and similarly (cid:101) p ( y, t ) = p ( x, t ). Gevrey-class norms.
We recall (cf. [KP, Le, LM]) the definition of the Gevrey-class s , denoted by G s . Definition 2.1.
A function v ∈ C ∞ ( D ) is said to be of Gevrey-class s on D , where s ≥ , written v ∈ G s ,if there exist positive constants M, τ > such that (cid:107) ∂ α v (cid:107) L ∞ ( D ) ≤ M | α | ! s τ | α | (2.9) NALYTICITY AND GEVREY CLASS REGULARITY FOR THE EULER EQUATIONS IN A BOUNDED DOMAIN 5 for all multi-indices α ∈ N . We refer to the constant τ in (2.9) as the radius of Gevrey-class regularity of v , or simply as the G s -radius of v . As opposed to the class of real analytic functions G , functions in G s with s > G s partitions of unity (cf. [KP]). The G s norms used in this paper are defined as follows. For a Gevrey-class s function (cid:101) v ( y, t ) denote[ (cid:101) v ( t )] m = (cid:88) | α | = m (cid:15) α sup <δ ≤ δ δ m − (cid:107) ∂ α (cid:101) v ( · , t ) (cid:107) L ( (cid:101) Ω t,δ ) , (2.10)for all m ≥
3. In this paper we work with the Lagrangian Gevrey-class s norm defined by (cid:107) (cid:101) v ( t ) (cid:107) X τ ( t ) = ∞ (cid:88) m =3 [ (cid:101) v ( t )] m τ ( t ) m − ( m − s , (2.11)where s ≥
1, and τ >
0. We also let (cid:107) (cid:101) v ( t ) (cid:107) Y τ ( t ) = ∞ (cid:88) m =4 [ (cid:101) v ( t )] m mτ ( t ) m − ( m − s . (2.12) Remark 2.2. If (cid:107) (cid:101) u ( y ) (cid:107) X τ < ∞ , it follows from the Sobolev inequality that (cid:101) u ∈ G s and that (cid:101) u ( y ) hasGevrey-class regularity radius at least τ . As opposed to the analytic case, if s >
1, the map θ − : y (cid:55)→ x possibly shrinks the radius by a constant factor 0 < a ∗ ≤
1, where a ∗ = a ∗ ( γ ). This fact may be provenusing the multi-dimensional generalization of the Fa´a di Bruno formula (cf. [CS, KP]). Thus, if (cid:101) u ( y ) hasGevrey-class radius τ , then u ( x ) has G s -radius at least a ∗ τ . Notation.
When it is clear from the context that we are working with a function on the flattened domain,we simply write v instead of (cid:101) v . In the present paper we set n ! = 1 whenever n ≤
0. Also we use the notation (cid:107) D k v (cid:107) L p = (cid:80) | α | = k (cid:107) ∂ α v (cid:107) L p and similarly (cid:107) D (cid:48) k v (cid:107) L p = (cid:80) | α | = k,α =0 (cid:107) ∂ α v (cid:107) L p . Lastly, C denotes a sufficientlylarge positive constant which may depend on the domain.3. Short time local Gevrey-class a priori estimates
The proof of Theorem 1.1 consists of a priori estimates. These estimates can be made rigorous by notingthat u ( · , t ) ∈ C ∞ ( ¯ D ) for all t < T ∗ (cf. [FFT]), and by performing all below estimates on truncated sums (cid:80) qm =3 [ (cid:101) v ] m τ m − / ( m − s . For q ≥ q , so we may let q → ∞ .Let d + f ( t ) /dt = lim sup h → ( f ( t + h ) − f ( t )) /h denote the right derivative of a function f ( t ), whichagrees with the usual derivative if the latter exists. Using the definitions (2.10)–(2.11) we obtain d + dt (cid:107) (cid:101) u ( t ) (cid:107) X τ ( t ) ≤ ˙ τ ( t ) (cid:107) (cid:101) u ( t ) (cid:107) Y τ ( t ) + ∞ (cid:88) m =3 (cid:88) | α | = m (cid:15) α d + dt sup <δ ≤ δ δ m − (cid:107) ∂ α (cid:101) u ( t ) (cid:107) L ( (cid:101) Ω δ,t ) τ ( t ) m − ( m − s . (3.1)In order to switch the d + /dt and the sup δ (cf. Lemma A.5) we need upper bounds for ( d/dt ) (cid:107) ∂ α (cid:101) u ( t ) (cid:107) (cid:101) Ω δ,t forall | α | ≥
3. The following lemma is a Lagrangian energy estimate in the straightened domain and providesthe desired upper bound.
Lemma 3.1.
For all α ∈ N , t > , and < δ ≤ δ , we have ddt (cid:107) ∂ α (cid:101) u ( · , t ) (cid:107) L ( (cid:101) Ω δ,t ) ≤ (cid:107) [ ∂ α , (cid:101) u j ∂ j θ k ∂ k ] (cid:101) u ( · , t ) (cid:107) L ( (cid:101) Ω δ,t ) + (cid:107) ∂ α ( ∂ j θ k ∂ k (cid:101) p ( · , t )) (cid:107) L ( (cid:101) Ω δ,t ) = M δ ( t ) , (3.2) where the bracket [ · , · ] represents a commutator.Proof. The standard Lagrangian energy estimate (cf. [AM1, Lemma 2.3] and [Lb, Section 2.b]) shows thata smooth solution v to ∂ t v + ( u · ∇ ) v = g satisfies ddt (cid:107) v ( t, · ) (cid:107) L (Ω δ,t ) = ddt (cid:90) Ω δ | v ( t, φ t ( x )) | dx = 2 (cid:90) Ω δ,t g ( t, x ) v ( t, x ) dx (3.3) IGOR KUKAVICA AND VLAD VICOL
Here we used the fact that div u = 0 implies det( ∂φ t ( x ) /∂x ) = 1. Let y = θ ( x ) and denote (cid:101) v ( y ) = v ( x ).Similarly define (cid:101) u ( y ) = u ( x ) and (cid:101) g ( y ) = g ( x ). Then (cid:101) v solves the equation ∂ t (cid:101) v + (cid:101) u j ∂ j θ k ∂ k (cid:101) v = (cid:101) g , and sincedet( ∂θ/∂x ) = 1, we have (cid:107) (cid:101) v ( t, · ) (cid:107) L ( (cid:101) Ω δ,t ) = (cid:107) v ( t, · ) (cid:107) L (Ω δ,t ) . The lemma follows from the above remarks with (cid:101) v = ∂ α (cid:101) u and (cid:101) g = [ ∂ α , (cid:101) u j ∂ j θ k ∂ k ] (cid:101) u + ∂ α ( ∂ j θ k ∂ k (cid:101) p ), and the H¨older inequality. (cid:3) Using the bound (3.2), from Lemma A.5 we obtain d + dt sup <δ ≤ δ δ m − (cid:107) ∂ α (cid:101) u ( t ) (cid:107) (cid:101) Ω δ,t ≤ sup <δ ≤ δ δ m − M δ ( t ) . Therefore, we may estimate the sum on the right of (3.1) as d + dt (cid:107) (cid:101) u ( t ) (cid:107) X τ ( t ) ≤ ˙ τ ( t ) (cid:107) (cid:101) u ( t ) (cid:107) Y τ ( t ) + ∞ (cid:88) m =3 (cid:88) | α | = m (cid:15) α sup <δ ≤ δ δ m − M δ ( t ) τ ( t ) m − ( m − s ≤ ˙ τ ( t ) (cid:107) (cid:101) u ( t ) (cid:107) Y τ ( t ) + C + P , (3.4)where C = ∞ (cid:88) m =3 (cid:88) | α | = m (cid:15) α sup <δ ≤ δ δ m − (cid:107) [ ∂ α , (cid:101) u j ∂ j θ k ∂ k ] (cid:101) u ( · , t ) (cid:107) L ( (cid:101) Ω δ,t ) τ ( t ) m − ( m − s , (3.5)and P = ∞ (cid:88) m =3 (cid:88) | α | = m (cid:15) α sup <δ ≤ δ δ m − (cid:107) ∂ α ( ∂ j θ k ∂ k (cid:101) p ( · , t )) (cid:107) L ( (cid:101) Ω δ,t ) τ ( t ) m − ( m − s . (3.6)The estimates for C and P are given in the following two lemmas. Lemma 3.2. If τ < τ ∗ , where τ ∗ is the Gevrey-class regularity radius of the boundary, then the followingestimate holds C ≤ C (1 + τ ) (cid:16) (cid:107) (cid:101) u (cid:107) W , ∞ ( (cid:101) Ω t ) + (cid:107) (cid:101) u (cid:107) H ( (cid:101) Ω t ) (cid:17) + C (cid:32) τ (cid:107) D (cid:101) u (cid:107) L ∞ ( (cid:101) Ω t ) + ( τ + τ ) (cid:16) (cid:107) (cid:101) u (cid:107) W , ∞ ( (cid:101) Ω t ) + (cid:107) (cid:101) u (cid:107) H ( (cid:101) Ω t ) (cid:17) + ( τ / + (1 + K ) τ ) (cid:107) (cid:101) u (cid:107) X τ (cid:33) (cid:107) (cid:101) u (cid:107) Y τ , (3.7) where C is a sufficiently large positive constant depending on γ , and K is as defined in (2.5) . The proof of Lemma 3.2 is given in Section 4, while the proof of Lemma 3.3 below is given in Section 5.
Lemma 3.3.
For (cid:15) > fixed, sufficiently small depending only on γ , if τ ≤ (cid:15)τ ∗ , where τ ∗ is the Gevrey-classregularity radius of the boundary, then we have P ≤ C (1 + τ ) (cid:16) (cid:107) (cid:101) u (cid:107) W , ∞ ( (cid:101) Ω t ) + (cid:107) (cid:101) u (cid:107) H ( (cid:101) Ω t ) + (1 + K ) (cid:107) (cid:101) p (cid:107) H ( (cid:101) Ω t ) + (cid:107) (cid:101) p (cid:107) W , ∞ ( (cid:101) Ω t ) (cid:17) + C (cid:32) τ (cid:107) (cid:101) u (cid:107) W , ∞ ( (cid:101) Ω t ) + ( τ + τ ) (cid:107) (cid:101) u (cid:107) W , ∞ ( (cid:101) Ω t ) + τ (cid:107) (cid:101) u (cid:107) H ( (cid:101) Ω t ) + ( τ / + (1 + K ) τ ) (cid:107) (cid:101) u (cid:107) X τ (cid:33) (cid:107) (cid:101) u (cid:107) Y τ , (3.8) for some sufficiently large constant C depending on γ , where K is as defined in (2.5) . By combining estimates (3.4), (3.7), and (3.8), with the Sobolev embedding, and the classical pressureestimate in Sobolev spaces (cid:107) p (cid:107) H m ( D ) ≤ C (cid:107) u (cid:107) H m − ( D ) (cf. [T, Lemma 1.2]), we obtain for r ≥ ddt (cid:107) (cid:101) u (cid:107) X τ ≤ C (1 + τ )(1 + K ) (cid:107) u (cid:107) H r ( D ) + (cid:16) ˙ τ + Cτ (cid:107) u (cid:107) W , ∞ ( D ) + C ( τ + τ ) (cid:107) u (cid:107) H r ( D ) + C ( τ / + (1 + K ) τ ) (cid:107) (cid:101) u (cid:107) X τ (cid:17) (cid:107) (cid:101) u (cid:107) Y τ . (3.9)for some fixed positive constant C depending on the domain D . Let M ( t ) = (cid:107) u ( · , t ) (cid:107) H r ( D ) (3.10) NALYTICITY AND GEVREY CLASS REGULARITY FOR THE EULER EQUATIONS IN A BOUNDED DOMAIN 7 and N ( t ) = (cid:107) u ( · , t ) (cid:107) W , ∞ ( D ) (3.11)for all 0 ≤ t < T ∗ . Note that K ( t ) = (cid:82) t N ( s ) ds . By possibly increasing the constant C = C ( D ), we have ddt (cid:107) (cid:101) u (cid:107) X τ ≤ C (1 + τ )(1 + K ) M + (cid:16) ˙ τ + Cτ N + C ( τ + τ ) M + C ( τ / + (1 + K ) τ ) (cid:107) (cid:101) u (cid:107) X τ (cid:17) (cid:107) (cid:101) u (cid:107) Y τ . (3.12)Let τ ( t ) be chosen such that τ ( t ) ≤ τ ≤ τ ∗ , where τ ∗ is the radius of Gevrey-class regularity of the boundary,and for all 0 = T ≤ t ≤ T let τ ( t ) be the solution of˙ τ + C τ N + C τ / L = 0 , (3.13)with the initial condition τ (0) = τ , where C is a sufficiently large fixed positive constant (for instance C / (2 + τ ∗ ) > C , the constant of (3.12)); we have denoted L ( t ) = C M ( t ) + (cid:16) C (cid:16) K ( t ) (cid:17)(cid:17) (cid:18) (cid:107) (cid:101) u (cid:107) X τ + C (cid:90) t (cid:16) K ( s ) (cid:17) M ( s ) ds (cid:19) . (3.14)Then τ is decreasing, and by (3.12) for short time we have (cid:107) (cid:101) u ( t ) (cid:107) X τ ( t ) ≤ (cid:107) (cid:101) u (cid:107) X τ + C (cid:90) t (cid:16) K ( s ) (cid:17) M ( s ) ds. (3.15)By (3.12), if (3.13) holds for all t ∈ [ T , T ], then (cid:101) u ( t ) ∈ X τ ( t ) and (3.15) is also valid for all t ∈ [ T , T ]. Theradius of Gevrey-class regularity τ ( t ) may be computed explicitly from (3.13) as τ ( t ) = exp (cid:16) − C K ( t ) (cid:17)(cid:32) τ − / + C (cid:90) t L ( s ) exp (cid:16) − C K ( s ) (cid:17) ds (cid:33) − , (3.16)where L is defined by (3.14). By further estimating the Sobolev norms in (3.16) using M ( t ) = (cid:107) u ( t ) (cid:107) H r ( D ) ≤ C (cid:107) u (cid:107) H r ( D ) exp (cid:18) C (cid:90) t (cid:107) u ( s ) (cid:107) W , ∞ ( D ) ds (cid:19) = C (cid:107) u (cid:107) H r ( D ) e CK ( t ) , for some positive constant C = C ( r, D ), we obtain a more compact lower bound for τ ( t ) given by τ ( t ) ≥ τ (cid:16) Ct (cid:107) (cid:101) u (cid:107) X τ + Ct (cid:107) u (cid:107) H r (cid:16) K ( t ) (cid:17)(cid:17) − exp (cid:16) − CK ( t ) (cid:17) ≥ τ (cid:0) Ct (cid:107) (cid:101) u (cid:107) X τ + Ct (cid:107) u (cid:107) H r (cid:1) − exp (cid:16) − CK ( t ) (cid:17) ; (3.17)we used (1 + x ) − ≥ exp( − x ) for all x ≥
0, where C = C ( D, r ) is a sufficiently large positive constant.Therefore we have proven the following theorem.
