AN INSTABILITY PROPERTY OF THE NONLINEAR SCHRÖDINGER EQUATION ON S^{d}

by

N. Burq, P. Gérard & N. Tzvetkov

Abstract. — We consider the NLS on spheres. We describe the nonlinear evolutions of the highest weight spherical harmonics. As a consequence, in contrast with the flat torus, we obtain that the flow map fails to be uniformly continuous for Sobolev regularity above the threshold suggested by a simple scaling argument.

Résumé. — On étudie l’équation de Schrödinger non linéaire sur les sphères. On décrit les évolutions non linéaires des harmoniques sphériques de poids principal. On en déduit que contrairement au cas du tore plat, le flot n’est pas uniformément continu pour des régularités Sobolev au dessus de l’indice critique donné par les arguments de changements d’échelles habituels.

1. Introduction

In [4], we obtained a set of Strichartz inequalities as well as local and global wellposedness results for the nonlinear Schrödinger equation (NLS) posed on a compact riemannian manifold (M, g)

i∂_{t}u + Δ_{g}u = f(|u|^{2})u, u(0, x) = u_{0}(x), (1.1)

where f is a suitably chosen real valued function. A natural question is whether the particular structure of (M, g) influences the critical threshold for the local wellposedness Sobolev regularity of the initial data. Our goal here is to give an affirmative answer to that question. Let us precise what we call local well-posedness in this paper.

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

Definition 1.1. — (see e.g. [9]) We say that the Cauchy problem (1.1) is locally well-posed for data in the Sobolev space H^{s}(M) if for any R > 0 there exist T > 0 and a functional space X_{T} continuously embedded in C([−T, T], H^{s}(M)) and invariant under the natural action of the isometries of M, such that for every

u_{0} ∈ B_{R} := {u_{0} ∈ H^{s}(M) : ∥u_{0}∥_{H^{s}(M)} < R}

the Cauchy problem (1.1) has a unique solution u ∈ X_{T}. Moreover 1. The map u_{0} ↦ u is uniformly continuous from B_{R} to C([−T, T], H^{s}(M)).

2. If u_{0} ∈ H^{1}(M), u ∈ C([−T, T], H^{1}(M)) and satisfies the usual conservation

laws

∥u(t)∥_{L^{2}(M)} = ∥u_{0}∥_{L^{2}(M)},
∥∇_{g}u(t)∥_{L^{2}(M)} + ∫_{M} F (|u(t, x)|^{2})dx = const,

where F is a primitive of f.

The assumption of uniform continuity of the flow map in the above definition seems to be natural for semilinear equations. Moreover it turns out to be the case in the local well-posedness results of [4]. However, as it was pointed to us by H. Koch, such an assumption is unrealistic in the context of quasilinear problems.

The main issue in our analysis is to study the nonlinear evolution by the NLS flow of some eigenfunctions of the Laplace-Beltrami operator on the d dimensional sphere S^{d} with the canonical metric. The situation turns out to be particularly simple in 1D. Consider the Cauchy problem

i∂_{t}u + ∂_{x}^{2}u = |u|^{2}u, u(0, x) = u_{0}(x), (1.2)

where x ∈ S^{1}, t ∈ ℝ. For s < 0, we set

u_{κ,n}(t, x) = κn^{−s} exp(−it(n^{2} + κ^{2}n^{−2s})) exp(inx).

It is easy to check that u_{κ,n} solves (1.2) with initial data κn^{−s} exp(inx). Moreover

∥u_{κ,n}(t, ·)∥_{H^{s}(S^{1})} ≤ |κ|.

Fix κ ∈]0, 1[. Let {κ_{n}} be a sequence of real numbers tending to κ which will be specified later. We observe that u_{κ,n}(0, ·) − u_{κ_{n},n}(0, ·) tends to zero in H^{s}(S^{1}) as n tends to infinity. Take now a positive t. Then there exist C > 0, independent of n and δ > 0 such that

∥u_{κ,n}(t, ·) − u_{κ_{n},n}(t, ·)∥_{H^{s}(S^{1})} ≥ C| exp(−itn^{−2s}(κ^{2} − κ_{n}^{2})) − 1| − Cn^{−δ}. (1.3)

If we suppose that (1.2) is locally well-posed in H^{s}(S^{1}), s < 0 then (1.3) would imply

lim_{n→+∞} | exp(−itn^{−2s}(κ^{2} − κ_{n}^{2})) − 1| = 0 (1.4)

But (1.4) easily fails by choosing {κ_{n}} so that

(κ^{2} − κ_{n}^{2})n^{−2s} = αn^{β}

for suitable α > 0, β > 0 satisfying 2s + β < 0. This leads to the following result.

