TWO SINGULAR DYNAMICS OF THE NONLINEAR SCHRÖDINGER EQUATION ON A PLANE DOMAIN

by

N. Burq, P. Gérard & N. Tzvetkov

Abstract. — We study the cubic, focusing nonlinear Schrödinger equation (NLS) posed on a bounded domain of ℝ^{2} with Dirichlet boundary conditions. We describe two types of nonlinear evolutions. First we obtain solutions which blow up with a minimal L^{2} norm in finite time at a fixed point of the interior of the domain. The argument can be performed equally well for the cubic NLS posed on the flat torus 𝕋^{2}. In the case when the domain is a disc, we also prove that the Cauchy problem is ill posed is the following sens : the flow map isnot uniformly continuous on bounded sets of the Sobolev space H^{s}, s < \frac{1}{3} , contrary to what is known on the square (recall that the scale invariant Sobolev space for the cubic NLS in 2D is L^{2}).

Résumé. — Nous étudions l’équation de Schrödinger non linéaire cubique focalisante sur un domaine borné de ℝ^{d} avec des conditions aux limites de Dirichlet. Nous décrivons deux types d’évolutions non linéaires. Nous obtenons d’abord des solutions qui explosent en temps fini, avec une norme L^{2} minimale, en un point fixé de l’intérieur du domaine. L’argument s’adapte aisément au cas du tore plat 𝕋^{2}. Lorsque le domaine est un disque, nous montrons également que le problème de Cauchy est mal posé au sens suivant: le flot (s’il est défini) n’est pas uniformément continu sur les bornés de l’espace de Sobolev H^{s}, s < \frac{1}{3} , contrairement à ce qui est connu sur un carré (rapppelons que l’espace de Sobolev invariant d’échelle pour NLS cubique en deux dimensions est L^{2}).

2000 Mathematics Subject Classification. — 35Q55, 35BXX, 37K05, 37L50, 81Q20 . Key words and phrases. — nonlinear Schrödinger, eigenfunctions, dispersive equations.

1. Introduction and statement of the results

Let Ω be a bounded domain of ℝ^{2}. In this paper, we are studying the cubic, focusing nonlinear Schrödinger equation (NLS), posed on Ω

(i∂_{t} + Δ)u = −|u|^{2}u, in ℝ × Ω (1.1)

with initial data

u(0, x) = u_{0}(x), x ∈ Ω

and Dirichlet boundary conditions

u(t, x) = 0, (t, x) ∈ ℝ × ∂Ω.

In (1.1), t denotes the time variable, Δ is the Laplace operator acting on the spatial variables and u is a complex valued function, defined on ℝ × Ω. When the NLS is posed on a bounded spatial domain the corresponding nonlinear dynamics is expected to differ from that of the NLS on ℝ^{d}. We refer to the recent book [1] and the references therein for results in the case of the flat torus (periodic boundary conditions) and to [3, 4] in the case of spheres or more general compact riemannian manifolds. The equation (1.1) can be written in a Hamiltonian format with respect to the canonical coordinates (u, ū)

iu_{t} = \frac{δH}{δū} , iū_{t} = −\frac{δH}{δu} .

where

H(u, ū) = \frac{1}{2} ∫_{Ω} |∇u|^{2} − \frac{1}{4} ∫_{Ω} |u|^{4}.

The quantity H(u, ū) is usually called the energy and is conserved by the time evolution (see e.g. [16] or [5, theorem 4.5.1] for an H^{1} existence result), as well as the L^{2} norm. Our goal is to give two qualitative results for the Cauchy problem associated to (1.1). During the last two decades, there has been a lot of work on the corresponding issue in the case of data defined on ℝ^{2} (or more generally on ℝ^{d}, d ≥ 1, with other power nonlinearities), starting from the work of Ginibre-Velo [7]. Notice that an important invariance of the NLS posed on ℝ^{d} is the scaling one. We now summarize some known facts concerning the theory of local and global existence of solutions of the Cauchy problem associated to (1.1), posed on the whole plane ℝ^{2}.

1. Strong local well-posedness down to the critical (scaling) Sobolev regularity of the initial data via the Strichartz inequalities for the free evolution and the Picard iteration scheme (see [6]).

2. Global well-posedness for initial data with L^{2} norm smaller than the L^{2} norm of the ground state (see (1.2) below) via the energy a priori bound and the sharp Gagliardo-Nirenberg interpolation inequality (see [19]).

3. Existence of smooth data developing singularities in finite time via the viriel identity (see [17, 20, 8]).

4. Classification of the blowing up solutions with L^{2} norm equal to the L^{2} norm of the ground state via the pseudo-conformal invariance of (1.1) (see [12]).

It is natural to study the above issues in the context of (1.1). In this paper, we show that if Ω is a unit disc of ℝ^{2} then the first of the above properties fails in the case of (1.1). It is interesting to mention that this is an effect of the boundary since when Ω is a square, the Cauchy problem associated to (1.1) is locally well-posed down to the critical Sobolev regularity of the initial data. This can be seen by an easy extension of the result of Bourgain on the flat torus 𝕋^{2} (see [1] and section 2 below). The property 2 can be extended to the case of (1.1), posed on a plane domain (see [2, 5] and section 2 below). Concerning the property 3, we refer to [10] where it is shown that under some geometric assumption on Ω, for smooth “large data” (for example data with negative energy) the solutions of (1.1) develop singularities in finite time. The proof uses again the classical viriel identity argument. The geometric assumption is imposed in order to evaluate the boundary terms. Whether the property 4 has a reasonable extension to the case of domains is an important open problem. In this paper, we however construct minimal L^{2} mass blowing up solutions with a precise description of the solutions near the blow-up time. Moreover our construction can be performed when Ω is not necessarily bounded.

Now we state our results.

Theorem 1. — Let Ω be a smooth bounded domain of ℝ^{2}. Let x_{0} ∈ Ω with ψ ∈ C_{0}^{∞}(Ω), ψ = 1 near x_{0}. Denote by Q the unique positive radial solution of the nonlinear elliptic equation^{(1)}

(−Δ_{ℝ^{2}} + 1)Q = |Q|^{2}Q (1.2)

(1)Q is usually called ground state.

(see [5]). Then there exist κ > 0, λ_{0} > 0 such that for every λ > λ_{0} there exist T_{λ} > 0 and a family {r_{λ}} of functions defined on [0, T_{λ}[×Ω satisfying

∥r_{λ}(t, ·)∥_{H^{2}(Ω)} ≤ ce^{− \frac{κ}{λ(T_{λ}−t)}} , t ∈ [0, T_{λ}[

such that

u_{λ}(t, x) = \frac{1}{λ(T_{λ} − t)}ψ(x)e^{\frac{i(4−λ^{2}(x−x_{0})^{2})}{4λ^{2}(T_{λ}−t)}} Q (\frac{x − x_{0}}{λ(T_{λ} − t)}) + r_{λ}(t, x), x ∈ Ω, t ∈ [0, T_{λ}[

are solutions of (1.1), satisfying the Dirichlet boundary conditions, which blow up at x_{0} in time T_{λ} in the energy space H^{1} with blow up speed (T_{λ} − t)^{−1}. Moreover ∥u_{λ}(t, ·)∥_{L^{2}(Ω)} = ∥Q∥_{L^{2}(ℝ^{2})}.

