Scattering for the 3D Gross-Pitaevskii equation

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Scattering for the 3D Gross-Pitaevskii equation

Zihua Guo Monash University

(joint with Zaher Hani, Kenji Nakanishi)

Jan 25, 2019

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Outline

1 Introduction and results

2 Proof of the theorem

Difficulty 1: singularity at zero frequency Difficulty 2: 3D quadratic term

Difficulty 3: weak low-frequency component of u2

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Outline

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Outline

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Gross-Pitaevskii (GP) equation

Consider the Gross-Pitaevskii (GP) equation

i ψ_t+ ∆ψ = (|ψ|²− 1)ψ, ψ : R¹⁺³→ C (1) with the boundary condition

|x|→∞lim ψ = 1. (2)

Formally, let φ = e^−itψ, then φ solves

i φ_t+ ∆φ = |φ|²φ. (3)

The non-vanishing boundary condition: very different. Physical contexts: Bose-Einstein condensates, superfluids and nonlinear optics, or in the hydrodynamic interpretation of NLS

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Gross-Pitaevskii (GP) equation

|x|→∞lim ψ = 1. (2)

i φ_t+ ∆φ = |φ|²φ. (3)

The non-vanishing boundary condition: very different. Physical contexts: Bose-Einstein condensates, superfluids and nonlinear optics, or in the hydrodynamic interpretation of NLS

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Gross-Pitaevskii (GP) equation

|x|→∞lim ψ = 1. (2)

i φ_t+ ∆φ = |φ|²φ. (3)

The non-vanishing boundary condition: very different.

Physical contexts: Bose-Einstein condensates, superfluids and nonlinear optics, or in the hydrodynamic interpretation of NLS

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Energy conservation

Let u = ψ − 1. Then u satisfies zero boundardy condition and

i ∂_tu + ∆u − 2 Re u = u²+ 2|u|²+ |u|²u (4) which is equivalent to (writing u = u₁+ iu₂)

˙

u₁ = −∆u₂+ 2(u₁+ |u|²/2)u₂,

− ˙u₂ = (2 − ∆)u₁+ 3u₁²+ u₂²+ |u|²u₁. (5)

Conservation of the energy: E (u) :=

Z

R³

|∇u|²+ (|u|²+ 2 Re u)²

2 dx

= Z

R³

|∇ψ|²+(|ψ|²− 1)²

2 dx = E (u₀).

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Note. We do not have conservation of kuk²₂, but ku(t)k2 ≤ Ce^Ct by Gronwall inequality.

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Energy conservation

Let u = ψ − 1. Then u satisfies zero boundardy condition and

i ∂_tu + ∆u − 2 Re u = u²+ 2|u|²+ |u|²u (4) which is equivalent to (writing u = u₁+ iu₂)

˙

u₁ = −∆u₂+ 2(u₁+ |u|²/2)u₂,

− ˙u₂ = (2 − ∆)u₁+ 3u₁²+ u₂²+ |u|²u₁. (5) Conservation of the energy:

E (u) :=

Z

R³

|∇u|²+ (|u|² + 2 Re u)²

2 dx

= Z

R³

|∇ψ|²+(|ψ|²− 1)²

2 dx = E (u₀).

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Note. We do not have conservation of kuk²₂, but ku(t)k2 ≤ Ce^Ct by Gronwall inequality.

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Well-posedness

In Zhidkov space X^k = {u ∈ L^∞ : ∂^α ∈ L², 1 ≤ |α| ≤ k}

d = 1: Zhidkov 1987;

d = 2, 3: Gallo 2004 In H¹, GWP

d = 2, 3: B´ethuel and Saut 1999 In energy space, GWP

d = 1, 2, 3 and small data for d = 4: G´erard;

d = 4: Killip-Oh-Pocovnicu-Visan Remark. No scattering in the above works.

