Scattering theory for nonlinear Schrödinger equation with inverse square potential
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1 Scattering theory for nonlinear Schrödinger equation with inverse square potential Université Nice Sophia-Antipolis Based on joint work with: Changxing Miao (IAPCM) and Junyong Zhang (BIT) February -6, 015 Saint-Étienne-de-Tinée
2 Introduction NLS inverse square potential Background Two basic tools Main result Let d 3, d < p d, consider (i t P a )u = u p 1 u, u(0, x) = u 0 (x), (t, x) R R d (1.1) where u(t, x) : R t R d x C, and we denote P a by the Friedrichs extension of + a x self-adjoint and σ(p a ) = σ ac (P a ) = [0, + ), with a > (d ) 4. The elliptic operator P a is the P 1 a f L x (R d ) ( ) 1 f L (R d ). The Schrödinger flows for the elliptic operator P a is of interest in quantum mechanics (see [4, 5]).
3 Background Two basic tools Main result The solution to (1.1) is left invariant by the scaling Furthermore, u λ (t, ) Ḣsc x u(t, x) λ p 1 u(λ t, λx) u λ (t, x), λ > 0. (1.) (Rd ) = u(λ t, ) Ḣsc x (Rd ), s c = d p 1. (1.3)
4 Background Two basic tools Main result The solution to (1.1) is left invariant by the scaling Furthermore, u λ (t, ) Ḣsc x u(t, x) λ p 1 u(λ t, λx) u λ (t, x), λ > 0. (1.) (Rd ) = u(λ t, ) Ḣsc x (Rd ), s c = d p 1. (1.3) Mass M[u(t)] = u(t, x) dx = M(u 0 ); R d ( Energy E[u(t)] = 1 u(t, x) + a u x + 1 p+1 u(t, x) p+1) dx = E(u 0 ). When p = d, i.e. s c = 1, (1.1) is called energy-critical; s c < 1, energy-subcritical; p = d, mass-critical.
5 Strichartz estimate NLS inverse square potential Background Two basic tools Main result (1) a = 0. where q + d r = d e it f q L L r t x (R R d ) C f Lx, and (q, r, d) (,, ). Dispersive estimate e it f L x C t d f L 1 x. () H = + V(x), V(x) is less singular than the inverse square potential at the origin, for instance, when it belongs to the Kato class; see D Ancona-Fanelli-Vega- Visciglia[8], Schlag[15], Schlag-Soffer-Staubach[16, 17], etc. (3) We do not have the dispersive estimate for e itpa. When a < 0, the classical dispersive estimate does not hold for the wave equation with inverse square potential.
6 Background Two basic tools Main result Strichartz estimate, Burq-Planchon-Stalker-Zadeh, 03 Since u(t, x) := e itpa f solves we have i t u + u = e itpa f q L L C f r t x Lx. (1.4) a x u, u(0, x) = f(x), t u(t, x) = e it f + ia e i(t s) ( x u ) (s)ds. 0
7 Background Two basic tools Main result Strichartz estimate, Burq-Planchon-Stalker-Zadeh, 03 Since u(t, x) := e itpa f solves i t u + u = e itpa f q L L C f r t x Lx. (1.4) a x u, u(0, x) = f(x), we have t u(t, x) = e it f + ia e i(t s) ( x u ) (s)ds. 0 Then, t e i(t s) ( x u ) (s)ds t e i(t s) ( x u ) (s)ds 0 d L t L d x 0 C x u L where we use local smoothing estimate (d 3) R R d P 1 4 α a u x 1+4α d t L d+, x d L t L d, x C x 1 L d, x dxdt C f, 0 < α < 1 Lx x 1 u L t,x (d ) + a. 4 C f L x,
8 Background Two basic tools Main result Background for Energy-subcritical NLS with a = 0 i t u + u = u p 1 u, (t, x) R R d, u(0, x) = u 0 (x) H 1 (R d ), (1.5) with d < p < d and d 3. lim u(t) e it H u ± = 0. 1 x (R d ) t ± (1) Ginibre-Velo[9],1985, Morawetz estimate R Rd u(t, x) p+1 x dtdx u L Ḣ 1 t C ( M(u 0 ), E(u 0 ) ). () There is another simple proof by using the following interaction Morawetz estimate (Colliander-Keel-Staffilani-Takaoka-Tao[6],003) d u C u L 4 t,x (I Rd ) 0 3 u Lx L t (I,Ḣ1 x (Rd )), d 3.