Theorem 1. — Let s < 0. Then the Cauchy problem (1.2) is not locally well-posed for data in H^{s}(S^{1}).

Remark 1.2. — 1. In [2] it is shown that (1.2) is locally well-posed for data in

H^{s}(S^{1}), s ≥ 0. Therefore the result of Theorem 1 is sharp.

2. The proof of Theorem 1 can be extended to equations of type

i∂_{t}u + ∂_{x}^{2}u = f(|u|^{2})u (1.5)

under weak assumptions on the nonlinearity, for example f(λ) = ±λ^{γ}, for some γ > 0. In the case of (1.5) one has to deal with the following explicit solution

u_{κ,n}(t, x) = κn^{−s} exp(−it(n^{2} + f(κ^{2}n^{−2s}))) exp(inx).

3. It is interesting to mention that if the gauge invariance (if u is a solution of (1.5) then so is e^{iθ}u, θ ∈ ℝ) of (1.5) is violated then one can obtain the local well-posedness of the corresponding Cauchy problem for data more singular than L^{2}(S^{1}). For instance, the Cauchy problem associated to the equation

i∂_{t}u + ∂_{x}^{2}u = u^{2}

is locally well-posed for data in H^{s}(S^{1}), s > −\frac{1}{2} (see [8]).

4. A result related to Theorem 1, due to Kenig-Ponce-Vega, when the spatial domain is ℝ is obtained in [9]. It is shown that the cubic \underline{focusing} NLS in 1D, posed on ℝ, is locally ill-posed for data in H^{s}(ℝ), s < 0. The construction performed in [9] involves some dilatation arguments. Therefore it seems that the technique of [9] is restricted to the case of data defined on ℝ.

We now turn to the higher dimensional case. Let (M, g) be a two dimensional riemannian manifold and Δ_{g} be the corresponding Laplace-Beltrami operator. We shall restrict our considerations to cubic defocusing nonlinearity but the arguments can be extended to some other nonlinearities. Thus consider the Cauchy problem

i∂_{t}u + Δ_{g}u = |u|^{2}u, u(0, x) = u_{0}(x), (1.6)

where x ∈ M, t ∈ ℝ. Let M = ℝ^{2} with the flat metric. Then (1.6) is invariant by a scaling transformation. Namely, if u(t, x) is a solution of (1.6) then so is

u_{λ}(t, x) = λu(λ^{2}t, λx)

with initial data λu_{0}(λx). Clearly λu_{0}(λx) has the same L^{2}(ℝ^{2}) norm as u_{0}(x). Heuristically this scaling argument suggests that (1.6) is locally well-posed for data in H^{s}(M), s > 0. Moreover this is the case when M = ℝ^{2} or M = 𝕋^{2} with the flat metrics, due to [5] in the case ℝ^{2} and [2] in the periodic case. In this paper we show that the above heuristics fail when M = S^{2}.

Theorem 2. — Let T > 0, s ∈]\frac{3}{20}, \frac{1}{4}[, κ ∈]0, 1[. Take M = S^{2} with the canonical metric in (1.6). For n ∈ ℕ, we denote by ψ_{n} : S^{2} → ℂ the restriction to S^{2} of the harmonic polynomial (x_{1} +ix_{2})^{n}. Then for t ∈ [0, T] the solution u_{n}(t) of (1.6) with initial data κϕ_{n}, where ϕ_{n} = n^{\frac{1}{4} −s}ψ_{n} can be represented as

u_{n}(t) = κ exp(−it(n(n + 1) + κ^{2}ω_{n}))(ϕ_{n} + r_{n}(t)), (1.7)

where ω_{n} ≈ n^{\frac{1}{2} −2s} and r_{n}(t) satisfies

∥r_{n}(t)∥_{H^{s}(S^{2})} ≤ C_{T} n^{−δ} (1.8)

where δ > 0 and C_{T} depends on T but not on n. Moreover there exists C > 0, independent of T and n such that

∥u_{n}∥_{L^{∞}(ℝ;H^{s}(S^{2}))} ≤ Cκ. (1.9)

As a consequence the Cauchy problem (1.6) is not locally well-posed for data in H^{s}(S^{2}).