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GWP in energy space

Energy space

E := {f ∈ ˙H¹(R³) : 2 Re f + |f |² ∈ L²(R³)} (7) with the distance d_E(f , g ) defined by

d_E(f , g )² = k∇(f − g )k²_L2 +1 2

|f |²+ 2 Re f − |g |²− 2 Re g

2 L². Note that (E, dE) is a complete metric space.

Theorem (G´ erard, 2006)

Unconditional GWP of (4) in the energy space E. Remark. GWP in H¹ ⊂ E, B´ethuel-Saut, 1999.

method: LWP by Strichartz analysis + energy conservation

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GWP in energy space

Energy space

E := {f ∈ ˙H¹(R³) : 2 Re f + |f |² ∈ L²(R³)} (7) with the distance d_E(f , g ) defined by

d_E(f , g )² = k∇(f − g )k²_L2 +1 2

|f |²+ 2 Re f − |g |²− 2 Re g

2 L². Note that (E, dE) is a complete metric space.

Theorem (G´ erard, 2006)

Unconditional GWP of (4) in the energy space E.

Remark. GWP in H¹ ⊂ E, B´ethuel-Saut, 1999.

method: LWP by Strichartz analysis + energy conservation

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Traveling wave solutions

There are a family of the solutions ψ(t, x ) = vc(x − ct) with finite energy for 0 < |c| <√

2.

Moreover, B´ethuel-Gravejat-Saut 2009 proved that in 3D E^∗ := inf{E (ψ − 1)|1 6= ψ(t, x ) = v (x − ct) solves (1) for some c} > 0, conjecturedthat E^∗ is the threshold for the global dispersive solutions.

Remark.

1. In 2D, there is no lower bound.

2. In the radial case, are there special non-scattering solutions in energy space? Not clear.

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Traveling wave solutions

There are a family of the solutions ψ(t, x ) = vc(x − ct) with finite energy for 0 < |c| <√

2.

Moreover, B´ethuel-Gravejat-Saut 2009 proved that in 3D E^∗ := inf{E (ψ − 1)|1 6= ψ(t, x ) = v (x − ct) solves (1) for some c} > 0, conjecturedthat E^∗ is the threshold for the global dispersive solutions.

Remark.

1. In 2D, there is no lower bound.

2. In the radial case, are there special non-scattering solutions in energy space? Not clear.

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Scattering

In 3D, scattering was proved by Gustafson-Nakanishi-Tsai (2006,2007,2009) for small data in weighted Sobolev space.

Question: What about the energy space?

Theorem (G.-Hani-Nakanishi, 2017)

For 3D GP equation, scattering holds for small radial data in E. Remark. Radial symmetry can be replaced by additional one order angular regularity.

Remaining questions: How about non-radial case? What about large data? Is smallness not needed?

Let’s state our theorem more precisely.

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Scattering

In 3D, scattering was proved by Gustafson-Nakanishi-Tsai (2006,2007,2009) for small data in weighted Sobolev space.

Question: What about the energy space?

Theorem (G.-Hani-Nakanishi, 2017)

For 3D GP equation, scattering holds for small radial data in E.

Remark. Radial symmetry can be replaced by additional one order angular regularity.

Remaining questions: How about non-radial case? What about large data? Is smallness not needed?

Let’s state our theorem more precisely.

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By the diagonalising transform

u = u₁+ iu₂ −→ v = v₁ + iv₂ := u₁+ iUu₂, (8) with

U :=p−∆(2 − ∆)⁻¹, we can rewrite the GP equation for v :

i ∂_tv − Hv = U(3u₁²+ u²₂+ |u|²u₁) + i (2u₁u₂+ |u|²u₂), (9) where (one can write u₁ = v₁, u₂ = U⁻¹v₂)

H :=p−∆(2 − ∆). (10)

Note. H has symbol |ξ|p2 + |ξ|². It behaves as Schr¨odinger equation for high frequency and as wave equation for low frequency.

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Theorem

There exists δ > 0 such that for any u₀ ∈ E, radial, with E (u0) ≤ δ, there exists a unique global solution u ∈ C (R : E) to (4). Moreover, there exists φ± ∈ H¹ such that

t→±∞lim

u₁+ (2 − ∆)⁻¹u²₂ + iUu₂− e^−itHφ±

H¹ = 0. (11) Remark.

We have the decay for quadratic terms of u₁ (not true for u₂):

t→±∞lim k(2 − ∆)⁻¹u₁²k_H¹ = 0.

We can transfer the asymptotic behaviour in the energy space E.

Indeed, we can prove

t→±∞lim d_E(u, T⁻¹(e^−itHφ±)) = 0, where T (u) = T (u₁+ iu₂) = u₁+ (2 − ∆)⁻¹u₂²+ iUu₂.