Remark 1.1. — 1. Theorem 1 can be easily generalized to the cubic NLS, posed

on the two dimensional torus 𝕋^{2}.

2. The proof of Theorem 1 is based on a perturbation argument used by Ogawa-Tsutsumi [14] in the spirit of an earlier work by Merle [11].

3. Our construction could be extended to the case of a large class of unbounded domains (for example the exterior of a smooth bounded domain)

We now come to our second result, which is a singular ansatz for the solution of (1.1) with particular high frequency data.

Theorem 2. — Let D = {x ∈ ℝ^{2} : |x| < 1} be the unit disc in ℝ^{2} and Δ_{D} the Laplace operator on D with Dirichlet boundary conditions. Fix κ > 0 and s ∈]\frac{1}{5}, \frac{1}{3}[. Then there exists a sequence ϕ_{n}(x) of eigenfunctions of −Δ_{D} with corresponding eigenvalues ζ_{n} (lim_{n→∞} ζ_{n} = +∞) such that ∥ϕ_{n}∥_{H^{s}(D)} ≈ 1 and the equation (1.1) with Cauchy data κϕ_{n}(x), has, for n ≫ 1, a unique global solution u_{n}(t, x) which can be represented as

u_{n}(t, x) = κe^{−it(ζ_{n}−κ^{2}ω_{n}))}(ϕ_{n}(x) + r_{n}(t, x)),

where ω_{n} ≈ n^{\frac{2}{3} −2s} and r_{n}(t, x) satisfies for any T > 0, n large enough and t ∈ [0, T]

∥r_{n}(t, ·)∥_{H^{s}(D)} ≤ C n^{−δ}

where δ > 0 and C is independent of n. Moreover

∥u_{n}∥_{L^{∞}(ℝ;H^{s}(D))} ≤ C|κ|.

As a consequence the Cauchy problem associated to (1.1) is not locally well-posed with uniformly continuous flow map for data in any ball of H^{s}(D).

Remark 1.2. — 1. In [4], we proved similar results for the NLS with spheres as

spatial domain.

2. See Lemma 2.1 below for the precise notion of local well-posedness with uniformly continuous flow map (see also [13], [4]).

3. Note that the scaling Sobolev regularity for (1.1) is s = 0.

4. Theorem 2 can be extended to the easier case of defocusing nonlinearity.

5. In [1], it is shown that the cubic NLS, posed on the torus 𝕋^{2} is locally wellposed with analytic flow map in H^{s}(𝕋^{2}), s > 0. The result of Theorem 2 shows that the results of Bourgain in the boundaryless case of the flat torus can not be extended to the case of the unit disc. This can be seen as an effect of the boundary to the initial value theory of NLS.

The rest of the paper is organized as follows. The next section is concerned with some well-posedness results for (1.1). Section 3 is devoted to the proof of Theorem 1. In section 4, we collect some properties of the Bessel functions, needed for the proof of Theorem 2. In section 5, we give the proof of Theorem 2. Finally in an appendix, we prove the results stated in section 4.

Acknowledgement: We thank Y. Tsutsumi for having drawn our attention to the reference [14].

2. Preliminary well-posedness results

Let Ω be a domain of ℝ^{2} and T > 0. Consider the initial boundary value problem

\{ \begin{matrix}(i∂_{t} + Δ)u = −|u|^{2}u, in [−T, T] × Ω, \\ u(0, x) = u_{0}(x), x ∈ Ω, \\ u(t, x) = 0, (t, x) ∈ [−T, T] × ∂Ω. \end{matrix} (2.1)

We now state a basic local well-posedness result for (2.1).

Lemma 2.1. — For any R > 0 there exists T > 0 such that, for every

u_{0} ∈ B_{R} := {u_{0} ∈ H^{2}(Ω) ∩ H_{0}^{1}(Ω) : ∥u_{0}∥_{H^{2}(Ω)} < R} ⊂ H^{2}(Ω),

the problem (2.1) has a unique solution in C([−T, T], H^{2}(Ω)). In addition the map u_{0} → u is uniformly continuous from B_{R} to C([−T, T], H^{2}(Ω)). Moreover, the following conservation laws hold

∥u(t, ·)∥_{L^{2}(Ω)} = ∥u_{0}∥_{L^{2}(Ω)},

and

\frac{1}{2}∥∇u(t, ·)∥_{L^{2}(Ω)}^{2} − \frac{1}{4}∥u(t, ·)∥_{L^{4}(Ω)}^{4} = \frac{1}{2}∥∇u_{0}∥_{L^{2}(Ω)}^{2} − \frac{1}{4}∥u_{0}∥_{L^{4}(Ω)}^{4},

where t ∈ [0, T].

The proof of Lemma 2.1 is classical and will be omitted. We only give several comments.

Remark 2.2. — 1. As it was already pointed out in [4] the uniform continuity of the flow map is typical for semilinear problems. This results from the fact that we solve (2.1) by performing the Picard iteration scheme which even gives the analyticity of the flow map.

2. The method of proof of Lemma 2.1 does not make use of the dispersive nature of (2.1). Well-posedness for (2.1) in Sobolev spaces of low order (smaller than 1) would probably require an appeal to Strichartz inequalities for the free evolution which are not known (see Remark 5.2 below).

It is a natural question to ask whether the local solutions obtained in Lemma 2.1 can be extended globally in time. We now state a classical result due to Brézis-Gallouet (see [2]) which gives an answer to that question for sufficiently small data u_{0}.

Lemma 2.3. — For every u_{0} ∈ H^{2}(Ω) ∩ H_{0}^{1}(Ω) satisfying

∥u_{0}∥_{L^{2}(Ω)} < ∥Q∥_{L^{2}(ℝ^{2})} (2.2)

the local solution of (2.1) with initial data u_{0} can be extended to the whole real line.

Remark 2.4. — 1. In the next section, we construct solutions of (2.1) with

∥u_{0}∥_{L^{2}(Ω)} = ∥Q∥_{L^{2}(ℝ^{2})}, developing singularities in finite time.

2. In [2], a stronger assumption on the L^{2} mass of the initial data is imposed. But using the sharp Gagliardo-Nirenberg inequality (see [19]), it is not difficult to see that the assumption (2.2) suffices to ensure that the energy controls the H^{1} norm.

3. Lemma 2.3 will be our starting point in the proof of Theorem 2.

4. Notice that Lemma 2.3 can be generalized to the case of H_{0}^{1} Cauchy data (see [16, 5]). However the uniform continuity of the flow on bounded sets of H^{1} is not clear in this case.

The result of Lemma 2.1 can be much improved if Ω is a square.

Lemma 2.5. — (compare with Theorem 2) Let Ω be a square and s ∈]0, \frac{1}{2}[. Then (2.1) is locally well-posed for data in H^{s}(Ω) with uniformly continuous flow map.

Remark 2.6. — 1. The flow map is actually analytic on H^{s}(Ω).