Remark 1.3. — 1. The existence of u ∈ C^{∞}(ℝ × S^{2}) is guaranteed by Theorem

2 of [4].

2. The condition s > \frac{3}{20} ensures that (1.7) is valid on an arbitrary time interval. If one is interested only in the local well-posedness, a slight modification of the proof of Theorem 2 gives the ill-posedness of the Cauchy problem (1.6) in H^{s}(S^{2}), s ∈ [0, \frac{1}{4}[ (see section 3.3 below).

3. In [4], we have proved the local well-posedness of (1.6) in H^{s}(M), s > \frac{1}{2}. Therefore, in the case M = S^{2} the critical Sobolev regularity for the local wellposedness of (1.6) is in the interval [\frac{1}{4}, \frac{1}{2}].

4. The choice of ψ_{n} is related to earlier works on spherical harmonics by Stanton-Weinstein [15] and Sogge [14]. In these references, it is proven that ψ_{n} maximizes the quotient ∥ψ∥_{L^{4}}/∥ψ∥_{L^{2}} among the spherical harmonics of degree n. Moreover, observe that ψ_{n} concentrates on the closed geodesic x_{1}^{2} + x_{2}^{2} = 1.

The method of proof of Theorem 2 can be further exploited in order to prove ill-posedness results in the energy space for H^{1} \underline{subcritical} NLS posed on S^{6}. These results can be regarded as a consequence of the infinite propagation speed of the Schrödinger operator since similar phenomenon does not occur in the context of H^{1} \underline{subcritical} semilinear wave equation, with power nonlinearity, posed on S^{6}. On the other hand, in the H^{1} \underline{supercritical} case some semilinear wave equations on ℝ^{3} are ill-posed in the energy space (see recent works of Brenner-Kumlin [3] and Lebeau [10]). Counterexamples for \underline{subcritical} semilinear wave equations containing derivative nonlinearities are constructed in [11].

Theorem 3. — Let α ∈]0, 1]. Then the Cauchy problem

i∂_{t}u + Δ_{S^{6}}u = ⟨u⟩^{α}u, u(0, x) = u_{0}(x), (1.10)

where x ∈ S^{6}, t ∈ ℝ, ⟨u⟩ := \sqrt{1 + |u|^{2}} is not locally well-posed for data in H^{1}(S^{6}).

Remark 1.4. — The question of extending our results to more general geometries than the sphere is still open. However, if (M, g) is a compact orientable ddimensional riemannian manifold with a closed stable (elliptic) geodesic, then by considering quasimodes for initial data as constructed in [13] (see also [1], [7], [6]), one easily shows that for s < \frac{d−1}{4}, d ≥ 2 the Picard iteration scheme applied to the integral formulation of the nonlinear Schrödinger equation

i∂_{t}u + Δ_{g}u = ±|u|^{2}u

sends at the second iteration any ball of H^{s}(M) into an unbounded set (see [12] and the references therein for related results in other contexts).

The rest of this note is organized as follows. In section 2, we collect some algebraic facts which are useful in the sequel. In Section 3, we first give the proof of Theorem 2. Then we extend the argument providing the ill-posedness for s ∈ [0, \frac{1}{4}[. Since the proof of Theorem 3 is very similar to that of Theorem 2, in section 4 we only give the main lines of the proof of Theorem 3. Finally, at the end of section 4, we discuss instability features of some weak solutions of (1.10).

Acknowledgements. We thank H. Koch for discussions on the notion of local well-posedness.

2. Some algebraic lemmas

In this section, n is a fixed positive integer, and d ≥ 2. For every α ∈ ℝ, we denote by R_{α} the rotation of ℝ^{d+1} defined by

R_{α}(x_{1}, x_{2}, x_{3}, .., x_{d+1}) = (cos α x_{1} − sin α x_{2}, sin α x_{1} + cos α x_{2}, x_{3}, .., x_{d+1})

and by R_{α}^{⋆} the associated unitary operator of L^{2}(S^{d}),

R_{α}^{⋆}u(x) = u(R_{α}(x)).

For x ∈ S^{d}, we set ψ(x) = (x_{1} + ix_{2})^{n}. Observe that R_{α}^{⋆}ψ = exp(inα)ψ for every α ∈ ℝ.

Lemma 2.1. — Let u ∈ L^{2}(S^{d}) satisfying, for every α ∈ ℝ

R_{α}^{⋆}u = exp(inα)u. (2.1)

Then the decomposition of u into spherical harmonics reads

u = ωψ + ∑_{j} g_{j}

where ω ∈ ℂ and each g_{j} is a spherical harmonic of degree > n.