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Outline

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Outline

Difficulty 3: weak low-frequency component of u₂

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Difficulty 1: singularity at zero frequency

At low frequency U⁻¹ ≈ |∇|⁻¹. Idea. Nonlinear transform

This was treated by Gustafson-Nakanishi-Tsai using some nonlinear transform

z = z₁ + iz₂ = u₁+u₁²+ u₂²

2 − ∆ + iUu₂ (12)

Under the transform (12)

iz_t− Hz = − 2iU(u₁²) − 4h∇i⁻²∇ · (u₁∇u₂)

+ [−iU(|u|²u₁) + U²(|u|²u₂)]. (13) Note. Quadratic terms have no zero frequency singularity.

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Outline

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Consider the 3D Schr¨odinger equation iu_t+ ∆u = |u|^pu

For scattering, with small data in H^s, Strichartz analysis requires p ≥ ⁴₃, can not work for u².

Scattering results for p < ⁴₃: 0 < p ≤ ²₃

Any nontrivial solution u, with φ ∈ S, does not scatter to linear solution in L². (Glassey 1974, Strauss 1974 )

2

3 < p < ⁴₃

For any φ ∈ H¹ with kx φk₂ < ∞, the solution u scatters to linear solution in L². (Tsutsumi-Yajima, 1984)

Nakanishi-Ozawa 2002, Masaki 2015 p = ²₃

Modified scatterings occur. Ozawa, Hayashi, Naumkin, etc

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Consider the 3D Schr¨odinger equation iu_t+ ∆u = |u|^pu

For scattering, with small data in H^s, Strichartz analysis requires p ≥ ⁴₃, can not work for u².

Scattering results for p < ⁴₃: 0 < p ≤ ²₃

Any nontrivial solution u, with φ ∈ S, does not scatter to linear solution in L². (Glassey 1974, Strauss 1974 )

2

3 < p < ⁴₃

For any φ ∈ H¹ with kx φk₂ < ∞, the solution u scatters to linear solution in L². (Tsutsumi-Yajima, 1984)

Nakanishi-Ozawa 2002, Masaki 2015 p = ²₃

Modified scatterings occur. Ozawa, Hayashi, Naumkin, etc

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Note. For quadratic terms 2/3 < 1 < 4/3

Weighted H^s is usually needed for 3D quadratic terms.

Our ideas. Replace weighted H^s by additional angular regularity. In particular, in the radial case, we can handle H^s.

Use earlier ideas in the works on 3D Zakharov system (G.-Nakanishi 2012, G.-Lee-Nakanishi-Wang 2015)

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To illustrate our ideas, we consider the classical 3D quadratic Klein-Gordon equation

u_tt − ∆u + u = u² (14)

To study the asymptotic problems, there are two well-known methods

Klainerman’s vector field method (1985)

Existence of enough vector fields + energy estimates Shatah’s normal form method (1985)

Non-resonant structure + normal form transform (transfer the quadratic term to a cubic or higher order term).

Key non-resonance: hξi + hηi − hξ + ηi&_hηi¹

Our approach: generalized Strichartz estimates+(partial) normal form

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u_tt − ∆u + u = u² (14)

To study the asymptotic problems, there are two well-known methods Klainerman’s vector field method (1985)

Existence of enough vector fields + energy estimates

Shatah’s normal form method (1985)

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u_tt − ∆u + u = u² (14)

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u_tt − ∆u + u = u² (14)

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(i ∂_t+ hDi)u = hDi⁻¹u² Let K (t) = e^ithDi. Then

Lemma 1 (radial case) (G.-Nakanishi-Wang 2013).

Let r > 10/3, for φ radial

ke^ithDiP_kφk_L²

tL^r_x.2^kβ^kkφk_L²

x, where P_k ≈ F⁻¹1_|ξ|∼2kF and

β_k =











1

2 − ³_r, k < 0;

1 −³_r, k ≥ 0;¹⁰₃ < r < 4;

1

4 + , k ≥ 0; r = 4;

1

r, k ≥ 0; r > 4.

Remark. The classical best non-radial estimates ke^ithDiP_kφk_L²

tL⁶_x.2^5k/6kφk_L²

x. Wave: Klainerman-Machedon, Sogge, Sterbenz 2005 Schr¨odinger: Sogge, Shao, G.-Wang 2010

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Lemma 2 (non-radial case) (G.-Hani-Nakanishi 2016).

