On the global Cauchy problem for the nonlinear Schrödinger equation

Abstract

We mainly survey recent results on the nonlinear Schrödinger equations with special emphasis on the Cauchy problem, local and global in time and for various regularity assumptions on the initial data.

1. The Equation

We consider nonlinear Schrödinger equations (NLSs) of the form

1.1

Most of our results and methods generalize to wider classes, however.

Eq. 1.1 is Hamiltonian and may be written as

1.2

with conserved Hamiltonian

1.3

The formal canonical coordinates are thus (Re u, Im u).

For equations such as Eq. 1.1, there is also conservation of the L² norm

1.4

A first important distinction between the different equations is the sign of λ

This short expose will mainly relate to the defocusing (i.e. nonblowup) case (more details and references may be found in ref. 1). Observe that if u(0) = φ ∈ H¹, then the Hamiltonian conservation implies an a priori bound of ∥u(t)∥_H¹ in the defocusing case.

2. Scale-Invariant Sobolev Spaces

Observe that the equation (with spatial domain ℝ^d)

2.1

is invariant under the scaling

2.2

Define the exponent s₀ by

2.3

Hence the (homogeneous) Sobolev space Ḣ^s₀ is scale-invariant. The particular cases s₀ = 0, s₀ = 1 are important, because they correspond to conserved quantities. Thus,

2.4

2.5

In case of Eq. 2.4, the NLS is called conformally invariant because of the additional symmetry

2.6

We distinguish the

2.7

2.8

2.9

Thus for d = 1, 2, Eq. 2.1 is always subcritical.

3. The Local Well Posedness Result

One should not expect a well posedness theory for the Cauchy problem below the scale-invariant threshold. For solutions local in time, the following classical result gives a satisfactory answer to the issue (cf. ref. 2).

Theorem.

Consider the initial value problem (IVP)

3.1

Assume s ≥ 0 and s ≥ s₀defined by Eq. 2.3.

We also assume p − 2 > [s] if p ∉ 2ℤ (a smoothness compatibility condition on the nonlinearity).

Then Eq. 3.1 is well posed on a time interval[0, T*[, T* > 0 and

3.2

3.3

3.4

Remarks:

• (i)  The result holds both in the focusing and defocusing case.

• (ii)  In the critical case s = s₀, T* = T*(φ) depends on possible concentrations of the H^s norm. This issue is particularly important in the global existence theory for the H¹-critical equation iu_t + Δu − u|u|^4/d−2 = 0, for instance.

4. Global Solutions

The local solution from the theorem in The Local Well Posedness Result extends to a global one (i.e. T* = ∞) in the following cases:

• (i)  Small data;

• (ii)  p < 2 + 4/d (L²-subcritical equation); and

• (iii)  Defocusing case, ϕ ∈ H^s (s ≥ 1)

  4.1

  i.e. H¹-subcritical.
• For radial data, also

  4.2

  i.e.

  

  There are other results involving decay conditions at infinity, thus

  4.3

  (see ref. 2).
• Also, recent results requiring in iii only a weaker assumption

  4.4

  This will be discussed next.

5. More About the Subcritical Case

Consider the IVP

5.1

There is s₁ < 1 such that (Eq. 5.1) is globally well posed for s > s₁, and moreover

5.2

Examples:

5.3

5.4

6. Ingredients of the Argument

1. Decomposition in low and high modes

Remark:

Inequality (Eq. 5.2) is not obtained as a stability result.

2. Refinements of Strichartz' theorem in terms of certain bilinear inequalities.

Example:

Assume ψ_i ∈ L²(ℝ^d), i = 1, 2, and

6.1

Then if M₁ ≤ M₂,

for d = 2

6.2

and for d = 3

6.3

Remark:

Bilinear inequalities in similar spirit may be proven for the wave equation as well (cf. 3, 4).

7. H¹-Critical Case

These are the equations:

7.1

7.2

Theorem.

For ϕ ∈ H^s(ℝ^d), s ≥ 1 and ϕ radial, there is global well posedness and scattering in H^s (cf. ref. 1).

Main Idea.

Proceed by “induction” on the size of H(φ) by “separation of localized energy” (see Fig. 1).

Fig. 1.

Spatial time energy concentration region.

Main Ingredients.

1. Strichartz' inequality in appropriate Besov spaces to study energy concentration phenomenon.

2. Morawetz inequality and variants. Thus, for d = 3, i.e.

   

       one has the inequality

   7.3
3. 3. The pseudoconformal conservation law

   7.4

       based on pseudoconformal transformation

   7.5

       For λ < 0, p ≥ 2 + 4/d, Eq. 7.4 implies the a priori decay estimate

   7.6

8. Scattering in Energy Space

Consider NLS

8.1

with

8.2

(also p = 2 + 4/d−2 in the radial case).

