Instability of non-constant harmonic maps for the 1 + 2-dimensional equivariant wave map system
Rapha¨el Cˆote
Abstract
In this paper we study (1+2)-dimensional equivariant wave maps to a surfaceN. This is, after reduction, solutions u:Rt×Rr→R to the following initial value problem :
utt−urr−1rur = −f(u)r2 , (u, ut)|t=0 = (u0, u1).
(f depends on N). We consider the existence of a finite energy har- monic mapQ(a stationary solution), and show that when it exists,Q is instable in the energy space.
Our result applies in particular to the case of wave maps to the sphereS2, and to the critical Yang-Mills equations in dimension 4.
1 Introduction
1.1 Recall of known results
Let us introduce a functiong ∈C1, such that g(0) = 0, andf =g·g0. We consider the following initial value problem on functionu:Rt×R+r →R:
utt−∆u = −f(u) r2 , (u, ut)|t=0 = (u0, u1).
(1) (We denote ∆u=urr+ur/r= 1/r(rur)r the radial Laplacian inR2). This problem has the following geometric interpretation. Let N be a surface of revolution with polar coordinates (ρ, θ)∈[0,∞)×S1. Letds2 be the metric onN :
ds2 =dρ2+g2(ρ)dθ2. (2)
A wave map is a functionU :R1+2→N satisfying the system : Uα+ Γαβγ(U)∂aUβ∂aUγ = 0,
(U, Ut)|t=0 = (U0, U1). (3)
Mathematical Subject Classification : 35L15.
Keywords : Instability, wave map, criticality.
(Γαβγ are the Christoffel symbols for (N, ds2), α = ρ or θ). Denote (r, φ) the usual polar coordinates onR2. We are concerned with the corotationnal equivariant case, that is, we impose that U has the form
ρ=u(t, r), θ=φ.
The wave map system (3) then simplifies to a single nonlinear scalar equation foru : Rt×R+r →R, which is (1) (see the book by Shatah and Struwe [4]
for further details). Of course, any result onu has a reformulation forU. At least formally, one has conservation of one quantity, that is energy :
E(u) = Z
u2t +u2r+g2(u) r2
rdr=E(u0, u1). (4) (In fact E(u) =kUtk2L2 +k∇Uk2L2). Let us introduce :
H =
(u, v)
k(u, v)k2H
def= E(u, v) = Z
v2+u2r+g(u)2 r2
rdr <∞
. (5) Happears as an energy space, in which it is natural to study the solutions to (1). In [6], Shatah and Tahvildar-Zadeh proved local in time existence and uniqueness of solutions to (1) arising from initial data in the energy space : Local existence in H [6, Theorem 1.1]. Let (u0, u1) ∈ H. Then there exist T >0 and a unique solution u to Problem (1) such that :
(u, ut)∈L∞([0, T), H), u∈Lq([0, T),B˙10/3,10/31/2 (R+, rdr)).
Let us define the usual notation : E(u, a, b)def=
Z b a
u2t +u2r+g2(u) r2
rdr.
One way to express the finite speed of propagation is the fact that the energy is decreasing on light cones :
∀R≥0, ∀|τ| ≤R, E(u(t),0, R− |τ|)≤E(u(t+τ),0, R). (6) One should also notice that Problem (1) has a natural scaling :
u(t, r) is a wave map ⇐⇒ uλ(t, r)def= u(λt, λr) is a wave map.
A straightforward computation gives : E(uλ) =
Z
λ2u2t +λ2u2r+λ2g2(u) (λr)2
rdr=E(u).
The energy remains unaffected by scaling : Problem (1) is thus said to be (scaling-)critical for the energy.
The main remaining open question for this problem is global well-pos- edness.
The result by Shatah and Tavildar-Zadeh [6] solves in particular the case of small energy data : there exists a constant ε0 > 0 such that if E(u) =E(u0, u1)< ε0, thenuis global in time. Another direct consequence of the proof of Theorem 1.1 of [6] is the following condition for blow-up :u blows-up at time T only if
lim inf
t↑T E(u(t),0, T −t)≥ε0. (7) Remind that due to radial symmetry, concentration of energy can only hap- pen at point r= 0.
Under some assumptions onN, the energy can not concentrate, and this is enough to ensure global existence in time : in [5], Shatah and Tavildar- Zadeh proved non-concentration under the assumption g0 ≥ 0 (geodesical convexity). This condition was later laxed by Grillakis tog(ρ) +g0(ρ)ρ >0, and finally by Struwe in [7], to g > 0 forρ >0. On the other hand, in [1], Bizo´n and al. gave strong evidence of blow-up for system (1), in the case N =S2, andg= sin.
In [7], Struwe proved further that ifu does blow up (say at timet= 0), then for a subsequencetn, there exists a scaling parameter λ(tn) such that λ(tn)|tn| → ∞ and :
u(tn+t/λ(tn), r/λ(tn))→Q(r) in Hloc1 (R1+2), (8) whereQis a non-constant harmonic map, i.e. a stationary solution to (1) :
∆Q= f(Q)
r2 . (9)
This proves in particular that if there is no harmonic map, then there is no blow up : this is the case wheng > 0 forρ > 0. Furthermore, [7, Theorem 1] has two corollaries :
1. Let Q be a non-constant harmonic map with least energy. Suppose E(u)≤E(Q) : thenu is global in time.
2. Blow-up (atT) is characterized by lim inft↑TE(u(t),0, T−t)≥E(Q).
In particular, if the initial data is such that E((u0, u1),0, R)≤E(Q), then the corresponding wave mapu(t) is defined at least up to time t=R (remind that the energy is decreasing on cones).
These results can be reformulated by saying that one can chooseε0 =E(Q).
1.2 Statement of the results
For a single function u, not depending on time, we shall often note kukH
instead ofk(u,0)kH, as well asE(u, a, b) forE((u,0), a, b).