2. A similar result holds for s ≥ 1/2 with a slight modification of the space for the initial data coming from the boundary condition (see for instance Lemma 2.1).

Proof of Lemma 2.5. We can suppose that Ω = (0, 1)×(0, 1). Let 𝕋^{2} = ℝ^{2}|(2ℤ)^{2} be a flat torus. We regard 𝕋^{2} as (−1, 1) × (−1, 1) with the usual identification on the boundary lines. Any function u, defined on 𝕋^{2} and satisfying

\{ \begin{matrix}u(x, y) = −u(−x, y), \\ u(x, y) = −u(x, −y) \end{matrix} (2.3)

can be restricted to a function on Ω. Conversely, any function on Ω can be extended to a function on 𝕋^{2} by (2.3). The equation (1.1) is invariant under the transformations (x, y) → (−x, y) and (x, y) → (x, −y) and therefore if the initial data of (1.1), posed on 𝕋^{2}, satisfies the symmetry property (2.3) then so does the solution. Hence the flow of (2.1) on H^{s}(Ω) can be regarded as a restriction of the flow on H^{s}(𝕋^{2}), established by Bourgain (see [1], Theorem 2.7), to the functions of H^{s}(𝕋^{2}) satisfying (2.3).

3. Proof of Theorem 1

Let T > 0 and λ > 0. Recall that

R̃_{λ}(t, x) = \frac{1}{λ(T − t)}e^{\frac{i(4−λ^{2}(x−x_{0})^{2})}{4λ^{2}(T −t)}} Q (\frac{x − x_{0}}{λ(T − t)}) , (t, x) ∈ ℝ × ℝ^{2}

is a solution of the cubic focusing NLS, posed on ℝ^{2} which blows up at x_{0} in time t = T (see [5]). The function R̃_{λ}(t, x) is solution of (1.1) but has its support on the whole space ℝ^{2}. For that purpose, we set R_{λ}(t, x) := ψ(x)R̃_{λ}(t, x). The function R_{λ}(t, x) has its support in Ω but is not solution of (1.1) anymore. The proof of Theorem 1 consists of constructing a smooth correction r_{λ}(t, x) such that with a suitable choice of T the function R_{λ}(t, x) + r_{λ}(t, x) is again a solution of (1.1). Write

(i∂_{t} + Δ)R_{λ} = −ψ|R̃_{λ}|^{2}R̃_{λ} + 2∇ψ ∇R̃_{λ} + Δψ R̃_{λ}.

We look for v ∈ C([0, T [, H^{2}(Ω) ∩ H_{0}^{1}(Ω)) such that

\{ \begin{matrix}(i∂_{t} + Δ)v = −|R_{λ} + v|^{2}(R_{λ} + v) + ψ|R̃_{λ}|^{2}R̃_{λ} − 2∇ψ ∇R̃_{λ} − Δψ R̃_{λ}, \\ v(t) ⟶ 0 as t ⟶ T (t < T) in H^{2}(Ω) ∩ H_{0}^{1}(Ω). \end{matrix} (3.1)

Write

−|R_{λ} + v|^{2}(R_{λ} + v) + ψ|R̃_{λ}|^{2}R̃_{λ} − 2∇ψ ∇R̃_{λ} − Δψ R̃_{λ} = Q_{0} + Q_{1}(v) + Q_{2}(v) + Q_{3}(v),

where

\{ \begin{matrix}Q_{0} = ψ(1 − |ψ|^{2})|R̃_{λ}|^{2}R̃_{λ} − 2∇ψ ∇R̃_{λ} − Δψ R̃_{λ}, \\ Q_{1}(v) = −R_{λ}^{2}v̄ − 2|R_{λ}|^{2}v, \\ Q_{2}(v) = −R̄_{λ}v^{2} − 2R_{λ}|v|^{2} \\ Q_{3}(v) = −|v|^{2}v. \end{matrix}

Due to the properties of the ground state of the NLS on ℝ^{2} (see [5]), there exists δ > 0 such that

∥Q_{0}(t, ·)∥_{H^{2}(Ω)} ≤ Ce^{− \frac{δ}{λ(T −t)}} . (3.2)

We shall look for solutions of (3.1) in the space

X_{T} = {v ∈ C([0, T [, H^{2}(Ω) ∩ H_{0}^{1}(Ω)) : ∥v∥_{X_{T}} < +∞},

where

∥v∥_{X_{T}} = sup_{t∈[0,T [} \{e^{\frac{δ}{2λ(T −t)}} ∥v(t)∥_{L^{2}(Ω)} + e^{\frac{δ}{3λ(T −t)}} ∥v(t)∥_{H^{2}(Ω)}\} .

Theorem 1 is then a consequence of the following existence result for (3.1).

Lemma 3.1. — There exists λ_{0} > 0 such that for any λ > λ_{0} there exists T > 0 depending on λ such that (3.1) has a unique solution v ∈ X_{T}.

Proof of Lemma 3.1. Let S(t) be the unitary group which defines the free evolution of the Schrödinger equation on Ω with Dirichlet boundary conditions. Set

∫_{t}^{T} S(t − τ)Q_{0}(τ)dτ + ∑_{j=1}^{3} ∫_{t}^{T} S(t − τ)Q_{j}(v(τ))dτ := ∑_{j=0}^{3} I_{j}(t). (3.3)

The goal is to solve the integral formulation of (3.1) by the contraction mapping principle in X_{T}, i.e. we need to show that the nonlinear operator defined by (3.3) acting on v ∈ X_{T} is a contraction in X_{T} for sufficiently large λ and small T depending on λ.

We now estimate separately ∥ · ∥_{X_{T}} norms of I_{j}(t), j = 0, 1, 2, 3.

\underline{Estimate for I_{0}(t).} Due to (3.2), we have ∥I_{0}∥_{X_{T}} ≤ CT. (3.4) \underline{Estimate for I_{1}(t).} Recall that Q_{1}(v) = −R_{λ}^{2}v̄ − 2|R_{λ}|^{2}v. We first bound ∥I_{1}(t)∥_{L^{2}}. Write for t ∈ [0, T [

∥I_{1}(t)∥_{L^{2}} ≤ C ∫_{t}^{T} ∥R_{λ}(τ)∥_{L^{∞}}^{2} ∥v(τ∥_{L^{2}}dτ
≤ C \{∫_{t}^{T} \frac{1}{(λ(T − τ))^{2}} e^{− \frac{δ}{2λ(T −τ)}} dτ\} ∥v∥_{X_{T}}
≤ \frac{C}{λ} e^{− \frac{δ}{2λ(T −t)}} ∥v∥_{X_{T}}

We next bound ∥I_{1}(t)∥_{H^{2}}. Write for t ∈ [0, T [

∥I_{1}(t)∥_{H^{2}} ≤ C ∫_{t}^{T} ∥∇^{2}Q_{1}(v(τ))∥_{L^{2}}dτ + ∥I_{1}(t)∥_{L^{2}}
≤ C ∫_{t}^{T} ∥∇^{2}(R_{λ}^{2})(τ)∥_{L^{∞}}∥v(τ)∥_{L^{2}}dτ +

C ∫_{t}^{T} ∥∇(R_{λ}^{2})(τ)∥_{L^{∞}}∥∇v(τ)∥_{L^{2}}dτ +

C ∫_{t}^{T} ∥R_{λ}^{2}(τ)∥_{L^{∞}}∥∇^{2}v(τ)∥_{L^{2}}dτ + \frac{C}{λ} e^{− \frac{δ}{2λ(T −t)}} ∥v∥_{X_{T}} .