Proof of Lemma 2.1. Since the family (R_{α}^{⋆})_{α∈ℝ} is a one-parameter group of unitary operators leaving invariant the space of spherical harmonics of degree l, one can find an orthonormal basis (h_{k}) of L^{2}(S^{d}) such that, for every k, h_{k} is a spherical harmonic satisfying, for some n_{k} ∈ ℤ, for every α ∈ ℝ,

R_{α}^{⋆}h_{k} = exp(in_{k}α)h_{k}. (2.2)

Comparing (2.1) and (2.2), the decomposition of u in the basis (h_{k}) reads

u = ∑_{k: n_{k}=n} c_{k}h_{k}. (2.3)

Let h be a spherical harmonic of degree l satisfying property (2.1) for every α ∈ ℝ. Denote by P the l-homogeneous polynomial on ℝ^{d+1} such that h = P_{|S^{d}}. Then (2.1) is equivalent to

∀x ∈ ℝ^{d+1}, P (R_{α}(x)) = exp(inα)P (x). (2.4)

Let us decompose P according to the powers of z = x_{1} + ix_{2} and z̄

P (x_{1}, x_{2}, x_{3}, .., x_{d+1}) = ∑_{p+q≤l} z^{p}z̄^{q}A_{pq}(x_{3}, .., x_{d+1}) (2.5)

where A_{pq} is a homogeneous polynomial of degree l − p − q. In view of (2.4), (2.5) and

P (R_{α}(x)) = ∑_{p+q≤l} z^{p}z̄^{q}A_{pq}(x_{3}, .., x_{d+1}), (2.6)

we conclude that A_{pq} = 0 unless p − q = n. As a consequence,

l ≥ p + q ≥ p − q = n

and, if l = n, then p = n and q = 0, so that P = cz^{n}, i.e. h = cψ. Coming back to decomposition (2.3), this completes the proof. The main advantage of property (2.1) is that it is invariant under the nonlinear transformations

u ⟼ f(|u|^{2})u.

Combining this remark with Lemma 2.1, we obtain the following two statements.

Lemma 2.2. — Let f ∈ L_{loc}^{∞}(ℝ_{+}). Then there exists ω ∈ ℂ such that

f(|ψ|^{2})ψ = ωψ + r,

where r is a sum of spherical harmonics of degree > n.

Lemma 2.3. — Let c ∈ ℂ and f ∈ C^{∞}(ℝ_{+}), such that (1.1) is locally well-posed (see Definition 1.1) in H^{s}(M). Let u ∈ C([−T, T], H^{s}(M)) ∩ X_{T} be the solution of (1.1) satisfying u(0) = cψ. Then for every t ∈ [−T, T],

u(t) = ω(t)ψ + ∑_{j} g_{j}(t),

where ω(t) ∈ ℂ and each g_{j}(t) is a spherical harmonic of degree > n.

Lemma 2.2 is an immediate consequence of Lemma 2.1 and the above remark. Lemma 2.3 follows from Lemma 2.1 and the observation that R_{α}^{⋆}u and exp(inα)u belong to C([−T, T], H^{s}(M)) ∩ X_{T} and solve (1.1) with the same Cauchy data c exp(inα)ψ.

3. Proof of Theorem 2

We drop the subscript n for conciseness. We first prove (1.9). Recall the following conservation laws for (1.6).

∥u(t)∥_{L^{2}}^{2} = κ^{2}∥ϕ∥_{L^{2}}^{2}, (3.1)

∥∇u(t)∥_{L^{2}}^{2} + \frac{1}{2}∥u(t)∥_{L^{4}}^{4} = κ^{2}∥∇ϕ∥_{L^{2}}^{2} + \frac{κ^{4}}{2} ∥ϕ∥_{L^{4}}^{4}. (3.2)

The identity (3.1) results from the gauge invariance of (1.6) while (3.2) is due to the time translation invariance of (1.6). Since one has explicit asymptotics with respect to n of all Sobolev and Lebesgue norms of ϕ one can obtain bounds for u. First a use of (3.1) yields

∥u∥_{L^{∞}(ℝ;L^{2})} ≤ Cκn^{−s}.