From now on,Qwill denote a non-constant harmonic map, i.e.Qsatisfies (9). Qis important for the qualitative study of (1). Indeed, notice that our first criterion (7) for blow-up states the impossibility to apply the local well- posedness result. Due to [7], it is in fact equivalent to the formation of a singularity with the well-defined blow-up profile Q, which is a descriptive result.
As we shall see, the existence of Qis equivalent to the vanishing of gat some point. The problematic is now : if Q exists, does blow-up occur ? In particular, for initial data in a neighborhood ofQ, does one has blow-up ?
Our goal in this paper is to prove the instability ofQin the energy space.
This result should be related to previous works on instability of stationary states for other equations. In particular, we refer to [3] for the critical non- linear Schr¨odinger equation :
iut+ ∆u+|u|4/Nu= 0, (t, x)∈R×RN, u(0, x) =u0(x), x∈RN,
and [2] for the critical generalized Korteweg-de Vries equation : ut+ (uxx+u5)x= 0, (t, x)∈R×R,
u(0, x) =u0(x), x∈R.
In this context, proving instability for Q should be viewed as a first step toward understanding the blow-up mechanism.
More precisely we can build wave maps, which are in an arbitrary neigh- borhood ofQ at time 0, and which change profile to a certain Q(λ·),λ6= 1 (in factλ >1) :
Theorem 1. – [Geometric instability of Q in H] Suppose g vanishes at some point C∗ > 0 (isolated zero). Then there exists a non-constant finite energy harmonic map Q for Problem (1). Moreover, for any λ0 >1, there exist a sequence of initial data (u0n, u1n) such that
k(u0n, u1n)−(Q,0)kH →0 as n→ ∞, (10) and if we denote the arising wave mapsun (solutions to (1)), un is defined at least up to some time tn such that
k(un, unt)(tn)−(Q(λ0·),0)kH →0 as n→ ∞. (11) We shall call this change of profile (from Q to Q(λ0·)) geometric insta- bility. Notice, as proven in Corollary 4, that one must have tn → ∞. As a direct consequence, we have :
Corollary 1. LetQbe a finite energy harmonic map for Problem (1). Then Qis unstable in H. For some constant C(λ0)>0 independent of n :
sup
t∈[0,tn]k(un, unt)−(Q,0)kH ≥C(λ0).
As Problem (1) has no symmetry besides scaling, this result also proves orbital instability.
Of course, as mentioned earlier, this result can be rewritten as geomet- rical instability of harmonic maps for the wave map system (3).
Theorem 1 deals in particular with the following two special cases.
Corollary 2 (Wave maps to the 2-sphere). It corresponds to the case g = sin :C∗ =π. The equation takes the form (cf. [1] for more details) :
utt−∆u=−sin 2u 2r2 .
And the harmonic solution is explicit : Q(r) = 2 arctan(r). Then Q is geo- metrically unstable.
Q can be seen as the minimal (in the sense of energy) connection be- tween the north and south poles. Theorem 1 is here in agreement with the numerical investigation of [1].
Corollary 3 (Critical Yang-Mills equation in dimension 4). It corresponds tog(ρ) = (1−ρ2) : C∗= 1, which gives the equation :
utt−∆u= 2u(1−u2) r2 .
Then the harmonic map Q(r) = tanh(lnr) = rr22−+11 is geometrically unstable.
Here, one should think of a slight modification of our setting, as ˜u=u+1 is the wave map - with greplaced by ˜g(ρ) =g(ρ−1) =ρ(2−ρ) : the main difference will be thatρ∈[−1,∞).
The proof is organized as follows : in Section 2.1 we characterize harmonic maps and their properties. In Section 2.2, we study special, infinite energy wave maps, which are in fact related to harmonic maps : the self similar solutions. Then, in Section 2.3, we regularize these solutions to obtain wave maps with initial data in a neighborhood ofQ. Finally in Section 2.4, using finite speed of propagation, we exhibit a family of initial data satisfying Theorem 1.
Acknowledgments.
I thoroughly thank my advisor Frank Merle for numerous fruitful discussions, and the Department of Mathematics of Princeton University, where this work has been done.
2 Proofs
2.1 On stationary solutions to (1)
Recall that the existence of a non constant harmonic map implies that g vanishes at some point (besides 0) : it is a consequence of [7]. In the fol- lowing, we suppose that there exists a least positive real number C∗ such that g(C∗) = 0. Without loss of generality, we can suppose that g(ρ) > 0 for ρ ∈ (0, C∗). Finally, as to avoid degeneracy of the energy, we assume that g has only isolated zeros. We shall denote G(ρ) = Rρ
0 |g(ρ0)|dρ0 : G is increasing.
Proposition 1. For any g such thatg(C∗) = 0, there exists a non constant finite energy harmonic map Q, i.e. a solution to (9). Furthermore :
1. Regularity for Q. Any harmonic map Q is of class C2 and satisfies one of the equations :
rQr=g(Q), or rQr =−g(Q).
As a consequence, Q is monotone, and joins 2 consecutive zeros ofg.
2. Variational characterization ofQ. Suppose thatQjoins 0toC∗. Then Q is of minimal energy for this property : for a function v, such that v(0) = 0, and v(r)→C∗ as r→ ∞, then
E(v)≤E(Q) =⇒v(r) =Q(λr) for some λ >0.
We will use many times the following simple inequality : Lemma 1 (Pointwise inequality). For a finite energy function v :
E(v, α, β) = Z β
α
vr2+g2(v) r2
rdr≥2 Z β
α |g|(v)vrdr
≥2|G(v(β))−G(v(α))|. (12) Proof. First, let us prove the existence of a harmonic map. Let Q be a maximal (in the sense of Cauchy-Lipschitz) solution to
rQr=g(Q), Q(1) =C∗/2. (13) SupposeQis defined on (a, b). Then :
Z b a
Q2r+g2(Q) r2
rdr= 2 Z b
a
Qrg(Q)dr= 2(G(Q(b))−G(Q(a))).
Now, as u = 0 and u = C∗ are solutions, by uniqueness, we always have Q(r)∈(0, C∗) forr >0. This proves both thatQdefined forr ∈[0,∞) and thatQis of finite energy 2G(C∗). If we differentiate (13), we obtain :
∆Q= 1
r(rQr)r= f(Q) r2 .