Theorem 3.4.
Let u be divergence-free and of Gevrey-class s , with s ≥ , on a Gevrey-class s , open,bounded domain D ⊂ R . Fix r ≥ , x ∈ ∂D , and r > sufficiently small. Let Ω be a neighborhood of x compactly embedded in B r ( x ) ∩ D , and let T be the maximal time such that φ t (Ω) ⊂ B r ( x ) ∩ D for all ≤ t < T . Then the unique H r -solution u ( φ t ( · ) , t ) to the initial value problem (E.1) – (E.4) is of Gevrey-class s for all t < T . Moreover, there exist constants (cid:15) = (cid:15) ( D ) , and τ ∗ = τ ∗ ( D ) , such that if (cid:101) u (0) ∈ X τ , and τ ≤ (cid:15)τ ∗ , then (cid:101) u ( · , t ) ∈ X τ ( t ) for all t ∈ [0 , T ) , where the Gevrey-class radius τ ( t ) of the solution u ( φ t ( · ) , t ) satisfies τ ( t ) ≥ τ (cid:0) Ct (cid:107) (cid:101) u (cid:107) X τ + Ct (cid:107) u (cid:107) H r (cid:1) − exp (cid:18) − C (cid:90) t (cid:107) u ( s ) (cid:107) W , ∞ ds (cid:19) (3.18) for all t < T , with C a sufficiently large constant depending only on the domain D . IGOR KUKAVICA AND VLAD VICOL
Remark 3.5.
Theorem 3.4 gives the local in time Gevrey-class persistence at the boundary of D . Theshort-time Gevrey-class persistence in the interior of D , with explicit bound on the radius of Gevrey-classregularity is obtained using similar arguments to the ones given in this section. Namely, given x ∈ D and r > x , with Ω ⊂ B r ( x ) ∩ D . The mainstep is to show that for all t > φ t (Ω) ⊂ B r ( x ) ∩ D , the analogous estimates to the ones givenin Lemmas 3.2 and 3.3 hold. The bound on the velocity commutator C is proven by repeating exactly thesame estimates as in Section 4 below. Since we are away from the boundary, the bound for the pressure term P is obtained from classical interior elliptic estimates for the pressure (cf. (5.19)) and arguments parallel tothe ones presented in Lemma 5.4. Since the interior pressure estimates are only simpler than the boundarycase, we omit further details. It follows that the solution u ( · , t ) is of Gevrey-class s on φ t (Ω), the radiusof Gevrey-class regularity τ ( t ) satisfies the lower bound (3.18), and that the Gevrey-class norm is boundedfrom below by the right side of (3.15).4. The velocity commutator estimate
Since in this section we work only for a fixed time t and on the straightened domain, we suppress the timedependence and the tilde for all functions and domains. The goal of this section is to prove Lemma 3.2, thatis to estimate C = ∞ (cid:88) m =3 (cid:88) | α | = m (cid:15) α sup <δ ≤ δ (cid:16) δ m − (cid:107) [ ∂ α , u j ∂ j θ k ∂ k ] u (cid:107) L (Ω δ ) (cid:17) τ m − ( m − s . Proof of Lemma 3.2.
The proof consists of two parts. First we estimate the ∂ j θ k coefficients from thedefinition of C and exploit the commutator (cf. (4.6) below). Then we estimate the Gevrey-class norm of u i ∂ j u k (cf. (4.15)–(4.18) below) for all 1 ≤ i, j, k ≤ δ (cid:80) n x n,δ ≤ (cid:80) n sup δ x n,δ for all sequences x n,δ , imply C ≤ ∞ (cid:88) m =3 (cid:88) | α | = m (cid:88) <β ≤ α (cid:88) ≤ γ ≤ β (cid:18) αβ (cid:19)(cid:18) βγ (cid:19) (cid:15) α (cid:107) ∂ γ ∂ j θ k (cid:107) L ∞ (Ω) sup <δ ≤ δ (cid:16) δ m − (cid:107) ∂ β − γ u j ∂ α − β ∂ k u (cid:107) L (Ω δ ) (cid:17) τ m − ( m − s , (4.1)where Ω = (cid:83) <δ ≤ δ Ω δ . Since the boundary is of Gevrey-class s , there exist constants C, τ ∗ > (cid:88) | β | = n (cid:107) ∂ β Dθ k (cid:107) L ∞ (Ω) ≤ C ( n − s τ n ∗ , (4.2)for all n ≥
0. Using (cid:0) αβ (cid:1)(cid:0) βγ (cid:1) = (cid:0) αγ (cid:1)(cid:0) α − γβ − γ (cid:1) ≤ (cid:0) αγ (cid:1)(cid:0) m − kj − k (cid:1) , we may rewrite the right side of (4.1) as C ≤ C ∞ (cid:88) m =3 m (cid:88) j =1 j (cid:88) k =0 (cid:18) mk (cid:19) ( k − s ( m − k − s ( m − s (cid:18) ττ ∗ (cid:19) k × (cid:88) | α | = m (cid:88) | β | = j, β ≤ α (cid:88) | γ | = k, γ ≤ β (cid:32)(cid:18) αγ (cid:19)(cid:18) mk (cid:19) − (cid:107) ∂ γ Dθ (cid:107) L ∞ (Ω) τ k ∗ ( k − s (cid:33) × (cid:18) (cid:15) α τ m − k − ( m − k − s (cid:18) m − kj − k (cid:19) sup <δ ≤ δ δ m − (cid:107) ∂ β − γ u (cid:107) L p (Ω δ ) (cid:107) ∂ α − β Du (cid:107) L p/ ( p − (Ω δ ) (cid:19) , (4.3)where p = 2 if j − k > m − j , and p = ∞ if j − k ≤ m − j . Observe that τ /τ ∗ <
1. Since s ≥
1, there existsa constant C such that (cid:18) mk (cid:19) ( k − s ( m − k − s ( m − s ≤ C ( m − k + 1) s − + Cχ { k =0 } (4.4) NALYTICITY AND GEVREY CLASS REGULARITY FOR THE EULER EQUATIONS IN A BOUNDED DOMAIN 9 for all 0 ≤ k ≤ m , where χ { k =0 } = 1 if k = 0, and χ { k =0 } = 0 if k ≥
1. By (4.2), (4.4), Lemma A.1, andusing (cid:0) αγ (cid:1) ≤ (cid:0) mk (cid:1) , we obtain C ≤ C ∞ (cid:88) m =3 m (cid:88) j =1 j (cid:88) k =0 (cid:18) ττ ∗ (cid:19) k (cid:88) | α | = m (cid:88) | β | = j, β ≤ α (cid:88) | γ | = k, γ ≤ β (cid:18) (cid:107) ∂ γ Dθ (cid:107) L ∞ (Ω) τ k ∗ ( k − s (cid:19) τ m − k − ( m − k − s (cid:18) m − kj − k (cid:19) × (cid:18) m − k + 1) s − + χ { k =0 } (cid:19) (cid:18) (cid:15) α sup <δ ≤ δ δ m − (cid:107) ∂ β − γ u (cid:107) L p (Ω δ ) (cid:107) ∂ α − β Du (cid:107) L p/ ( p − (Ω δ ) (cid:19) ≤ C ∞ (cid:88) m =3 m (cid:88) j =1 j (cid:88) k =0 (cid:18) ττ ∗ (cid:19) k (cid:88) | α | = m − k (cid:88) | β | = j − k, β ≤ α τ m − k − ( m − k − s (cid:18) m − kj − k (cid:19) × (cid:18) m − k + 1) s − + χ { k =0 } (cid:19) (cid:18) (cid:15) α sup <δ ≤ δ δ m − (cid:107) ∂ β u (cid:107) L p (Ω δ ) (cid:107) ∂ α − β Du (cid:107) L p/ ( p − (Ω δ ) (cid:19) . (4.5)Due to the definition of the Gevrey-class norm, in (4.5) we need to consider the cases m − k < m − k ≥ C ≤ C (cid:107) u (cid:107) W , ∞ (Ω) (cid:107) u (cid:107) H (Ω) + Cτ (cid:107) u (cid:107) L ∞ (Ω) (cid:107) u (cid:107) Y τ + C ∞ (cid:88) m =3 m (cid:88) j =1 (cid:18) mj (cid:19) τ m − ( m − s (cid:88) | α | = m (cid:88) | β | = j, β ≤ α (cid:15) α sup <δ ≤ δ δ m − (cid:107) ∂ β u (cid:107) L p (Ω δ ) (cid:107) ∂ α − β Du (cid:107) L p/ ( p − (Ω δ ) , (4.6)where C is a constant depending on the domain, and on τ /τ ∗ <
1. We rewrite the estimate (4.6) as
C ≤ C (cid:107) u (cid:107) H (Ω) + C (cid:107) u (cid:107) W , ∞ (Ω) + Cτ (cid:107) u (cid:107) L ∞ (Ω) (cid:107) u (cid:107) Y τ + C ( C + C + C low + C high + C + C + C ) , (4.7)where for 1 ≤ j ≤ C = ∞ (cid:88) m =3 m (cid:88) | α | = m (cid:88) | β | =1 ,β ≤ α sup <δ ≤ δ (cid:16) (cid:15) β (cid:107) ∂ β u (cid:107) L ∞ (Ω δ ) (cid:17)(cid:16) (cid:15) α − β δ m − (cid:107) ∂ α − β Du (cid:107) L (Ω δ ) (cid:17) τ m − ( m − s , (4.8) C = ∞ (cid:88) m =4 (cid:18) m (cid:19) (cid:88) | α | = m (cid:88) | β | =2 ,β ≤ α sup <δ ≤ δ (cid:16) (cid:15) β δ (cid:107) ∂ β u (cid:107) L ∞ (Ω δ ) (cid:17)(cid:16) (cid:15) α − β δ m − (cid:107) ∂ α − β Du (cid:107) L (Ω δ ) (cid:17) τ m − ( m − s , (4.9)for 3 ≤ j ≤ m − C low = ∞ (cid:88) m =6 [ m/ (cid:88) j =3 (cid:18) mj (cid:19) (cid:88) | α | = m (cid:88) | β | = j,β ≤ α sup <δ ≤ δ (cid:16) (cid:15) β δ j − (cid:107) ∂ β u (cid:107) L ∞ (Ω δ ) (cid:17)(cid:16) (cid:15) α − β δ m − j − (cid:107) ∂ α − β Du (cid:107) L (Ω δ ) (cid:17) τ m − ( m − s , (4.10) C high = ∞ (cid:88) m =7 m − (cid:88) j =[ m/ (cid:18) mj (cid:19) (cid:88) | α | = m (cid:88) | β | = j,β ≤ α sup <δ ≤ δ (cid:16) (cid:15) β δ j − (cid:107) ∂ β u (cid:107) L (Ω δ ) (cid:17)(cid:16) (cid:15) α − β δ m − j (cid:107) ∂ α − β Du (cid:107) L ∞ (Ω δ ) (cid:17) τ m − ( m − s , (4.11)and for m − ≤ j ≤ m C = ∞ (cid:88) m =5 (cid:18) mm − (cid:19) (cid:88) | α | = m (cid:88) | β | = m − ,β ≤ α sup <δ ≤ δ (cid:16) (cid:15) β δ m − (cid:107) ∂ β u (cid:107) L (Ω δ ) (cid:17)(cid:16) (cid:15) α − β δ (cid:107) ∂ α − β Du (cid:107) L ∞ (Ω δ ) (cid:17) τ m − ( m − s , (4.12) C = ∞ (cid:88) m =4 (cid:18) mm − (cid:19) (cid:88) | α | = m (cid:88) | β | = m − ,β ≤ α sup <δ ≤ δ (cid:16) (cid:15) β δ m − (cid:107) ∂ β u (cid:107) L (Ω δ ) (cid:17)(cid:16) (cid:15) α − β δ (cid:107) ∂ α − β Du (cid:107) L ∞ (Ω δ ) (cid:17) τ m − ( m − s , (4.13) C = ∞ (cid:88) m =3 (cid:88) | α | = m sup <δ ≤ δ (cid:16) (cid:15) α δ m − (cid:107) ∂ α u (cid:107) L (Ω δ ) (cid:17) (cid:107) Du (cid:107) L ∞ (Ω δ ) τ m − ( m − s . (4.14) These seven terms are bounded as in the proof of [KV2, Lemma 3.2]. Namely, letting Ω = (cid:83) <δ ≤ δ Ω δ , and j = | β | , for the cases j = 1 and j = m we have C + C ≤ C (cid:107) Du (cid:107) L ∞ (Ω) (cid:107) u (cid:107) H (Ω) + Cτ (cid:107) Du (cid:107) L ∞ (Ω) (cid:107) u (cid:107) Y τ , (4.15)for the cases j = 2 and j = m − C + C ≤ Cτ (cid:107) D u (cid:107) L ∞ (Ω) (cid:107) u (cid:107) H (Ω) + Cτ (cid:107) D u (cid:107) L ∞ (Ω) (cid:107) u (cid:107) Y τ , (4.16)when j = m − C ≤ Cτ (cid:107) u (cid:107) H (Ω) + τ (cid:107) u (cid:107) H (Ω) (cid:107) u (cid:107) Y τ , (4.17)and when 3 ≤ j ≤ m − C low + C high ≤ C (cid:16) τ / + (1 + K ) τ (cid:17) (cid:107) u (cid:107) X τ (cid:107) u (cid:107) Y τ , (4.18)for some sufficiently large constant C , where K is as defined in (2.5). We sketch the proof of the C low estimateand refer the reader to [KV2] for further details on the other five terms. Modulo multiplying by a smoothcut-off function η supported on Ω δ − r and which is identically 1 on Ω δ , (2.8) and the three-dimensionalAgmon inequality give that (cid:107) v (cid:107) L ∞ (Ω δ ) ≤ C (cid:107) v (cid:107) / L (Ω δ − r ) (cid:107) ∆ v (cid:107) / L (Ω δ − r ) + C + CK r / (cid:107) v (cid:107) L (Ω δ − r ) , (4.19)where C > δ , and K ( t ) = (cid:107) u (cid:107) L t (0 ,t ) W , ∞ x ( D ) is as in(2.5). Letting r = δ/j , for j ≥
3, we havesup <δ ≤ δ (cid:16) (cid:15) β δ j − (cid:107) ∂ β u (cid:107) L ∞ (Ω δ ) (cid:17) ≤ C (1 + K ) sup <δ ≤ δ ( j/δ ) / (cid:16) (cid:15) β ( δ − δ/j ) j − (cid:107) ∂ β u (cid:107) L (Ω δ − δ/j ) (cid:17) δ + sup <δ ≤ δ (cid:16) (cid:15) β ( δ − δ/j ) j − (cid:107) ∂ β u (cid:107) L (Ω δ − δ/j ) (cid:17) / sup <δ ≤ δ (cid:16) (cid:15) β ( δ − δ/j ) j − (cid:107) ∂ β u (cid:107) L (Ω δ − δ/j ) (cid:17) / δ / (4.20)In the above inequality we used (1 + 1 / ( j − j − ≤ e for all j ≥
1. By the H¨older inequality, and [KV2,Lemma 4.2], we obtain from the definition of C low and the above inequality C low ≤ C ∞ (cid:88) m =6 [ m/ (cid:88) j =3 (cid:18) mj (cid:19) [ u ] / j [ u ] / j +2 [ u ] m − j +1 τ m − ( m − s + C (1 + K ) ∞ (cid:88) m =6 [ m/ (cid:88) j =3 (cid:18) mj (cid:19) [ u ] j j / [ u ] m − j +1 τ m − ( m − s , (4.21)where we have denoted [ v ] m = (cid:88) | α | = m (cid:15) α sup <δ ≤ δ δ m − (cid:107) ∂ α v (cid:107) L (Ω δ ) (4.22)for all smooth v . In the above estimate (4.21) we used the fact that [ Dv ] m ≤ C [ v ] m +1 and [∆ v ] m ≤ C [ v ] m +2 ,where C > (cid:15) , which is fixed. The right side of (4.21) is then bounded by C (cid:16) τ / + (1 + K ) τ (cid:17) (cid:107) u (cid:107) X τ (cid:107) u (cid:107) Y τ . Here we used the definitions (2.11)–(2.12), the discrete Young and H¨older inequalities, and the combinatorialestimate (cid:18) mj (cid:19) ( j − s/ ( j − s/ ( m − j − s ( m − s ( m − j + 1) + (cid:18) mj (cid:19) ( j − s j / ( m − j − s ( m − s ( m − j + 1) ≤ C, (4.23)which holds for all m ≥
6, 3 ≤ j ≤ [ m/ s ≥
1, where
C > j and m − j , similar estimates give the bound on C high , thereby proving (4.18). This concludesthe proof of the Lemma 3.2. (cid:3) NALYTICITY AND GEVREY CLASS REGULARITY FOR THE EULER EQUATIONS IN A BOUNDED DOMAIN 11 The pressure estimate
The goal of this section is to prove Lemma 3.3. This is achieved in several steps: First we use an H regularity estimate on the flattened domain to estimate all tangential derivatives of the pressure; next, weobtain a recursion formula to bootstrap to an estimate with higher number of normal derivatives, whichleads to an estimate in terms of the velocity; lastly, we prove a product-type estimate for the LagrangianGevrey-class norms defined in Section 2 which concludes the proof.For the rest of this section all functions depend on y = θ ( x ), hence we shall further suppress all tildes,and since there is no time evolution for the pressure we also suppress the time dependence. Semi-norms and a decomposition of the pressure term.
The following semi-norms are useful whentreating the pressure term. Namely, define (cid:104) v (cid:105) l,γ,δ = δ l + | γ |− (cid:107) ∂ γ ∂ γ v (cid:107) L (Ω δ ) (5.1)for all γ ∈ N , l ∈ Z with | γ | + l ≥
3, and all 0 < δ ≤ δ . Also let (cid:104) v (cid:105) l,n = (cid:88) | α | = n, α =0 sup <δ ≤ δ (cid:104) v (cid:105) l,α (cid:48) ,δ = (cid:88) | α | = n, α =0 sup <δ ≤ δ δ l + n − (cid:107) ∂ α ∂ α v (cid:107) L (Ω δ ) (5.2)for all n ≥ n + l ≥
3. Note that we have the inequality (cid:104) D (cid:48) k v (cid:105) l,n ≤ (cid:104) v (cid:105) l − k,n + k , where (cid:104) D (cid:48) k v (cid:105) l,n = (cid:80) | α | = n,α =0 sup <δ ≤ δ δ l + n − (cid:107) D (cid:48) k v (cid:107) L (Ω δ ) , and (cid:107) D (cid:48) k v (cid:107) L is defined above.Next, we shall estimate the pressure term arising in (3.4), i.e., P = ∞ (cid:88) m =3 (cid:88) | α | = m (cid:15) α sup <δ ≤ δ δ m − (cid:13)(cid:13) ∂ α (cid:0) ∂ j θ k ∂ k p (cid:1)(cid:13)(cid:13) L (Ω δ ) τ m − ( m − s . (5.3)Similarly to (4.1)–(4.6), we let C, τ ∗ > (cid:80) | β | = j (cid:107) ∂ β Dθ (cid:107) L ∞ (Ω) ≤ C ( j − s /τ j ∗ for all j ≥ τ < τ ∗ , it follows from the Leibniz rule and the bound (cid:0) mj (cid:1) ( m − j − s ( j − s ( m − − s ≤ C that the pressure term is bounded as P ≤ C (cid:107) Dp (cid:107) W , ∞ (Ω) + C ∞ (cid:88) m =3 (cid:88) | α | = m (cid:15) α sup <δ ≤ δ δ m − (cid:107) ∂ α Dp (cid:107) L (Ω δ ) τ m − ( m − s , (5.4)for some positive constant C = C ( D, η ), where η = τ /τ ∗ < P ≤ C (cid:107) Dp (cid:107) W , ∞ (Ω) + C ∞ (cid:88) m =3 (cid:32) m (cid:88) α =0 (cid:15) α (cid:104) ∂ α Dp (cid:105) α ,m − α (cid:33) τ m − ( m − s ≤ C (cid:107) p (cid:107) W , ∞ (Ω) + C (cid:16) (1 + (cid:15) ) P + P + P (cid:17) , (5.5)where we have denoted the term with at most one normal derivative by P = ∞ (cid:88) m =3 (cid:104) Dp (cid:105) ,m τ m − ( m − s , (5.6)and the terms with at least two normal derivatives (according to D = ∂ + D (cid:48) ) by P = ∞ (cid:88) m =3 m (cid:88) α =1 (cid:15) α (cid:104) ∂ α +13 p (cid:105) α ,m − α τ m − ( m − s (5.7)and P = ∞ (cid:88) m =3 m (cid:88) α =2 (cid:15) α (cid:104) ∂ α p (cid:105) α − ,m − α +1 τ m − ( m − s . (5.8) The elliptic Neumann problem for the pressure.
Under the change of variables θ : x (cid:55)→ y , the ellipticNeumann problem for the pressure (cf. [T]) becomes (omitting tildes) − ∆ p = A ij ∂ ij p + B j ∂ j p + D ijkl ∂ i u j ∂ k u l , in Ω (5.9) − ∂ p = C j ∂ j p + Φ ij u i u j , on ∂ Ω , (5.10)where we denoted A ij = 1Γ − ( ∂ γ ) − ( ∂ γ ) − ∂ γ − ( ∂ γ ) − ( ∂ γ ) − ∂ γ − ∂ γ − ∂ γ , (5.11) B j = 1Γ − ∂ γ − ∂ γ , (5.12) C j = 1 + Γ / (2 − Γ + Γ / )Γ / ∂ γ∂ γ , (5.13) D ijkl = 1Γ δ ik δ jl + 1Γ δ k ∂ l γ ∂ l γ ∂ l γ ) ∂ γ∂ γ ∂ l γ , (5.14)Φ ij = 1Γ / ∂ γ ∂ γ ∂ γ ∂ γ
00 0 0 , (5.15)with Γ = 1 + ( ∂ γ ) + ( ∂ γ ) . (5.16)The precise form of the above matrices is not essential; what is important for the following arguments isthat A = C = 0, and that the coefficients A ij , C j are small. We also denote f = D ijkl ∂ i u j ∂ k u l , (5.17) g = Φ ij u i u j . (5.18) The interior H -regularity estimate. Let p be the smooth solution of the elliptic Neumann problem − ∆ p = A ij ∂ ij p + B j ∂ j p + f, in Ω , (5.19) − ∂ p = C j ∂ j p + g, on ∂ Ω , (5.20)where all coefficients are of Gevrey-class s . We have the following interior H -regularity estimate. Lemma 5.1.
There exists a sufficiently small positive dimensional constant ε such that if A = C = 0 , (cid:107) A ij (cid:107) L ∞ (Ω) ≤ ε , and (cid:107) C j (cid:107) L ∞ (Ω) ≤ ε , the smooth solution p of (5.19) – (5.20) satisfies (cid:107) D p (cid:107) L (Ω δ + r ) ≤ C (cid:18) (cid:107) f (cid:107) L (Ω δ ) + (cid:107) Dg (cid:107) L (Ω δ ) + 1 + K r (cid:107) Dp (cid:107) L (Ω δ ) (cid:19) , (5.21) for all δ ∈ (0 , δ ) and for all < r << , where K is as defined in (2.5) , and C = C ( A ij , B j , C j ) is apositive constant depending on the domain. The proof is standard and thus omitted. It relies on the fact that the elliptic operator acting on p in (5.19) is a small/lower-order perturbation of the Laplacian, and on the the fact that by (2.8) we have C dist(Ω δ + r , Ω cδ ) ≥ r/ (1 + K ), for some sufficiently large constant C . NALYTICITY AND GEVREY CLASS REGULARITY FOR THE EULER EQUATIONS IN A BOUNDED DOMAIN 13
The estimation of tangential derivatives.