Next using (3.2), we infer that

∥∇u∥_{L^{∞}(ℝ;L^{2})} ≤ Cκ(∥∇ϕ∥_{L^{2}} + ∥ϕ∥_{L^{4}}^{2})
≤ Cκn^{1−s}.

An interpolation argument gives

∥u∥_{L^{∞}(ℝ;H^{s})} ≤ ∥u∥_{L^{∞}(ℝ;L^{2})}^{1−s}∥u∥_{L^{∞}(ℝ;H^{1})}^{s}
≤ Cκ

which proves (1.9). Further we set c(t) := exp(−it(n(n + 1) + κ^{2}ω)), where

ω := \frac{∥ϕ∥_{L^{4}}^{4}}{∥ϕ∥_{L^{2}}^{2}} ≈ n^{\frac{1}{2} −2s}.

Then c(t) satisfies the equation

ic_{t} − n(n + 1)c = κ^{2}ω|c|^{2}c. (3.3)

Write u(t) = κc(t)(ϕ+w(t)). Using Lemma 2.2 and (3.3), we obtain that the equation for w(t) is

iw_{t} + (Δ_{S^{2}} + n(n + 1) + κ^{2}ω)w = κ^{2}(|ϕ + w|^{2}(ϕ + w) − |ϕ|^{2}ϕ + r). (3.4)

with w(0) = 0 and

r ∈ \overline{⊕_{l≥n+1}Ker(Δ_{S^{2}} + l(l + 1))}.

Further we decompose w(t) = z(t)ϕ + q(t) for some z : ℝ → ℂ and q : ℝ → C^{∞}(S^{2}). Then due to Lemma 2.3, we can suppose

q(t) ∈ \overline{⊕_{l≥n+1}Ker(Δ_{S^{2}} + l(l + 1))} (3.5)

The proof of Theorem 2 will be completed^{(1)} once we prove the following two lemmas.

Lemma 3.1. — There exists a constant C > 0, independent of T and n such that

∥q(t)∥_{H^{s}} ≤ Cn^{− \frac{1}{4} −s}.

Lemma 3.2. — There exists a constant C_{T} > 0, which depends on T but not on n such that

sup_{t∈[0,T]} |z(t)| ≤ C_{T} n^{\frac{1}{4} −3s}.

3.1. Proof of Lemma 3.1. — We first rewrite the conservation laws (3.1), (3.2) in terms of z(t) and q(t). Since ϕ is orthogonal to q(t) in L^{2}(S^{2}) as well as ∇ϕ to ∇q(t), we can rewrite (3.1) and (3.2) as

|1 + z(t)|^{2}∥ϕ∥_{L^{2}}^{2} + ∥q(t)∥_{L^{2}}^{2} = ∥ϕ∥_{L^{2}}^{2}, (3.6)

|1 + z(t)|^{2}∥∇ϕ∥_{L^{2}}^{2} + ∥∇q(t)∥_{L^{2}}^{2} + \frac{1}{2κ^{2}} ∥u(t)∥_{L^{4}}^{4} = ∥∇ϕ∥_{L^{2}}^{2} + \frac{κ^{2}}{2} ∥ϕ∥_{L^{4}}^{4}. (3.7)

Observe that

∥∇ϕ∥_{L^{2}}^{2} = (ϕ | − Δ_{S^{2}} ϕ) = n(n + 1)∥ϕ∥_{L^{2}}^{2},

where (·|·) denotes the L^{2}(S^{2}) scalar product. Therefore multiplying (3.6) with −n(n + 1) and adding it to (3.7) gives

∥∇q(t)∥_{L^{2}}^{2} − n(n + 1)∥q(t)∥_{L^{2}}^{2} ≤ \frac{κ^{2}}{2} ∥ϕ∥_{L^{4}}^{4}
≤ Cn^{\frac{1}{2} −4s}.

(1)The ill-posedness in H^{s}(S^{2}) follows from an argument similar to the one in 1D already performed in the introduction.

Due to (3.5), we can write q(t) = ∑_{l≥n+1} q_{l}(t) where q_{l} ∈ Ker(Δ_{S^{2}} +l(l+1)). Hence

∥∇q(t)∥_{L^{2}}^{2} − n(n + 1)∥q(t)∥_{L^{2}}^{2} = ∑_{l≥n+1}(l(l + 1) − n(n + 1))∥q_{l}(t)∥_{L^{2}}^{2}. (3.8)

If l ≥ n + 1, the following inequalities hold

l(l + 1) − n(n + 1) ≥ Cn

and

\frac{l(l + 1) − n(n + 1)}{l(l + 1)} ≥ Cn^{−1}.

