# Blow-up on manifolds with symmetry for the nonlinear Schröding

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1 Blow-up on manifolds with symmetry for the nonlinear Schrödinger equation March, Université de Nice

2 Euclidean L 2 -critical theory Consider the one dimensional equation i t u + u = u 4 u, t > 0, x R. Local well-posedness: This equation is well-posed in H 1 : if u 0 H 1 (R), there exists a maximal time T > 0 and a solution u C([0, T ), H 1 (R)) C 1 ([0, T ), H 1 (R)) with u(0) = u 0 and with the criteria T < implies u(t) L 2.

3 Goal Blow up problematic: Estimate blow-up rate u(t) L 2, Find singularities (points of concentration of mass), How much mass is concentrated?

4 Self-similar regime (in Ḣ 1 ): Blow-up regimes known u(t) L 2 (R) 1 (T t). Pseudo-conformal regime (in H 1 ): u(t) L 2 (R) 1 T t. Log-log regime (Merle-Raphaël, Perelman) (in H 1 ): ( ) log log(t t) 1/2 u(t) L 2 (R). T t

5 Classification near the ground state Let Q be the 1D ground state: yy Q + Q = Q 4 Q, x R. Theorem (Merle-Raphaël) There exists α > 0 such that for all u 0 H 1 (R) such that Q L 2 (R) u 0 L 2 (R) Q L 2 (R) + α and E(u 0 ) < 0, u(t) blows up in H 1 with the speed ( ) log log(t t) 1/2 u(t) L 2 (R). T t

6 Questions Do these regimes persist in other geometries? - lack of strong dispersion (losses in Strichartz estimates) - no smoothing effect - loss of symmetry (translation in space) and remarkable identity (Virial relation) We treat the log-log regime.

7 Quintic Schrödinger equation on surfaces i t u + u = u 4 u, t > 0, x M, where (M, g) is a two dimensional complete Riemannian manifold. Hamiltonian structure: L 2 and H 1 conservation laws: M(u(t)) := u(t) 2 = M(u(0)), E(u(t)) = 1 2 M M u(t) M u(t) 6 = E(u(0)). Local well-posedness in H 1 (Burq-Gérard-Tzvetkov): if u 0 H 1 (M), there exists a maximal time T > 0 and a solution u C([0, T ), H 1 (M)) C 1 ([0, T ), H 1 (M)) with u(0) = u 0 and with the criteria T < implies u(t) L 2.

8 Scaling lower bound Proposition If u 0 H 1 (M) is such that T (u 0 ) < then Key points: scaling symmetry u(t) L 2 (M) uniform Strichartz estimate C (T (u 0 ) t) 1/4.

9 Proof. Let L 1, M L = L.M and consider the equation on M L. 1) Uniform in L Strichartz estimates: if 1/q + 1/r = 1/2 and q > 2 then there exists C > 0 such that for all L 1 and u L 0 H1/q (M L ): e it M L u L 0 L q ([0,1])L r (M L ) C u L 0 H 1/q (M L ). if u0 L H1 (M L ) then T (u0 L) F ( ul 0 H1) for a function F independent of L. 2) Use of the scaling symmetry: if u is a solution on M then v τ (t, x) = λ 1/2 (τ)u(λ 2 (τ)t + τ, λ(τ)x) is a solution on M 1/λ(τ) for all τ and λ(τ). 3) Apply 1) to v τ with the good rescaling λ(τ) = u(τ) 2 L 2.

10 Log-log regime with radial symmetry Theorem Let (M, g) be a rotationally symmetric surface with g = dr 2 + h 2 (r)dθ 2, 0 < r < ρ. Assume in the non-compact case (ρ = ) h (r) Ch(r) for r large. Consider the equation on M (dim(m) = 2): i t u + u = u 4 u, t > 0. Then there exists an open set P of Hrad 2 (M) such that if u(0) P then u blows up with the log log speed on a set {x M, r(x) = r 0 } for some r 0 = r(pole) > 0. Prototypes. Compact: sphere Non-compact: hyperbolic space Euclidean space (treated in H 1 rad (R2 ) by P. Raphaël).

11 Heuristic In radial coordonates and for a radial solution, the equation becomes i t u + 2 r u + h (r) h(r) r u = u 4 u, u(t, x) = u(t, r), Outside poles h (r) h(r) r 2 r. The equation is almost L 2 -critical outside poles: we prove a radial Sobolev embedding on M: for a radial function u on M: u L p (ε<r(x)<ρ ε) C(ε) u H s (ε<r(x)<ρ ε), p p = 2 1 2s. Near poles: we are outside the blow-up set better estimates on u: good H 1/2 estimate.

12 Modulation Splitting the dynamic: we choose u 0 such that u 0 L 2 (M) Q L 2 (R) so that until a time t 1 > 0: u(t, r) = 1 ( ( ) r r(t) Q b(t) + ε λ(t) λ(t) ( t, r r(t) λ(t) )) e iγ(t), with if y = (r r(t))/λ(t) and for k = 0, 1, 2: y k ε(t) 2 µ(y)dy <, µ(y) = h(λ(t)y+r(t))1 { } r(t) ρ r(t) (y), y λ(t) λ(t) with orthogonality conditions: Re(ε(t), y 2 Q b(t) ) = Re(ε(t), yq b(t) ) = 0, Im(ε(t), ΛQ b(t) ) = Im(ε(t), Λ 2 Q b(t) ) = 0. where Λ = y y. Q b is a troncated version of refined profiles: 2 y Q b Q b + ibλq b + Q b 4 Q b = O(e π b ) 1.