Using the explicit formula for R̃_{λ}(t, x), it is not difficult to derive the following bound for R_{λ}

∥∇^{k}((R_{λ}^{2})(τ))∥_{L^{∞}} ≤ \frac{C}{(λ(T − τ))^{2+k}} (1 + λ^{k}), k = 0, 1, 2, τ ∈ [0, T [. (3.5)

Now using the elementary interpolation inequality

∥∇v(τ)∥_{L^{2}} ≤ ∥∇^{2}v(τ)∥L^{2}^{\frac{1}{2}} ∥v(τ)∥L^{2}^{\frac{1}{2}}, v ∈ X_{T} ,

10 N. BURQ, P. GÉRARD & N. TZVETKOV

and (3.5), we obtain for λ ≥ 1 and T ≤ 1,

∥I_{1}(t)∥_{H^{2}} ≤ C∥v∥_{X_{T}} \{\frac{1}{λ}e^{− \frac{δ}{2λ(T −t)}} + ∫_{t}^{T} \frac{1 + λ^{2}}{(λ(T − τ))^{4}} e^{− \frac{δ}{2λ(T −τ)}} dτ +

∫_{t}^{T} \frac{1 + λ}{(λ(T − τ))^{3}} e^{− \frac{1}{2} ( \frac{δ}{2λ(T −τ)} + \frac{δ}{3λ(T −τ)} )}dτ +

∫_{t}^{T} \frac{1}{(λ(T − τ))^{2}} e^{− \frac{δ}{3λ(T −τ)}} dτ\}

≤ C(\frac{1}{λ} + λ^{2}T)e^{− \frac{δ}{3λ(T −t)}} ∥v∥_{X_{T}}

Therefore

∥I_{1}∥_{X_{T}} ≤ C\{\frac{1}{λ} + T^{\frac{1}{2}} \}∥v∥_{X_{T}} , (3.6)

provided λ^{2}T^{\frac{1}{2}} ≤ 1, λ ≥ 1. \underline{Estimate for I_{2}(t).} We shall directly estimate ∥I_{2}(t)∥_{H^{2}}. Recall that

Q_{2}(v) = −R̄_{λ}v^{2} − 2R_{λ}|v|^{2}.

Using again the explicit representation of R̃_{λ}, we get

∥R_{λ}(τ)∥_{H^{2}} ≤ \frac{C(1 + λ^{2})}{(λ(T − τ))^{2}} . (3.7)

Since H^{2}(Ω) is an algebra, we easily obtain via (3.7),

∥I_{2}(t)∥_{H^{2}} ≤ C ∫_{t}^{T} ∥R_{λ}(τ)∥_{H^{2}}∥v(τ)∥_{H^{2}}^{2}dτ
≤ C \{∫_{t}^{T} \frac{1 + λ^{2}}{(λ(T − τ))^{2}} e^{− \frac{2δ}{3λ(T −τ)}} dτ\} ∥v∥_{X_{T}}^{2}
≤ CT (1 + λ^{2})e^{− \frac{δ}{2λ(T −t)}} ∥v∥_{X_{T}}^{2}

Hence ∥I_{2}∥_{X_{T}} ≤ CT^{\frac{1}{2}} ∥v∥_{X_{T}}^{2} , (3.8) provided λ^{2}T^{\frac{1}{2}} ≤ 1, λ ≥ 1. \underline{Estimate for I_{3}(t).} Recall that Q_{3}(v) = −|v|^{2}v. We then have that

∥I_{3}(t)∥_{H^{2}} ≤ C ∫_{t}^{T} ∥v(τ)∥_{H^{2}}^{3}dτ
≤ \{∫_{t}^{T} e^{−3 \frac{δ}{3λ(T −τ)}} dτ\} ∥v∥_{X_{T}}^{3}
≤ CT e^{− \frac{δ}{3λ(T −t)}} ∥v∥_{X_{T}}^{3}

Therefore

∥I_{3}∥_{X_{T}} ≤ CT ∥v∥_{X_{T}}^{3} . (3.9)

In view of (3.4), (3.6), (3.8), (3.9), it is a standard procedure to solve the integral formulation of (3.1) by the contraction mapping principle in X_{T} for λ ≫ 1, T ≪ 1 and λ^{2}T^{\frac{1}{2}} ≤ 1. This completes the proof of Lemma 3.1.

4. Some properties of the Bessel functions

In this section, we state some results concerning the Bessel functions of integer order. All these results are well known in the literature (see [18]) but for the sake of completeness we shall give the needed proofs in an appendix of the present paper. Let n be a positive integer. The Bessel function of order n is defined as follows

J_{n}(z) = (\frac{z}{2})^{n} ∑_{k=0}^{∞} \frac{(−1)^{k}(\frac{z}{2})^{2k}}{k!(n + k)!}

(see [15],[18]). The function J_{n}(z) is analytic in ℂ and satisfies the following second order ordinary differential equation

Jn^{′′}(z) + \frac{1}{z}J′_{n}(z) + (1 − \frac{n^{2}}{z^{2}} )J_{n}(z) = 0. (4.1)

Now we give estimates for J_{n}(z) in different regimes for z.

Lemma 4.1. — Let β_{1} and β_{2} be two positive real numbers and γ ∈]0, \frac{1}{2}[. Then for

ρ ∈ [1 − β_{1}n^{−γ}, 1 + β_{2}n^{− \frac{2}{3}} ,], n ≫ 1

one has

J_{n}(nρ) = (\frac{2}{nρ})^{\frac{1}{3}} Ai(n^{\frac{2}{3}} (1 − ρ)(2/ρ)^{\frac{1}{3}} ) + r_{n}(ρ),

where Ai is the classical Airy function (see [9], page 213, Definition 7.6.8) and the following estimate holds uniformly in ρ,

|r_{n}(ρ)| ≤ Cn^{−γ_{1}}, γ_{1} < 3γ − 1.

Lemma 4.2. — Let β > 0 and γ ∈]0, \frac{1}{2}[. Then for any k > 0 there exists a positive constant c_{k} such that for ρ ∈ [0, 1 − βn^{−γ}], n ≫ 1, one has |J_{n}(nρ)| ≤ c_{k}n^{−k}.

Let 0 < z_{n1} < z_{n2} < z_{n3} < ... be the sequence of the positive zeros of J_{n}(z). It is not difficult to see from equation (4.1) that n < z_{n1} and that the zeros are simple. In the next lemma, we give more precise information about the the localization of z_{n1} and the gap between z_{n1} and z_{n2}.

Lemma 4.3. — Let n ≥ 1. Then there exists a constant α, independent of n, such that z_{n1} = n + αn^{\frac{1}{3}} + O(n^{λ}), for any λ > \frac{1}{6}. Moreover |z_{n2} − z_{n1}| ≥ Cn^{\frac{1}{3}} .

