Scattering for cubic-quintic nonlinear Schrödinger equation on R 3

Transcription

1 Scattering for cubic-quintic nonlinear Schrödinger equation on R 3 Oana Pocovnicu Princeton University March 9th 2013 Joint work with R. Killip (UCLA), T. Oh (Princeton), M. Vişan (UCLA) SCAPDE UCLA 1 / 23

2 Cubic-quintic nonlinear Schrödinger equation on R 3 : { i t u + u = u 4 u u 2 u (CQ) u(0) = u 0 H 1 (R 3 ) Model in various physical problems in : nonlinear optics, plasma physics, Bose-Einstein condensation Conserved quantities : mass, energy, and momentum : u M(u) = u 2 2 dx, E(u) = + u u 4 dx, P (u) = Im 4 u udx Globally well-posed in H 1 (R 3 ) Scattering : there exist u + and u in H 1 (R 3 ) such that lim u(t) t ± eit u ± H1 (R 3 ) = 0 (CQ) admits solitons e itω P ω (x) : solutions which do not scatter Goal : Scattering in a large region 2 / 23

3 Nonlinear Schrödinger equations Nonlinear Schrödinger equation (NLS) on R d : (NLS) i t u + u = ± u p 1 u defocusing with the sign + and focusing with the sign - Scaling invariance : u(t, x) u λ (t, x) := λ 2 p 1 u(λ 2 t, λx). NLS is energy-critical (Ḣ1 -critical) if u λ (0) Ḣ1 (R d ) = u(0) Ḣ1 (R d ) λ 2 p 1 +1 d 2 u(0) Ḣ1 (R d ) = u(0) Ḣ1 (R d ) p = d 2 NLS is energy-subcritical if p < d 2 NLS is energy-supercritical if p > d 2 (CQ) on R 3 : quintic nonlinearity is energy-critical and defocusing cubic nonlinearity is subcritical and focusing 3 / 23

4 Energy-critical NLS Defocusing energy-critical NLS : i tu + u = u d 2 u on R d, d 3 Global well-posedeness (GWP) and scattering Bourgain (1999) d = 3, 4 radial case Grillakis (2000) d = 3, radial case, no scattering Colliander-Keel-Staffilani-Takaoka-Tao (2008) d = 3 Ryckman-Vişan, Vişan (2007, 2012) d 4 Focusing energy-critical NLS : i tu + u = u d 2 u on R d, d = 3, 4, 5 : GWP and scattering versus finite time blowup solutions Kenig-Merle (2006) radial case Threshold for scattering : (kinetic) energy of the stationary solution W W + W 4 W = 0 Specific features of the cubic-quintic NLS on R 3 : GWP for all data in H 1 (R 3 ) Scattering versus soliton behavior (e itω P ω (x)) Threshold for scattering : the energies of a branch of (rescaled) solitons / 23

5 Cubic-Quintic NLS on R 3 Tao, Vişan, Zhang (2007) : global well-posedness in H 1 (R 3 ) : treat (CQ) as a perturbation of defocusing energy-critical NLS scattering for small mass : M(u 0) c( u 0 L 2) Crucial space-time norm for scattering : L 10 t,x(r R 3 ) u L 10 t,x (R R3 ) < = solution u scatters Virial V (u) := u u 4 + u 6 dx E min defined by E min(m) := inf { E(f) : f H 1 (R 3 ), M(f) = m, V (f) = 0 } Theorem (R. Killip, T. Oh, O.P., M. Vişan 2012) If u 0 H 1 (R 3 ) is such that 0 < E(u 0 ) < E min (M(u 0 )), then the corresponding solution scatters both forward and backward in time. 5 / 23

6 Strategy of proof Variational part describe the region R given by 0 E(f) < E min(m(f)) find the minimizers of E min when they exist : (rescaled) solitons Dispersive part to specify the region R, introduce M(f) + E(f) D(f) := E(f) + dist((m(f), E(f)), epigraph E min) D 1 near the origin and D near the graph of E min L(D) := sup { u L 10 t,x (R R3 ) : u solution, D(u) D } Scattering in the region R L(D) < for all 0 < D < By small data theory : D 1 = L(D) < Argue by contradiction : there exists a critical 0 < D c < such that L(D) < if D < D c L(D) = if D > D c 6 / 23