Therefore we arrive at

∥q(t)∥_{L^{2}}^{2} ≤ Cn^{− \frac{1}{2} −4s}

and

∥∇q(t)∥_{L^{2}}^{2} ≤ Cn^{\frac{3}{2} −4s}.

An interpolation argument yields

∥q(t)∥_{H^{s}} ≤ ∥q(t)∥_{L^{2}}^{1−s}∥q(t)∥_{H^{1}}^{s}
≤ Cn^{− \frac{1}{4} −s}.

This completes the proof of Lemma 3.1.

3.2. Proof of Lemma 3.2. — We project the equation (3.4) on ϕ and obtain the following equation for z(t)

iz_{t} + ωκ^{2}z = \frac{κ^{2}}{∥ϕ∥_{L^{2}}^{2}} ((|ϕ + w|^{2}(ϕ + w) | ϕ) − (|ϕ|^{2}ϕ | ϕ))
= \frac{κ^{2}}{∥ϕ∥_{L^{2}}^{2}} \{∫ (2|ϕ|^{2}w + ϕ^{2}w̄)ϕ̄ + ∫ (2Re(ϕ̄w)wϕ̄ + |w|^{2}|ϕ|^{2} + |w|^{2}wϕ̄)\}.

Recall that w(t) = z(t)ϕ + q(t) and |ϕ|^{2}ϕ = ωϕ + r. Therefore the equation for z(t) can be rewritten as

iz_{t} + ωκ^{2}z = 2ωκ^{2}z + ωκ^{2}z̄ + \frac{κ^{2}}{∥ϕ∥_{L^{2}}^{2}} O(|z|^{2} ∫ |ϕ|^{4} + |z|^{3} ∫ |ϕ|^{4}
+ ∫ |q|^{3}|ϕ| + ∫ |q|^{2}|ϕ|^{2} + |(q|r)|).

We first estimate the source terms. Write

\frac{∫ |q|^{3}|ϕ|}{∥ϕ∥_{L^{2}}^{2}} ≤ Cn^{2s}∥q∥_{L^{3}}^{3}∥ϕ∥_{L^{∞}}
≤ Cn^{2s}∥q∥_{L^{2}}^{2}∥∇q∥_{L^{2}}∥ϕ∥_{L^{∞}}
≤ Cn^{2s} n^{− \frac{1}{2} −4s} n^{\frac{3}{4} −2s} n^{\frac{1}{4} −s}
= Cn^{\frac{1}{2} −5s}.

Further we have

\frac{∫ |q|^{2}|ϕ|^{2}}{∥ϕ∥_{L^{2}}^{2}} ≤ Cn^{2s}∥q∥_{L^{2}}^{2}∥ϕ∥_{L^{∞}}^{2}
≤ Cn^{2s} n^{− \frac{1}{2} −4s} n^{\frac{1}{2} −2s}
= Cn^{−4s}.

and

\frac{|(q|r)|}{∥ϕ∥_{L^{2}}^{2}} ≤ Cn^{2s}∥q∥_{L^{2}}∥r∥_{L^{2}}
≤ Cn^{2s} n^{− \frac{1}{4} −2s} n^{\frac{1}{2} −3s}
= Cn^{\frac{1}{4} −3s}.

Therefore if s > \frac{1}{8} the equation for z(t) can be written as iz_{t} = 2ωκ^{2}Re(z) + O(ω|z|^{2} + ω|z|^{3} + n^{\frac{1}{4} −3s}) (3.9) with z(0) = 0. Moreover using once again the L^{2} conservation law (3.6), we have

1 − |1 + z|^{2} = \frac{∥q(t)∥_{L^{2}}^{2}}{∥ϕ∥_{L^{2}}^{2}} = O(n^{− \frac{1}{2} −2s}).

Therefore

|2Re(z) + |z|^{2}| = O(n^{− \frac{1}{2} −2s})

and the equation (3.9) takes the form

iz_{t} = O(ω|z|^{2} + ω|z|^{3} + n^{\frac{1}{4} −3s}),

with ω = O(n^{\frac{1}{2} −2s}). Hence if we set

M(T) = sup_{0≤t≤T} |z(t)|,

we obtain M(T) ≤ CT (n^{\frac{1}{2} −2s}([M(T)]^{2} + [M(T)]^{3}) + n^{\frac{1}{4} −3s}). (3.10)

In view of (3.10), we set

M̃(T) = n^{3s− \frac{1}{4}} M(T)

and therefore (3.10) yields

M̃(T) ≤ CT (1 + n^{\frac{3}{4} −5s}([M̃(T)]^{2} + n^{\frac{1}{4} −3s}[M̃(T)]^{3})).

