Local ill-posedness of the Euler equations in B 1 ∞,1
aa r X i v : . [ m a t h . A P ] M a y LOCAL ILL-POSEDNESS OF THE EULER EQUATIONS IN B ∞ , GERARD MISIO LEK AND TSUYOSHI YONEDA
Abstract.
We show that the incompressible Euler equations on R are notlocally well-posed in the sense of Hadamard in the Besov space B ∞ , . Ourapproach relies on the technique of Lagrangian deformations of Bourgain andLi. We show that the assumption that the data-to-solution map is continuousin B ∞ , leads to a contradiction with a well-posedness result in W ,p of Katoand Ponce. Introduction
The study of the Cauchy problem for the Euler equations u t + u ·∇ u + ∇ π = 0 , t ≥ , x ∈ R n (1.1) div u = 0(1.2) u (0) = u (1.3)has a long history going back to the works of Gyunter [9], Lichtenstein [11] andWolibner [19] in the late 1920’s and 1930’s. Tremendous progress has been madesince those pioneering papers and we refer to several excellent monographs andsurveys for example Majda and Bertozzi [12], Constantin [6] or Bahouri, Cheminand Danchin [1] for detailed accounts. Nevertheless, the problems related to thephenomenon of turbulence and persistence of smooth solutions in 3D for all timeremain open. Furthermore, despite extensive studies of local well-posedness for theEuler equations our understanding of this problem especially in the cases of impor-tant borderline spaces including C , Lip, B p,q , W n/p +1 ,p etc. has also remainedincomplete. However, this picture is changing fast.Recall that a Cauchy problem is locally well-posed in a Banach space X (in thesense of Hadamard) if for any initial data in X there exist T > C ([0 , T ) , X ) and which depends continuouslyon the data. Otherwise, the problem is said to be ill-posed.A few years ago Bardos and Titi [2] used a shear flow of DiPerna and Majda[7] to construct solutions in 3D with an instantaneous loss of regularity in H¨older C α and Zygmund B ∞ , ∞ spaces. More precisely, they found C α initial data forwhich the corresponding (weak) solution does not belong to C β for any 1 > β > α and any t >
0. This technique has also been used to obtain similar results in theTriebel-Lizorkin F ∞ , space by Bardos, Lemarie and Titi and in the logarithmicLipschitz spaces logLip α by the authors [13]. Date : October 26, 2018.2000
Mathematics Subject Classification.
Primary 35Q35; Secondary 35B30.
Key words and phrases.
Euler equations, ill-posedness, Lagrangian flow, Besov space.The second author was partially supported by JSPS KAKENHI Grant Number 25870004.
More recently, in a breakthrough paper Bourgain and Li [3] employed both La-grangian and Eulerian techniques to obtain strong local ill-posedness results in bor-derline Sobolev spaces W n/p +1 ,p for any 1 ≤ p < ∞ and in Besov spaces B n/p +1 p,q with 1 ≤ p < ∞ and 1 < q ≤ ∞ and n = 2 or 3. In [14] the authors adaptedthe approach of [3] to settle (in the 2D case) a long standing open question of localill-posedness in the classical C space by showing that the assumption on continuityof the data-to-solution map in C leads to a contradiction with a results of Katoand Ponce [10] for W ,p . Almost simultaneously Elgindi and Masmoudi [8] andBourgain and Li [4] produced similar results using different methods. It now seemsthat a complete resolution of local ill-posedness questions for the Euler equationsincluding various borderline spaces is fully within reach. In fact, the main resultsof [4] show that the Euler equations are ill-posed in the C m spaces for any integer m ≥ C ,α for any 0 < α < B ∞ , ( R n ). It is interesting to observe that in order to establishuniqueness they first show that the data-to-solution map u → u is continuous (evenLipschitz) into B ∞ , (cf. [15], Section 4). They do not prove that it is continuousinto B ∞ , (and consequently that the Euler equations are locally well-posed in thesense of Hadamard in B ∞ , ). Our main result is Theorem 1.
