Blowup on manifolds with symmetry for the nonlinear Schröding


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1 Blowup 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 wellposedness: This equation is wellposed 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 blowup rate u(t) L 2, Find singularities (points of concentration of mass), How much mass is concentrated?
4 Selfsimilar regime (in Ḣ 1 ): Blowup regimes known u(t) L 2 (R) 1 (T t). Pseudoconformal regime (in H 1 ): u(t) L 2 (R) 1 T t. Loglog regime (MerleRaphaël, 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 (MerleRaphaël) 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 loglog 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 wellposedness in H 1 (BurqGérardTzvetkov): 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 Loglog 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 noncompact 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 Noncompact: 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 blowup 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 cutoff 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 pseudoenergy 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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