7 Variational part : Solitons Soliton solutions of the form u(t, x) = e iωt P ω (x), where P ω satisfies P ω P ω 4 P ω + P ω 2 P ω ωp ω = 0 Berestycki-Lions (1983) : radial, positive solitons P ω exist ω (0, 3 16 ) Serrin-Tang (2000) : uniqueness of non-negative radial solitons P ω Vanishing virial : V (P ω ) = 0 for all ω (0, 3 16 ). Desyatnikov et al. (Phys. Rev. E, 2000) and Mihalache et al. (Phys. Rev. Lett., 2002) show the existence of two branches of solitons We can prove analytically the asymptotic behavior of M(P ω ) and E(P ω ) as ω 0 and as ω 3 16, but NOT their monotonicity 7 / 23

8 β-gagliardo-nirenberg inequality on R 3 3β u 4 L C 1+β β u 4 L 2 u L u 3 6 L 2 Proof : u 4 L u 4 L 2 u 3 1+β L u 6 L 2 u L u 3 6 L 2 The optimal constant is achieved modulo scaling by positive, radial Q β satisfying Q β Q 5 β + Q 3 β ω β Q β = 0 In particular, Q β is equal to the soliton P ωβ for some ω β (0, 3 16 ) 3β 1+β 1+β 8 / 23

9 β-gagliardo-nirenberg inequality on R 3 3β u 4 L C 1+β β u 4 L 2 u L u 3 6 L 2 Proof : u 4 L u 4 L 2 u 3 1+β L u 6 L 2 u L u 3 6 L 2 The optimal constant is achieved modulo scaling by positive, radial Q β satisfying Q β Q 5 β + Q 3 β ω β Q β = 0 In particular, Q β is equal to the soliton P ωβ for some ω β (0, 3 16 ) Uniqueness of Q β is NOT known The optimal constant can be expressed as 4(1 + β) C β = 3β β 2(1+β) 3β 1 1+β 1+β 1 β 1+β Q β L 2 Q β L 2 For β = 1 and any Q 1, we have unique mass M(Q 1 ) = ( 8 3C 1 ) 2 Fix an optimizer Q 1 and define R(x) := 1 2 Q 1 ( 3 2 x) 8 / 23

10 Feasible pairs (M(u), E(u)) for u H 1 (R 3 ) All the feasible pairs (M(u), E(u)) lie above or on the graph of E with the following properties : E(m) := inf { E(f) : f H 1 (R 3 ), M(f) = m }, E is continuous, concave, non-increasing We have that E(m) = 0 if 0 m M(Q 1) E(m) < 0 if m > M(Q 1) m M(Q 1 ) : the infimum E(m) is achieved by a soliton P ω Indeed, by the Euler-Lagrange equation : de du = ω dm 2 du u u5 + u 3 ωu = 0 m < M(Q 1 ) : the infimum E(m) = 0 is not achieved 9 / 23

11 Positive virial for mass less than M(R) Virial V (u) := u u 4 + u 6 dx Lemma If M(u) < M(R), then V (u) > 0. 8 Proof : Since C 1 = 3 Q 1, Gagliardo-Nirenberg inequality for β = 1 and L 2 Young s inequality yield : 3 4 u 4 dx 2 u ( ) 1/4 ( 3/4 L 2 u 6 dx u dx) 2 Q 1 L 2 33/4 2 33/4 2 u L 2 Q 1 L 2 u L 2 Q 1 L 2 ( 4 ) 1/4 ( u 6 4 dx 3 ( u 6 + u 2) dx. Thus, if u L 2 < 2 3 3/4 Q 1 L 2 = R L 2, then V (u) > 0. ) 3/4 u 2 dx Remark : R has the smallest positive mass with vanishing virial. 10 / 23

12 The region R : 0 E(f) < E min (M(f)) E min (m) := inf { E(f) : f H 1 (R 3 ), M(f) = m, V (f) = 0 } E min (m) = if m < M(R) E min (m) = E(m) if m M(Q 1 ) For M(R) m < M(Q 1 ), E min (m) is finite strictly decreasing attained by solitons P ω or rescaled solitons ap ω(λx) Indeed, the Euler-Lagrange equation yields de du = µ dv du ν dm du u + u5 u 3 = µ( 2 u + 6u 5 3u 3 ) 2νu u is a soliton if µ = 0 u is a rescaled soliton for µ (0, ] : ap ω(λx) 1 + 3µ a = 1 + 6µ, λ = 1 + 3µ ν(1 + 6µ), ω = (1 + 2µ)(1 + 6µ) (1 + 3µ) 2 11 / 23