The 2D incompressible Euler equations are locally ill-posed in theBesov space B ∞ , . As in our previous paper [14] we will work with the vorticity equations. Recallthat in two dimensions the vorticity of a vector field u is a 2-form ω = du ♭ whichis identified with the function ω = rot u = − ∂u ∂x + ∂u ∂x . In this case the Cauchy problem (1.1)-(1.3) can be rewritten as ω t + u ·∇ ω = 0 , t ≥ , x ∈ R (1.4) ω (0) = ω (1.5)where the velocity is recovered from ω using the Biot-Savart law(1.6) u = K ∗ ω = ∇ ⊥ ∆ − ω with kernel K ( x )=(2 π ) − ( − x / | x | , x / | x | ) and where ∇ ⊥ =( − ∂∂x , ∂∂x ) denotesthe symplectic gradient of a function.Our strategy is similar to that adopted for the C case in [14]. Namely, following[3] we first choose an initial vorticity ω such that the Lagrangian flow of the corre-sponding velocity field retains a large gradient on a (possibly short) time interval.We then perturb ω to get a sequence of initial vorticities in W ,p . Finally, weshow that the assumption that the Euler equations are well-posed in B ∞ . ( R ) (inparticular, that the solutions depend continuously on the initial data) leads to acontradiction with the following result of Kato and Ponce OCAL ILL-POSEDNESS OF THE EULER EQUATIONS IN B ∞ , Theorem (Kato-Ponce [10]) . Let
p . For any ω ∈ W s − ,p ( R ) and any T > there exists a constant K = K ( T, ω, s, p ) > such that sup ≤ t ≤ T k ω ( t ) k W s − ,p ≤ K. Theorem 1 will be a consequence of the following result
Theorem 2.
Let < p < ∞ . Assume that the vorticity equations (1.4) - (1.5) arewell-posed in B ∞ , ( R ) . There exist T > and a sequence ω ,n in C ∞ c ( R ) withthe following properties1. there exists a constant C > such that k ω ,n k W .p ≤ C for all n ∈ Z + and2. for any M ≫ there is < t ≤ T such that k ω n ( t ) k W ,p ≥ M / for allsufficiently large n and all p sufficiently close to . In Section 2 we provide some technical tools and construct an initial vorticitywhose Lagrangian flow has a large gradient. Since some of the constructions areanalogous to those in [14] some details are omitted. The proof of Theorem 2 isgiven in Section 3.
Remark . In this paper we do not employ the ”patching” argument of [3] whichleaves open the question of strong ill-posedness in B ∞ ,q in the sense of Bourgain-Li. Remark . Since we treat here non-decaying data, an analogous local ill-posednessresult in 3D follows immediately from our 2D construction. The details will beelaborated elsewhere.2.