Since M̃(0) = 0 and s > \frac{3}{20}, we obtain that M̃ ≤ CT uniformly with respect to n. This completes the proof of Lemma 3.2.

3.3. Ill-posedness for s ∈ [0, \frac{3}{20}].— In order to prove that (1.6) is not locally wellposed for data in H^{s}(S^{2}), s ∈ [0, \frac{3}{20}] one needs to slightly extend the argument of section 3.2. Suppose that (1.6) is locally well-posed for data in H^{s}(S^{2}), s ∈ [0, \frac{3}{20}]. Using the above method, we need to show only that (1.7) holds on a small time interval [0, T_{n}] such that

lim_{n↦+∞} n^{\frac{1}{2} −2s} T_{n} = +∞ (3.11)

since the argument providing the ill-posedness performed in the introduction works equally well provided that (3.11) holds. Therefore, with the notation of the previous section, one needs a bound for M(T) only for T ∈ [0, T_{n}] with T_{n} satisfying (3.11). One now defines M̃(T) as

M̃(T) = T_{n}^{−1} n^{3s− \frac{1}{4}} M(T)

and therefore for T ∈ [0, T_{n}] one has

M̃(T) ≤ C(1 + T_{n}^{2}n^{\frac{3}{4} −5s}([M̃(T)]^{2} + T_{n} n^{\frac{1}{4} −3s}[M̃(T)]^{3})).

Let ε > 0 be a small number to be fixed later. For s ≥ 0, we choose T_{n} as follows

T_{n} := n^{\frac{5}{2} s− \frac{3}{8} −ε}.

Clearly with the above choice of T_{n} the relation (3.11) holds provided ε ≪ 1. Moreover, we have a uniform bound for M̃(T), T ∈ [0, T_{n}]. This implies that for t ∈ [0, T_{n}] one has |z(t)| ≤ cn^{− \frac{1}{2} s− \frac{1}{8} −ε} which proves the validity of (1.7) the interval [0, T_{n}].

4. Sketch of the proof of Theorem 3

Suppose that (1.10) is locally well-posed in H^{1}(S^{6}). Let ψ : S^{6} → ℂ be the restriction to S^{6} of the harmonic polynomial (x_{1} + ix_{2})^{n}. Then it is easy to see that ∥ψ∥_{L^{q}} ≈ n^{− \frac{5}{2q}}. Consider (1.10) with Cauchy data u(0) = κϕ := κn^{\frac{1}{4}} ψ, where κ ∈ [\frac{1}{2}, 1]. Then ∥ϕ∥_{H^{1}} ≈ 1 and due to Lemma 2.1 there exists ω ∈ ℂ and r so that

(κ^{−2} + |ϕ|^{2})^{α/2}ϕ = ωϕ + r,

where r is a sum of spherical harmonics of degree > n. Moreover, we can check that uniformly in κ ∈ [\frac{1}{2}, 1],

ω = c^{⋆}n^{α/4} + O(n^{−δ}),

where δ > 0 and c^{⋆} is independent of κ. Now we set c(t) := exp(−it(n(n+5)+κ^{α}ω)). Let u(t) be the solution of (1.10) subject to initial data ϕ. Writing u(t) = κc(t)(ϕ + w(t)) we obtain that the equation for w(t) is

iw_{t} + (Δ_{S^{6}} + n(n + 5) + κ^{α}ω)w = κ^{α}((κ^{−2} + |ϕ + w|^{2})^{α/2}(ϕ + w) − (κ^{−2} + |ϕ|^{2})^{α/2}ϕ + r).

Let w(t) = z(t)ϕ + q(t). We now give the estimates of the quantities involved in the argument, performed in details in the previous section.

Estimate for ∥r∥_{L^{2}}. Clearly

∥r∥_{L^{2}} ≤ C∥ϕ^{α+1}∥_{L^{2}} ≤ Cn^{\frac{α−4}{4}} .

Estimate for ∥q(t)∥_{L^{2}}. Similarly to the previous section one can check that

n∥q(t)∥_{L^{2}}^{2} ≤ C∥ϕ∥_{L^{α+2}}^{α+2}

and therefore

∥q(t)∥_{L^{2}} ≤ Cn^{\frac{α−12}{8}} .