AsQr(1)6= 0,Q is not constant : it is the desired harmonic map.
Let us now prove the properties for any harmonic map. We denoter=ex, so thatr∂/∂r =∂/∂x, and thus equation 9 writes :
Qxx=f(Q). (14)
The Theorem of Cauchy-Lipschitz allows us to solve (14) ; this ensures that Q is C2 where it is defined. Multiply (14) by Qx and integrate between a and b:
[Q2x]ba= [g2(Q(x))]ba. (15) On another side, for any finite energy function v(r), in view of (12), G(v) satisfies the Cauchy criterion as α, β → 0, and as α, β → ∞. Thus G(v) admits limitsG(v)(0) andG(v)(∞), atr = 0 andr → ∞respectively. AsG is increasing and continuous, it is a homeomorphism, and in fact v admits limits in 0 and∞,v(0) andv(∞). Of course, asvis of finite energy, we must have :
g(v(0)) =g(v(∞)) = 0.
Therefore,Qadmits limits atr = 0 and at∞ (or in thexvariable, at±∞), that are zeros ofg. In (15), fixb, and leta→ −∞. The left hand side has a limit, so thatQx has a limit`at −∞. If `6= 0, then Qhas no limit at−∞, a contradiction. Hence Qx→ 0, and finally we get
Q2x(b) =g2(Q(b)).
We already know thatQ joins two zeros of g, say α and β : if they are not consecutive, then for a certain c,g(Q(c)) = 0, so Qx(c) = 0. By uniqueness, Q = Q(c) = constant : a contradiction, so a and b are two consecutive zeros. The same argument shows that for allx∈R, Q(x)∈(a, b), and that g(Q) has a constant sign : as a consequence, Qx does not vanish, henceQis monotone. We then deduce (for sign reasons) that either :
∀x, Qx =g(Q) or ∀x, Qx=−g(Q).
Finally, let us prove the the minimizing property. Up to rescaling of v, we can suppose thatv(1) =Q(1). Now, using (12) :
E(Q,0,1) = 2|G(Q(1))−G(Q(0))|= 2|G(v(1))−G(v(0)| ≤E(v,0,1).
In the same way, E(Q,1,∞) ≤ E(v,1,∞), and we obtain E(v) ≥ E(Q).
Thus E(v) = E(Q) (note that this already shows that Q is minimizing).
More precisely, using 12, we have for allr : E(v) =E(v,0, r) +E(v, r,∞)
≥(|G(v(r))−G(0)|+|G(C∗)−G(v(r))|)≥E(Q) =E(v).
So there is in fact equality everywhere. As a consequence,vis non-decreasing (so as not to lose any energy), and takes its values in (0, C∗) for r >0. The energy equality also gives :
v2r =g2(v)/r2.
As before, using the fact that v is non-decreasing, we get : vr =g(v)/r. As v(1) =Q(1), v=Q.
From now on, Q will denote a harmonic map joining 0 toC∗. Asg >0 on (0, C∗) :
rQr=g(Q). (16)
Proposition 2 (Decomposition). There exists α0 > 0 and an increasing function δ : [0, α0]→ R+, with δ(α) →0 as α→ 0, such that the following is true.
Suppose v is a function of finite energy, withv(0) = 0andv(r)→C∗ as r→ ∞ and such that :
E(v) =E(Q) +α < E(Q) +α0. Then there exist λ∈R+
∗, ∈H, such that :
v(r) =Q(λr) +(r), kkH ≤δ(α).
Proof. Follows from the variational characterization of Q : cf. Appendix A.
Let u be a wave map with energy lower than E(Q) +α0. Such a de- composition exists for all u(t) : this gives two functionsλ(t) and(t, r) (the proof of Proposition 2 shows that they are continuous functions of t) such that
u(t, r) =Q(λ(t)r) +(t, r).
Now blow up occurs only when there is concentration of at leastE(Q) energy.
As(t, r) has a small energy, it cannot lead to blow up : only the Q(λ(t)r) can do it, which it equivalent toλ(t)→ ∞. On the other side, ifλ(t)→ ∞, then the initial blow-up criterion is fulfilled, and the result of [7] applies.
Finally u blows up at time T if and only if λ(t) → ∞ as t ↑ T. In this setting, Theorem 1 gives the existence of a wave map whose scaling parameterλ(t) goes fromλ(0) = 1 toλ(tn) =λ0>0 : this can be seen as a first step toward existence of blow up.
Another consequence of Proposition 2 is the following :
Corollary 4. Let T > 0 and ε >0. Then there exists η > 0 such that for all initial data (u0, u1) satisfyingk(u0, u1)−(Q,0)kH ≤η, the arising wave map u is defined at least up to time T and :
sup
t∈[0,T]k(u, ut)−(Q,0)kH ≤ε.
In particular, this shows that one can not expect instability of Q for bounded times, and in Theorem 1, tn → ∞. The proof is postponed to Appendix A.
2.2 Self-similar solutions on light cones
The proof of Theorem 1 relies on the study of self-similar solutions. A self similar solution is a solutionu to (1), defined fort <0, with the ansatz :
u(t, r) =v r
|t|
.
It of course blows up at time t = 0. Let us first exhibit self similar wave maps.
Corollary 5. Define, forξ <1 :
Pα(ξ) =Q 2αξ 1 +p
1−ξ2
! .
ThenPα generates a self-similar solution (on the light cone r <|t|).
Proof. It is mainly computations with change of variables. We plug the ansatz u(t, r) = v(−r/t) (and −r/t=ξ) in (1), and write the equation for v(ξ) :
ut = r
t2v0(ξ), utt= r2
t4v00(ξ)− 2r
t3v0(ξ), ur =−1
tv0(ξ), urr = 1 t2v00(ξ).
Sov satisfies : r2
t4 − 1 t2
v00(ξ) +
−2r t3 + 1
rt
v0(ξ) =−f(v(ξ)) r2 .
