DISCRETE SCHR¨ODINGER EQUATION AND ILL-POSEDNESS FOR THE EULER EQUATION

. We consider the 2 D Euler equation with periodic boundary con-ditions in a family of Banach spaces based on the Fourier coeﬃcients, and show that it is ill-posed in the sense that ‘norm inﬂation’ occurs. The proof is based on the observation that the evolution of certain perturbations of the ‘Kolmogorov ﬂow’ given in velocity by can be well approximated by the linear Schr¨odinger equation, at least for a short period of time.

1. Introduction. We consider the 2D Euler equation in vorticity formulation on the torus: ∂ t Ω + U · ∇Ω = 0, U = ∇ ⊥ ∆ −1 Ω, x ∈ T 2 (1) and we study its local well-posedness in different topologies. Without loss of generality, we will restrict ourselves to mean zero functions on T 2 , and define the norms Ω X s p := |k| s · Ω(k) p (Z 2 ) , where the Fourier transform is defined by Our main result is the following Theorem 1.1. Let s > 2, 1 < p ≤ ∞ and p = 2. There exists a sequence of times t n → 0 and a sequence of initial data Ω 0,n ∈ C ∞ (T 2 ) with associated solution Ω n ∈ C ∞ (T 2 × [0, ∞)) such that Ω 0,n X s p → 0 and Ω n (t n ) X s p → +∞.
Thus the flow of (1) is ill-posed in any neighborhood of 0 in X s p . Remark 1. For s > 1, (1) is locally well-posed in H s , thus the above result is false when p = 2 and another way to interpret Theorem 1.1 is that for p > 1, s > 2, the Euler equation (1) is well-posed in X s p if and only if p = 2. Let us briefly comment on the norms introduced above. The regularity, s > 2 is natural to make sense of the equation when p = ∞; for lower values of p, this could be lowered, but for simplicity, we did not pursue this since we consider the result strongest for large s. As for the summability exponent, the case p = 2 simply corresponds to classical Sobolev spaces H s , and the global well-posedness of 2D Euler and Navier-Stokes equations with vorticity in H s (s > 1) is well-known; this goes back to the work of Ebin and Marsden [4] (on compact manifolds) and Kato [7,8] on R d . In the case p = ∞, X s ∞ has a geometric flavor, as the unit ball consists of functions whose Fourier coefficients lie in an infinite-dimensional rectangle. Firmly based on this geometric fact, Mattingly and Sinai [9] gave a simple proof of the well-posedness for 2D Navier-Stokes equation by first showing existence and uniqueness in the space L ∞ ([0, ∞) : X s ∞ ) for s large enough. Incidentally, the result of Cheskidov and Shyvdkoy [3] can be adapted to show that for the Euler equation, there are initial data from X s ∞ whose solutions cannot be contained in C 0 ([0, T ] : X s ∞ ) (again for s large enough), but this method fails for p < ∞. Hence the spaces based on other values of p can be viewed as interpolation/extrapolation spaces, which avoid certain pathologies 1 of the case p = ∞ and it seems natural to ask the question of well-posedness in them.
We refer to [6] for a use of similar norms for a Schrödinger-type equation.
There is a large literature on the well-posedness problem of the Euler equation (see [2] for instance), but we only mention a few. Cheskidov-Shvydkoy [3] work in Besov spaces based on ∞ and show existence of initial data whose solutions cannot be continuous in the same norm. While the initial data are explicit, it is unclear how their data evolve, since the proof is based on a contradiction argument. In particular, their proof does not show 'norm inflation'. Nevertheless, this work has partially motivated ours; indeed, their ill-posedness is due to the fast and non-uniform oscillations in time of certain Fourier modes, a behavior which our approximate solutions explicitly show. On the other hand, the papers of Bourgain and Li [1,2], and Elgindi and Masmoudi [5] consider 2 -based spaces (and are thus beyond the scope of our analysis), but rely on the fact that the Riesz transform fails to be bounded on L ∞ , while all our spaces are trivially invariant by the Riesz transform (in x). In these works, the initial data (and the evolution) are not so explicitly given. We refer to the aforementioned works for the precise notion of well-posedness/ill-posedness as well as a more extensive list of references on the topic.
Our approach is to study flows which are small perturbations of a 'Kolmogorov state', given in vorticity byΩ (x, y) = sin y, and to show that the linear Schrödinger equation is somehow embedded in the Euler equation, even for arbitrarily short times (see Lemma 1.3 below). Here, a key point is that as we place the initial perturbation further away in the frequency plane, we get to follow the Schrödinger dynamics for a longer period of time. From this, we deduce easily that only the norms which remain controlled along a Schrödinger evolution lead to local well-posedness, thus forcing p = 2. Incidentally, we have shown that the stationary solutionΩ is highly unstable in the spaces X s p with s > 2 and 1 < p ≤ ∞ with p = 2.
We note in passing that studying other equilibriums, one can realize other equations by the same mechanism; for a simple example, withΩ(x, y) = cos y, one can get the linear wave equation.
In our opinion, the main interests of the present work are threefold: (i) it shows a strong version of ill-posedness, even for p = ∞, (ii) the observation that p = 2 is an isolated index for local well-posedness for the Euler equation in X s p , (iii) it exhibits solutions to linear dispersive equations inside solutions of the Euler equation. It would be interesting to know if one can also exhibit a classical nonlinear dispersive equation.
1.1. Notations. We define the left and right discrete difference operators on sequences {c p } p∈Z , and we notice the integration by parts formula: We will work in the Fourier space and we introduce norms on sequence spaces. Sequences indexed by Z will often be denoted by lower case letter, while sequence indexed by Z 2 will be denoted by capitalized letters. We define When the summation domain is Z, we will often omit it. We write A B if there exists an absolute constant C > 0 such that A ≤ CB.
If such a constant depends on some parameters, we will write it explicitly by using a subscript, thus A s B means that the constant C = C(s) depends on the parameter s.