Vorticity and the Lagrangian flow
We first recall some basic harmonic analysis. Let ψ be a smooth radial bumpfunction on R which is supported in the unit ball B (0 ,
1) and equal to 1 on theball of radius 1/2. Set ψ − = ψ and let(2.1) ψ ( ξ ) = ψ (2 − ξ ) − ψ ( ξ ) and ψ ℓ ( ξ ) = ψ (2 − ℓ ξ ) ∀ ℓ ≥ . Each ψ ℓ is supported in a shell { ℓ − ≤ | ξ | ≤ ℓ +1 } with ψ ℓ ( ξ ) = 1 when | ξ | = 2 ℓ .For any f ∈ S ′ ( R ) define the frequency restriction operators by d ∆ ℓ f ( ξ ) = ψ ℓ ( ξ ) ˆ f ( ξ ) ∀ ℓ ≥ − f = X ℓ ≥− ∆ ℓ f where ∆ ℓ f ( x ) = X ξ ∈ R ψ ℓ ( ξ ) ˆ f ( ξ ) e i h ξ,x i , x ∈ R . For any s ∈ R and 1 ≤ p, q ≤ ∞ the Besov space B sp,q ( R ) is defined as the setof all f ∈ S ′ ( R ) such that the number(2.2) k f k B sp,q = X ℓ ≥− sqℓ k ∆ ℓ f k qL p ! /q if 1 ≤ q < ∞ sup ℓ ≥− sℓ k ∆ k f k L p if q = ∞ is finite. Among many special cases of interest are the Sobolev spaces W s,p = B sp,p and the H¨older-Zygmund class C s = B s ∞ , ∞ both defined for any 1 ≤ p < ∞ andany non-integer s > GERARD MISIO LEK AND TSUYOSHI YONEDA
Next, given a radial bump function 0 ≤ ϕ ≤ B (0 ,
1) define ϕ ( x , x ) = X ε ,ε = ± ε ε ϕ ( x − ε , x − ε )(2.3)and for a fixed positive integer N ∈ Z + and any M ≫
1, set ω ( x ) = ω M,N ( x ) = M − N − p X N ≤ k ≤ N + N ϕ k ( x ) , N = 1 , , . . . (2.4)where 2 < p < ∞ and where ϕ k ( x ) = 2 ( − p ) k ϕ (2 k x ) . Observe that by construction ϕ is an odd function in both x , x and for any k ≥ ϕ k ⊂ [ ε ,ε = ± B (cid:0) ( ε − k , ε − k ) , − ( k +2) (cid:1) . Combined with the uniform (in time) L ∞ control of the vorticity in R this ensuresthe existence of a unique solution of the Cauchy problem (1.4)-(1.5) with the initialdata (2.4); e.g., by a result of Yudovich [20], see also Majda and Bertozzi [12].The construction so far parallels that of our previous paper [14] and thereforethe proofs of Lemma 5 and Proposition 6 below will be omitted. Lemma 5.
We have k ω k W ,p . M − (2.6) with the bound independent of N > and < p < ∞ .Proof. See [14]; Lemma 3. (cid:3)
Since p > n = 2 the results of Kato and Ponce [10] (cf. Lemma 3.1; Thm. III)imply that there exists a unique velocity field u ∈ C ([0 , ∞ ) , W ,p ( R )) solving theproblem (1.1)-(1.2) and whose vorticity function ω ∈ C ([0 , ∞ ) , W ,p ( R )) satisfiesthe initial condition (2.4).The associated Lagrangian flow η ( t ) of u = ∇ ⊥ ∆ − ω is a solution of the initialvalue problem ddt η ( t, x ) = u ( t, η ( t, x )) (cid:0) = F u ( η ( t, x )) (cid:1) (2.7) η (0 , x ) = x (2.8)and defines a curve in the group of volume-preserving diffeomorphisms such that ω ◦ η ∈ C ([0 , ∞ ) , W ,p ( R )), see e.g., [10] or [5]. It can be readily checked thatthe odd symmetry of ω is preserved by η and thus (by conservation of vorticityin 2D) is retained by ω for all time. In this case the Biot-Savart law (1.6) impliesthat the velocity field v must be symmetric in the variables x and x so that bothcoordinate axes are invariant under η with the origin x = x = 0 a hyperbolicstagnation point.Observe that if ξ : R → R is a volume-preserving diffeomorphism then theJacobi matrix of its inverse ξ − can be computed from Dξ − = ( Dξ ) − ◦ ξ − = (cid:18) ∂ ξ ◦ ξ − − ∂ ξ ◦ ξ − − ∂ ξ ◦ ξ − ∂ ξ ◦ ξ − (cid:19) OCAL ILL-POSEDNESS OF THE EULER EQUATIONS IN B ∞ , so that for any smooth function f : R → R we have(2.9) ∇ ( f ◦ ξ − ) = (cid:0) −∇ f ◦ ξ − · ∇ ⊥ ξ ◦ ξ − , ∇ f ◦ ξ − · ∇ ⊥ ξ ◦ ξ − (cid:1) where ∇ ⊥ is the symplectic gradient as in (1.6). Proposition 6.