9 Interaction Morawetz estimate Background Two basic tools Main result Proposition 1 (Morawetz-type estimates) Let u be an H 1 -solution to (1.1) on the spacetime slab I R d, the dimension d 3 and a > 1 4 (d ) 4, then we have 3 d 4 u L 4 t,x (I Rd ) C u(t 0) L sup u(t) Ḣ 1. (1.6) t I Proof: Consider the quadratic Morawetz quantity M(t) := t u, x u. ( 3 d 4 Rd 4 u L 4 t,x (I Rd ) C u(t 0) sup u(t) u(t, x) ) + dxdt. L t I Ḣ 1 x 3 And consider the Virial quantity V(t) = Im ū x u dx, R x d we deduce that Rd u(t, x) dxdt sup u(t). (1.7) x 3 t I Ḣ 1 I I
10 Sobolev norm equivalence Background Two basic tools Main result From Hardy s inequality x 1 f L x (R d ) d f Ḣ 1 (R d ) and interpolation, it is easy to see that for a > (d ) 4 P s a f L x (R d ) f Ḣ s (R d ), 1 s 1. P 1 a ( u p 1 u) L q Theorem 1 (Equivalence of Sobolev norm) Let a > (d ) 4, σ = d (d ) 4 + a. Let 0 s < and max { } d 1, d σ < p <, then, we have for (s + σ)p < d ( ) s L f p (R d ) d,p,s P s a f L p (R d ), f C c (R d ), (1.8) t L r x. and for (s + σ)p < d and sp < d P s a f L p (R d ) d,p,s ( ) s L f p (R d ), f C c (R d ), (1.9) (1.8) with s = 1 corresponds to the boundedness of Riesz transform in Hassell-Lin [10].
11 Background Two basic tools Main result Main result for energy-subcritical NLS Theorem (Zhang-Z, 14, JFA) Let d 3, λ d = (d ) 4 and let p ( d, 1 + d ) 4. Assume that a > 4p λ (p+1) d and u 0 H 1 (R d ). Then the solution u to (1.1) is global. 4 Moreover, if a λ (p+1) d for d 4, and a 0 for d = 3, then, the solution u scatters in the sense that there exists a unique u ± H 1 (R d ) such that Remark 1 lim u(t) t ± e itp a u ± H 1 x (R d ) = 0. This result allows some negative inverse-square potential when d 4.
12 Background Two basic tools Main result Energy-critical with a = 0,u 0 Ḣ1 (R d ) u scatters. The main difficulty in energy-critical case stems from the fact that none of the known monotonicity formulas (i.e. Morawetz estimates) for NLS scale like the energy (Ḣ1 x (R d )). d = 3 d = 4 d 5 radial Bourgain [1] Tao [18] Tao [18] non-radial I-term[7], Killip-Visan [11] Ryckman-Visan [14], Visan [1] 1 (Bourgain, 1999): induction on energy technique paved the way for how to proceed in such a scenario: by finding a bubble of concentration inside a solution, one can introduce a characteristic length scale into the problem, bringing the available Morawetz estimates back into play (despite their non-critical scaling). space localized Morawetz : I x C I 1 u(t, x) 6 x ( ) dtdx I 1 C u L Ḣ ; 1 t x (Colliander Keel Staffilani Takaoka Tao,004): Induction on energy frequency-localized interaction Morawetz: I P Nu(t,x) P Nu(t,y) x y 3 dxdy.
13 Background Two basic tools Main result Main result for energy-critical NLS (i t P a )u = u 4 u, (t, x) R R 3 u(0, x) = u 0 (x) Ḣ1 (R 3 ). (1.10) Theorem 3 (Miao-Zhang-Z, 14) Let a > 1 4. Given u 0 Ḣ1 (R 3 ), and assume in addition that u 0 is radial when a < 0, then there exists a unique global solution u to (1.10) satisfying u(t, x) 10 dxdt < +. (1.11) R R 3 Thus, the solution u scatters. We will show by the concentration-compactness approach to induction on energy.