Main reduction. In Fourier space, (1) becomes
We first need a lemma to control our ansatz.
then σ is defined for all times. In addition, σ remains close to its continuous analogue: defining c ∈ C ∞ (R x × R t ) to be the solution of and defining its sample by with a constant C s,ϕ,ε > 0 depending only on s, ϕ and ε. In addition, there exist constants C 1 s,ϕ , C 2 s,ϕ > 0 depending only on s and ϕ such that so long as Remark 2. Note that by conservation of mass, we also have for all time Next, given parameters , k and a profile ϕ ∈ C ∞ c (−1, 1), we build an approximate solution A as follows: and all other modes equal to 0, where η > 0 is a normalization constant to be determined later, χ ∈ C ∞ c (−2, 2) is a bump function satisfying χ(x) = 1 for |x| ≤ 1, τ := (k − 1/k) 2 and σ is the solution of (4) given by Lemma 1.2.
The following stability Lemma will let us to produce an exact solution close to A: and assume that B satisfies Let W be the solution of (1) with initial data W (0) = B(0). There exists a constant C s > 0 such that, if we have The key observation in Lemma 1.4 is that one only needs β 0 T + εβ 1 T 1 to have a good approximate solution.
Finally, combining these results, we may prove Theorem 1.1.
Proof of Theorem 1.1. We first treat the case 2 < p ≤ ∞. We observe that, using the scaling Ω → Ω λ , Ω λ (x, t) = λΩ(x, λt), which trivially changes the norm of the initial data, and using the time reversibility nature of our equation, it suffice to prove the following: There exists a sequence of solutions Ω n and a sequence of times t n → 0 satisfying that Indeed, one may then set λ n := max{ Ω n (0) To prove Claim b, we fix a nonzero, nonnegative ϕ ∈ C ∞ c (−1, 1), a sequence k → +∞, and for fixed k, we let for 1/3 < α < 1/2. With σ defined as in (4), we may produce A = A k by (9) which satisfies according to Lemma 1.3. We may now use Lemma 1.4 with and, letting Ω k be the solution of (1) with initial data Ω k (0) = A k (0), we see that :h s (Z 2 )) → 0 and therefore Ω k satisfies Claim b.
We now treat the case 1 < p < 2. This time, we will construct Ω n and t n → 0 in a way that and again a simple scaling argument will establish the result. We let δ = 1/2p and fix α > 0 a small parameter such that 2δ + 5sα ≤ 1.
This time, we set and again using σ in (4), we may produce A = A k by (9) which satisfies 1, A L ∞ ([0,t k ]:hs(Z 2 )) 1 + ηks 1 + ks −s 1 according to Lemma 1.3. We may now control the difference between A and the actual solution in L ∞ ([0, t k ] : h s+2δ (Z 2 )) by using Lemma 1.4 with and we deduce that which establishes (14).