13 Positive virial in the region R : 0 E(f) < E min (M(f)) Lemma We have V (u) > 0 for all (M(u), E(u)) R. Proof : By the definition of E min, V (u) 0 for all u R. Suppose there is u R such that V (u) < 0. Then, there is λ 0 > 1 such that u λ 0 (x) := λ u(λ 0x) satisfies V (u λ 0 ) = 0, M(u λ 0 ) = M(u), 0 < E(u λ 0 ) < E(u) < E min(m(u)), = contradiction 12 / 23

14 Positive virial in the region R : 0 E(f) < E min (M(f)) Lemma We have V (u) > 0 for all (M(u), E(u)) R. Proof : By the definition of E min, V (u) 0 for all u R. Suppose there is u R such that V (u) < 0. Then, there is λ 0 > 1 such that u λ 0 (x) := λ u(λ 0x) satisfies V (u λ 0 ) = 0, M(u λ 0 ) = M(u), 0 < E(u λ 0 ) < E(u) < E min(m(u)), = contradiction d dλ E(uλ ) = V (uλ ) = λ λ u 2 dx 3 4 λ2 u 4 dx + λ 5 u 6 dx V (u) < 0 and lim λ V (u λ ) = = there exists λ 0 > 1 with V (u λ 0 ) = 0 take the first such λ 0 = V (u λ ) < 0 for λ [1, λ 0) = E(u λ 0 ) < E(u) 12 / 23

15 Dispersive part We use the concentration-compactness and rigidity method developed by Kenig-Merle. 1 Concentration-compactness : Using a profile decomposition theorem for bounded sequences in H 1 (R 3 ), we prove the existence of a minimal blowup solution u : D(u) = D c < and u L 10 t,x ((,0] R3 ) = u L 10 t,x ([0, ) R3 ) = u is well-localized in physical space (almost periodic modulo translations) : for all η > 0 there exists C(η) and x(t) = o(t) as t such that ( u 2 + u 6 + u 4 + u 2) dx η sup t R x x(t) C(η) 2 Rigidity : such an almost periodic blowup solution cannot occur key element : V (f) > 0 for all f in the region R 13 / 23

16 Dispersive part : Profile decomposition Strichartz admissible pair (p, q) : = 3 p q 2 Strichartz + Sobolev inequality : 30 = (10, ) is admissible 13 e it f L 10 t,x (R R3 ) e it f L t L 13 x Theorem (Profile decomposition theorem) f L 2 Let {f n} n N be bounded in H 1 (R 3 ). Then, there exist J {0, 1, 2,...}, φ j Ḣ1 (R 3 ) \ {0}, {λ j n} n N (0, 1], {t j n} n N R, and {x j n} n N R 3 λ j n 1 or λ j n 0 as n, and t j n 0 or t j n ± as n. Moreover, with we have with f n(x) = P j nφ j := j=1 { φ j, if λ j n 1, P (λ j n ) θ φ j, if λ j n 0, 0 < θ < 1, J ( (λ j n) 2 1 e itj n Pnφ j j)( ) x x j n + w λ j n J in H 1 (R 3 ), n lim J J lim sup n e it w J n L 10 t,x (R R3 ) = / 23

17 Decoupling of mass and energy e itj n (λ j n) 1 2 w J n (λ j n +x j n) 0 in Ḣ1 (R 3 ), for all 1 j J ( J ) sup lim M(f J n n) M(gn) j M(wn) J = 0 sup J lim n [ λ j lim n n λ j n ( E(f n) + λj n λ j n j=1 J j=1 ) E(gn) j E(wn) J = 0 + xj n x j n 2 + λ j nλ j n t j n (λ j n) 2 t j n (λ j λ j nλ j n where gn j is defined by ( gn(x) j := (λ j n) 2 1 e itj n Pnφ j j)( ) x x j n λ j n n ) 2 ] =, for all j j, 15 / 23