Let η ( t ) be the flow of the velocity field u = ∇ ⊥ ∆ − ω with initialvorticity given by (2.4) . Given M ≫ we have sup ≤ t ≤ M − k Dη ( t ) k ∞ > M for any sufficiently large integer N > in (2.4) and any < p < ∞ sufficientlyclose to .Proof. See [14]; Section 3. (cid:3)
In what follows it can be assumed without loss of generality that 2 < p ≤
3. Inthis case all estimates on the flow η or its derivative Dη can be made independentof the Lebesgue exponent 2 < p < ∞ .We will also need a comparison result for solutions of the Lagrangian flow equa-tions, namely Lemma 7.
Let u and v be smooth divergence-free vector fields on R and let η and ξ be the solutions of (2.7) - (2.8) with the right-hand sides given by F u and F u + v respectively. Then sup ≤ t ≤ (cid:0) k ξ ( t ) − η ( t ) k ∞ + k Dξ ( t ) − Dη ( t ) k ∞ (cid:1) ≤ C sup ≤ t ≤ ( k v ( t ) k ∞ + k Dv ( t ) k ∞ ) for some C > depending only on u and its derivatives.Proof. See e.g. [3]; Lemma 4.1. (cid:3) Proof of Theorem 2
Let M ≫ T = 1 ≤ M − . Recall thatby assumption 2 < p < ∞ and set s = 2.Given the initial vorticity ω defined in (2.4) let ω ( t ) be the corresponding solu-tion of the vorticity equations (1.4)-(1.5) and let η ( t ) be the associated Lagrangianflow of u = ∇ ⊥ ∆ − ω .If there exists 0 < t ≤ M − such that k ω ( t ) k W ,p > M / then there is nothingto prove. We will therefore assume that(3.1) k ω ( t ) k W ,p ≤ M / , ≤ t ≤ M − . Using Proposition 6 we can then find a point x ∗ = ( x ∗ , x ∗ ) such that at least oneof the entries ∂η i /∂x j of the Jacobi matrix (for example, the i = j =2 entry) satisfies | ∂ η ( t , x ∗ ) | > M . Therefore, by continuity, there is a δ > (cid:12)(cid:12)(cid:12)(cid:12) ∂η ∂x ( t , x ) (cid:12)(cid:12)(cid:12)(cid:12) > M for all | x − x ∗ | < δ. (3.2)Consider a smooth bump function ˆ χ ∈ C ∞ c ( R ) in Fourier variables with support inthe unit ball B (0 ,
1) and such that 0 ≤ ˆ χ ≤ R R ˆ χ ( ξ ) dξ = 1. Let ξ = (2 , ρ ( ξ ) = ˆ χ ( ξ − ξ ) + ˆ χ ( ξ + ξ ) , ξ ∈ R Observe that the support of ˆ ρ is contained in B ( − ξ , ∪ B ( ξ ,
1) and(3.4) ρ (0) = Z R ˆ ρ ( ξ ) dξ = 2 . For any k ∈ Z + and λ > β k,λ ( x ) = λ − p √ k X ε ,ε = ± ε ε ρ ( λ ( x − x ∗ ǫ )) sin kx where x ∗ ǫ = ( ε x ∗ , ε x ∗ ).To proceed we will need two technical lemmas. Lemma 8.