14 Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness for Sobolev-norm equivalence Main difficulty: Gaussian bound of the heat kernel e tpa fail to hold when a is negative. Lemma 1 (Heat kernel boundedness, Liskevich-Sobol[1], Milman-Semenov[13]) Assume a > (d ) 4 and σ = d (d ) 4 + a, let H(t, x, y) be the kernel of the operator e tpa. Then, for all t > 0 and all x, y R d \ {0} ( ) t σ ( ) t σ C t d e x y c 1 t x y ( ) t σ ( ) t σ H(t, x, y) C 1 1 t d e x y c t. x y Lemma (Riesz kernels) Let s (0, d) and let L s a (x, y) be the kernel of the operator s P a. Then, we have for x, y R d \{0} L s 1 ( a (x, y) = e tpa (x, y)t s dt 4 x x y s d Γ(s/) 0 t x y 4 y ) σ x y 1, (.1) when d s σ {0,, 4, }. Here A B := min{a, B}, and A B := max{a, B}.
15 Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness Lemma 3 (Hardy inequality for P a ) Let p > max { 1, d d σ }, (σ + s)p < d and 0 < s < d. Then there exists a constant C such that for all f such that P s a f L p (R d ), x s f(x) C s P L p (R d a f L p ) (R d ). (.) The estimate (.) is sharp in sense that it fails for (s + σ)p d or p d d σ with σ > 0.
16 Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness Lemma 3 (Hardy inequality for P a ) Let p > max { 1, d d σ }, (σ + s)p < d and 0 < s < d. Then there exists a constant C such that for all f such that P s a f L p (R d ), x s f(x) C s P L p (R d a f L p ) (R d ). (.) The estimate (.) is sharp in sense that it fails for (s + σ)p d or p Theorem 4 (Multiplier theorem) d d σ with σ > 0. Let σ be as in Lemma 1 and let m : [0, ) C satisfy k λ m(λ) λ k, 0 k [ d ] + 1. (.3) Then m( P a ), which we define via the L functional calculus, extends uniquely from L p (R d ) L (R d ) to a bounded operator on L p (R d ), for all 1 < p < when σ 0, and for all r 0 < p < r 0 := d σ when 0 < σ < ν := [ ] d d + 1.
17 Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness Littlewood Paley theory for P a Let φ : [0, ) [0, 1] be a smooth positive function obeying φ(λ) = 1 for 0 λ 1 and φ(λ) = 0 forλ. For each dyadic number N Z, we define φ N (λ) := φ(λ/n) and ψ N (λ) := φ N (λ) φ N/ (λ); It is clear that {ψ N (λ)} N Z is a partition of unity for (0, ). Define the Littlewood Paley projections by P a N := φ N ( Pa ), P a N := ψ N( P a ), and P a >N := I Pa N. We define another family of Littlewood Paley projections as follows P a N := e Pa /N, Pa N := e Pa /N e 4Pa /N, and Pa >N := I P a N.
18 Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness Lemma 4 (Bernstein estimates) For s R, 1 < p q when a 0 and r 0 < p q < r when a < 0. Then, 0 (1) All operators P a N, Pa N, P a N and P a N are bounded in L p ; () P a N, Pa N, P a N and P a N are bounded from L q to L p with O(N d( 1 p 1 ) q ); (3) N s P a N f L p (R d ) (Pa ) s P a N f L p (R d ), f C c (R d ). Theorem 5 (Square function estimates) Fix s 0, 1 < p < when a 0 and r 0 < p < r 0 when a < 0. Then for any f C c (R d ), ( )1 N s P a N f N Z L p (R d ) ( s P a f )1 L p (R d ) N s ( P a N )k f N Z L p (R d ), k 1, k > s. ( ) s L f p ( s P a f L p x x N Z N s PN f ) 1 ( N s N Z Pa N f ) 1 L p (R d ) x s f(x) L p (R d ).
19 Morawetz-type inequality Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness Lemma 5 (Modified Morawetz inequality) For a Let I be a time interval and let u be a Ḣ1 x (R 3 )-solution to (1.10) on I. Then for any C 1, we have I x C I 1 u(t, x) 6 dx dt C 1+γ I 1+γ x 1 γ sup u(t) Ḣ1, γ = x t I 7 1, (.4) 3 where the implicit constant depends only on the energy of u. Proof: Consider the Morawetz quantity M(t) := Im v x γ x R x vdx, 3 v(t, x) = u(t, x)χ R(x). Error term [ ] a + γ γ (1 + γ) v dx 0. R 3 r 3 γ [ γ max γ 0 γ (1 + γ) ] =
20 Local smoothing estimate Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness Lemma 6 (Local smoothing) For 1 4 < a < 0. Let u = e itpa u 0. Then T T uniformly for T > 0 and R > 0. For a 0, T T u(t, x) dx dt R u 0 L x u 0 L x + u 0, (.5) Lx x R u(t, x) dx dt R u 0 L x u 0 L x, z R 3. (.6) x z R Error term can be controlled by local smoothing estimate R3 u x dxdt C u 0 Lx (R 3 ), a > 1 4. R Remark: The local smoothing estimate does guarantee local energy decay, it falls short of fulfilling the role of a dispersive estimate.