18 Compactness of minimizing sequences Theorem If D(u n ) D c and lim n u n L 10 t,x ([tn, ) R3 ) = lim n u n L 10 t,x ((,tn] R3 ) =, then there exists {x n } n N R 3 such that, on a subsequence, u n (t n, +x n ) φ in H 1 (R 3 ). Proof : t n 0, D(u n(0)) D c + 1 < = {u n(0)} n N bounded in H 1 (R 3 ) : M(u n(0)) M 0 < M(Q 1), E(u n(0)) E 0, u n(0) 2 L 2 E(un(0)) Profile decomposition : u n(0) = ( ) ( ) J j=1 (λj n) 2 1 e itj n Pnφ j j x x j n + w λ j n J n Nonlinear profiles for each j : if λ j n 1, t j n 0 = v j (0) = φ j and vn(t, j x) = v j (t + t j n, x x j n) if λ j n 1, t j n ± = v j scatters forward/backward to e it φ j ) : ( if λ j n 0 = vn(0) j = (λ j n) 1 2 e it j n Pnφ j j)( x x j n λ j ( n vn(t, j x) (λ j n) 2 1 w j t (λ j n) 2 + tj n, x xj n λ j n w j is the solution of quintic NLS with w j (0) = φ j 16 / 23 ),

19 By mass and energy decoupling : lim sup n lim sup n J j=1 J j=1 ) (M(vn(0)) j + M(wn J ) M 0 E(v j n(0)) + E(w J n ) E 0 1 Case 1 : lim sup n sup j M(v j n(0)) = M 0 and lim sup n sup j E(v j n(0)) = E 0 = only one profile φ 1 and λ 1 n 1, t 1 n 0 : u n(0, x) = φ 1 (x x 1 n) + w 1 n and w 1 n 0 in H 1 (R 3 ) 2 Case 2 : lim sup n sup j M(v j n(0)) < M 0 or lim sup n sup j E(v j n(0)) < E 0 D(v j n) D c ε for all j Introduce : u J n(t) := J vn(t) j + e it wn, J j=1 J large, finite 17 / 23

20 By mass and energy decoupling : lim sup n lim sup n J j=1 J j=1 ) (M(vn(0)) j + M(wn J ) M 0 E(v j n(0)) + E(w J n ) E 0 1 Case 1 : lim sup n sup j M(v j n(0)) = M 0 and lim sup n sup j E(v j n(0)) = E 0 = only one profile φ 1 and λ 1 n 1, t 1 n 0 : u n(0, x) = φ 1 (x x 1 n) + w 1 n and w 1 n 0 in H 1 (R 3 ) 2 Case 2 : lim sup n sup j M(v j n(0)) < M 0 or lim sup n sup j E(v j n(0)) < E 0 D(v j n) D c ε for all j Introduce : u J n(t) := J vn(t) j + e it wn J, J large, finite }{{}}{{} 0 j=1 scatter u J n approximates well u n (in the L 10 t,x-norm) By perturbation theory = u n scatters = contradiction 17 / 23

21 By mass and energy decoupling : lim sup n lim sup n J j=1 J j=1 ) (M(vn(0)) j + M(wn J ) M 0 E(v j n(0)) + E(w J n ) E 0 1 Case 1 : lim sup n sup j M(v j n(0)) = M 0 and lim sup n sup j E(v j n(0)) = E 0 = only one profile φ 1 and λ 1 n 1, t 1 n 0 : u n(0, x) = φ 1 (x x 1 n) + w 1 n and w 1 n 0 in H 1 (R 3 ) 2 Case 2 : lim sup n sup j M(v j n(0)) < M 0 or lim sup n sup j E(v j n(0)) < E 0 D(vn) j D c ε for all j Introduce : J u J n(t) := vn(t) j + e it wn J, J large, finite }{{}}{{}}{{} j=1 scatters scatter 0 u J n approximates well u n (in the L 10 t,x-norm) By perturbation theory = u n scatters = contradiction 17 / 23