Now we multiply byt2, replace−r/tbyξ, and again multiply byξ2 : we get the equation for vin the ξ variable :
ξ2(ξ2−1)d2v
dξ2 + 2ξ3−ξdv
dξ =−f(v). (17)
We would like to make the first order term vanish. For this, let us now do the change of variable χ = ln
ξ/(1 +p
1−ξ2)
i.e. ξ = 1/coshχ (diffeo- morphism forξ ∈(0,1), that isr <|t|), then :
dv
dξ = 1
ξp 1−ξ2
dv dχ, d2v
dξ2 = 2ξ2−1 ξ2(1−ξ2)3/2
dv
dχ+ 1
ξ2(1−ξ2) d2v dχ2. We plug this last relation in (17), the equation simplifies to
d2v
dχ2 =f(v).
But this is simply equation (14), whose solutions are Q(2αexpχ) =Q 2αξ
1 +p 1−ξ2
! ,
according to Proposition 1 (recall x= expr).
This computation motivates the definition, for b >0 : S(b;r)def= P1/b(br) =Q
2r 1 +√
1−b2r2
forr≤ 1
b. (18) Indeed if we define
S(b;t, r)def= P1/b r
|t|
=S
b; r b|t|
, (19)
S(b;t, r) satisfies system (1) with initial data (at time t=−1/b) : S(b;−1b, r) =S(b;r),
St(b;−1b, r) =brSr(b;r). (20) S(b;t, r) is defined in the interior of the cone{(t, r)|r≤b|t|}; it blows up at timet= 0, and its life-span is 1/b. Observe that from the proof of Corollary 5, it follows thatS(b;r) satisfies the equation :
(b2r2−1)∆S(b;r) +b2rSr(b;r) =−f(S(b;r)) r2 . One important thing to notice is the following :
Claim.
∀A≥0, limb→0k(S(b;r), brSr(b;r))−(Q,0)kH([0,A]) →0, but∀b >0, kS(b, r)kH = +∞.
This is due to the specific form ofPα. If the convergence were inH(R+), we would obtain a family of blowing up wave maps, whose initial data would
converge toQ(inH). But these blowing up wave maps are always of infinite energy. Indeed, if we compute at point ξ= 1 (c.f. (45)) :
Sr
b;1 b −ε
∼Q0 1
b
· rb
ε,
(as Q0(1/b) 6= 0), and Sr2(b) ∼ Cbε gives a logarithmic divergence when integrating inrdr, and thus an infinite energy. However, if we were to forget this infinite energy tail, S(b;t, r) = S(b;r/(b|t|) has the “profile” Q(λ(t)r) with λ(t) = 1/(b|t|). We recover the fact that λ(t) → ∞ as t ↑ 0, that is, blow-up. Furthermore 1/|t| is the self-similar blow-up rate. Notice that this rate can never be the blow-up rate of a finite energy wave map (cf. (8)).
In higher dimension, self-similar solutions are some examples of blowing up wave maps with smooth initial conditions (see [4, ch. 7]). In dimension 2, no blowing up wave map of finite energy is known.
The next step is to regularize S(b;r) to obtain initial conditions in H.
2.3 Construction of approximation of self-similar profiles in the energy space
For simplicity in writing expressions in throughout this section, let us first introduce the following notations.
Letc0<1. Notation.We define :
• C(c0), a constant that may change form line to line, but which does only depend onc0 <1 (as c0 ↑1,C(c0 may tend to +∞).
• the intervalI(b, r) = [r,2r/(1 +√
1−b2r2)],
• the functions h(r) = supρ∈[Q(r),C∗]|g(ρ)| = supθ∈[r,∞)|g(Q)(θ)|, and h2(r) = r12
Rr
0 h2(s)sds.
Observe that if g is decreasing in a neighborhood of C∗, (which is always the case ifg0(C∗)<0), h(r) =g(Q)(r) in this neighborhood. In any case,h decreases to 0 asr → ∞. In particular,h2(r)→0 as r→ ∞. This gives the existence of a constantA≥10 such that forα≥A:
h(α)≤0.01, and h2(α)≤0.01.
This section is devoted to the proof of the following proposition :
Proposition 3. Let α, b >0such that bα≤c0<1, andα≥A. There exist C2 functions S0(α, b;r) andS1(α, b;r), defined for r∈R+, and satisfying :
S0(α, b;r) = S(b;r) if 0≤r≤α,
S0(α, b;r) = Q(r) if r ≥α(1 +h(α)), (21) S1(α, b;r) = brSr(b;r) if 0≤r≤α,
S1(α, b;r) = 0 if r ≥α(1 +h(α)). (22)
With the following estimate :
k(S0(α, b;r), S1(α, b;r))−(Q(r),0)k2H ≤C(c0)(b2α2h2(α) +h(α)). (23) Remark.
1. These modified profiles are simply truncated self-similar wave maps (at pointα), that were smoothly reconnected toQ (at pointα(1 +h(α)).
2. As we shall see through the proof, the contribution before truncation is inh2(α), and the contribution for joining is inh(α), which is often worse (one should think ash2 ∼ h2). So the global contribution is in h(α).
Proof. The proof goes as follows. As the values of (S0(α, b;r), S1(α, b;r)) are given on [0, α], we only have to compute the desired estimate on this interval : this is done in lemma 2 (pointwise estimate) and 3 (Hestimate). On [α, α(1+
h(α))], we need both the construction of a smooth reconnection, and the estimates that goes along with it : this is lemma 4. The part [α(1+h(α)),∞) does not add any contribution. The proofs are mainly computational, and are postponed to Appendix B.