21 Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness Outline of energy-critical: concentration-compactness For 0 E < +, define L(E) := sup { S I (u) : u : I R 3 C s.t E(u 0 ) E }, wite S I (u) = u L 10 t,x (I R3 ). By the small data theory, we know that L(E) is finite for E sufficiently small. Define E c = sup{e 0, L(E 0 ) < + }, then, Theorem 3 is equivalent to E c = +.
22 Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness Outline of energy-critical: concentration-compactness For 0 E < +, define L(E) := sup { S I (u) : u : I R 3 C s.t E(u 0 ) E }, wite S I (u) = u L 10 t,x (I R3 ). By the small data theory, we know that L(E) is finite for E sufficiently small. Define E c = sup{e 0, L(E 0 ) < + }, then, Theorem 3 is equivalent to E c = +. We argue by contradiction. If E c < +, then we know that there exist a sequence {u n (t)} such that as n E(u n ) E c, and S In (u n ) =.
23 Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness e itpa : P 1 a L x (R 3 ) L 10 t,x (R R3 ). Theorem 6 (Ḣ1 a(r 3 ) linear profile decomposition for a < 0) Assume a < 0. Let {f n } be a bounded radial sequence in Ḣ1 a(r 3 ). After passing to a subsequence, there exist J {0, 1,,..., }, {φ j } J j=1 Ḣ1 a(r 3 ), {λ j n }J (0, ), and j=1 {t j n }J j=1 R such that for each finite 0 J J, we have the decomposition f n = J φ j n + wj n, φ j n (x) = n[ Gj e it j n Pa φ j] (x), j=1 with [G j n f](x) := (λj n ) 1 f ( ) x and w J λ j n Ḣ1 a(r 3 ) satisfying n lim lim e itpa wn J n L 10 t,x (R R3 ) = 0, J J asymptotic orthogonality : { lim n J } f n Ḣa 1 (R 3 ) φ j n Ḣa 1 (R 3 ) wj n = 0, Ḣa 1 (R 3 ) j=1 lim log λj n n + t j n (λj n ) tn k (λ k n) =, j k. λ k n λ j n λk n
24 Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness Lemma 7 (Refined Strichartz estimate for a < 0) Assume 1 4 < a < 0, 6 < r < r 0 = 3 σ = 6 1 and 1+4a q + 3 r = 1. Let f Ḣ1 a(r 3 ). Then we have e itpa f q L L r t x (R R 3 ) f 1 Ḣ1 sup e itpa P a N f 1 N Z L q L r t x (R R 3 ). Lemma 8 (Inverse Sobolev embedding) Assume 1 4 < a < 0 and the function f is radial. Let 6 < r < 3 σ and P a N f L r x ηn 1 1 r, and f Ḣ1 A. (.7) Then, we have x 1 N ( η ) 36 f(x) 6 r 6 dx η. (.8) A
25 Extraction of a critical element Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness Proposition (critical element) Assume that E c < +. Then, there exists a radial maximal life-span solution u c : I R 3 C satisfying E(u c ) = E c, and S I (u c ) = +. (.9) Moreover, there exist N(t) 1 such that the set { 1 K := N(t) 1 ( u t, ) x t I} N(t) (.10) is precompact in Ḣ1 x (R 3 ). Our goal is to show that the above critical element does not exist.
26 Property of critical element Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness Remark The Arzelà Ascoli theorem tells us that a family of functions F is precompact in L x (R 3 ) if and only if it is norm-bounded and there exists a compactness modulus function C(η) such that x C(η) f(x) dx + ξ C(η) ˆf(ξ) dξ η { ( ) 1 x } uniformly for f F. Thus, from the fact that the set K := u t, t I is N(t) 1 N(t) precompact in Ḣ1 x (R 3 ), there exists a function C : R + R + such that x C(η) N(t) u(t, x) dx + ξ C(η)N(t) ξ û(t, ξ) dξ η (.11) for all t I. By the Sobolev embedding Ḣ1 x (R 3 ) L 6 x (R3 ) and inf N(t) 1, inf u(t, x) 6 dx u 1. (.1) t I x C(u)
27 Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness Extinction of the global critical solutions From the above remark, we have R 1 u(t, x) 6 dx 1, uniformly for t R. (.13) x R Integrating over a time interval of length I 1, we obtain I R 1 γ I x R u(t, x) 6 x 1 γ dx dt R 1 γ I x R I 1 u(t, x) 6 dx dt. x 1 γ The Morawetz inequality (Lemma 5) gives I x R I 1 u(t, x) 6. x 1 γ dx dt R 1+γ I 1+γ I R I 1+γ, γ = < 1. Taking I sufficiently large depending on R and E c, we derive a contradiction.