Fix k ≥
3, and let α (cid:48) = ( α , α , ∈ N be such that | α (cid:48) | = k .Consider the system (5.19)–(5.20). The function ∂ α (cid:48) p satisfies the elliptic Neumann problem − ∆( ∂ α (cid:48) p ) = A ij ∂ ij ∂ α (cid:48) p + B j ∂ j ∂ α (cid:48) p + ∂ α (cid:48) f + [ ∂ α (cid:48) , A ij ∂ ij ] p + [ ∂ α (cid:48) , B j ∂ j ] p, in Ω , (5.22) − ∂ ( ∂ α (cid:48) p ) = C j ∂ j ∂ α (cid:48) p + ∂ α (cid:48) g + [ ∂ α (cid:48) , C j ∂ j ] p, on ∂ Ω . (5.23)We apply the H -estimate (5.21) to the solution of (5.22)–(5.23), and bound the commutators using theLeibniz rule as (cid:107) [ ∂ α (cid:48) , A ij ∂ ij ] p (cid:107) L (Ω δ ) ≤ (cid:88) <β (cid:48) ≤ α (cid:48) (cid:18) α (cid:48) β (cid:48) (cid:19) (cid:107) ∂ β (cid:48) A ij (cid:107) L ∞ (Ω δ ) (cid:107) ∂ α (cid:48) − β (cid:48) ∂ ij p (cid:107) L (Ω δ ) . (5.24)The terms involving [ ∂ α (cid:48) , B j ∂ j ] and [ ∂ α (cid:48) , C j ∂ j ] are estimated similarly. Letting r = δ/k ≤ δ/
3, we obtain (cid:107) ∂ α (cid:48) D p (cid:107) L (Ω δ + δ/k ) ≤ C (cid:18) (cid:107) ∂ α (cid:48) f (cid:107) L (Ω δ ) + (cid:107) ∂ α (cid:48) Dg (cid:107) L (Ω δ ) + (1 + K ) kδ (cid:107) ∂ α (cid:48) Dp (cid:107) Ω δ (cid:19) + C k (cid:88) j =1 (cid:18) kj (cid:19) (cid:88) | β (cid:48) | = j, β (cid:48) ≤ α (cid:48) max {(cid:107) ∂ β (cid:48) A ij (cid:107) L ∞ (¯Ω δ ) , (cid:107) ∂ β (cid:48) B j (cid:107) L ∞ (¯Ω δ ) , (cid:107) ∂ β (cid:48) C j (cid:107) L ∞ (¯Ω δ ) }× (cid:16) (cid:107) ∂ α (cid:48) − β (cid:48) DD (cid:48) p (cid:107) L (Ω δ ) + (cid:107) ∂ α (cid:48) − β (cid:48) Dp (cid:107) L (Ω δ ) (cid:17) , (5.25)where we used A = 0. Denote ψ β (cid:48) ,δ = δ | β (cid:48) |− max {(cid:107) ∂ β (cid:48) A ij (cid:107) L ∞ (¯Ω δ ) , (cid:107) ∂ β (cid:48) B j (cid:107) L ∞ (¯Ω δ ) , (cid:107) ∂ β (cid:48) C j (cid:107) L ∞ (¯Ω δ ) } , (5.26)for all | β (cid:48) | ≥
2, and ψ β (cid:48) ,δ = max {(cid:107) ∂ β (cid:48) A ij (cid:107) L ∞ (¯Ω δ ) , (cid:107) ∂ β (cid:48) B j (cid:107) L ∞ (¯Ω δ ) , (cid:107) ∂ β (cid:48) C j (cid:107) L ∞ (¯Ω δ ) } , (5.27)if | β (cid:48) | = 1. Also let ψ j = (cid:88) | β (cid:48) | = j sup <δ ≤ δ ψ β (cid:48) ,δ , (5.28)Note that since γ , and hence A ij , B j , C j , is of Gevrey-class s , there exist C, τ ∗ > ψ j ≤ C ( j − s τ j ∗ , (5.29)for all j ≥
1, where recall ( − C and the Gevrey-class s radius τ ∗ of theboundary are not functions of time. Multiplying estimate (5.25) by ( δ + δ/k ) k − , and using (1 + 1 /k ) k − ≤ e for k ≥
3, we obtain (cid:104) D p (cid:105) ,α (cid:48) ,δ + δ/k ≤ C (cid:104) f (cid:105) ,α (cid:48) ,δ + C (cid:104) Dg (cid:105) ,α (cid:48) ,δ + Cψ α (cid:48) ,δ (cid:0) (cid:107) DD (cid:48) p (cid:107) L (Ω δ ) + (cid:107) Dp (cid:107) L (Ω δ ) (cid:1) + C (1 + K ) k (cid:88) | β (cid:48) | = k − ,β (cid:48) ≤ α (cid:48) ψ β (cid:48) ,δ (cid:16) δ (cid:107) DD (cid:48) p (cid:107) L (Ω δ ) + δ (cid:107) DD (cid:48) p (cid:107) L (Ω δ ) (cid:17) + Ck ( k − (cid:88) | β (cid:48) | = k − ,β (cid:48) ≤ α (cid:48) ψ β (cid:48) ,δ δ (cid:107) DD (cid:48) p (cid:107) L (Ω δ ) + Cχ k ≥ k − (cid:88) j =2 (cid:18) kj (cid:19) (cid:88) | β (cid:48) | = j,β (cid:48) ≤ α (cid:48) ψ β (cid:48) ,δ (cid:104) DD (cid:48) p (cid:105) ,α (cid:48) − β (cid:48) ,δ + Cχ k ≥ k − (cid:88) j =2 (cid:18) kj (cid:19) (cid:88) | β (cid:48) | = j,β (cid:48) ≤ α (cid:48) ψ β (cid:48) ,δ (cid:104) Dp (cid:105) ,α (cid:48) − β (cid:48) ,δ + Ck (cid:104) Dp (cid:105) ,α (cid:48) ,δ + Ck (cid:88) | β (cid:48) | =1 ,β (cid:48) ≤ α (cid:48) ψ β (cid:48) ,δ (cid:16) (cid:104) DD (cid:48) p (cid:105) ,α (cid:48) − β (cid:48) ,δ + (cid:104) Dp (cid:105) ,α (cid:48) − β (cid:48) ,δ (cid:17) . (5.30) By taking the supremum over 0 < δ ≤ δ ≤ | α (cid:48) | = k ≥ (cid:104) ∂ Dp (cid:105) ,k + (cid:104) Dp (cid:105) ,k +1 ≤ C (cid:16) (cid:104) f (cid:105) ,k + (cid:104) Dg (cid:105) ,k (cid:17) + C (1 + K ) (cid:16) ψ k (cid:107) Dp (cid:107) L (Ω) + ( ψ k + kψ k − ) (cid:107) D p (cid:107) L (Ω) + ( kψ k − + k ψ k − ) (cid:107) D p (cid:107) L (Ω) (cid:17) + C k − (cid:88) j =1 (cid:18) kj (cid:19) ψ j (cid:104) Dp (cid:105) ,k − j +1 + Cχ k ≥ k − (cid:88) j =1 (cid:18) kj (cid:19) ψ j (cid:104) Dp (cid:105) ,k − j , (5.31)where as usual we write Ω = (cid:83) <δ ≤ δ Ω δ . Estimate (5.31) is used to bound the term P in the decomposition(5.5) of P . Furthermore, using the bound (5.29) on ψ j , estimate (5.5) implies (cid:104) ∂ Dp (cid:105) ,k + (cid:104) Dp (cid:105) ,k +1 ≤ C (cid:16) (cid:104) f (cid:105) ,k + (cid:104) Dg (cid:105) ,k (cid:17) + C (1 + τ ∗ + τ ∗ )(1 + K ) (cid:16) (cid:107) Dp (cid:107) L (Ω) + (cid:107) D p (cid:107) L (Ω) + (cid:107) D p (cid:107) L (Ω) (cid:17) ( k − s τ k ∗ + C k − (cid:88) j =1 (cid:18) kj (cid:19) ( j − s τ j ∗ (cid:104) Dp (cid:105) ,k − j +1 + Cχ k ≥ k − (cid:88) j =1 (cid:18) kj (cid:19) ( j − s τ j ∗ (cid:104) Dp (cid:105) ,k − j , (5.32)for all k ≥
3, where C depends on C and δ ≤
1, while τ ∗ is fixed, depending only on γ . The transfer of normal to tangential derivatives.
We use the special structure of the coefficients A ij and B j to rewrite (5.19) as − ∂ p = ( a ∂ + a ∂ ) ∂ p + b ∂ p + c ( ∂ + ∂ ) p + f = ( a · ∇ (cid:48) ) ∂ p + b ∂ p + c ∆ (cid:48) p + f, (5.33)where, as above, (cf. (5.11),(5.12), and (5.16)) a i = − ∂ i γ Γ , b = − ∂ γ + ∂ γ Γ , c = 1Γ . (5.34)Since γ , and hence a, b , and c , is a function of ( y , y ) only, we obtain from (5.33) that for k ≥ − ∂ k p = ( a · ∇ (cid:48) ) ∂ k − p + b ∂ k − p + c ∆ (cid:48) ∂ k − p + ∂ k − f. (5.35)Note that in the case of the half-space (cf. [KV2]), identity (5.33) simplifies to − ∂ p = ∆ (cid:48) p + f , which allowsone to obtain an explicit formula for ∂ k p in terms of f and ( − ∆ (cid:48) ) k p . The combinatorial structure of thistransfer of normal to tangential derivatives is encoded in the coefficients M α of [KV2]. In the case of thepresent paper, it is highly inconvenient use the recursion formula (5.35) to explicitly calculate ∂ k p in termsof f and tangential derivatives of p . Instead we use the fact that we may choose (cid:15) << p . By applying ∂ α (cid:48) , where | α (cid:48) | = n ,to (5.35), using the Leibniz rule, the H¨older inequality, we obtain (cid:107) ∂ k ∂ α (cid:48) p (cid:107) L (Ω δ ) ≤ (cid:107) ∂ k − ∂ α (cid:48) f (cid:107) L (Ω δ ) + C n (cid:88) j =0 (cid:18) nj (cid:19) (cid:88) | β (cid:48) | = j,β (cid:48) ≤ α (cid:48) ψ β (cid:48) ,δ (cid:16) (cid:107) ∂ α (cid:48) − β (cid:48) D (cid:48) ∂ k − p (cid:107) L (Ω δ ) + (cid:107) ∂ α (cid:48) − β (cid:48) ∂ k − p (cid:107) L (Ω δ ) + (cid:107) ∂ α (cid:48) − β (cid:48) ∆ (cid:48) ∂ k − p (cid:107) L (Ω δ ) (cid:17) , (5.36)where we have denoted ψ β (cid:48) ,δ and ψ j similarly to (5.26)–(5.28) (replace A ij , B j , C j by a, b, c ). Since a, b , and c are of Gevrey-class s (they only depend on γ ), as in (5.29), there exist C, τ ∗ > ψ j ≤ C ( j − s /τ j ∗ .Multiplying the bound (5.36) by δ n + k − , it follows that (cid:104) ∂ k p (cid:105) k − ,α (cid:48) ,δ ≤ (cid:104) ∂ k − f (cid:105) k − ,α (cid:48) ,δ + n (cid:88) j =0 (cid:18) nj (cid:19) (cid:88) | β (cid:48) | = j,β (cid:48) ≤ α (cid:48) ψ β (cid:48) ,δ (cid:16) (cid:104) ∂ k − D (cid:48) p (cid:105) k +1 ,α (cid:48) − β (cid:48) ,δ + (cid:104) ∂ k − ∆ (cid:48) p (cid:105) k +1 ,α (cid:48) − β (cid:48) ,δ + (cid:104) ∂ k − p (cid:105) k +1 ,α (cid:48) − β (cid:48) ,δ (cid:17) , (5.37) NALYTICITY AND GEVREY CLASS REGULARITY FOR THE EULER EQUATIONS IN A BOUNDED DOMAIN 15 for all n + k ≥
4, 0 ≤ δ ≤ δ , and α ∈ N with α = 0, and | α (cid:48) | = n . Estimate (5.37) above will be used tobound the terms with high number of normal derivatives in the pressure estimate, namely P , and P . Bounds for P , P , and P . For the term P with a low number of tangential derivatives we have thebound P ≤ Cη − η P + C (1 + K ) (cid:107) p (cid:107) H (Ω) + C ∞ (cid:88) m =4 (cid:16) (cid:104) f (cid:105) ,m − + (cid:104) Dg (cid:105) ,m − (cid:17) τ m − ( m − s , (5.38)where η = τ /τ ∗ < τ ∗ is the Gevrey-class radius of the boundary, C = C ( γ ) and C = C ( γ, η, (cid:15) ) aresufficiently large constant positive constants. As usual, Ω = (cid:83) <δ ≤ δ Ω δ . Note that the condition η < η <
1, we also have the bounds P ≤ (cid:15) C − η ( P + P ) + C (cid:107) p (cid:107) H (Ω) + C ∞ (cid:88) m =3 m (cid:88) α =1 (cid:15) α (cid:104) ∂ α − f (cid:105) α ,m − α τ m − ( m − s , (5.39)and P ≤ + (cid:15) C − η ( (cid:15) P + P + P ) + C (cid:107) p (cid:107) H (Ω) + C ∞ (cid:88) m =3 m (cid:88) α =2 (cid:15) α (cid:104) ∂ α − f (cid:105) α − ,m − α +1 τ m − ( m − s , (5.40)where C = C ( γ ) is a fixed sufficiently large constant, while C = C ( γ, η, (cid:15) ) has additional dependence onthe Gevrey-class norm and radius of γ cf. (5.29) and the parameter (cid:15) . First we prove the bound for P . Proof of (5.38) . Letting k = m − P , we obtain P ≤ (cid:104) Dp (cid:105) , + C ∞ (cid:88) m =4 (cid:16) (cid:104) f (cid:105) ,m − + (cid:104) Dg (cid:105) ,m − (cid:17) τ m − ( m − s + C (1 + K )(1 + τ ∗ ) (cid:107) p (cid:107) H (Ω) ∞ (cid:88) m =4 ( m − s τ m − ∗ τ m − ( m − s + C ∞ (cid:88) m =4 m − (cid:88) j =1 (cid:18) m − j (cid:19) ( j − s τ j ∗ (cid:104) Dp (cid:105) ,m − j τ m − ( m − s + C ∞ (cid:88) m =5 m − (cid:88) j =1 (cid:18) m − j (cid:19) ( j − s τ j ∗ (cid:104) Dp (cid:105) ,m − j − τ m − ( m − s . Using the fact that for all s ≥ m ≥
4, and 1 ≤ j ≤ m − (cid:18) m − j (cid:19) ( j − s ( m − j − s ( m − s ≤ C, (5.41)and recalling that we have η = τ /τ ∗ <
1, we estimate the discrete convolution and obtain P ≤ η C − η P + C (1 + K ) (cid:107) p (cid:107) H (Ω) + C ∞ (cid:88) m =4 (cid:16) (cid:104) f (cid:105) ,m − + (cid:104) Dg (cid:105) ,m − (cid:17) τ m − ( m − s , (5.42)where C is a dimensional constant and C = C ( γ, τ , η ), concluding the proof. (cid:3) The estimates for P and P are symmetric, and so to avoid redundancy we only give the proof of (5.39). Proof of (5.39) . Let k = α + 1 and n = m − α (so that n + k ≥