5. Proof of Theorem 2

Let Δ_{D} be the Laplace operator on D with Dirichlet boundary conditions. It is classical that Δ_{D} is a self-adjoint operator acting on L^{2}(D) with domain H^{2}(D) ∩ H_{0}^{1}(D). By introducing polar coordinates x_{1} = r cos θ, x_{2} = r sin θ and using that

∂_{x_{1}}^{2} + ∂_{x_{2}}^{2} = \frac{∂^{2}}{∂r^{2}} + \frac{1}{r} \frac{∂}{∂r} + \frac{1}{r^{2}} \frac{∂^{2}}{∂θ^{2}} , (5.1)

it can be seen that f_{nk}(r, θ) := J_{n}(z_{nk}r) exp(inθ) is an an orthogonal basis of L^{2}(D) of eigenfunctions of −Δ_{D} with corresponding eigenvalue z_{nk}^{2}. For s ∈ [0, \frac{1}{2}[, we can express the norm in H^{s}(D) in terms of f_{nk} and z_{nk}. Let u ∈ L^{2}(D). Then there exists c_{nk} ∈ ℂ such that u = ∑_{n,k} c_{nk}f_{nk} and u ∈ H^{s}(Ω) if and only if

∥u∥_{H^{s}(D)} ≈ \{∑_{n,k} z_{nk}^{2s}|c_{nk}|^{2}\}^{\frac{1}{2}} < ∞. (5.2)

Further, if s ≥ \frac{1}{2}, the space defined by (5.2) will be denoted by H_{0}^{s}(D). By complex interpolation, elements of H_{0}^{1/2}(D) can be extended by 0 to the whole of ℝ^{2} as elemnts of H^{1/2}(ℝ^{2}), and therefore, by the Sobolev embedding, H0^{\frac{1}{2}} (D) ⊂ L^{4}(D). Moreover, given a positive integer n, for u ∈ L^{2} one has

(5.3)  ∀x ∈ D, θ ∈ S^{1}, u(e^{iθx}) = e^{inθ}u(x) ⇔ u = ∑_{k≥1} c_{k}f_{nk}

Next, using the results of the previous section, we give the asymptotics for the L^{p} norms of f_{n1}.

Lemma 5.1. — Let p ∈ [2, +∞]. Then ∥f_{n1}∥_{L^{p}(D)} ≈ n^{− \frac{2}{3p} − \frac{1}{3}} .

Remark 5.2. — As a consequence of Lemma 5.1, notice that usual Strichartz estimates on the disc fail. More precisely, by testing on eigenfunctions f_{n1}, one can show that for s < \frac{1}{6}

∥e^{itΔ_{D}}u_{0}∥_{L^{4}([0,1]×D)} ≤ C∥u_{0}∥_{L^{2}}

cannot hold.

Proof. Write

∥f_{n1}∥_{L^{p}(D)}^{p} = ∫_{0}^{1} ∫_{0}^{2π} |f_{n1}(r, θ)|^{p}rdθdr
= 2π ∫_{0}^{1} |J_{n}(z_{n1}r)|^{p}rdr.

Next, we perform a change of variables ρ = \frac{z_{n1}}{n} r. Then

∥f_{n1}∥_{L^{p}(D)}^{p} = 2π (\frac{n}{z_{n1}} )^{2} ∫0^{\frac{z_{n1}}{n}} |J_{n}(nρ)|^{p}ρdρ.

In view of Lemma 4.3, we can write

\frac{z_{n1}}{n} = 1 + αn^{− \frac{2}{3}} + O(n^{−λ}),

for any λ < \frac{5}{6} and for some α > 0. Thus, we can write

∥f_{n1}∥_{L^{p}(D)}^{p} ≈ ∫_{0}^{1−n^{−γ}} |J_{n}(nρ)|^{p}ρdρ + ∫_{1−n^{−γ}}^{1+αn^{− \frac{2}{3}} +O(n^{−λ})} |J_{n}(nρ)|^{p}ρdρ,

where γ < \frac{1}{2}. A use of Lemma 4.2 yields that for any k > 0 there exists c_{k} such that

∫_{0}^{1−n^{−γ}} |J_{n}(nρ)|^{p}ρdρ ≤ c_{k}n^{−k}. (5.4)

Then, with the notation of Lemma 4.1, we have

∫_{1−n^{−γ}}^{1+αn^{− \frac{2}{3}} +O(n^{−λ})} |r_{n}(ρ)|^{p}ρdρ ≤ Cn^{−γ}n^{−pγ_{1}},

where γ_{1} < 3γ − 1. Clearly n^{−γ}n^{−pγ_{1}} ≪ n^{− \frac{2}{3}} n^{− \frac{p}{3}} by taking γ < \frac{1}{2} sufficiently close to \frac{1}{2} and γ_{1} sufficiently close to 3γ − 1 ∼ \frac{1}{2}. On the other hand a change of variables ρ − 1 = −n^{− \frac{2}{3}} y leads to

∫_{1−n^{−γ}}^{1+αn^{− \frac{2}{3}} +O(n^{−λ})} (\frac{2}{nρ})^{\frac{p}{3}} |Ai(n^{\frac{2}{3}} (1 − ρ)(2/ρ)^{\frac{1}{3}} )|^{p} ρdρ = c(n)n^{− \frac{2}{3}} n^{− \frac{p}{3}} ,

where

c(n) = 2^{p/3} ∫_{−n^{\frac{2}{3} −γ}}^{α+O(n^{\frac{2}{3} −λ})} \frac{1}{(1 + n^{− \frac{2}{3}} y)^{\frac{p}{3}}} |Ai ((\frac{2}{(1 + n^{− \frac{2}{3}} y)^{1/3}} )^{\frac{1}{3}} y)|^{p} (1 + n^{− \frac{2}{3}} y)dy.

The proof of Lemma 5.1 is completed by observing that

c(n) → 2^{p/3} ∫_{−∞}^{α} |Ai(2^{\frac{1}{3}} y)|^{p}dy.

using the Lebesgue Theorem and the well known bounds on the Airy function (see e.g. [9, § 7.6, p 215]).

We now turn to the proof of Theorem 2. We can suppose that κ > 0 since the case κ < 0 can be treated in the same fashion. Set ϕ_{n} = n^{\frac{2}{3} −s}f_{n1}. Due to Lemma 5.1 and Lemma 4.3, we have that ∥ϕ_{n}∥_{H^{s}} ≈ 1. Set ω_{n} = ∥ϕ_{n}∥_{L^{4}}^{4}/∥ϕ_{n}∥_{L^{2}}^{2}. Another use of Lemma 5.1 gives ω_{n} ≈ n^{\frac{2}{3} −2s}. Since ∥ϕ_{n}∥_{L^{2}} ≈ n^{−s}, Lemma 2.3 ensures that if n ≫ 1 then there exists a unique global solution of (1.1) with Cauchy data κϕ_{n}(x), x ∈ D. Denote it by u_{n}(t, x). Due to the L^{2} conservation law, we obtain that ∥u_{n}(t, ·)∥_{L^{2}} ≤ Cκn^{−s}. Then the energy conservation yields the bound ∥u_{n}(t, ·)∥_{H_{0}^{1}} ≤ Cκn^{1−s}. Therefore, by interpolation, ∥u_{n}(t, ·)∥_{H^{s}(D)} ≤ Cκ.