28 Sobolev-norm equivalence Morawetz estimate and local smoothing estimate concentration-compactness Extinction of finite time blowup solution We argue by contradiction. Assume that T max := sup I < +. lim inf N(t) =, t Tmax lim sup u(t, x) dx = 0, R > 0. (.14) t Tmax x R For t I, define ( ) x 1, r 1, M R (t) = φ u(t, x) dx, φ(r) = R R d 0, r, (.15) We derive u 0 Tmax t 1. Letting t 1 T max, we obtain u 0 0. Therefore, u(t) 0, which contradicts with u L 10 t,x (I R3 ) =.
29 Outlook: Let d 3, we consider (i t P a )u = u 4 d u, u(0, x) = u 0 (x) Ḣ1 (R d ). (t, x) R R d (3.1) It has been proved that in the defocusing case, globally well-posed and scattering for any u 0 Ḣ1. In the focusing case, there are known counterexamples to global well-posedness and scattering for (3.1). Let W be the positive solution to the elliptic equation (S. Terracini [19]) W + a x W = W 4 d W. Then u(t, x) = W(x) is a solution to (3.1) that blows up both forward and backward in time. What is the threshold for blowup and GWP?
30 Reference I NLS inverse square potential J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case. J. Amer. Math. Soc., 1 (1999), MR N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential. J. Funct. Anal., 03(003), N.Burq, F. Planchon, J. Stalker and S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Indiana Univ. Math. J., 53 (004), H. E. Camblong, L. N. Epele, H. Fanchiotti and C. A. Garcia Canal, Quantum anomaly in molecular physics. Phys. Rev. Lett., 87(001), 040(4). K. M. Case, Singular potentials. Physical Rev.(), 80: , J. Colliander, M.Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equations on R 3. Comm. Pure. Appl. Math., 57(004), J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R 3. Ann. Math., 167 (008),
31 Reference II NLS inverse square potential P. D Ancona, L. Fanelli, L. Vega and N. Visciglia, Endpoint Strichartz estimates for the magnetic Schrödinger equation. J. Funct. Anal., 58(010), J. Ginibre and G. Velo, Scattering theory in energy space for a class nonlinear Schrödinger equations. J. Math. Pure Appl., 64(1985), A. Hassell and P. Lin, The Riesz transform for homogeneous Schrödinger operators on metric cones. Revista Mat. Iberoamericana, 30 (014), R. Killip and M. Visan, Global well-posedness and scattering for the defocusing quintic NLS in three dimensions. Analysis and PDE 5 (01), V. Liskevich and Z. Sobol, Estimates of integral kernels for semigroups associated with second order elliptic operators with singular coefficients. Potential Anal., 18(003), P. D. Milman and Yu. A. Semenov, Global heat kernel bounds via desingularizing weights. J. Funct. Anal., 1(004), E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4. Amer. J. Math. 19 (007), MR88737
32 Reference III W. Schlag, Dispersive estimates for Schrödinger operators: a survey. Ann. of Math., 163(007), W. Schlag, A. Soffer and W. Staubach, Decay for the wave and Schrödinger evolutions on manifolds with conical ends, I. Trans. Amer. Math. Soc., 36(010), W. Schlag, A. Soffer and W. Staubach, Decay for the wave and Schrödinger evolutions on manifolds with conical ends, II. Trans. Amer. Math. Soc., 36(010), T. Tao, Global well-posedness and scattering for higher-dimensional energy-critical non-linear Schrödinger equation for radial data. New York J. of Math. 11 (005), MR S. Terracini, On positive entire solutions to a clsss of equations with a singular coefficient and critical exponent. Advances in Differential Equations, 1(1996), J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. J. Funct. Anal., 173(000), M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. Duke Math. J., 138 (007) MR31886.
33 Thanks for your attention!
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