For any k ∈ Z + and λ > we have k ∂ j ∆ − β k,λ k ∞ . k − / λ − p k ˆ ρ k L k ∂ i ∂ j ∆ − β k,λ k ∞ . k − / λ − p k ˆ ρ k L k β k,λ k W ,p . (cid:0) k − / + k / λ − + k − / λ − (cid:1) k ˆ ρ k L p ′ where i, j = 1 , and < p ′ < is the conjugate exponent of < p < ∞ .Proof. Let ξ ± = ( ξ ± k/ π, ξ ) and first compute the Fourier transformˆ β k,λ ( ξ ) = λ − p √ k X ε ,ε ε ε i (cid:18) e − πi h ξ − ,x ∗ ε i λ ˆ ρ (cid:16) ξ − λ (cid:17) −− e − πi h ξ + ,x ∗ ε i λ ˆ ρ (cid:16) ξ + λ (cid:17)(cid:19) . (3.6)Next, using the change of variable formula we estimate (cid:12)(cid:12) ∂ j ∆ − β k,λ ( x ) (cid:12)(cid:12) ≃ (cid:12)(cid:12)(cid:12) F − F (cid:0) ∂ j ∆ − β k,λ (cid:1) ( x ) (cid:12)(cid:12)(cid:12) . Z R | ξ | − (cid:12)(cid:12) ˆ β k,λ ( ξ ) (cid:12)(cid:12) dξ . k − / λ − p Z R | ξ | − λ (cid:16)(cid:12)(cid:12) ˆ ρ ( λ − ξ − ) (cid:12)(cid:12) + (cid:12)(cid:12) ˆ ρ ( λ − ξ + ) (cid:12)(cid:12)(cid:17) dξ ≃ k − / λ − p X j =1 , Z R | ξ | − (cid:12)(cid:12) ˆ ρ ( ξ + ( − j π λ − k, ξ ) (cid:12)(cid:12) dξ. Since by construction for any sufficiently large k ≫
10 we havesupp ˆ ρ (cid:0) · ± π λ − k, · (cid:1) ∩ B (0 ,
1) = ∅ we can further estimate the above expression by ≃ k − / λ − p X j =1 , Z | ξ |≥ (cid:12)(cid:12) ˆ ρ ( ξ + ( − j π λ − k, ξ ) (cid:12)(cid:12) dξ ≃ k − / λ − p k ˆ ρ k L . For the second assertion we similarly obtain (cid:12)(cid:12) ∂ j ∆ − β k,λ ( x ) (cid:12)(cid:12) . k ˆ β k,λ k L . k − / λ − p k ˆ ρ k L . Finally, using the triangle inequality and the change of variables formula we get (cid:13)(cid:13)(cid:13) ∂β k,λ ∂x (cid:13)(cid:13)(cid:13) L p . √ k (cid:13)(cid:13)(cid:13) λ /p X ε ,ε ε ε ∂ρ∂x (cid:0) λ ( · − x ∗ ε ) (cid:1)(cid:13)(cid:13)(cid:13) L p + √ kλ (cid:13)(cid:13)(cid:13) λ /p X ε ,ε ε ε ρ (cid:0) λ ( · − x ∗ ε ) (cid:1)(cid:13)(cid:13)(cid:13) L p ≃ √ k Z R (cid:12)(cid:12)(cid:12) X ε ,ε ε ε ∂ρ∂x ( x ) (cid:12)(cid:12)(cid:12) p dx ! /p + √ kλ Z R (cid:12)(cid:12)(cid:12) X ε ,ε ε ε ρ ( x ) (cid:12)(cid:12)(cid:12) p dx ! /p OCAL ILL-POSEDNESS OF THE EULER EQUATIONS IN B ∞ , and using Hausdorff-Young with 1 /p ′ + 1 /p ′ = 1 we find . k − / (cid:13)(cid:13)(cid:13) ∂ρ∂x (cid:13)(cid:13)(cid:13) L p + k / λ − k ρ k L p . k − / (cid:13)(cid:13)(cid:13) d ∂ρ∂x (cid:13)(cid:13)(cid:13) L p ′ + k / λ − k ˆ ρ k L p ′ . (cid:0) k − / + k / λ − (cid:1) k ˆ ρ k L p ′ . Similarly, we also obtain (cid:13)(cid:13)(cid:13) ∂β k,λ ∂x (cid:13)(cid:13)(cid:13) L p . k − / (cid:13)(cid:13)(cid:13) ∂ρ∂x (cid:13)(cid:13)(cid:13) L p . k − / k ˆ ρ k L p ′ and k β k,λ k L p . λ − √ k Z R (cid:12)(cid:12)(cid:12) λ /p X ε ,ε ε ε ρ (cid:0) λ ( x − x ∗ ε ) (cid:1)(cid:12)(cid:12)(cid:12) p dx ! /p . k − / λ − k ˆ ρ k L p ′ which combined yield the lemma. (cid:3) Observe that choosing(3.7) k = λ and λ = 3 n, n ≫ β n = β k,λ in Lemma 8 we immediately obtain k∇ ⊥ ∆ − β n k ∞ → k D ∇ ⊥ ∆ − β n k ∞ → n → ∞ (3.8)and k β n k W ,p ≃ k β n k L p + k∇ β n k L p . k ˆ ρ k L p ′ < ∞ for any n ∈ Z + . (3.9)The second lemma we need is Lemma 9.