According to 5.3, we can clearly write |ϕ_{n}|^{2}ϕ_{n} = ω_{n}ϕ_{n} +r_{n}, where r_{n} = ∑_{k≥2} c_{k}f_{nk}. Set c_{n}(t) = exp(−it(z_{n1}^{2} − κ^{2}ω_{n})) and u_{n}(t, x) = κc_{n}(t)(ϕ_{n}(x) + w_{n}(t, x)). Then the equation satisfied by w_{n} is

(5.5) \{ \begin{matrix}(i∂_{t} + Δ + z_{n1}^{2} − κ^{2}ω_{n})w_{n} & = −κ^{2}(|ϕ_{n} + w_{n}|^{2}(ϕ_{n} + w_{n}) − |ϕ_{n}|^{2}ϕ_{n} + r_{n}) \\ w_{n}(0, x) & = 0, x ∈ D. \end{matrix}

According to gauge invariance of NLS and (5.3) we now further decompose w_{n}(t, x) as w_{n}(t, x) = λ_{n}(t)ϕ_{n}(x) + q_{n}(t, x), where

q_{n}(t, x) = ∑_{k≥2} c_{k}(t)f_{nk}(x), t ∈ ℝ, x ∈ D.

Since w_{n}(0, x) = 0, we have λ_{n}(0) = 0 and q_{n}(0, x) = 0. The L^{2} conservation law now can be rewritten as follows, |1 + λ_{n}(t)|^{2}∥ϕ_{n}∥_{L^{2}}^{2} + ∥q_{n}(t)∥_{L^{2}}^{2} = ∥ϕ_{n}∥_{L^{2}}^{2}, (5.6) while the energy conservation becomes

|1 + λ_{n}(t)|^{2}∥∇ϕ_{n}∥_{L^{2}}^{2} + ∥∇q_{n}(t)∥_{L^{2}}^{2} − \frac{1}{2κ^{2}} ∥u_{n}(t)∥_{L^{4}}^{4} = ∥∇ϕ_{n}∥_{L^{2}}^{2} − \frac{κ^{2}}{2} ∥ϕ_{n}∥_{L^{4}}^{4}.

(5.7) Since ϕ_{n} is an eigenfunction of Δ_{D}, we obtain that ∥∇ϕ_{n}∥_{L^{2}}^{2} = z_{n1}^{2}∥ϕ_{n}∥_{L^{2}}^{2}. We now multiply (5.6) with −z_{n1}^{2} and add it to (5.7). This yields

∥∇q_{n}(t)∥_{L^{2}}^{2} − z_{n1}^{2}∥q_{n}(t)∥_{L^{2}}^{2} = \frac{1}{2κ^{2}} ∥u_{n}(t)∥_{L^{4}}^{4} − \frac{κ^{2}}{2} ∥ϕ_{n}∥_{L^{4}}^{4}
≤ \frac{1}{2κ^{2}} ∥u_{n}(t)∥_{L^{4}}^{4}.

In view of (5.6), we readily obtain that |λ_{n}(t)| ≤ 2. Therefore we have the pointwise estimate |u_{n}(t, x)|^{4} ≤ Cκ^{4}(|ϕ_{n}(x)|^{4} + |q_{n}(t, x)|^{4}), t ∈ ℝ, x ∈ D. (5.8) Due to Lemma 4.3, For k ≥ 2 one has z_{nk}^{2} − z_{n1}^{2} ≥ Cn^{\frac{1}{3}} z_{nk}. (5.9) Moreover, ∥ϕ_{n}∥_{L^{4}}^{4} = ω_{n}∥ϕ_{n}∥_{L^{2}}^{2} ≈ n^{\frac{2}{3} −4s}. (5.10) Now using (5.8), (5.9), (5.10), the definition of Sobolev spaces in terms of the spectral decomposition with respect to Δ_{D} and the Sobolev embedding H0^{\frac{1}{2}} (D) ⊂ L^{4}(D) give the estimates

n^{\frac{1}{3}} ∥q_{n}(t, ·)∥_{H \frac{1}{2} 0 (D)}^{2} ≤ Cκ^{2}n^{\frac{2}{3} −4s} + Cκ^{2}∥q_{n}(t, ·)∥_{L^{4}(D)}^{4}
≤ Cκ^{2}n^{\frac{2}{3} −4s} + Cκ^{2}∥q_{n}(t, ·)∥_{H \frac{1}{2} 0 (D)}^{4}.

Since q_{n}(0, ·) = 0, by a classical bootstrap argument we infer ∥q_{n}(t, ·)∥_{H \frac{1}{2} 0 (D)} ≤ Cκn^{\frac{1}{6} −2s}. (5.11)

Using once again the Sobolev embedding H0^{\frac{1}{2}} (D) ⊂ L^{4}(D), we obtain that

∥q_{n}(t, ·)∥_{L^{4}(D)}^{4} ≤ Cκ^{4}n^{\frac{2}{3} −8s}.

Coming back to (5.11) we also get the bound

∥q_{n}(t, ·)∥_{L^{2}(D)} ≤ Cκn^{− \frac{1}{3} −2s}. (5.12)

An interpolation between (5.11) and (5.12) yields the estimate

∥q_{n}(t, ·)∥_{H^{s}(D)} ≤ Cκn^{− \frac{1}{3} −s}. (5.13)

In the next lemma, we prove the needed bound for |λ_{n}(t)|.

Lemma 5.3. — There exists a constant C > 0, such that for any T > 0, and any n ≥ N(T),

sup_{t∈[0,T]} |λ_{n}(t)| ≤ CT n^{\frac{1}{3} −3s}.

Proof of Lemma 5.3. We project the equation (5.5) on ϕ_{n}(x). Using that u_{n}(t, x) = κc_{n}(t)(ϕ_{n}(x) + w_{n}(t, x)), we obtain the following equation for λ_{n}(t)

\{ \begin{matrix}(i∂_{t} − ω_{n}κ^{2})λ_{n} = −\frac{κ^{2}}{∥ϕ_{n}∥_{L^{2}}^{2}} ((|ϕ_{n} + w_{n}|^{2}(ϕ_{n} + w_{n}) | ϕ_{n}) − (|ϕ_{n}|^{2}ϕ_{n} | ϕ_{n})) \\ λ_{n}(0) = 0. \end{matrix}

An easy computation shows that we can rewrite the right-hand side of the equation for λ_{n} as follows

−\frac{κ^{2}}{∥ϕ_{n}∥_{L^{2}}^{2}} \{∫ (2|ϕ_{n}|^{2}w_{n} + ϕ_{n}^{2}w̄_{n})ϕ̄_{n} + ∫ (2Re(ϕ̄_{n}w_{n})w_{n}ϕ̄_{n} + |w_{n}|^{2}|ϕ_{n}|^{2} + |w_{n}|^{2}w_{n}ϕ̄_{n})\}.