13 Control of the finite dimensional part Modulation equations : Time rescaling: s = t dτ 0 λ 2 (τ). Then λ s λ + b + b s + r s CE(s) + Γ 1 λ b, γ s 1 (Re ε, L +(Λ 2 Q)) ΛQ 2 δe 1/2 (s) + Γ 1 b, L 2 where E(s) = δ 1, Γ b e π b, as b 0, y ε(s, y) 2 h(λ(s)y +r(s))dy + ε(s, y) 2 e y dy, y 10 b(s) and (L +, L ) is the linearized operator near Q: L = 2 y + 1 5Q 4, L + = 2 y + 1 Q 4. Extra terms due to h h r are controled by the smallness of λ(s): λ Γ b.

14 Control of the infinite dimensional part, I Goal: use a virial type argument to control the rest ε Heuristic: u satisfies: For the 1D Euclidean equation i t u + rr u u 4 u. i t u + u = u 4 u, x R, we have the Virial relation: if u Σ := {u H 1, xu L 2 } then d 2 dt 2 x 2 u 2 dx = 4 d dt Im ( ) x uu = 16E(u 0 ).

15 Control of the infinite dimensional part, II Thus, we expect: ( d dt Im ) φ(r)r r uu 4E 0, where φ is a cut-off function avoiding poles. We expand in term of ε and switch in (s, y) variables to obtain where A 0 d ds Im A 0 + A 1 + A 2 = 4λ 2 E 0, y y Q b Q b Db s, D > 0, A 1 = 2 d ds Im(ΛQ b, ε) = 0 (orthogonality condition), A 2 = d ds Im y εεψ(y)µ(y)dy, ψ(y) = φ(µ(y)). The symbol means modulo O(Γ b ) + o(e(s)).

16 Control of the infinite dimensional part, III Control of the A 2 term. The equation on ε is roughly: so that A 2 becomes i s ε + Lε = Ψ = O(Γ b ), A 2 H(ψε, ψε) for a quadratic form H. Finally, with λ 2 E 0 Γ b, we get with δ 1. Cb s H(ψε, ψε) Γ 1 b δe(s)

17 Control of the infinite dimensional part, IV The form H is almost coercive: ( H(ψε, ψε) C E(s) (Re(ψε), Q) 2 (Re(ψε), y 2 Q) 2 (Re(ψε), yq) 2 (Im(ψε), ΛQ) 2 (Im(ψε), Λ 2 Q) 2 (Im(ψε), y Q) 2). Control of negative directions. Linearization of conservation laws (energy (needs an H 1/2 control), localized momentum) Orthogonality conditions Conclusion. first Virial estimate: (ψε i,...) 2 δe(t), δ 1. Db s E(s) Γ 1 b. In particular, this is well defined in H 2 (M).

18 Control of the infinite dimensional part, V Refined virial estimate: Virial estimate in terms of ε = ε ζ b where ζ b is a troncated version of ζ b : 2 y ζ b ζ b + ibλζ b = Ψ b, lim y ζ b(y) 2 = Γ b, y implies ( ) d ds J cb(s) Γ b(s) + Ẽ(s) + ε 2, A y 2A with J (s) d 0 b 2 δb 2, δ 1. Once again, additional terms due to h /h r u controled by the smallness of λ: λ Γ b.

19 Smallness of the critical norm near poles, I To obtain a control from the conservation of energy, we need to prove: ũ 6 ũ(t) 2. ũ(t, r) = 1 ( e iγ(t) ε t, r r(t) ). λ(t) λ(t) Outside poles: this is the conservation of mass Near poles: H 1/2 control. Derivation of a pseudo-energy E 2 at level H 2 with u(t) 2 H 2 (M) CE 2(u(t)), and d dt E 2(u(t)) C( u(t) H 1/2, u(t) H 1, u(t) H 3/2)E 2 (u(t)) 1 θ, θ (0, 1)

20 Smallness of the critical norm near poles, II This implies an H 2 estimate u(t) H 2 (M) 1, for some η > 0. λ(t) 2+η Moreover, for all 0 < a 2 < a 1 < b 1 < b 2, and t > 0, D s u L [0,t) L 2 (a 1,b 1 ) C ( D s u(0) L 2 (a 2,b 2 ) + u L 2 [0,t) H max(1,s+ 1 2 ) (a 2,b 2 ) + D s (u u 4 ) L 1 [0,t) L 2 (a 2,b 2 )), we get u(t) H 1/2 ( r 1 >1/2) 1.

21 Integration of modulation equations The integration of gives with s = t 0 b s e π b, dτ λ 2 (τ) : 1 λ(t) u(t) L 2 λ s λ b, r s 1, λ ( ) T t 1/2 C, log log(t t) and r t integrable thus r(t) r(poles).

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