Let r > 10/3,

ke^ithDiP_kφk_L²

tL^r_xL²_σ.Bk(2, r )kφk_L²

x, where P_k ≈ F⁻¹1_|ξ|∼2^kF and

B_k(2, r ) =











2^k(¹²⁻³^r⁾, k < 0;

2^k(1−³^r⁾, k ≥ 0;¹⁰₃ < r < 4;

hki2^k⁴, k ≥ 0; r = 4;

2^k^r, k ≥ 0; r > 4.

Remark.

Wave: Sterbenz 2005

Schr¨odinger: G.-Lee-Nakanishi-Wang 2014, G. 2016

Theorem. Scattering for 3D quadratic KG with small data in H¹× L² and with 1-order angular regularity.

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Generalized Strichartz for GP

Lemma 3 (G.-Hani-Nakanishi 2016). For GP, we have: for r > 10/3,

ke^−itHP_kφk_L²

tL^r_xL²_σ.Ck(2, r )kφk_L²

x(R³), (15)

where

C_k(2, r ) =











2^k(¹²⁻³^r⁾, k ≥ 0;

2^k(2−⁷^r⁾, k < 0,¹⁰₃ < r < 4;

2^k(1−³^r⁾, k < 0, r > 4;

hki2^k⁴, k < 0, r = 4.

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Note. H := p−∆(2 − ∆).

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Intermediate Theorem

With this estimate and the nonlinear transform by (12), we get

Theorem

Scattering holds for small radial data u(0) ∈ H¹.

Question: What about energy space E? For small data, we can think E = {u : Re u ∈ H¹, Im u ∈ ˙H¹∩ L⁴}

Difficulty. Im u 6∈ L².

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Outline

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Difficulty. For u(0) ∈ E, we can only have ∇u2(0) ∈ L². Recall under the nonlinear transform

z = z₁ + iz₂ = u₁+u₁²+ u₂²

2 − ∆ + iUu₂ (17)

iz_t− Hz = − 2iU(u₁²) − 4h∇i⁻²∇ · (u₁∇u₂)

+ [−iU(|u|²u₁) + U²(|u|²u₂)]. (18) Problematic terms: u₂²∆u₂ or u₂²u₁ when u₂ has very low frequency.

Key new ingredients: “Null-structure” achieved by new nonlinear transform. Let

m = m₁+ im₂ = u₁+ 2u₁²+ u₂²

2 − ∆ + iUu₂ (19)

Then

i ∂_tm − Hm = N₂(m) + N₃(m, u) + N₄(m, u) + N₅(m, u)

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where

N₂(m, u) = U(m₁²) + 2i

2 − ∆[−3m₁∆u₂− 2∇m₁· ∇u₂], N₃(m, u) = U(2m₁R) + iN₃¹(u) + 2i

2 − ∆[4u₁m₁u₂− m₁²u₂], N₄(m, u) = U(R²− |u|⁴/4) + 2i

2 − ∆[4u₁Ru₂− 2u₂m₁R], N₅(m, u) = 2i

2 − ∆[−u₂R²+ u₂|u|⁴/4], with

R = −∆u₂²

2(2 − ∆) − (2 + ∆)u₁² 2(2 − ∆), N₃¹(u) =(2 − ∆)⁻¹

−2u₂|∇u₂|²+ 3∆u²₂

2 − ∆∆u₂+ 2∇∆u₂² 2 − ∆ ∇u₂

+ 4

2 − ∆

2u₁² 2 − ∆∆u2

− 2 + ∆ 2 − ∆u₁²

u2.

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Remark

Our nonlinear transform can achieve cancellation and is actually natural.

The GP equation (4) for u = u1 + iu2 can be rewritten as follows

˙

u₁ = −∆u₂+ 2(u₁+ |u|²/2)u₂,

− ˙u₂ = (2 − ∆)u₁+ 3u₁²+ u₂²+ |u|²u₁

= (2 − ∆)(u₁) + 2u²₁+ u₂²+ (2u₁+ |u|²)²/4 − |u|⁴/4.

Note that 2u1+ |u|² ∈ L² is bounded by the conserved energy.

In view of the equation of u₂, we first make the following change of variables

z₁ := u₁+2u₁²+ u₂²

2 − ∆ , z₂ = u₂.

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Thank you very much!