Using that w_{n}(t, x) = λ_{n}(t)ϕ_{n}(x) + q_{n}(t, x) and |ϕ_{n}(x)|^{2}ϕ_{n}(x) = ω_{n}ϕ_{n}(x) + r_{n}(t, x), we can split the above expression into the sum of the following three terms:

\begin{aligned}L_{1} & = −(2ω_{n}κ^{2}λ_{n} + ω_{n}κ^{2}λ̄_{n}), \\ L_{2} & = \frac{κ^{2}}{∥ϕ_{n}∥_{L^{2}}^{2}} O(|λ_{n}|^{2} ∫ |ϕ_{n}|^{4} + |λ_{n}|^{3} ∫ |ϕ_{n}|^{4}), \\ L_{3} & = \frac{κ^{2}}{∥ϕ_{n}∥_{L^{2}}^{2}} O(∫ |q_{n}|^{3}|ϕ_{n}| + ∫ |q_{n}|^{2}|ϕ_{n}|^{2} + |(q_{n}|r_{n})|). \end{aligned}

NLS ON A PLANE DOMAIN 17

We now study the decay of the source term, L_{3}. Write

\frac{∫ |q_{n}|^{3}|ϕ_{n}|}{∥ϕ_{n}∥_{L^{2}}^{2}} ≤ Cn^{2s}∥|q_{n}|^{3}∥_{L^{\frac{4}{3}}} ∥ϕ_{n}∥_{L^{4}}
≤ Cn^{2s}∥q_{n}∥_{L^{4}}^{3}∥ϕ_{n}∥_{L^{4}}
≤ Cn^{2s} n^{\frac{1}{2} −6s} n^{\frac{1}{6} −s}
= Cn^{\frac{2}{3} −5s}.

Further we have

\frac{∫ |q_{n}|^{2}|ϕ_{n}|^{2}}{∥ϕ_{n}∥_{L^{2}}^{2}} ≤ Cn^{2s}∥q_{n}∥_{L^{2}}^{2}∥ϕ_{n}∥_{L^{∞}}^{2}
≤ Cn^{2s} n^{− \frac{2}{3} −4s} n^{\frac{2}{3} −2s}
= Cn^{−4s}

and moreover

\frac{|(q_{n}|r_{n})|}{∥ϕ_{n}∥_{L^{2}}^{2}} ≤ Cn^{2s}∥q_{n}∥_{L^{2}}∥r_{n}∥_{L^{2}}
≤ Cn^{2s} n^{− \frac{1}{3} −2s} ∥ϕ_{n}∥_{L^{6}}^{3}
≤ Cn^{2s} n^{− \frac{1}{3} −2s} n^{\frac{2}{3} −3s}
= Cn^{\frac{1}{3} −3s}.

For s > \frac{1}{6}, the source term is bounded by n^{\frac{1}{3} −3s} and consequently the equation for λ_{n}(t) can be written as

\{ \begin{matrix}i∂_{t}λ_{n} = −2ω_{n}κ^{2}Re(λ_{n}) + O(ω_{n}|λ_{n}|^{2} + ω_{n}|λ_{n}|^{3} + n^{\frac{1}{3} −3s}), \\ λ_{n}(0) = 0. \end{matrix} (5.14)

The L^{2} conservation law yields

1 − |1 + λ_{n}(t)|^{2} = \frac{∥q_{n}(t)∥_{L^{2}}^{2}}{∥ϕ_{n}∥_{L^{2}}^{2}} = O(n^{− \frac{2}{3} −2s})

and hence (5.14) now becomes

\{ \begin{matrix}i∂_{t}λ_{n} = O(ω_{n}|λ_{n}|^{2} + ω_{n}|λ_{n}|^{3} + n^{\frac{1}{3} −3s}), \\ λ_{n}(0) = 0. \end{matrix} (5.15)

Set γ_{n} := n^{\frac{1}{3} −3s} and M_{n}(T) := sup_{t∈[0,T]} γ_{n}^{−1}|λ_{n}(t)|. Then by integrating (5.15), we obtain the estimate

M_{n}(T) ≤ CT (1 + γ_{n}ω_{n}[M_{n}(T)]^{2} + γ_{n}^{2}ω_{n}[M_{n}(T)]^{3})
≤ CT (1 + n^{1−5s}([M_{n}(T)]^{2} + n^{\frac{1}{3} −3s}[M_{n}(T)]^{3})).

Since s > \frac{1}{5} and M(0) = 0, we obtain that M(T) ≤ CT for n large enough with respect to T. Therefore

|λ_{n}(t)| ≤ CT n^{\frac{1}{3} −3s}.

This completes the proof of Lemma 5.3.

It remains to prove that the flow map fails to be uniformly continuous on any ball of H^{s}. For that purpose, we fix κ > 0 and a sequence {κ_{n}} tending to κ. Denote by u_{κ,n} the solution of (1.1) with Cauchy data κϕ_{n} and by u_{κ_{n},n} the solution of (1.1) with data κ_{n}ϕ_{n}. Clearly (κ − κ_{n})ϕ_{n} tends to zero in H^{s}(D) as n tends to infinity. Hence, if we suppose that the Cauchy problem associated to (1.1) (with Ω = D) is locally well-posed with uniformly continuous flow map then the sequence

∥u_{κ,n}(t, ·) − u_{κ_{n},n}(t, ·)∥_{H^{s}(D)}

should tend to zero as n tends to infinity. But due to the above considerations one easily gets the estimate

∥u_{κ,n}(t, ·) − u_{κ_{n},n}(t, ·)∥_{H^{s}(D)} ≥ C| exp(itω_{n}(κ^{2} − κ_{n}^{2})) − 1| − Cn^{−δ},

where δ > 0. Therefore one should have

lim_{n→+∞} | exp(itω_{n}(κ^{2} − κ_{n}^{2})) − 1| = 0. (5.16)

Since ω_{n} ≈ n^{\frac{2}{3} −2s} → ∞, we easily obtain that with a suitable choice of \{κ_{n}\}, (5.16) fails for every t ≠ 0. This completes the proof of Theorem 2.

Remark 5.4. — Similarly to [4], if one is interested only on the ill-posedness result, an adaptation of the proof of Theorem 2, involving the length of the time interval on which our ansatz is valid, can provide ill-posedness for s in the whole interval ]0, \frac{1}{3}[.