Let t > be as in (3.1) and let k, λ and n be as in (3.7) . Then k ∂ β k,λ ∂ η ( t ) k L p . n − C ,T k ˆ ρ k L p ′ −−−−→ n →∞ k ∂ β k,λ ∂ η ( t ) k L p & M (cid:0) ǫπ + O ( n − / ) (cid:1) − n − C ,T k ˆ ρ k L p ′ −−−−→ n →∞ M where C ,T = exp T sup ≤ t ≤ T k D ∇ ⊥ ∆ − ω ( t ) k ∞ < ∞ .Proof. From (3.5) we have (cid:18) Z R (cid:12)(cid:12)(cid:12) ∂β k,λ ∂x ( x ) ∂η ∂x ( t , x ) (cid:12)(cid:12)(cid:12) p dx (cid:19) /p = Z R (cid:12)(cid:12)(cid:12)(cid:12) √ k λ p X ε ,ε ε ε ∂ρ∂x ( λ ( x − x ∗ ε )) sin kx ∂η ∂x ( t , x ) (cid:12)(cid:12)(cid:12)(cid:12) p dx ! /p (3.10) ≤ √ k X ε ,ε (cid:18)Z R λ (cid:12)(cid:12)(cid:12) ∂ρ∂x ( λ ( x − x ∗ ε )) (cid:12)(cid:12)(cid:12) p dx (cid:19) /p sup x ∈ R (cid:12)(cid:12)(cid:12) ∂η ∂x ( t , x ) (cid:12)(cid:12)(cid:12) . Differentiating the flow equations (2.7)-(2.8) in the spatial variable and taking the L ∞ norms we obtain a differential inequality which with the help of Grownwall’slemma gives k Dη ( t ) k ∞ ≤ e R t k Du ( τ ) k ∞ dτ ≤ C ,T . Using this bound the right hand side of (3.10) can be estimated further by ≃ k − / k ∂ η ( t ) k ∞ k ρ k L p . k − / C ,T k ˆ ρ k L p ′ . GERARD MISIO LEK AND TSUYOSHI YONEDA which gives the first assertion. For the second one we have (cid:18) Z R (cid:12)(cid:12)(cid:12) ∂β k,λ ∂x ( x ) ∂η ∂x ( t , x ) (cid:12)(cid:12)(cid:12) p dx (cid:19) /p == Z R (cid:12)(cid:12)(cid:12)(cid:12) √ k λ p − X ε ,ε ε ε (cid:16) kρ ( λ ( x − x ∗ ε )) cos kx ++ λ ∂ρ∂x ( λ ( x − x ∗ ε )) sin kx (cid:17) ∂η ∂x ( t , x ) (cid:12)(cid:12)(cid:12)(cid:12) p dx ! /p ≥ Z B ( x ∗ ,δ ) (cid:12)(cid:12)(cid:12)(cid:12) √ kλ − p cos kx ρ ( λ ( x − x ∗ )) ∂η ∂x ( t , x ) ++ 1 √ k λ p sin kx ∂ρ∂x ( λ ( x − x ∗ )) ∂η ∂x ( t , x ) (cid:12)(cid:12)(cid:12)(cid:12) p dx ! /p . Using the