Denote the wave operator

Then

8.3

and

8.4

For d ≥ 3: Classical (Ginibre–Velo, Lin–Strauss).

For d ≤ 2: Recent (Nakanishi), based on separation of the localized energy argument.

9. IVP for NLS with Periodic Boundary Conditions

Theorem.

d = 1, 2, 3.

Consider the defocusing IVP with periodic bc

If d = 1, 2 : p arbitrary.

If d = 3 : p < 6.

Then there is global well posedness and

9.1

9.2

Problem.

True growth of Sobolev norms for s > 1?

Remark:

Possible powerlike growth in IVP for 1D nonlinear wave equations of the form

9.3

where B = + o(1) is a selfadjoint multiplier.

No such examples are known for NLSs.

10. Remark on the Case of a Linear Schrödinger Equation with Time-Dependent Potential

Consider the Schrödinger equation

10.1

V = real potential, smooth in x, t; periodic in x (no specified behaviour in t).

Denote S(t) the flowmap. Thus,

10.2

Theorem.

For all s < ∞, ɛ > 0,

10.3

Remarks:

• (i)  Examples show that slow growth of ∥S(t)φ∥_H^s, s > 0 is possible when t → ∞ even for time periodic V.

• (ii)  Case of time-periodic potential was established by T. Spencer (personal communication) (later extended to quasiperiodic potential), conjectured also for smooth potentials with random behavior in time.

• (iii)  One may construct real potentials V that are smooth and periodic in x, random in time with limited smoothness, such that ∥S(t)∥_H¹→H¹ has powerlike growth.

• (iv)  Nonlinear analogues?

The theorem is proven by constructing flow up to time T → ∞ from “approximate Bloch waves” well localized in Fourier space.

Main Arithmetical Ingredient.

Separated cluster structures. More precisely:

Lemma (Granville–Spencer).

Fix some 0 < ρ < 1/10. Then there is partition of ℤ^d

such that if

the following properties hold

for some ρ₁ = ρ₁(ρ, d) > 0.

This fact plays also an important role in the Kolmogorov–Arnold–Moser theory for NLS.

11. Invariant Measures for NLS

Construction of invariant measures from normalized Gibbs measure

11.1

(we assume periodic boundary conditions).

1D Case.

Defocusing case.

11.2

11.3

Theorem.

∀p

• (i)  dμ ∼ Wiener measure.

• (ii)  ∃ unique global dynamics that leave μ invariant.

Thus IVP is globally well posed for almost all data given by the random Fourier series

11.4

Focusing case.

11.5

Normalization of Gibbs measure by L² truncation:

11.6

where

• (i)  B is arbitrary for p < 6.[11.7]

• (ii)  B is sufficiently small for p = 6 (related to blowup phenomena).[11.8]

• (iii)  This construction and result are due to Lebowitz–Rose–Speer.

• (iv)  Same theorem on invariant dynamics.

D > 1 Case.

Defocusing Case.

Normalization of Gibbs measure by Wick ordering of nonlinearity. Assume d = 2 and consider the equation

11.9

11.10

11.11

Example:

11.12

defines invariant measure for Wick-ordered cubic NLS

11.13

Moreover, ∃s > 0 such that for φ = u(0) almost surely

11.14

Hartree-type equations.

11.15

V = real interaction potential such that

11.16

11.17

(possibly focusing).

Theorem.

Assume V̂(0) = 0 and

11.18

11.19

Then truncated Gibbs measure

11.20

is invariant measure for Eq. 11.15.

The Cauchy problem is globally well posed for almost all datau(0) = φ ∈ supp μ.

Problems.

• (i)  Existence of invariant measures on smooth phase spaces. Unknown except in integrable case(1D cubic NLS) and Kolmogorov–Arnold–Moser-type results.

• (ii)  Ergodic properties of Gibbs measures?

• (iii)  Invariant measures on the line. Thus we let the period → ∞ in the preceding.

Example:

11.21

11.22

periodic bc with period L → ∞.

Normalized Gibbs measure

11.23

This equation satisfies uniform distributional estimates from Brasscamp–Lieb inequality.

For p ≥ 4 in Eq. 11.21, existence of a unique invariant dynamics in the limit for L → ∞ may be shown.

Abbreviations

• NLS, nonlinear Schrödinger equation

• IVP, initial value problem