Lemma 2. If br≤c0, then :
|S(b;r)−Q(r)| ≤C(c0)b2r2h2(r), (24)
|Sr(b;r)−Qr(r)| ≤C(c0)b2rh(r). (25) Lemma 3. For bα≤c0 andα≥A, then :
kS(b;r)−Q(r)k2H([0,α])≤C(c0)b4α4h2(α), (26) kbrSr(b;r)k2L2([0,α])≤C(c0)b2α2h2(α), (27) kQkH([α;∞))≤2|C∗−Q(α)|h(α). (28) Lemma 4 (Joining lemma). Let v : I = [0, a] ∪[b,∞) → R. Then there existsv˜: R+→Rextending v, as smooth as v, such that :
kv˜k2H ≤ kvk2H(I)+ (v(b)−v(a))2b+a
b−a+ 2 max(v(a), v(b))2lnb
a. (29) To construct the Si(α, b;r), we only have choice on the interval r ∈ [α, α(1 +h(α))]. For S0(α, b;r) we use the joining Lemma 4 to obtain a smooth reconnection. For S1(α, b;r), the density of smooth functions in L2 allows us to have a smooth reconnection such that :
kS1(α, b;r)k2L2([α,α(1+h(α))]) ≤h(α).
Let us now turn to estimate (23). First, by estimate (27) we have :
kS1(α, b;r)k2L2 ≤C(c0)b2α2h2(α) +h(α). (30) Let us now focus on S0(α, b;r) −Q(r). For r ≥ α(1 +h(α), there is no contribution. On [0, α], we use lemma 3 : it gives the bound
C(c0)b4α4h2(α)≤C(c0)b2α2h2(α). (31) On [α, α(1 +h(α))], in order to use estimate (29), we compute (with Rolle’s theorem, using h(α)≤0.01)) :
|S(b;α)−Q(α(1 +h(α))| =
Q
2α 1 +√
1−b2α2
−Q(α(1 +h(α)))
≤α
2 1 +√
1−b2α2 −(1 +h(α))
sup
ϑ≥α|Qr(ϑ)|
≤α(1 +h(α))h(α)/α≤h(α).
So the second term on the right hand side of (29) can be estimated by (α≥10):
4h(α)2α(2 +h(α))
αh(α) ≤4h(α).
And for the third term of (29), it remains to notice that S(b;α)2, Q(α(1 + h(α))2 ≤ C∗2 and the well known : ln(1 +h(α)) ≤ h(α). Let’s also allow h(α) for the time-derivative estimate on this interval (using the density of regular function in L2). So the contribution of kS0(α, b;r)−Q(r)kH on [α, α(1 +h(α))] is bounded by :
(2C∗2+ 4)h(α). (32)
Summing up the contributions (31) and (32), we get :
kS0(α, b;r)−Q(r)kH ≤C(c0)b2α2h2(α) +Ch(α). (33) (33) and (30) give the estimate (23).
2.4 Finite speed of propagation and conclusion
Before proving Theorem 1, let us remind a consequence of the finite speed of propagation for system (1).
Proposition 4. Let (u0, u1) and(v0, v1)be a couple of initial data (at time t = 0) with finite energy on the interval [0, R). Let u, v be the respective solutions to (1) arising from them. Suppose that :
∀r∈[0, R), (u0, u1)(r) = (v0, v1)(r).
Suppose that u is defined up to time T >0, and let T0 = min{R, T}. Then v does not blow up before time T0, and fort∈[0, T0] we have :
∀r∈[0, R− |t|), u(t, r) =v(t, r).
Remark. Of course, if (u0, u1) and (v0, v1) coincide on (a, b), with a > 0, then at time t∈[(a−b)/2,(b−a)/2],u(t) andv(t) coincide on the interval r∈(a+|t|, b− |t|).
Proof. It relies on the proof of Theorem 1.1 of [6] : more precisely, this Theorem is a direct consequence of the following claim, which is what is proved indeed in [6]
Claim : There exists ε0 > 0 such that the following is true. Let (u0, u1), initial data at timet0, have energy less thanε0onB(R0, C) = (R0−C, R0+ C). Then exists a unique solution u to (1) on the full cone of dependence K(R0, C) ={(t, r)||r−R0|< C− |t−t0|}.
Proposition 4 is also a direct consequence of this claim. Let T1 ≥0 be the greatest time for which uand v coincide on the truncated cone
{(t, r)|t∈[0, T1] and 0≤r < R−t}.
First,T1>0 : indeed, we divide [0, R] into finitely many overlapping inter- vals (an, bn) such that on every interval, the energy of (u0, u1) is less thanε0. Thanks to the claim, on every cone of dependence associated to (an, bn), v (exist and) coincides withu. As the (an, bn) overlap and are in finite number, there exists δ >0 such that u and v coincide on the truncated cone
{(t, r)|t∈[0, δ] and 0≤r < R−t}.
Thus, T1 ≥ δ > 0. Now, if T1 < T0 : u(T1) = v(T1), ut(T1) = vt(T1) on [0, R−T1), we can repeat the same argument at timeT1 to obtain a greater time for the truncated cone on whichuand vcoincide, and this contradicts the maximality ofT1. Hence T1 =T0.
Proof of Theorem 1. The idea is the following. We know the evolution of S(b) : given initial dataSi(α, b;r), we deduce from Proposition 4 what hap- pens at time ton the space interval r∈[0, α−t].
The problem is now to chooseα =α(b) large enough so that our control takes place for large enough times (so that the scaling parameterλchanges), but not too large so that the initial data is close to (Q,0) in H : there we need the estimates of Proposition 3.
Choose a fixedc0 <1, and set : λ0= 1
1−c20 >1 and for b >0, α=α(b) = c0 b .
Define R(b, t;r) as the wave map arising from the regularized initial condi- tions withα(b) = 2c0/b :
R(b; 0, r) =S0(c0/b, b;r),
Rt(b,0, r) =S1(c0/b, b;r). (34) First, (R(b; 0, r), Rt(b; 0, r)) coincide with (S(b,−1/b, r),St(b,−1/b, r)) on the interval [0, c0/b], and is of finite total energy. Thus, Proposition 4 ensures thatR(b;t, r) is defined at least up to timeT =c0/b(asS(b;t, r) is defined on an interval of length 1/b≥2c0/b), and :
∀t∈h 0;c0
b
i,∀r≤ c0
b −t, R(b;t, r) =S
b;−1 b +t, r
=S
b; r 1−bt
.