triangle inequality, (3.2) and the change of variables formula we estimatethe above integral from below by & M √ kλ − (cid:18) Z B ( x ∗ ,δ ) λ (cid:12)(cid:12) cos kx ρ ( λ ( x − x ∗ )) (cid:12)(cid:12) p dx (cid:19) /p −− √ k (cid:18) Z B ( x ∗ ,δ ) λ (cid:12)(cid:12)(cid:12) sin kx ∂ρ∂x ( λ ( x − x ∗ )) ∂η ∂x ( t , x ) (cid:12)(cid:12)(cid:12) p dx (cid:19) /p (3.11) ≥ M √ kλ − (cid:18) Z B (0 ,λδ ) (cid:12)(cid:12) cos ( kλ − x + kx ∗ ) (cid:12)(cid:12) p | ρ ( x ) | p dx (cid:19) /p −− √ k k ∂ ρ k L p k ∂ η ( t ) k L ∞ ( B ( x ∗ ,δ )) . We now focus on the integral term on the right hand side of (3.11). From (3.4) wehave ρ (0) = 2 so that by continuity there exists an ǫ > | ρ ( x ) | ≥ , for any x ∈ B (0 , ǫ ) . Therefore, if we set δ = ǫ/λ then the integral term can be bounded below by Z B (0 ,ǫ ) (cid:12)(cid:12) cos ( kλ − x + kx ∗ ) (cid:12)(cid:12) p dx ! /p ≥ Z ǫπ/ − ǫπ/ Z ǫπ/ − ǫπ/ cos ( λx + λ x ∗ ) dxdy ! / ≃ ǫπ √ O ( λ − / )by a straightforward calculation using the assumption on p > & M (cid:0) ǫπ + O ( n − / ) (cid:1) − C ,T n k ˆ ρ k L p ′ which is the required estimate. (cid:3) Consider the following sequence of initial vorticities(3.12) ω ,n ( x ) = ω ( x ) + β n ( x ) , n ∈ Z + . OCAL ILL-POSEDNESS OF THE EULER EQUATIONS IN B ∞ , From equations (3.9) and (2.6) of Lemma 5 it follows that ω ,n belongs to W ,p forany n ∈ Z + . Let ω n ( t ) be the corresponding solutions of the vorticity equations(1.4)-(1.5). For each n ∈ Z + (sufficiently large if necessary) let η n ( t ) be the flow ofvolume-preserving diffeomorphisms of the velocity fields u n = ∇ ⊥ ∆ − ω n .(A). Let 1 ≤ q < ∞ and assume that the data-to-solution map for the Eulerequations is continuous from bounded sets in B ∞ , ( R ) to C ([0 , , B ∞ , ( R )). Lemma 10.