4) in (5.37), to obtain that for all | α (cid:48) | = m − α we have (cid:104) ∂ α +13 p (cid:105) α ,α (cid:48) ,δ ≤ (cid:104) ∂ α − f (cid:105) α ,α (cid:48) ,δ + m − α (cid:88) j =0 (cid:18) m − α j (cid:19) (cid:88) | β (cid:48) | = j,β (cid:48) ≤ α (cid:48) ψ β (cid:48) ,δ ×× (cid:16) (cid:104) ∂ α D (cid:48) p (cid:105) α +2 ,α (cid:48) − β (cid:48) ,δ + (cid:104) ∂ α − D (cid:48) p (cid:105) α +2 ,α (cid:48) − β (cid:48) ,δ + (cid:104) ∂ α p (cid:105) α +2 ,α (cid:48) − β (cid:48) ,δ (cid:17) . Taking the supremum over 0 < δ ≤ δ <
1, and summing over all | α (cid:48) | = m − α , the above estimate implies P ≤ C ∞ (cid:88) m =3 m (cid:88) α =1 (cid:15) α (cid:104) ∂ α − f (cid:105) α ,m − α τ m − ( m − s + C ∞ (cid:88) m =3 m (cid:88) α =1 m − α (cid:88) j =0 (cid:18) m − α j (cid:19) ψ j (cid:15) α ×× (cid:16) (cid:104) ∂ α p (cid:105) α +1 ,m − j − α +1 + (cid:104) ∂ α − p (cid:105) α ,m − j − α +2 + (cid:104) ∂ α p (cid:105) α +2 ,m − j − α (cid:17) τ m − ( m − s . Using the bound (5.29) on ψ j and the combinatorial estimate (cid:18) m − α j (cid:19) ( j − s ( m − j − s ( m − s ≤ C, (5.43)which holds for all m ≥
3, 1 ≤ α ≤ m , and 0 ≤ j ≤ m − α , we obtain P ≤ C ∞ (cid:88) m =3 m (cid:88) α =1 (cid:15) α (cid:104) ∂ α − f (cid:105) α ,m − α τ m − ( m − s + (cid:15)C ∞ (cid:88) m =3 m (cid:88) α =1 m − α (cid:88) j =0 η j (cid:18) (cid:15) α − (cid:104) ∂ α p (cid:105) α +1 ,m − j − α +1 τ m − j − ( m − j − s (cid:19) + τ (cid:15)C ∞ (cid:88) m =3 m (cid:88) α =1 m − α (cid:88) j =0 η j +1 (cid:18) (cid:15) α − (cid:104) ∂ α p (cid:105) α +2 ,m − j − α τ m − j − ( m − j − s (cid:19) + (cid:15) C ∞ (cid:88) m =3 m (cid:88) α =1 m − α (cid:88) j =0 η j (cid:18) (cid:15) α − (cid:104) ∂ α − p (cid:105) α ,m − j − α +2 τ m − j − ( m − j − s (cid:19) . (5.44)Here, as before we denoted η = τ /τ ∗ <
1. It is convenient to reverse the summation order in the aboveestimate and write P ≤ C ∞ (cid:88) m =3 m (cid:88) α =1 (cid:15) α (cid:104) ∂ α − f (cid:105) α ,m − α τ m − ( m − s + (cid:15)C ∞ (cid:88) m =3 m − (cid:88) j =0 η j m − j − (cid:88) α =0 (cid:18) (cid:15) α (cid:104) ∂ α +13 p (cid:105) α +2 ,m − j − α τ m − j − ( m − j − s (cid:19) + τ (cid:15)C ∞ (cid:88) m =3 m (cid:88) j =1 η j m − j (cid:88) α +3 ,α =0 (cid:18) (cid:15) α (cid:104) ∂ α +13 p (cid:105) m − j − α τ m − j − ( m − j − s (cid:19) + (cid:15) C ∞ (cid:88) m =3 m − (cid:88) j =0 η j m − j − (cid:88) α =0 (cid:18) (cid:15) α (cid:104) ∂ α +13 p (cid:105) α +2 ,m − j − α τ m − j − ( m − j − s (cid:19) + (cid:15)C ∞ (cid:88) m =3 m − (cid:88) j =0 η j (cid:104) p (cid:105) ,m − j +1 τ m − j − ( m − j − s ≤ C ∞ (cid:88) m =3 m (cid:88) α =1 (cid:15) α (cid:104) ∂ α − f (cid:105) α ,m − α τ m − ( m − s + T + T + T + T . (5.45)The terms T , T , T , and T are bounded by estimating the discrete convolution (cid:80) m (cid:80) j x j y m − j , and usingthe fact that since η < (cid:80) j ≥ η j = 1 / (1 − η ). We have the following estimate T ≤ (cid:15) C − η P + (cid:15) C − η P + C (cid:107) p (cid:107) H (Ω) , (5.46) NALYTICITY AND GEVREY CLASS REGULARITY FOR THE EULER EQUATIONS IN A BOUNDED DOMAIN 17 where C = C ( γ ) is a positive constant, and C has additional dependance on η , and (cid:15) . Similarly we obtain T ≤ (cid:15) C − η P + (cid:15) C − η P + C (cid:107) p (cid:107) H (Ω) , (5.47) T ≤ (cid:15) C − η P + (cid:15) C − η P + C (cid:107) p (cid:107) H (Ω) , (5.48)and T ≤ (cid:15) C − η P + C (cid:107) p (cid:107) H (Ω) , (5.49)with C = C ( γ ) >
0, and C = C ( γ, (cid:15), η ) >
0. The proof is concluded by combining (5.45)–(5.49). (cid:3)
Gevrey-class estimates for the pressure.Lemma 5.2.
There exists a sufficiently small constant (cid:15) > depending only on γ , such that if τ ≤ (cid:15)τ ∗ , thenwe have P ≤ C ∞ (cid:88) m =3 m (cid:88) α =1 (cid:15) α (cid:104) ∂ α − f (cid:105) α ,m − α τ m − ( m − s + C ∞ (cid:88) m =3 m (cid:88) α =2 (cid:15) α (cid:104) ∂ α − f (cid:105) α − ,m − α +1 τ m − ( m − s + C ∞ (cid:88) m =3 (cid:104) Dg (cid:105) ,m − τ m − ( m − s + C (1 + K ) (cid:107) p (cid:107) H (Ω) + C (cid:107) p (cid:107) W , ∞ (Ω) , (5.50) where C = C ( γ ) is a fixed positive constant.Proof of Lemma 5.2. By combining estimates (5.38)–(5.40) we obtain that for η < P + P + P ≤ ( (cid:15) + η ) C ∗ − η P + (cid:15) C ∗ − η P + (cid:15) C ∗ − η P + C (1 + K ) (cid:107) p (cid:107) H (Ω) + C ∞ (cid:88) m =4 (cid:16) (cid:104) f (cid:105) ,m − + (cid:104) Dg (cid:105) ,m − (cid:17) τ m − ( m − s + C ∞ (cid:88) m =3 m (cid:88) α =2 (cid:15) α (cid:104) D∂ α − f (cid:105) α ,m − α τ m − ( m − s , (5.51)for a sufficiently large, fixed constant C ∗ = C ∗ ( γ ) >
0, and C = C ( γ, (cid:15), η ) >
0. Define (cid:15) = (cid:15) ( γ ) by (cid:15) = 11 + 4 C ∗ . (5.52)It is clear that (cid:15) may be fixed for all time, as it only depends on the boundary of the domain. Whenever τ ≤ (cid:15)τ ∗ , we have η = τ /τ ∗ ≤ (cid:15) , and therefore ( (cid:15) + η ) / (1 − η ) ≤ (cid:15)/ (1 − (cid:15) ) ≤ / (2 C ∗ ), by the choice of (cid:15) (5.52). Thus the terms involving P , P , and P on the right side of (5.51), may be absorbed on the leftside of (5.51) and the proof of the lemma is completed. (cid:3) Remark 5.3.
The condition τ < (cid:15)τ ∗ is not restrictive; it is a manifestation of the fact that the velocity fieldcannot have arbitrarily large Gevrey-class radius close to the boundary, it must be bounded from above bythe Gevrey-class radius of the boundary.In the following lemma we use the definitions of f and g (cf. (5.17), (5.18)) to bound the right side of(5.50) in terms of the velocity. Lemma 5.4.
For (cid:15) = (cid:15) ( γ ) > as in Lemma 5.2, if τ < (cid:15)τ ∗ , then we have P ≤ C ∞ (cid:88) m =3 (cid:104) D ( uu ) (cid:105) ,m − τ m − ( m − s + C ∞ (cid:88) m =3 m − (cid:88) α =0 (cid:15) α +1 (cid:104) ∂ α ( DuDu ) (cid:105) α +3 ,m − − α τ m − ( m − s + C (cid:107) DuDu (cid:107) H (Ω) + C (cid:107) uu (cid:107) H (Ω) + C (1 + K ) (cid:107) p (cid:107) H (Ω) + C (cid:107) p (cid:107) W , ∞ (Ω) , (5.53) where C = C ( γ ) > is a sufficiently large constant. Proof of Lemma 5.4.
Denote the right side of (5.50) by T f + T f + T g + C (1 + K ) (cid:107) p (cid:107) H (Ω) + C (cid:107) p (cid:107) W , ∞ (Ω) .First we estimate the term T g = C ∞ (cid:88) m =3 (cid:104) Dg (cid:105) ,m − τ m − ( m − s . (5.54)Recall that g = u i u j Φ ij (cf. (5.18)). As in the proof of (5.38) and (5.39), we denote ψ β,δ = δ | β | (cid:107) ∂ β Φ ij (cid:107) L ∞ (Ω δ ) ,and ψ j = (cid:80) | β | = j sup <δ ≤ δ ψ β,δ . Since Φ ij is of Gevrey-class s (cf. (5.15)) there exist C, τ ∗ such that ψ j ≤ C ( j − s /τ ∗ , for all j ≥ n ! = 1 if n ≤ T g ≤ C ∞ (cid:88) m =3 m (cid:88) j =0 (cid:88) | α | = m,α ≤ (cid:88) | β | = j,β ≤ α (cid:18) αβ (cid:19) sup <δ ≤ δ δ m − (cid:107) ∂ β Φ (cid:107) L ∞ (Ω δ ) (cid:107) ∂ α − β ( uu ) (cid:107) L (Ω δ ) τ m − ( m − s . (5.55)We split this sum into four pieces according to j = m , j = m − j = m −
2, and 0 ≤ j ≤ m −
3. We obtain T g ≤ C ∞ (cid:88) m =3 m − (cid:88) j =0 (cid:18) mj (cid:19) ψ j (cid:104) D ( uu ) (cid:105) ,m − j − τ m − ( m − s + C ∞ (cid:88) m =3 (cid:16) ψ m (cid:107) uu (cid:107) L (Ω) + mψ m − (cid:107) D ( uu ) (cid:107) L (Ω) + m ψ m − (cid:107) D ( uu ) (cid:107) L (Ω) (cid:17) τ m − ( m − s (5.56)Using the bound ψ j ≤ C ( j − s /τ j ∗ , the combinatorial estimate (cid:0) mj (cid:1) ( j − s ( m − j − s / ( m − s ≤ C ,and η = τ /τ ∗ <
1, we obtain T g ≤ C ∞ (cid:88) m =3 m − (cid:88) j =0 η j (cid:16) (cid:104) D ( uu ) (cid:105) ,m − j − τ m − j − ( m − j − s (cid:17) + C (cid:107) uu (cid:107) H (Ω) ≤ C ∞ (cid:88) m =3 (cid:104) D ( uu ) (cid:105) ,m − τ m − ( m − s + C (cid:107) uu (cid:107) H (Ω) , (5.57)for some sufficiently large constant C = C ( γ ). We now estimate the terms T f and T f . We have T f = C ∞ (cid:88) m =3 m (cid:88) α =1 (cid:15) α (cid:104) ∂ α − f (cid:105) α ,m − α τ m − ( m − s ≤ C ∞ (cid:88) m =3 (cid:88) | α | = m − (cid:15) α +1 sup <δ ≤ δ δ m − (cid:107) ∂ α f (cid:107) L (Ω δ ) τ m − ( m − s , and similarly T f = C ∞ (cid:88) m =3 m (cid:88) α =2 (cid:15) α (cid:104) ∂ α − f (cid:105) α − ,m − α +1 τ m − ( m − s ≤ C ∞ (cid:88) m =3 (cid:88) | α | = m − (cid:15) α +2 sup <δ ≤ δ δ m − (cid:107) ∂ α f (cid:107) L (Ω δ ) τ m − ( m − s . Recall that cf. (5.17) we have f = ∂ i u j ∂ k u l D ijkl , where D ijkl is of Gevrey-class s (cf. (5.14)), and thereforewe have ψ j ≤ C ( j − s /τ j ∗ , for all j ≥
0. Here we have denoted ψ β,δ = δ max {| β |− , } (cid:107) ∂ β D ijkl (cid:107) L ∞ (Ω δ ) , andalso ψ j = (cid:80) | β | = j sup <δ ≤ δ ψ β,δ . From the above estimates and the Leibniz rule we obtain that T f + T f is NALYTICITY AND GEVREY CLASS REGULARITY FOR THE EULER EQUATIONS IN A BOUNDED DOMAIN 19 bounded by C ∞ (cid:88) m =3 m − (cid:88) j =0 (cid:18) m − j (cid:19) (cid:88) | α | = m − (cid:88) | β | = j,β ≤ α (cid:15) α +1 sup <δ ≤ δ δ m − (cid:107) ∂ β D ijkl (cid:107) L ∞ (Ω δ ) (cid:107) ∂ α − β ( DuDu ) (cid:107) L (Ω δ ) τ m − ( m − s ≤ C ∞ (cid:88) m =3 m − (cid:88) j =0 (cid:18) m − j (cid:19) ψ j m − j − (cid:88) α =0 (cid:15) α +1 (cid:104) ∂ α ( DuDu ) (cid:105) α +3 ,m − j − − α τ m − ( m − s + C ∞ (cid:88) m =3 ( ψ m − + mψ m − ) (cid:107) DuDu (cid:107) H (Ω) τ m − ( m − s . (5.58)Using the bound ψ j ≤ C ( j − s /τ j ∗ , the combinatorial estimate (cid:18) m − j (cid:19) ( j − s ( m − j − s ( m − s ≤ C, (5.59)and the fact that η = τ /τ ∗ <
1, from (5.58) we obtain T f + T f ≤ C ∞ (cid:88) m =3 m − (cid:88) j =0 η j (cid:32) m − j − (cid:88) α =0 (cid:15) α +1 (cid:104) ∂ α ( DuDu ) (cid:105) α +3 ,m − j − − α τ m − j − ( m − j − s (cid:33) + C (cid:107) DuDu (cid:107) H (Ω) ≤ C ∞ (cid:88) m =3 m − (cid:88) α =0 (cid:15) α +1 (cid:104) ∂ α ( DuDu ) (cid:105) α +3 ,m − − α τ m − ( m − s + C (cid:107) DuDu (cid:107) H (Ω) , (5.60)for some sufficiently large C = C ( γ ) >
0. This concludes the proof of the lemma. (cid:3)
Proof of Lemma 3.3.