In particular, fort(b) =c0(1−c0)/b :
∀r≤ c20
b, R(b;t(b), r) =S(b;λ0r). (35) Now, we use the estimate (23) (fort= 0) and we obtain :
k(R(b; 0, r), Rt(b; 0, r)−(Q(r),0)k2H ≤C(h(c0/b) +h2(c0/b))→0 as b→0.
This is the first condition (10). It remains to estimate what happens at time t(b) =c0(1−c0)/b, i.e. to bound :
k(R(b;t(b), r), Rt(b;t(b), r))−(Q(λ0r),0)kH.
Consider separately the contributions on [0, c20/b] and on [c20/b,∞). On the first interval, estimates of Lemma 3 give the bound
k(R(b;t(b), r), Rt(b;t(b), r)−(Q(λ0r),0)k2H([0,c20/b]) ≤Ch2(c20/b). (36) On the second interval, we work out separately
k(R(b;t(b), r), Rt(b;t(b), r)kH([c20/b,∞)) and kQ(λ0r)kH([c20/b,∞)). (37) Let us focus on the first term of (37). In view of the initial conditions (34),
E(R(b))≤E(Q) +C(h2(c0/b) +h(c0/b)).
On the other side, we use our control on the interval [0, c0/b] : k(S(b;λ0r), bλ0rS(b;λ0r))kH([0,c20/b])≥ kQkH([0,c20/b])
≥E(Q)−2(C∗−Q(c20/b))h(c20/b).
(First inequality because S(b;λ0r) ≥ Q(r) and the pointwise inequality, second inequality due to (28)). Hence we have (ash is decreasing) :
k(R(b;t(b), r), Rt(b;t(b), r))kH([c20/b,∞))
=E(R(b))− k(S(b;λ0r), bλ0rS(b;λ0r))kH[0,c20/b]
≤E(Q) +C(h(c20/b) +h2(c20/b))−(E(Q)−Ch(c20/b))
≤C(h(c20/b) +h2(c20/b)).
(38) For the second term of (37), using (28), we obtain :
kQ(λ0r)kH([c20/b,∞))≤2(C∗−Q(λ0c20/b))h(c20/b) =o(h(c20/b)). (39) Finally adding up (36), (38) and (39), we have :
k(R(b;t(b), r), Rt(b;t(b), r))−(Q(λ0r),0)kH
≤C(h(c20/b) +h2(c0/b) +h2(c20/b))→0 as b→0.
This exactly the second condition (11). To conclude, choose a sequence (bn) decreasing to 0, and define the sequence :
un(t, r) =R(bn;t, r), tn= (1−c0)c0/bn.
The previous computations show that the initial data (un(0), unt(t)) → (Q,0) in H and that (un(tn), unt(tn))→ (Q(λ0·),0) in H asn → ∞. This is true for anyc0 <1, and thus for any λ0 = 1/(1−c20)>1. The instability inH is then straightforward.
Remark.
1. As k(R(b; 0, r), Rt(b,0, r)−(Q,0)kH → 0, E(R(b)) → E(Q) : for b small enough,R(b) admits a decomposition and a scaling factor λb(t) for allt up to an possible blow-up time. Theorem 1 simply says that λb(c0/b)→λ0.
2. The proof of Theorem 1 also gives the time to leave a neighborhood of Q. Suppose we start in aδ-neighborhood ofQ: then the corresponding b is such that h(c0/b) +h2(c0/b) ∼δ. We have to wait time c0/2b to leave the neighborhood, that is approximatelyh−1(δ). Let us give an example.
In the case of Corollary 2 (g= sin), studied in [1], we have : C∗ =π, andg0(π) =−1, soh(r)∼1/r, h2(r)∼lnr/r2, sokQ˜b−QkH ∼1/√
b.
Thus the time to exit aδ-neighborhood is O(1/δ2).
In the case of Corollary 3 (g(ρ) = (1−ρ2), we have :C∗ = 1, g0(1) =
−2, so h(r)∼1/r2, h2(r)∼1/r2, sokQ˜b−QkH ∼1/b, and the time to exit aδ-neighborhood is O(1/δ).
Appendix A : Decomposition of a wave map with energy close to E(Q)
Proposition 2 (Decomposition). There exist α0 > 0 and an increasing function δ : [0, α0]→ R+, with δ(α) →0 as α→ 0, such that the following is true.
Suppose v is a function of finite energy, withv(0) = 0andv(r)→C∗ as r→ ∞ and such that :
E(v) =E(Q) +α < E(Q) +α0 Then there exist λ∈R+
∗ and ∈H such that :
v(r) =Q(λr) +(r), and kkH ≤δ(α).
Proof. Recall thatG(s) =Rs
0 |g|,E(Q) = 2G(C∗), and thatGis increasing.
DenoteD∗the unique point such thatG(Q(D∗)) =G(C∗)/2 (or equivalently E(Q,0, D∗) =E(Q)/2.
We proceed by contradiction. Suppose that there exist δ0 > 0 and a sequence of finite energy functionsvn, such that
E(vn)≤E(Q) + 1
n, vn(0) = 0, vn(r)→C∗ as r→ ∞, and :
∀n∈N, ∀λ >0, kvn−Q(λ·)kH ≥δ0. (40) Set wn(r) = vn(r/λn), where λn is such that wn(D∗) = Q(D∗) (this is possible because vn is continuous). Using scaling invariance, we have :
E(wn) =E(vn)≤E(Q) + 1 n.
(wnr) isL2(rdr)-bounded, and so, on (0,∞), (wn) is locallyC1/2-continuous, and so locally equicontinuous. Furthermore :
|G(wn(a))−G(wn(b))| ≤ Z b
a |g(wn(ρ)wnr(ρ)|dρ≤E(wn, a, b).