For any ≤ q < ∞ we have k∇ ⊥ ∆ − β n k B ∞ ,q ≃ k∇ ⊥ ∆ − β n k ∞ + k D ∇ ⊥ ∆ − β n k ∞ for any sufficiently large n ∈ Z + .Proof. It will be sufficient to show the equalities k D α ∇ ⊥ ∆ − β n k B ∞ ,q ≃ k D α ∇ ⊥ ∆ − β n k ∞ for | α | = 0 and 1.As in the proof of Lemma 8 recall that for any large enough integer n ∈ Z + wehave supp ˆ β n ∩ B (0 ,
1) = ∅ so that only need to establish k β n k ∞ ≃ k β n k B ∞ ,q . Let ψ ℓ be the family of Paley-Littlewood functions as in (2.1) supported in the shell { ℓ − ≤ | ξ | ≤ ℓ +1 } . From (3.6) and the definition of ρ we find thatsupp ˆ β n =2 j ⊂ B ((2 j , , j )since supp ˆ ρ ⊂ B (0 , j ∈ Z + we can find an ℓ j ∈ Z + such that supp ˆ β j ⊂ supp ψ ℓ j which implies that k β j k B ∞ ,q = (cid:18) X ℓ ≥− k ∆ ℓ β j k q ∞ (cid:19) /q ≃ k ˆ ψ ℓ j ∗ β j k ∞ ≃ k β j k ∞ . The other equality can be shown analogously. (cid:3)
Combining (3.8), (3.12) and Lemma 10 it follows now from the assumption (A)on the continuity of the solution map that(3.13) sup ≤ t ≤ k∇ ⊥ ∆ − ( ω n ( t ) − ω ( t )) k B ∞ , −→ n → ∞ and consequently by an elementary embedding B ∞ , ⊂ C we also havesup ≤ t ≤ k∇ ⊥ ∆ − ( ω n ( t ) − ω ( t )) k C −→ n → ∞ . Applying the comparison Lemma 7 we now findsup ≤ t ≤ (cid:0) k η n ( t ) − η ( t ) k ∞ + k Dη n ( t ) − Dη ( t ) k ∞ (cid:1) = θ n −→ n → ∞ (3.14)where η ( t ) is the flow of the velocity field u = ∇ ⊥ ∆ − ω with the initial vorticity ω given by (2.4) as in Proposition 6.Using conservation of vorticity, formula (2.9) and the invariance of the L p normsunder volume-preserving Lagrangian flows η n ( t ) we have k ω n ( t ) k W ,p ≥ k∇ ( ω ,n ◦ η − n ( t )) k L p ≃k dω ,n ◦ η − n ( t )( ∇ ⊥ η n, ( t ) ◦ η − n ( t )) k L p + k dω ,n ◦ η − n ( t )( ∇ ⊥ η n, ( t ) ◦ η − n ( t )) k L p ≃ k dω ,n ( ∇ ⊥ η n, ( t )) k L p + k dω ,n ( ∇ ⊥ η n, ( t )) k L p (3.15) & k dω ,n ( ∇ ⊥ η n, ( t )) k L p . Since from the comparison estimate (3.14) we have k dω ,n ( ∇ ⊥ η − ∇ ⊥ η n, )( t ) k L p . k D ( η − η n, )( t ) k ∞ k∇ ω ,n k L p ≤ θ n k∇ ω ,n k L p applying the triangle inequality and (3.12) we can further bound the right side ofthe expression in (3.15) below by k dω ,n ( ∇ ⊥ η ( t )) k L p − θ n k∇ ω ,n k L p (3.16) & k dβ n ( ∇ ⊥ η ( t )) k L p − k dω ( ∇ ⊥ η ( t )) k L p − θ n k∇ ω ,n k L p . Observe that by the assumption (3.1) we can bound the middle term on the rightside of (3.16) as in (3.15) above by k dω ( ∇ ⊥ η ( t )) k L p ≤ k∇ ω ◦ η − ( t ) · Dη − ( t ) k L p ≃ k∇ ( ω ◦ η − ( t )) k L p ≤ k ω ( t ) k W ,p ≤ M / . (3.17)It therefore remains to find a lower bound on the β -term in (3.16). This howeverfollows from the the two estimates in Lemma 9. Namely, we have k dβ n ( ∇ ⊥ η ( t )) k L p = (cid:13)(cid:13) − ∂ β n ∂ η ( t ) + ∂ β n ∂ η ( t ) (cid:13)(cid:13) L p ≥ k ∂ β n ∂ η ( t ) k L p − k ∂ β n ∂ η ( t ) k L p & M (cid:0) ǫπ + O ( n − / ) (cid:1) − n − C ,T k ˆ ρ k L p ′ − n − C ,T k ˆ ρ k L p ′ (3.18) & M provided that n is sufficiently large. Theorem 2 follows now by combining (3.15)with (3.16), (3.17) and (3.18). References
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