Therefore, using the above bounds for ∥r∥_{L^{2}} and ∥q(t)∥_{L^{2}}, we obtain

\frac{|(q(t) | r)|}{∥ϕ∥_{L^{2}}^{2}} ≤ Cn^{\frac{3α−4}{8}} .

and

\frac{∥q(t)∥_{L^{2}}^{2}}{∥ϕ∥_{L^{2}}^{2}} ≤ Cn^{\frac{α−4}{4}} .

Estimate for ∥q(t)∥_{H^{1}}. Clearly

∥q(t)∥_{H^{1}} ≤ Cn^{1+ \frac{α−12}{8}} ≤ Cn^{−δ},

where δ > 0 provided α < 4.

Analysis on the nonlinear term. Set

F_{ϕ}(w, w̄) = (κ^{−2} + |ϕ + w|^{2})^{α/2}(ϕ + w) − (κ^{−2} + |ϕ|^{2})^{α/2}ϕ.

Then F_{ϕ}(0, 0) = 0 and

\frac{∂F_{ϕ}}{∂w} (0, 0) = (κ^{−2} + |ϕ|^{2})^{α/2} + \frac{α}{2}|ϕ|^{2}(κ^{−2} + |ϕ|^{2})^{α/2−1},
\frac{∂F_{ϕ}}{∂w̄} (0, 0) = \frac{α}{2}ϕ^{2}(κ^{−2} + |ϕ|^{2})^{α/2−1}.

Since α ≤ 1, the second derivatives of F_{ϕ} are bounded and therefore, similarly to the previous section, the equation for z(t) can be written as

iz_{t} = O(n^{\frac{α}{4}} |z|^{2} + n^{\frac{3α−4}{8}} ) + \frac{1}{∥ϕ∥_{L^{2}}^{2}} O(∫ |zϕ + q|^{2}|ϕ|).

Estimate for ∫ |zϕ + q|^{2}|ϕ|. Write

\frac{∫ |zϕ + q|^{2}|ϕ|}{∥ϕ∥_{L^{2}}^{2}} ≤ \frac{C}{∥ϕ∥_{L^{2}}^{2}} (|z|^{2} ∫ |ϕ|^{3} + ∥q∥_{L^{2}}^{2}∥ϕ∥_{L^{∞}})
≤ O(n^{\frac{1}{4}} |z|^{2}) + O(n^{\frac{α−3}{4}} ).

Since α ∈]0, 1] the equation for z becomes

iz_{t} = O(n^{\frac{1}{4}} |z|^{2} + n^{\frac{3α−4}{8}} )

with z(0) = 0. Now we set

M(T) := sup_{0≤t≤T} |z(t)|.

We look for a T_{n} such that

lim_{n↦+∞} n^{\frac{α}{4}} T_{n} = +∞ (4.1)

with a bound for M(T), T ∈ [0, T_{n}]. Now we set

M̃(T) = T_{n}^{−1} n^{\frac{4−3α}{8}} M(T)

and therefore for T ∈ [0, T_{n}] one has

M̃(T) ≤ C(1 + T_{n}^{2}n^{\frac{3α−2}{8}} [M̃(T)]^{2}).

Since α is positive we can take T_{n} so that n^{− \frac{α}{4}} ≪ T_{n} ≪ n^{\frac{2−3α}{16}} and therefore the representation u(t) = κc(t)ϕ + O(n^{−γ}) in H^{s} holds for t ∈ [0, T_{n}] for some γ > 0. Now it remains to perform the argument of the introduction in order to prove the ill-posedness.

Remark 4.1. — Global weak solutions of (1.10) in the class H^{1} can be obtained by a standard compactness argument. The method consists of considering a regularized version of (1.10), involving a small parameter. The Cauchy problem associated to the regularized equation possesses unique global solution, due to control on high Sobolev norms. Then one passes into the limit as the small parameter tends to zero. This is possible due to the H^{1} bound on the solutions of (1.10). Note that if the regularized version of (1.10) is gauge invariant then one may apply Lemma 2.3 to the regularized equation and therefore there exists a global weak solution of (1.10) with data ϕ (see the previous section for the definition of ϕ) such that its spectral decomposition at any fixed time t contains only spherical harmonics of degree ≥ n. Due to the argument of the previous section we obtain that this solution has strongly instability features when letting n tend to infinity.

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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