Apply with a → 0, b = r, and then a= r, b → ∞. As g is not uniformly 0 outside [0, C∗], we deduce that for N large enough, the (wk)k≥N are uni- formly bounded in C0([0,∞)) by a constant K. So one can apply Ascoli’s theorem : for any compact setX of (0,∞), (wn|X)n has a compact closure inC0(X). Let us construct a diagonal extraction. Let m ∈N, and suppose we already constructed an extraction φ1 ◦ · · · ◦φm : N → N to obtain a converging subsequence
wφ1◦···◦φm(n)|[1/m,m]→w inC0([1/m, m]) as n→ ∞.
We can then construct φm+1 so that the convergence of wφ1◦···◦φm◦φn+1(n)
takes place inC0([1/(m+1), m+1]) (by applying Ascoli’s theorem on [1/(m+
1), m+ 1]). Now, define :
ϕ(n) =φ1◦ · · · ◦φn(n).
For any m, and n ≥ m, wϕ(n) is a subsequence of wφ1◦···◦φm(n), and hence converges in the space C0([1/m, m]). Let us denote again w ∈C0(R+∗) the common limit. For convenience, we can consider the subsequence as the initial sequence, and thus drop theϕ. We obtained :
∀m∈N, wn→w C0([1/m, m]).
We can also suppose that wnx* wx weakly inL2(rdr). In particular,w is continuous,w(D∗) =Q(D∗). By weak limit :
Z
w2rrdr≤lim infwn2 rrdr.
And by Fatou lemma (of course, there is a.e. convergence) : Z
g2(w)dr/r≤lim inf Z
g2(wn)dr/r.
So that E(w) ≤E(Q). Moreover w(D∗) =Q(D∗). As w is of finite energy, wadmits limits inr= 0 andr→ ∞, where gvanishes : using the pointwise inequality between 0, D∗ and ∞, we deduce that the only possibility for these limits are 0 andC∗. Let us now prove that :
∀r≥D∗, w(r)≥Q(D∗), and ∀r≤D∗, w(r)≤Q(D∗). (41) We prove only one of these inequalities, the second one can be deduced in the same way. We again proceed by contradiction. Letr < D∗such that w(r)>
Q(D∗) : denoteε=G(w(r))−G(C∗)/2>0. Due to uniform convergence on compact sets, there existsN such that forn≥N,G(wn(r))−G(C∗)/2≥ε.
Then : E(Q) + 1
n ≥E(wn)≥E(wn,0, r) +E(wn, r, D∗) +E(wn, D∗,∞)
≥2
G(wn(r)) +|G(wn(r))−G(C∗)/2|+G(C∗)/2
≥2(G(C∗) + 2ε) ≥E(Q) + 4ε.
This is impossible if n ≥ 1/(4ε), and proves (41). Thus, we conclude that w(0) = 0 andw→C∗ asr→ ∞. Together withE(w)≤E(Q) andw(D∗) = Q(D∗), the variational characterization ofQallows to conclude thatw=Q.
Let us now prove that kwn−QkH = E(wn−Q) → 0. First, E(wn) → E(Q), so that limkwnrkL2(rdr) = kQrkL2(rdr), and the weak convergence wnr * Qr is in fact strong-L2(rdr) :
kwnr−QrkL2(rdr)→0. (42)
Let us now consider R
g2(wn−Q)dr/r. Letε >0. Definec, d >0 such that E(Q,0, c) ≤ ε/16 and E(Q, d,∞) ≤ ε/16. The convergence of wn to Q in C0([c, d]) gives the existence of N ≥4/εsuch that
∀n≥N, Z d
c
g2(wn−Q)dr
r ≤ε/2.
(because g is continuous at 0). Again by convergence in C0([c, d]), we can choose N such that :
∀n≥N, |G(wn(c))−G(Q(c)|+|G(wn(d))−G(Q(d))| ≤ε/16.
Hence (pointwise inequality),E(wn, c, d) ≥E(Q, c, d)−ε/8≥E(Q)−ε/4.
In view ofN ≥4/ε, we obtain
E(wn,0, c) +E(wn, d,∞) ≤ε/2.
So that forn≥N : Z
g2(wn−Q)dr r ≤
Z c 0
+ Z d
c
+ Z ∞
d ≤E(wn,0, c) +ε/2 +E(wn, d,∞)≤ε.
This together with (42) proves that kwn−QkH =kvn−Q(λn·)kH → 0 as n→ ∞ : a contradiction with (40).
Corollary 4. Let T > 0 and ε >0. Then there exists η > 0 such that for all initial data (u0, u1) satisfyingk(u0, u1)−(Q,0)kH ≤η, the arising wave map u is defined at least up to time T and :
sup
t∈[0,T]k(u, ut)−(Q,0)kH ≤ε.
Proof. We will choose η >0 small enough later.
First, let us notice that we can use the decomposition. Observe indeed that the energyE(u0, u1)≤E(Q) +Cη2 ≤E(Q) +α0, ifη is so small that Cη2 ≤ α0. Hence, u0 has limits at r = 0 and as r → ∞, that are zeros for g. As 0 and C∗ are isolated zeros of g, we necessarily have u0(0) = 0, u0(r)→C∗ asr→ ∞, if we choose η >0 small enough.
The local existence Theorem gives a maximal time T∗ >0 of existence : we can consideru(t) the arising wave map. By conservation of energyE(u)≤ E(Q) +Cη2, we get :
Z
u2trdr≤Cη2. (43)
In particular, asu(t,0) and limr→∞u(t, r) are always zeros forg, we obtain that for all t < T∗, u(t,0) = u0(0) = 0 and limru(t, r) = limru0(r) = C∗ (see [7, Lemma 1]). Therefore, we can apply Proposition 2 : for allt < T∗, there exists λ(t), (t, r) such that
u(t, r) =Q(λ(t)r) +(t, r), and k(t)kH ≤δ(Cη2).
We can choose λ(0) = 1. Now, as noticed earlier, blow-up for u is charac- terized byλ(t)→ ∞. From now on, we will consider T0 ≤T∗ maximal such that :
∀t < T0, λ(t)≤2 +ε.