Here we use the estimate obtained in Lemma 5.4 to bound P in terms of the Gevrey-class norm of the velocity, and prove the estimate (3.8). In view of Lemma 5.4, we need to estimate theterms ∞ (cid:88) m =3 m − (cid:88) α =0 (cid:15) α +1 (cid:104) ∂ α ( DuDu ) (cid:105) α +3 ,m − − α τ m − ( m − s ≤ C ∞ (cid:88) m =3 m − (cid:88) j =0 (cid:88) | α | = m − (cid:88) | β | = j,β ≤ α (cid:18) αβ (cid:19) (cid:15) α +1 sup <δ ≤ δ δ m − (cid:107) ∂ β Du∂ α − β Du (cid:107) L (Ω δ ) τ m − ( m − s = R , (5.61)and the lower order term ∞ (cid:88) m =3 (cid:104) D ( uu ) (cid:105) ,m − τ m − ( m − s ≤ C ∞ (cid:88) m =3 m (cid:88) j =0 (cid:88) | α | = m,α ≤ (cid:88) | β | = j,β ≤ α (cid:18) αβ (cid:19) sup <δ ≤ δ δ m − (cid:107) ∂ β u∂ α − β u (cid:107) L (Ω δ ) τ m − ( m − s = S . (5.62)Similarly to the estimate for the the commutator term C (cf. Proof of Lemma 3.2), bounding R and S isachieved by splitting the above sums according to the relative sizes of j and m − j . This idea was introduced inour previous work [KV2]. Namely, we write the right side of (5.61) as R + R + R + R low + R high + R + R ,according to j = 0 , ,
2, 3 ≤ j ≤ [( m − / m − /
2] + 1 ≤ j ≤ m − j = m −
2, and respectively j = m −
1. Note that by symmetry (replace j by m − j ) the terms R and R , R and R , and also R low and R high , have the same upper bounds. We have the estimates R + R ≤ C (cid:107) Du (cid:107) L ∞ (Ω) (cid:107) u (cid:107) H (Ω) + Cτ (cid:107) Du (cid:107) L ∞ (Ω) (cid:107) u (cid:107) Y τ (5.63) R + R ≤ C (cid:107) D u (cid:107) L ∞ (Ω) (cid:107) u (cid:107) H (Ω) + Cτ (cid:107) D u (cid:107) L ∞ (Ω) (cid:107) u (cid:107) H (Ω) + Cτ (cid:107) D u (cid:107) L ∞ (Ω) (cid:107) u (cid:107) Y τ (5.64) R ≤ Cτ (cid:107) u (cid:107) H (Ω) (cid:107) u (cid:107) Y τ , (5.65)and also R low + R high ≤ C ( τ / + (1 + K ) τ ) (cid:107) u (cid:107) X τ (cid:107) u (cid:107) Y τ . (5.66) The proofs of (5.63)–(5.66) are similar to those in [KV2, Section 5] and those in Section 4 of the presentpaper, and are thus omitted. Combined they give the desired estimate on P . To estimate S one proceedssimilarly. Note though that this is a lower order term. We have the following bound S ≤ C (cid:16) τ (cid:107) u (cid:107) L ∞ (Ω) + τ (cid:107) Du (cid:107) L ∞ (Ω) + τ (cid:107) D u (cid:107) L ∞ (Ω) + ( τ / + (1 + K ) τ ) (cid:107) u (cid:107) X τ (cid:17) (cid:107) u (cid:107) Y τ + C (1 + τ ) (cid:16) (cid:107) u (cid:107) W , ∞ (Ω) + (cid:107) u (cid:107) H (Ω) (cid:17) , (5.67)where C > γ . The proof of (5.67) is omitted (see [KV2, Section 5] fordetails). By collecting the above estimates, and the lower order terms from (5.53), we conclude the proof ofthe pressure estimate. 6. Global Gevrey-class persistence
In this section we prove that the local, short time estimates of Section 3 may be combined together toobtain global (in space) Gevrey-class a priori estimates that are valid for all t < T ∗ , the maximal time ofexistence of the Sobolev solution.Let T < T ∗ be fixed. We shall prove that the solution u ( t ) is of Gevrey-class s on [0 , T ] and give a lowerbound on the radius of Gevrey-class regularity. For this purpose let { x λ } Nλ =1 be points on ∂D determinedas follows. In a small neighborhood of x λ the boundary of D is the graph of a Gevrey-class function γ λ ,i.e., there exists r λ > D λ = D ∩ B r λ ( x λ ) = { x ∈ B r λ ( x λ ) : x > γ λ ( x , x ) } .Moreover, we can pick r λ small enough so that after composing with a rigid body rotation about x λ we have (cid:107) ∂ γ λ (cid:107) L ∞ + (cid:107) ∂ γ λ (cid:107) L ∞ ≤ ε , where ε > λ ∈ { , . . . , N } we let Ω λ = D ∩ B r λ / ( x λ ). We take N large enough so that there exists a compactly embedded open setΩ ⊂ D with analytic boundary, such that Ω ∪ (cid:83) ≤ λ ≤ N Ω λ = D . To obtain Gevrey-class regularity in theinterior of D , we cover Ω with finitely many, sufficiently small, analytic charts { D λ } N + N (cid:48) N +1 , chosen as follows.Denote by Ω λ a ball inside D λ , and let r λ = dist( ¯Ω λ , ( D λ ) c ), where λ ∈ { N + 1 , . . . , N + N (cid:48) } . We let N (cid:48) belarge enough so that 1 ≤ N (cid:88) λ =1 χ Ω λ ( x ) + N + N (cid:48) (cid:88) λ = N +1 χ Ω λ ( x ) ≤ C (6.1)for all x ∈ D , where C ≥ s ≥ φ t,s ( a ) the particle trajectory with initial condition φ s,s ( a ) = a , i.e., the uniquesmooth solution to ddt X ( t ) = u ( X ( t ) , t ) X ( s ) = a. Note that φ t, ( a ) = φ t ( a ), where φ t is as defined in (2.1)–(2.2). Since the flow map φ t,s : D (cid:55)→ D is abijection, cf. (6.1), we also have 1 ≤ (cid:80) N + N (cid:48) λ =1 χ φ t,s (Ω λ ) ( x ) ≤ C for all 0 ≤ s ≤ t and all x ∈ D .Let T = 0, and define T as the maximal time 0 = T < T ≤ T such that for all T ≤ t ≤ T wehave that φ t,T (Ω λ ) ⊂ D λ for all λ ∈ { , . . . , N + N (cid:48) } . Note that if T < T , then by the maximality of T , there exists λ ∈ { , . . . , N + N (cid:48) } with φ T ,T (Ω λ ) ∩ ( D λ ) c (cid:54) = ∅ . Thus there exists and x ∈ Ω λ suchthat | φ T ,T ( x ) − x | ≥ r λ / ≥ r ∗ , where r ∗ = min ≤ λ ≤ N + N (cid:48) { r λ / } is a fixed constant. We obtain that if T < T , then T may be estimated from below via (cid:90) T T (cid:107) u ( · , t ) (cid:107) W , ∞ ( D ) dt ≥ r ∗ . (6.2)For each λ ∈ { , . . . , N + N (cid:48) } , let θ λ ( x , x , x ) = ( x , x , x − γ λ ( x , x )) = ( y , y , y ) be a boundarystraightening map and define (cid:101) Ω λ = θ λ (Ω λ ). Note that this is exactly the setup from Section 2. Let u λ ( x, t ) = u ( x, t ) χ D λ ( x ) and for y = θ λ ( x ) ∈ θ λ ( D λ ) define (cid:101) u λ ( y, t ) = u λ ( x, t ).Let τ = τ ( T ) be the uniform radius of Gevrey-class regularity of the initial data u . By possiblydecreasing τ by a factor, we may assume that τ ≤ (cid:15)τ ∗ , where (cid:15) = (cid:15) ( D ) > τ ∗ is NALYTICITY AND GEVREY CLASS REGULARITY FOR THE EULER EQUATIONS IN A BOUNDED DOMAIN 21 the uniform radius of Gevrey-class regularity of ∂D . Since u λ ( T ) has Gevrey-class radius τ , we have that (cid:107) (cid:101) u λ ( T , y ) (cid:107) X a ∗ τ < ∞ for all λ ∈ { , . . . , N + N (cid:48) } , where 0 < a ∗ ≤ θ λ (cf. Remark 2.2). Therefore,on [ T , T ] we can apply Theorem 3.4 for each chart { Ω λ } Nλ =1 , respectively Remark 3.5 for { Ω λ } N + N (cid:48) λ = N +1 , toobtain that for all λ ∈ { , . . . , N + N (cid:48) } we have (cf. (3.15)) (cid:107) (cid:101) u λ ( · , t ) (cid:107) X τ ( t ) ≤ Q + C (cid:90) tT (cid:16) K ( s ) (cid:17) M ( s ) ds (6.3)where Q = max λ ∈{ ,...,N + N (cid:48) } (cid:107) (cid:101) u λ ( · , T ) (cid:107) X a ∗ τ , C = C ( D ) is a positive constant, and the radius of Gevrey-class regularity τ ( t ) is bounded from below (cf. (3.17)) by τ ( t ) ≥ a ∗ τ (cid:16) CtQ + Ct M ( T ) (cid:17) − exp (cid:16) CK ( T ) − CK ( t ) (cid:17) (6.4)for all T ≤ t ≤ T . Here we recall that K ( t ) = (cid:82) t (cid:107) u ( · , s ) (cid:107) W , ∞ ( D ) ds , and M ( t ) = (cid:107) u ( · , t ) (cid:107) H r ( D ) . Therefore,modulo composing with ( θ λ ) − we obtain that the localized velocity u λ ( x, t ) is of Gevrey-class s on [ T , T ]for each λ ∈ { , . . . , N + N (cid:48) } . By (6.1) we obtain that u ( · , t ) is of Gevrey-class s on [ T , T ], with uniformradius of Gevrey-class regularity bounded from below by a ∗ times the right side of (6.4).We proceed inductively. Let k ≥ φ T k ,T k − ( D ) = D , as above for t = 0 we cover D withlocal charts { Ω λ } N + N (cid:48) λ =1 and define T k +1 as the maximal time T k +1 ≤ T such that φ t,T k (Ω λ ) ⊂ D λ for all λ ∈ { , . . . , N + N (cid:48) } . Similarly to (6.2) we obtain that if T k +1 < T , then T k +1 may be estimated from (cid:90) T k +1 T k (cid:107) u ( · , t ) (cid:107) W , ∞ ( D ) dt ≥ r ∗ . (6.5)The induction assumption is that u ( x, T k ) if of Gevrey-class s , the uniform (over x ∈ D ) radius of Gevrey-class regularity of u ( x, T k ) is bounded from below by τ k = a ∗ τ k − (cid:16) C ( T k − T k − ) Q k − + C ( T k − T k − ) M ( T k − ) (cid:17) − exp (cid:16) CK ( T k − ) − CK ( T k ) (cid:17) , (6.6)and that the Gevrey-class norm at t = T k , given by Q k = max λ ∈{ ,...,N + N (cid:48) } (cid:107) (cid:101) u λ ( · , T k ) (cid:107) X a ∗ τk , is bounded as Q k ≤ Q k − + C (cid:90) T k T k − (cid:16) K ( s ) (cid:17) M ( s ) ds. (6.7)We apply Theorem 3.4, respectively Remark 3.5, on each local chart Ω λ , and conclude that (cid:101) u λ ( y, t ) is ofGevrey-class s on ∈ [ T k , T k +1 ] for all λ ∈ { , . . . , N + N (cid:48) } , with Gevrey-class norm bounded as (cid:107) (cid:101) u λ ( · , t ) (cid:107) X τ ( t ) ≤ Q k + C (cid:90) tT k (cid:16) K ( s ) (cid:17) M ( s ) ds, (6.8)and radius of Gevrey-class regularity τ ( t ) bounded from below by a ∗ τ k (cid:16) C ( t − T k ) Q k + C ( t − T k ) M ( T k ) (cid:17) − exp (cid:16) CK ( T k ) − CK ( t ) (cid:17) . (6.9)Modulo composing with the inverse map of θ λ , if follows from (6.1) and (6.8) that u ( x, t ) is of Gevrey-class s for all t ∈ [ T k , T k +1 ] with radius bounded from below by a ∗ times the quantity in (6.9). Moreover, letting t = T k +1 in (6.8)–(6.9) we obtain that the induction assumptions (6.6)–(6.7) hold for the next iteration step.We claim that for each fixed T < T ∗ the inductive argument described above stops after finitely manysteps, i.e., there exists a k ≥ T k = T . To see this, note that if T k < T , then from (6.2) and (6.5)we obtain kr ∗ ≤ (cid:90) T k (cid:107) u ( · , t ) (cid:107) W , ∞ ( D ) dt ≤ (cid:90) T (cid:107) u ( · , t ) (cid:107) W , ∞ ( D ) dt < ∞ , (6.10)which cannot hold for all k ≥