6. Appendix

In this appendix, we give the proofs of Lemmas 4.1, 4.2, 4.3. Proof of Lemma 4.1. For a fixed z ≥ 0, the function J_{n}(z) can be seen as the n-th coefficient in the Fourier series expansion of the periodic smooth function exp(iz sin θ). Therefore

J_{n}(nρ) = \frac{1}{2π} ∫_{S^{1}} e^{inϕ_{ρ}(θ)}dθ, (6.1)

where ϕ_{ρ}(θ) := ρ sin θ − θ. Take a real number λ < \frac{γ}{2} . Let χ ∈ C_{0}^{∞}(ℝ) be real valued, function such that χ(x) = 1 for |x| < 1 and χ(x) = 0 for |x| > 2. Then for n ≫ 1, we define χ_{n} ∈ C^{∞}(S^{1}) as χ_{n}(θ) = χ(n^{λ}θ). Set J_{n}(nρ) = J_{n}^{1}(nρ) + J_{n}^{2}(nρ), where J_{n}^{1}(nρ) = (2π)^{−1} ∫_{−π}^{π} χ_{n}(θ) exp(inϕ_{ρ}(θ))dθ. Since λ < γ/2, we can write, for θ ∈ supp(1 − χ_{n}) and |1 − ρ| ≤ βn^{−γ},

|ϕ′_{ρ}(θ)| = |ρ(cos θ − 1) + ρ − 1|
≥ 2ρ sin^{2} \frac{θ}{2} − |1 − ρ|
≥ Cn^{−2λ}.

Note that |χ_{n}^{(j)}(θ)| ≤ Cn^{λj}, j ∈ ℕ. Hence the non-stationary phase estimate (see [9], page 216, Theorem 7.7.1) applied to J_{n}^{2}(nρ) gives |J_{n}^{2}(nρ)| ≤ c_{k}n^{(4λ−1)k}. for any k. Hence for λ < \frac{1}{4}, we obtain that |J_{n}^{2}(nρ)| is rapidly decreasing for n ≫ 1 uniformly on ρ. It remains to estimate J_{n}^{1}(nρ). Since for |θ| ≤ Cn^{−λ}, we have sin θ = θ − θ^{3}/6 + r_{n}(θ) with |r_{n}(θ)| ≤ Cn^{−5λ}, we obtain that

J_{n}^{1}(nρ) = \frac{1}{2π} ∫_{−π}^{π} χ_{n}(θ)e^{in((ρ−1)θ− \frac{ρθ^{3}}{6} )}dθ + r̃_{n}, (6.2)

where |r̃_{n}| ≤ Cn|r_{n}(θ)|n^{−λ} ≤ Cn^{1−6λ}. Therefore |r̃_{n}| ≤ Cn^{−γ_{1}} for any γ_{1} < 3γ − 1 by choosing λ < \frac{1}{4} sufficiently close to γ/2. Hence one only needs to evaluate the first term in the right-hand side of (6.2). Writing χ_{n}(θ) = (χ_{n}(θ) − 1) + 1, and observing that the expression (ρ − 1)θ − θ^{3}/6, has no critical points θ such that |θ| ≥ n^{−λ}, another use of the non-stationary phase estimate gives for any k ∈ ℕ,

\frac{1}{2π} ∫_{−π}^{π} χ_{n}(θ)e^{in((ρ−1)θ− \frac{ρθ^{3}}{6} )}dθ = \frac{1}{2π} ∫_{−∞}^{∞} e^{in((ρ−1)θ−ρθ^{3}/6)}dθ + O(n^{−k}).

Now a simple change of variables provides the formula

\frac{1}{2π} ∫_{−∞}^{∞} e^{in((ρ−1)θ−ρθ^{3}/6)}dθ = (\frac{2}{nρ})^{\frac{1}{3}} Ai(n^{\frac{2}{3}} (1 − ρ)(2/ρ)^{\frac{1}{3}} ), (6.3)

which completes the proof of Lemma 4.1. Proof of Lemma 4.2. Write |ϕ′_{ρ}(θ)| = |1−ρ+ρ(1−cos θ)|. Since 1−ρ ≥ βn^{−γ}, we have that |ϕ′_{ρ}(θ)| ≥ βn^{−γ}. Hence an application of the non-stationary phase estimate to J_{n}(nρ) yields |J_{n}(nρ)| ≤ c_{k}n^{−(1−2γ)k} which completes the proof of Lemma 4.2. Proof of Lemma 4.3. We first observe that due to Lemma 4.1, there is no zero of J_{n}(nρ) for 0 ≤ ρ − 1 ≪ n^{− \frac{2}{3}} . Denote by ρ̃_{n1} < ρ̃_{n2} < ⋯ the zeros of

Ai(n^{\frac{2}{3}} (1 − ρ)(2/ρ)^{\frac{1}{3}} ).

Denote by z_{k}, k = 1, 2 the first zeros of Ai(−z). Then a straightforward computation shows that for k = 1, 2 one has

ρ̃_{nk} = 1 + \frac{z_{k}}{2^{\frac{1}{3}}} n^{− \frac{2}{3}} + r_{nk}, (6.4)

where |r_{nk}| ≤ Cn^{− \frac{4}{3}} . Let now ρ_{n1} < ρ_{n2} < ⋯ be the zeros of J_{n}(nρ), bigger than 1. Fix δ ∈]0, \frac{1}{6}[. Let ξ_{nk}^{±} solve the equations

n^{\frac{2}{3}} (1 − ξ_{nk}^{±})(2/ξ_{nk}^{±})^{\frac{1}{3}} = −z_{k} ∓ n^{−δ}.

Note that the restriction on δ is imposed in order to keep the first term in the expansion of J_{n}(nρ) from Lemma 4.1 leading near ξ_{nk}^{±}. For k = 1, 2 one has

ξ_{nk}^{±} = 1 + \frac{z_{k}}{2^{\frac{1}{3}}} n^{− \frac{2}{3}} + r̃_{nk}

where |r̃_{nk}| ≤ Cn^{−δ− \frac{2}{3}} . For n ≫ 1, the numbers J_{n}(nξ_{nk}^{+}) and J_{n}(nξ_{nk}^{−}) have different signs and hence there is at least one zero of J_{n}(nρ) between ξ_{nk}^{−} and ξ_{nk}^{+}. Using the method of proof of Lemma 4.1, we obtain the following representation for J′_{n}(nρ) showing that J′_{n}(nρ) has a constant sign on [ξ_{nk}^{−}, ξ_{nk}^{+}].

J′_{n}(nρ) = (\frac{2}{nρ})^{\frac{2}{3}} Ai′ (n^{\frac{2}{3}} (1 − ρ)(2/ρ)^{\frac{2}{3}} ) + O(n^{−γ}), γ < \frac{3}{4}

Hence, we conclude that the only zero of J_{n}(nρ) between ξ_{nk}^{−} and ξ_{nk}^{+} is ρ_{nk} and therefore

|ρ_{nk} − ρ̃_{nk}| ≤ Cn^{−δ− \frac{2}{3}} , k = 1, 2. (6.5)

Since z_{nk} = nρ_{nk}, the proof of Lemma 4.3 is achieved thanks to (6.4) and (6.5) by choosing δ close to \frac{1}{6}.

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N. Burq, Université Paris Sud, Mathématiques, Bât 425, 91405 Orsay Cedex

    E-mail : Nicolas.burq@math.u-psud.fr • Url : http://www.math.u-psud.fr/~burq

P. Gérard, Université Paris Sud, Mathématiques, Bât 425, 91405 Orsay Cedex

    E-mail : Patrick.gerard@math.u-psud.fr

N. Tzvetkov, Université Paris Sud, Mathématiques, Bât 425, 91405 Orsay Cedex

    E-mail : Nikolay.tzvetkov@math.u-psud.fr