Then (if δ(Cη2) is small enough) :
ku(t)−QkH ≤C|λ(t)−1|. (44) Now fixa∈(0,1/2), such that forr ∈[1−a,1 +a],Qr(r)≥Qr(1)/2. Using (43), we have :
Z t
0
Z 1+a
1−a
utdrdt≤ Z t
0
s Z
u2trdr s
Z 1+a
1−a
dr
r dt≤Cηt.
On the other side : Z t
0
Z 1+a 1−a
utdrdt= Z 1+a
1−a
(u(t, r)−u(0, r))dr
= Z 1+a
1−a
(Q(λ(t)r)−Q(r) +(t, r)−(0, r))dr.
Now, ask(t)kL∞ ≤C(k(t)kH)≤c(η) (for some functioncsuch thatc→0 at 0) :
Z 1+a 1−a
((t, r)−(0, r))dr ≤2c(η).
And :
Z 1+a 1−a
Q(λ(t)r)−Q(r)dr ≥ 1
2|λ(t)−1| Z 1+a
1−a
Qr(1)rdr≥C|λ(t)−1|. Combining our two expressions for the integral, we deduce :
|λ(t)−1| ≤C(ηt+c(η)).
And finally, using (44) andkutk2L≤η :
k(u(t), ut(t)−(Q,0)kH ≤C(ηt+η+c(η)).
It is enough to choose η ≤ α0/C so that C(ηT +η+c(η)) < ε. Indeed, the previous computations are then valid up toT0. Now ifT0 < T, we have that for all t ∈ [0, T0], λ(t) ≤ 1 +ε < 2 +ε, therefore the solution can be continued (blow-up hasn’t occurred) and the continuity of λ(t) contradicts the maximality ofT0.
Remark. The proof gives a more accurate result : the time to leave a η neighborhood is at leastO(1/η). This is coherent with the computations for the sequence un of Theorem 1.
Appendix B
Here, we prove the computational lemmas needed for Proposition 3.
Proof of Lemma 2. Let us note ϑ= 2r
1+√
1−b2r2. We compute explicitly : Sb(b;r) =Qr(ϑ) 2br3
1 +√
1−b2r22√
1−b2r2 .
So if we plug inbr≤0.01 and Qr=g(Q)/r :
|Sb(b;r)| ≤C(c0)Qr(ϑ)br3 ≤C(c0)g(Q)(ϑ)br2. Now
|S(b;r)−Q(r)| ≤ Z b
0 |Sb(b;r)|db≤C(c0) sup
θ∈I(r,b)|(g(Q))(θ)|b2r2. For the second estimate, we again compute explicitly :
Sr(b;r) =Qr(ϑ)
2 1 +√
1−b2r2 + 2r (1 +√
1−b2r2)2
b2r
√1−b2r2
. (45) Now, we can compute :
Sr,b(b;r) =Qrr(ϑ) 4r3b
1 +√
1−b2r23
(1−b2r2)
−Qr(ϑ) 2br2
2b2r2−3−3√
1−b2r2 (1−b2r2)3/2
1 +√
1−b2r23. (46) Recall Qr = g(Q)/r and Qrr = ((g0 −1)g)(Q)/r2. We plug in again br ≤ c0<1, and we get :
|Sr,b(b;r)| ≤C(c0)rb |((1−g0)g)(Q)(ϑ)|+g(Q)(ϑ)
=C(c0)((1 +|1−g0|)g)(Q)(ϑ)br.
So when we integrate inb, we have :
|Sr(b;r)−Qr(r)| ≤ Z b
0 |Sr,b(b;r)|db
≤C(c0)b2r sup
θ∈I(b,r)|(g+|1−g0|g)(Q)(θ)|
≤C(c0)b2r 1 + sup
θ∈I(b,r)|g0|(Q)(θ)
! sup
θ∈I(b,r)|g(Q)(θ)|. And as Qtakes its values in [0, C∗], we get the second estimate.
Proof of Lemma 3. We integrate the pointwise estimate of the previous lem- ma.
kS(b;r)−Q(r)k2H([0,α]) = Z α
0
(S(b;r)−Q(r))2r+ (S(b;r)−Q(r))2 r2
rdr
≤ Z α
0
D0h(r)b2r2
+
h(r)b2r2 r
2! rdr
≤C(c0)b4 Z α
0
h2(r)r3dr≤C(c0)b4α2·α2h2(α).
For the time-derivative estimates, we use directly (45), and we plug inbr≤ bα≤c0 <1. This leads to :
|Sr(b;r)| ≤ |Qr(ϑ)|(2 +C(c0)b2r2)≤C(c0)h(r)/r, so that :
kbrSr(b;r)k2L2[0,α]≤C(c0)b2 Z α
0
h2(r)rdr.
For the third bound, we compute, usingrQr=g(Q) : kQkH([α;∞)) = 2
Z ∞
α
g(Q)Qrdr= 2 Z C∗
Q(α)
g≤2|C∗−Q(α)|h(α).
Proof of Lemma 4. Let us first do the computations for an affine interpola- tion :
For r∈[a, b],v(r) = (v(b)˜ −v(a))r−a
b−a+v(a), i.e. ˜vr(r) = v(b)−v(a) b−a . Then :
Z b a
˜
vr2rdr= 1 2
v(b)−v(a) b−a
2
(b2−a2)≤ 1
2(v(b)−v(a))2b+a b−a, Z b
a
˜ v2
r dr≤max(v(b), v(a))2 Z b
a
dr
r ≤max(v(b), v(a))2lnb a.
This already gives an extension inH. For a smoother extension, it is enough to regularize ˜vr(locally nearaandb) with a small variation of theL2 norm, while keeping constant ˜v(b)−v(a) =˜ Rb
av˜rdr. This is possible thanks to the density of smooth functions with given mean among L2 functions with the same given mean.
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Rapha¨el Cˆote
D´epartement de Math´ematiques et Applications Ecole normale sup´erieure´
45, rue d’Ulm, 75230 Paris Cedex 05, France raphael.cote@ens.fr