22 Extraction of the minimal blowup solution Consider solutions {u n } n N of (CQ) such that D(u n ) D c and lim n u n L 10 t,x ((,0] R3 ) = lim n u n L 10 t,x ([0, ) R3 ) = By the compactness property of minimizing sequences : u n (0, + x n ) u 0 in H 1 (R 3 ). Consider u the solution of (CQ) with u(0) = u 0. Then : D(u n ) D(u) = D c and u L 10 t,x ((,0] R3 ) = u L 10 t,x ([0, ) R3 ) = = u is a minimal blowup solution u is almost periodic modulo translations : for any {t n } n N, there exists {x(t n )} n N such that u(t n, + x(t n )) u in H 1 (R 3 ) 18 / 23

23 Virial bounded away from zero for the minimal blowup solution For the minimal blowup solution u : D(u) = D c < (M(u), E(u)) R = V (u(t)) > 0 for all t R 19 / 23

24 Virial bounded away from zero for the minimal blowup solution For the minimal blowup solution u : D(u) = D c < (M(u), E(u)) R = V (u(t)) > 0 for all t R Actually, there is δ > 0 such that V (u(t)) > δ for all t R. Indeed, assume V (u(t n )) 0. By compactness of the minimal blowup solution u, there exists u H 1 such that u(t n, + x(t n )) u in H 1 (R 3 ). By Sobolev embeddings, it follows that M(u(t n )) = M(u ), E(u(t n )) = E(u ) = u R V (u(t n )) V (u ) = 0, = contradiction 19 / 23

25 Rigidity argument Almost periodic minimal blowup solution : for all η > 0 there exists C(η) and x(t) = o(t) as t s.t. ( u 2 + u 6 + u 4 + u 2) dx η sup t R x x(t) C(η) Consider φ(r) = 1 if r 1 and φ(r) = 0 if r 2 and U R(t) := 2Im φ ( x ) u(t)x u(t)dx R U R(t) C R and ( ) tu R(t) = 4V (u(t)) + O ( u 2 + u 6 + u 4 + u 2 )dx x R Taking R = C(η) + sup t [T0,T 1 ] x(t), we get tur(t) > V (u(t)) > δ Integrating from T 0 to T 1 : δ(t 1 T 0) < U R(T 1) U R(T 0) 2C R C (η) + δ 2 T1 Taking T 1 = contradiction 20 / 23

26 Maximality of the connected region R Lemma The region R is a maximal connected component of the origin with positive virial. Proof : For any u minimizer of E min (m), there exist two branches of continuous curves emanating from u, situated above the graph of E min such that : for u λ (x) := λ 3 2 u(λx) with 1 < λ 1 + ε, we have V (u λ ) > 0 for u λ (x) := λ 1 2 u(λx) with 1 ε λ < 1, we have V (u λ ) < 0 21 / 23

27 Comparison to other NLS with combined-power nonlinearity Miao, Xu, Zhao (2011) : focusing quintic, defocusing cubic NLS on R 3, radial data e := inf{e(u) : K(u) = 0} is not achieved e = E c (W ) where E c is a modified energy and W is the stationary solution for focusing energy-critical NLS GWP and scattering if E(u 0) < e and K(u 0) 0 Akahori, Ibrahim, Hikuchi, Nawa (2012) : both nonlinearities are focusing, one energy critical, the other energy-subcritical, on R d for d 5 s := inf{s ω(u) = E(u) + ωm(u) : K(u) = 0} s is achieved by solitons GWP and scattering if S ω(u 0) < s and K(u 0) > 0 In both the above cases : there are finite time blowup solutions : the equations have a different nature than (CQ) less precise understanding of the scattering region 22 / 23

28 Open issues Uniqueness of the optimizers Q β of the β-gagliardo-nirenberg inequality Any optimizer Q β is a soliton P ω. Is any soliton P ω an optimizer Q β(ω)? On the curve E min, M(rescaled solitons) < M(P ω ) M(solitons) Scattering above the graph of E min (m) for M(R) m < M(P ω ) Continuity of m E min (m) for M(R) m < M(Q 1 ) Stability/instability of solitons : related to the monotonicity of ω M(P ω ) lower branch is conjectured to be stable upper branch is conjectured to be unstable Behavior near unstable solitons after a transitory period of time, perturbations of unstable solitons are expected to approach stable solitons Buslaev, Grikurov (2001), LeMesurier, Papanicolaou, Sulem, Sulem (1988) 23 / 23