1, proving the claim. Moreover, we proved that it takes at most [ K ( T ) /r ∗ ] + 1applications of Theorem 3 to show that u ( · , T ) is uniformly of Gevrey-class s , where [ · ] denotes the integerpart, and K ( t ) is as usual defined by (2.5). It is left to prove that the uniform radius of Gevrey-class regularity τ ( T ) of u ( · , T ) depends explicitly onthe initial data and K ( T ). Let k = [ K ( T ) /r ∗ ] + 1 and hence T = T k . It follows form the above paragraphthat τ ( T ) ≥ τ k . By the induction assumptions (6.6)–(6.7) we bound τ k from below as τ k ≥ a k ∗ τ k (cid:89) j =1 exp (cid:16) CK ( T j − ) − CK ( T j ) (cid:17)(cid:16) C ( T j − T j − ) Q j − + C ( T j − T j − ) M ( T j − ) (cid:17) − . (6.11)Since a k ∗ ≤ exp( − k log(1 /a ∗ )) ≤ exp( − K ( T ) log(1 /a ∗ ) /r ∗ ) we obtain that τ k ≥ τ exp (cid:16) − CK ( T ) (cid:17) k (cid:89) j =1 (cid:16) C ( T j − T j − ) Q j − + C ( T j − T j − ) M ( T j − ) (cid:17) − (6.12)for a sufficiently large constant C depending only on the domain. To estimate the product term in the aboveinequality we note that by (6.7) we have that Q j − ≤ Q + CM (0) exp( CK ( T j − )), while from the Sobolevenergy estimate we obtain M ( T j − ) ≤ M (0) exp( CK ( T j − )). Therefore we have τ k ≥ τ exp (cid:16) − CK ( T ) (cid:17) k (cid:89) j =1 (cid:16) C ( T j − T j − ) Q + C ( T j − T j − )(1 + T ) M (0) exp (cid:16) CK ( T j − ) (cid:17)(cid:17) − ≥ τ exp (cid:16) − CK ( T ) (cid:17) exp (cid:16) − C k (cid:88) j =1 K ( T j − ) (cid:17) k (cid:89) j =1 (cid:16) C ( T j − T j − ) Q + C ( T j − T j − )(1 + T ) M (0) (cid:17) − . By using the inequality between the arithmetic and the geometric mean, and the fact that k = CK ( T ), weobtain τ ( T ) ≥ τ exp (cid:16) − C k (cid:88) j =1 K ( T j ) (cid:17) (cid:18) CT Q + CT M (0) k (cid:19) − k ≥ τ exp (cid:16) − CK ( T ) (cid:17) exp (cid:16) − CT Q − CT M (0) (cid:17) . (6.13)Therefore we have proven the following statement, which is the main theorem of this paper. Theorem 6.1.
Let u be divergence-free and of Gevrey-class s , with s ≥ , on a Gevrey-class s , open,bounded domain D ⊂ R , and r ≥ . Then the unique solution u ( · , t ) ∈ C ([0 , T ∗ ); H r ( D )) to the initial valueproblem (E.1) – (E.4) is of Gevrey-class s for all t < T ∗ , where T ∗ ∈ (0 , ∞ ] is the maximal time of existencein H r ( D ) . Moreover, the radius τ ( t ) of Gevrey-class regularity of the solution u ( · , t ) satisfies τ ( t ) ≥ Cτ exp (cid:32) − C (cid:18)(cid:90) t (cid:107) u ( s ) (cid:107) W , ∞ ds (cid:19) (cid:33) exp (cid:16) − Ct (cid:107) u (cid:107) X τ − Ct (cid:107) u (cid:107) H r (cid:17) , (6.14) for all t < T ∗ , where C is a sufficiently large constant depending only on the domain D , τ is the radius ofGevrey-class regularity of the initial data u , and (cid:107) u (cid:107) X τ is its Gevrey-class norm. Remark 6.2.
Theorem 6.1 also holds in the case of a two-dimensional Gevrey-class domain. In 2D it isknown that (cid:107) u ( s ) (cid:107) W , ∞ ≤ C exp( Ct ) for some positive constant C = C ( D, u ), and therefore estimate (6.14)shows that the radius of Gevrey-class regularity of the solution is bounded from below by C exp( − C exp( Ct ))for some C >
0, depending on the domain and on the initial data. We note that such a lower bound on τ ( t ) was obtained in the 2 D analytic case s = 1 by Bardos, Benachour, and Zerner [BBZ], whereas in thenon-analytic Gevrey-class case on domains with generic boundary, Theorem 6.1 is the first such result (seealso [KV1] for the periodic domain, and [KV2] for the half-plane). Appendix A. Lemma A.1.
Let { a λ } , and { b λ,µ } be sequences of positive numbers, where λ, µ ∈ N . The identity (cid:88) | α | = m (cid:88) | β | = j, β ≤ α (cid:88) | γ | = k, γ ≤ β a γ b α − β,β − γ = (cid:88) | γ | = k a γ (cid:88) | α | = m − k (cid:88) | β | = j − k, β ≤ α b α − β,β NALYTICITY AND GEVREY CLASS REGULARITY FOR THE EULER EQUATIONS IN A BOUNDED DOMAIN 23 holds for positive integers j, k, m such that k ≤ j ≤ m .Proof. (cid:88) | α | = m (cid:88) | β | = j, β ≤ α (cid:88) | γ | = k, γ ≤ β a γ b α − β,β − γ = (cid:88) | α | = m (cid:88) | γ | = k, γ ≤ α a γ (cid:88) | β | = j, γ ≤ β ≤ α b ( α − γ ) − ( β − γ ) ,β − γ = (cid:88) | α | = m (cid:88) | γ | = k, γ ≤ α a γ (cid:88) | λ | = j − k, λ ≤ α − γ b ( α − γ ) − λ,λ = (cid:88) | α | = m (cid:88) | γ | = k, γ ≤ α a γ d α − γ , (A.1)where we let d α − γ = (cid:88) | λ | = j − k, λ ≤ α − γ b ( α − γ ) − λ,λ . By [KV2, Lemma 4.2] the far right side of (A.1) may be written as (cid:88) | γ | = k a γ (cid:88) | α | = m − k d α = (cid:88) | γ | = k a γ (cid:88) | α | = m − k (cid:88) | β | = j − k, β ≤ α b α − β,β , which concludes the proof of the lemma. (cid:3) Lemma A.2.
Let < η < and { a m,j } m ≥ ,j ≥ be a sequence of positive numbers. Then we have ∞ (cid:88) m =3 m (cid:88) j =1 j (cid:88) k =0 η k a m − k,j − k = η − η a , + η − η ( a , + a , ) + η − η ( a , + a , + a , )+ η − η ∞ (cid:88) m =3 a m, + 11 − η ∞ (cid:88) m =3 m (cid:88) j =1 a m,j . (A.2) Proof.
By re-indexing we have ∞ (cid:88) m =3 m (cid:88) j =1 j (cid:88) k =1 η k a m − k,j − k = ∞ (cid:88) m =3 m (cid:88) k =1 η k m (cid:88) j = k a m − k,j − k = ∞ (cid:88) m =3 m (cid:88) k =1 η k m − k (cid:88) j =0 a m − k,j = ∞ (cid:88) m =3 m (cid:88) k =1 η k b m − k , where we denoted b l = (cid:80) lj =0 = a l,j . By summing the geometric series in η the far right side of the aboveequality may be re-written as ( η b + η b + η (cid:80) ∞ j =2 b l ) / (1 − η ). Therefore we obtain ∞ (cid:88) m =3 m (cid:88) j =1 j (cid:88) k =0 η k a m − k,j − k = ∞ (cid:88) m =3 m (cid:88) j =1 a m,k + η − η a , + η − η ( a , + a , ) + η − η ∞ (cid:88) m =2 m (cid:88) j =0 a m,j , and (A.2) follows by grouping appropriate terms. (cid:3) Lemma A.3.
Let { F δ ( t ) } δ ∈ [0 ,δ ] be a family of nonnegative C functions, where δ ≤ is a fixed constant.Assume that (i) { ˙ F δ ( t ) } δ ∈ [0 ,δ ] is a uniformly equicontinuous family, (ii) for every fixed t , the functions F δ ( t ) and ˙ F δ ( t ) depend continuously on δ .Then for every fixed t ∈ (0 , ∞ ) we have d + dt sup δ ∈ [0 ,δ ] F δ ( t ) = lim sup h → + h (cid:32) sup δ ∈ [0 ,δ ] F δ ( t + h ) − sup δ ∈ [0 ,δ ] F δ ( t ) (cid:33) ≤ sup δ ∈ [0 ,δ ] ˙ F δ ( t ) . (A.3) Proof.
Fix t ∈ (0 , ∞ ). For a fixed δ ∈ [0 , δ ], and h >
0, we have F δ ( t + h ) = (cid:90) t + ht ˙ F δ ( s ) ds + F δ ( t ) ≤ (cid:90) t + ht (cid:32) sup δ ∈ [0 ,δ ] ˙ F δ ( s ) (cid:33) ds + sup δ ∈ [0 ,δ ] F δ ( t ) . Therefore 1 h (cid:32) sup δ ∈ [0 ,δ ] F δ ( t + h ) − sup δ ∈ [0 ,δ ] F δ ( t ) (cid:33) ≤ h (cid:90) t + ht (cid:32) sup δ ∈ [0 ,δ ] ˙ F δ ( s ) (cid:33) ds, and if we can prove that sup δ ∈ [0 ,δ ] ˙ F δ ( t ) is a continuous function of t , then (A.3) holds, concluding the proofof the lemma. The fact that sup δ ∈ [0 ,δ ] ˙ F δ ( t ) is a continuous function of t follows directly from the definitionof uniform equicontinuity and the inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup δ ∈ [0 ,δ ] a ( δ ) − sup δ ∈ [0 ,δ ] b ( δ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup δ ∈ [0 ,δ ] | a ( δ ) − b ( δ ) | , which holds for all functions a, b : [0 , δ ] → R . (cid:3) Lemma A.4.