Remark 3.
Inspecting the proof, the problem when p = 1 comes form the fact that the growth in the approximate solution is provided in terms of τ t k k 2 t k → +∞, while to control the difference between the approximate solution and the real solution via Sobolev inequality, we need to control the h s+s -norm, for s > 2(1/p − 1/2) which leads to use stability in h s+s . Then β 0 k s · 1/2 has to satisfy β 0 t k = k s 1 2 t k → 0. When s ≥ 1, this is no longer covered by our simple analysis.

Discrete Schrödinger equations.
Proof of Lemma 1.2. The difference d := σ − c satisfies where e(p, t) = i and a Taylor expansion gives that Similarly, we get where ∇ α denotes some α copies of ∇ R and ∇ L .
We then obtain (7) from (6) and (22), (23). To see this, we first note that Indeed, from the expression for c in (22), we see directly that the main term gives above estimate, while the contribution from the remainder term r is smaller by a factor of 1/(1 + t) up to a constant depending only on s and ϕ, and hence this contribution can be made arbitrarily small by choosing C 1 s,ϕ large if necessary. Therefore, for 2 ≤ q ≤ ∞, by choosing a smaller constant C 2 s,ϕ if necessary.
Lemma 2.1. Let d be the solution of (15) with error e satisfying

IN-JEE JEONG AND BENOIT PAUSADER
for t ≥ 0 and α * ∈ N, where ∇ α means any application of α copies of ∇ R , ∇ L . Then, for all integer 0 ≤ s ≤ α * − 1 and small > 0, there holds that Proof of Lemma 2.1. We may assume κ = 1. For N ≥ 1, we observe that for some constant C = C(α).

DISCRETE SCHRÖDINGER EQUATION AND THE EULER EQUATION 289
Therefore, since E α (0) = 0, using (20), we successively find that Similarly, we find that In particular, for N ≥ −1 and for integer 0 ≤ α ≤ α * , and by convexity, we may extend this for real 0 ≤ α. Letting N range over dyadic integers bigger than −1 , and applying the above estimate for α = s + , we find that so that we finally obtain (16).
Finally, we recall the well-known Fraunhofer formula: Lemma 2.2. Assume that ϕ ∈ C ∞ c (−1, 1) and let p, w ∈ N. Then we have the Fraunhofer formula for c, the solution of (5) with initial data ϕ: with estimates and (24) Proof of Lemma 2.2. If 0 ≤ t ≤ 1, the bound for c follows directly from computing F −1 e it|ξ| 2 ϕ. Assume now that t ≥ 1. Using the explicit integral kernel for the Schrödinger equation, we observe that 1] e −i xy 2t · ϕ(y)(e i y 2 4t − 1)dy and since, on the support of integration integrations by parts give the result.
Proof of Lemma 1.4. We consider D = W − B, which satisfies the equation We first estimate the H s−1 -norm of D. We will use the following two simple estimates (direct in Fourier space) where |∇| α corresponds to multiplication of the Fourier coefficients by |k| α . Letting and from (30) we obtain that Multiplying both sides of the first line of (31) by D s−1 and integrating over T 2 gives Since D = 0 at t = 0, we may take the maximal time 0 < T 1 ≤ T where D H s−1 ≤ β 0 on [0, T 1 ]. Then for 0 < t ≤ T 1 , we have the bound and by Gronwall's inequality we obtain on 0 < t ≤ T 1 the bound but from our assumption we deduce D(T 1 ) H s−1 < β 0 . Hence, we conclude that the estimate (33) is valid for 0 < t ≤ T . Next, we estimate D in H s . We proceed as before except that we decompose for 0 < t ≤ T 2 , and with our assumptions, we can show that T 2 = T as previously.