Let (cid:101) v = ∂ α (cid:101) u , for some α ∈ N , and (cid:101) u as in Lemma 3.1. Let f δ ( t ) = δ | α | (cid:107) (cid:101) v ( t, · ) (cid:107) L ( (cid:101) Ω δ,t ) , (A.4) and F δ ( t ) = f δ ( t ) . Then the family { F δ ( t ) } δ ∈ [0 ,δ ] satisfies the conditions (i) and (ii) of Lemma A.3.Proof. Let (cid:101) Ω t = (cid:83) δ ∈ [0 ,δ ] (cid:101) Ω δ,t , and φ t ( x ) be the particle trajectory with initial data x . Without loss ofgenerality assume that (cid:101) Ω t ∈ (cid:101) D for all t >
0, and that (cid:107) (cid:101) v (cid:107) L ( (cid:101) Ω) (cid:54) = 0.The fact that for a fixed t the family F δ ( t ) depends continuously on δ , follows from the continuity of theintegral with respect to the Lebesgue measure, and the fact that (cid:101) v ∈ L ∞ ( (cid:101) D ). Also, from (E.2) and the factthat det( ∂θ/∂x ) = 1, we have ddt (cid:107) (cid:101) v ( t, · ) (cid:107) L ( (cid:101) Ω δ,t ) = 2 (cid:90) Ω δ g ( t, φ t ( x )) v ( t, φ t ( x )) dx, (A.5)and since gv ∈ L ∞ ( D ), the continuity of the integral implies that ˙ F δ ( t ) depends continuously on δ , so thatcondition (ii) holds.To show that the family { ˙ F δ ( t ) } is uniformly equicontinuous, let (cid:15) >
0. We need to show that there exists τ = τ ( (cid:15) ) > | ˙ F δ ( t ) − ˙ F δ ( s ) | < (cid:15) for all | t − s | ≤ τ and all δ ∈ [0 , δ ]. By (A.5) and the mean valuetheorem we have (cid:12)(cid:12)(cid:12)(cid:12) ddt (cid:107) (cid:101) v ( t, · ) (cid:107) L ( (cid:101) Ω δ,t ) − ddt (cid:107) (cid:101) v ( s, · ) (cid:107) L ( (cid:101) Ω δ,s ) (cid:12)(cid:12)(cid:12)(cid:12) = 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω δ (cid:16) g ( t, φ t ( x )) v ( t, φ t ( x )) − g ( s, φ s ( x )) v ( s, φ s ( x )) (cid:17) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | t − s | sup z ∈ ( t,s ) sup x ∈ Ω | ∂ t ( gv )( z, φ z ( x )) | + | u j ( z, φ z ( x )) ∂ j ( gv )( z, φ z ( x )) |≤ C | t − s | (cid:0) (cid:107) ∂ t ( gv ) (cid:107) L ∞ ( D ) + (cid:107) u (cid:107) L ∞ ( D ) (cid:107) gv (cid:107) W , ∞ ( D ) (cid:1) , since (cid:101) Ω t ∈ (cid:101) D for all t >
0. Recall that (cid:101) g = [ ∂ α , (cid:101) u j ∂ j θ k ∂ k ] (cid:101) u + ∂ α ( ∂ j θ k ∂ k (cid:101) p ), and so the right side of theabove is bounded by C | t − s |(cid:107) u (cid:107) H r + | α | ( D ) for some sufficiently large r . To conclude the proof of the lemmaone follows standard arguments. (cid:3) Lemma A.5.
Let (cid:101) v = ∂ α (cid:101) u , for some α ∈ N , and (cid:101) u as in Lemma 3.1. Let M δ ( t ) ≥ be an upper bound δ | α | ddt (cid:107) (cid:101) v ( t, · ) (cid:107) L ( (cid:101) Ω δ,t ) ≤ M δ ( t ) (A.6) NALYTICITY AND GEVREY CLASS REGULARITY FOR THE EULER EQUATIONS IN A BOUNDED DOMAIN 25 which holds for all t ≥ and all δ ∈ [0 , δ ] such that (cid:107) (cid:101) v ( t, · ) (cid:107) L ( (cid:101) Ω δ,t ) > . Furthermore, assume that sup δ ∈ [0 ,δ ] M δ ( t ) is continuous in t . Then we have d + dt sup δ ∈ [0 ,δ ] δ | α | (cid:107) (cid:101) v ( t, · ) (cid:107) L ( (cid:101) Ω δ,t ) ≤ sup δ ∈ [0 ,δ ] M δ ( t ) , (A.7) where we denote by d + a ( t ) /dt = lim sup h → ( a ( t + h ) − a ( t )) /h the right derivative of a function.Proof. Let f δ = δ | α | (cid:107) (cid:101) v ( t, · ) (cid:107) L ( (cid:101) Ω δ,t ) and let F δ = f δ . Note that by assumption we have ˙ f δ ( t ) ≤ M δ ( t ) for all δ ∈ [0 , δ ] and t >
0. It follows from Lemmas A.3 and A.4 that d + dt sup δ ∈ [0 ,δ ] F δ ( t ) ≤ sup δ ∈ [0 ,δ ] ˙ F δ ( t ) . Due to the continuity in δ of the family f δ ,sup δ ∈ [0 ,δ ] F δ = (cid:32) sup δ ∈ [0 ,δ ] f δ (cid:33) . Therefore, d + dt sup δ ∈ [0 ,δ ] f δ ( t ) = d + dt sup δ ∈ [0 ,δ ] F δ ( t )2 sup δ ∈ [0 ,δ ] f δ ( t ) ≤ sup δ ∈ [0 ,δ ] f δ ( t ) M δ ( t )sup δ ∈ [0 ,δ ] f δ ( t ) ≤ sup δ ∈ [0 ,δ ] M δ ( t ) , (A.8)for all t such that sup δ ∈ [0 ,δ ] f δ ( t ) >
0. This concludes the proof of the lemma by noting that if g ( t ) is anonnegative function such that ( d + /dt ) g ( t ) ≤ G ( t ) for all t such that g ( t ) >
0, with G ( t ) continuous, then g ( t ) ≤ g ( t ) + (cid:82) tt G ( s ) ds , for all 0 ≤ t < t . (cid:3) Acknowledgment.
The work of both authors was supported in part by the NSF grant DMS-1009769.
References [AM1] S. Alinhac and G. Metivier,
Propagation de l’analycit´e locale pour les solutions de l’´equation d’Euler . Arch. RationalMech. Anal. (1986), 287–296.[AM2] S. Alinhac and G. Metivier, Propagation de l’analyticit´e des solutions de syst`emes hyperboliques non-lin´eaires . In-vent. Math. (1984), no. 2, 189–204.[B] C. Bardos, Analycit´e de la solution de l’´equation d’Euler dans un ouvert de R n . C. R. Acad. Sci. Paris (1976),255–258.[BB] C. Bardos and S. Benachour, Domaine d’analycit´e des solutions de l’´equation d’Euler dans un ouvert de R n .Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (1977), 647–687.[BBZ] C. Bardos, S. Benachour, and M. Zerner, Analycit´e des solutions p´eriodiques de l’´equation d’Euler en deux dimensions .C. R. Acad. Sci. Paris (1976), 995–998.[BT1] C. Bardos and E.S. Titi,
Euler equations of incompressible ideal fluids . Russian Mathematical Surveys (3) (2007),409–451.[BT2] C. Bardos and E.S. Titi, Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations .Discrete Contin. Dyn. Syst. (2010), no. 2, 185–197.[BG1] M.S. Baouendi and C. Goulaouic, Solutions analytiques de l’´equation d’Euler d’un fluide compressible . S´eminaireGoulaouic-Schwartz 1976/1977, Exp. No. , ´Ecole Polytech., Paliseau, 1977.[BG2] M.S. Baouendi and C. Goulaouic, Sharp estimates for analytic pseudodifferential operators and application to Cauchyproblems . J. Differential Equations (1983), no. 2, 241–268.[Be] S. Benachour, Analycit´e des solutions p´eriodiques de l’´equation d’Euler en trois dimension . C. R. Acad. Sci. Paris (1976), 107–110.[BKM] J.T. Beale, T. Kato, and A. Majda,
Remarks on the breakdown of smooth solutions for the -D Euler equations .Comm. Math. Phys. (1984), no. 1, 61–66.[Bi] A. Biswas, Local existence and Gevrey regularity of 3-D Navier-Stokes equations with l p initial data . J. DifferentialEquations (2005), 429–447.[BGK1] J.L. Bona, Z. Gruji´c, and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for thegeneralized KdV equation . Ann. Inst. H. Poincar´e Anal. Non Lin´eaire (2005), no. 6, 783–797.[BGK2] J.L. Bona, Z. Gruji´c, and H. Kalisch, Global solutions of the derivative Schr¨odinger equation in a class of functionsanalytic in a strip . J. Differential Equations (2006), no. 1, 186–203. [BoB] J.P. Bourguignon and H. Brezis,
Remarks on the Euler equation . J. Functional Analysis (1974), 341–363.[Ch] J.Y. Chemin, Fluides parfaits incompressibles . Ast´erisque , Soc. Math. France, Paris, 1995.[C1] P. Constantin,
An Eulerian-Lagrangian approach for incompressible fluids: local theory . J. Amer. Math. Soc. (2001),no. 2, 263–278.[C2] P. Constantin, On the Euler equations of incompressible fluids . Bull. Amer. Math. Soc. (N.S.) (2007), no. 4, 603–621.[CTV] P. Constantin, E.S. Titi, and J. Vukadinovi´c, Dissipativity and Gevrey regularity of a Smoluchowski equation . IndianaUniv. Math. J. (2005), no. 4, 949–969.[CS] G.M. Constantine and T.H. Savits, A multivariate Fa di Bruno formula with applications . Trans. Amer. Math. Soc. (1996), no. 2, 503–520.[D] J.-M. Delort,
Estimations fines pour des op´erateurs pseudo-diff´erentiels analytiques sur un ouvert `a bord de R n .Application aux ´equations d’Euler . Comm. Partial Differential Equations (1985), no. 12, 1465–1525.[DM] R. DiPerna and A. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations .Comm. Math. Phys. (1987), 667-689.[EM] D.G. Ebin and J.E. Marsden,
Groups of diffeomorphisms and the solution of the classical Euler equations for a perfectfluid . Bull. Amer. Math. Soc. (1969), 962–967.[F] A.B. Ferrari, On the blow-up of solutions of the -D Euler equations in a bounded domain . Comm. Math. Phys. (1993), no. 2, 277–294.[FTi] A.B. Ferrari and E.S. Titi, Gevrey regularity for nonlinear analytic parabolic equations . Comm. Partial DifferentialEquations (1998), no. 1–2, 1–16.[FFT] C. Foias, U. Frisch, and R. Temam, Existence de solutions C ∞ des ´equations d’Euler . (French) C. R. Acad. Sci. ParisS´er. A-B (1975), A505–A508.[FT] C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier–Stokes equations . J. Funct. Anal. (1989), 359–369.[GK1] Z. Gruji´c and I. Kukavica, Space analyticity for the nonlinear heat equation in a bounded domain . J. DifferentialEquations (1999), no. 1, 42–54.[GK2] Z. Gruji´c and I. Kukavica,
Space analyticity for the Navier–Stokes and related equations with initial data in L p .J. Funct. Anal. (1998), no. 2, 447–466.[HKR] W.D. Henshaw, H.-O. Kreiss, and L.G. Reyna, Smallest scale estimates for the Navier-Stokes equations for incom-pressible fluids . Arch. Rational Mech. Anal. (1990), no. 1, 21–44.[Ka] T. Kato,
Nonstationary flows of viscuous and ideal fluids in R . J. Functional Analysis (1972), 296–305.[KP] S. Krantz and H. Parks, A primer of real analytic functions . Second edition. Birkh¨auser Advanced Texts: BaselTextbooks, Birkh¨auser Boston, Inc., Boston, MA, 2002.[K1] I. Kukavica,
Hausdorff length of level sets for solutions of the Ginzburg-Landau equation . Nonlinearity (1995), no. 2,113–129.[K2] I. Kukavica, On the dissipative scale for the Navier-Stokes equation . Indiana Univ. Math. J. (1999), no. 3, 1057–1081.[KTVZ] I. Kukavica, R. Temam, V. Vicol, and M. Ziane, Local Existence and Uniqueness for the Hydrostatic Euler Equationson a Bounded Domain . C. R. Acad. Sci. Paris, to appear.[KV1] I. Kukavica and V. Vicol,
On the radius of analyticity of solutions to the three-dimensional Euler equations .Proc. Amer. Math. Soc. (2009), 669-677.[KV2] I. Kukavica and V. Vicol,
The domain of anlyticity of solutions to the three-dimensional Euler equations in a half-space .Discrete Contin. Dyn. Syst., to appear.[Lb] D. Le Bail,
Analyticit´e locale pour les solutions de l’´equation d’Euler . Arch. Rational Mech. Anal. (1986), no. 2,117–136.[Le] N. Lerner, R´esultats d’unicit´e forte pour des op´erateurs elliptiques `a coefficients Gevrey . Comm. Partial DifferentialEquations (1981), no. 10, 1163-1177.[LO] C.D. Levermore and M. Oliver, Analyticity of solutions for a generalized Euler equation . J. Differential Equations (1997), no. 2, 321–339.[LM] J.-L. Lions and E. Magenes,
Problem`es aux limites non homog`enes et applications . Vol. 3, S.A. Dunod, Paris, 1968.[LCS] M.C. Lombardo, M. Cannone, and M. Sammartino, Well-posedness of the boundary layer equations, SIAMJ. Math. Anal. 35 (2003), no. 4, 987–1004.[MB] A.J. Majda and A.L. Bertozzi,
Vorticity and incompressible flow . Cambridge Texts in Applied Mathematics, vol. 27,Cambridge University Press, Cambridge, 2002.[MN] C.B. Morrey and L. Nirenberg,
On the analyticity of the solutions of linear elliptic systems of partial differentialequations . Comm. Pure Appl. Math. (1957), 271–290.[OT] M. Oliver and E.S. Titi, On the domain of analyticity of solutions of second order analytic nonlinear differentialequations . J. Differential Equations (2001), no. 1, 55–74.[SC] M. Sammartino and R.E. Caflisch,
Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on aHalf-Space I. Existence for Euler and Prandtl equations . Commun. Math. Phys. (1998), 433-461.[T] R. Temam,
On the Euler equations of incompressible perfect fluids . J. Functional Analysis (1975), no. 1, 32–43.[Y] V.I. Yudovich, Non stationary flow of an ideal incompressible liquid . Zh. Vych. Mat. (1963), 1032–1066. NALYTICITY AND GEVREY CLASS REGULARITY FOR THE EULER EQUATIONS IN A BOUNDED DOMAIN 27
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