[Paper Review] Dispersive Estimates for Nonlinear Schr\"odinger Equations with External Potentials

Charlotte Dietze|arXiv (Cornell University)|Apr 12, 2021

Advanced Mathematical Physics Problems38 references7 citations

TL;DR

This paper establishes optimal L∞ decay estimates for solutions to the Hartree-type nonlinear Schrödinger equation with external potentials in dimensions d ≥ 3, under small initial data. Using high-order Sobolev control in Duhamel's formula, it proves that the solution decays at the same rate as free solutions, ||u(t)||∞ ≤ C₀(1 + |t|)⁻ᵈᐟ², extending prior results to the case with external potentials and confirming the dispersive behavior expected in the weak-coupling regime.

ABSTRACT

We consider the long time dynamics of nonlinear Schr\"odinger equations with an external potential. More precisely, we look at Hartree type equations in three or higher dimensions with small initial data. We prove an optimal decay estimate, which is comparable to the decay of free solutions. Our proof relies on good control on a high Sobolev norm of the solution to estimate the terms in Duhamel's formula.

Motivation & Objective

• To establish dispersive decay estimates for the Hartree-type nonlinear Schrödinger equation with external potentials in dimensions d ≥ 3.
• To extend known decay results—previously valid only for V = 0 or large initial data—to the case of small initial data and non-zero external potentials.
• To provide a rigorous framework for understanding long-time dynamics in weakly interacting quantum systems with external traps.
• To confirm that the presence of a potential does not alter the optimal decay rate of the solution's L∞ norm.

Proposed method

• Derives dispersive estimates for the linear propagator e⁻ⁱᵗᴴ using assumptions on the potential V ∈ Wᵏ,∞(ℝᵈ) with k > d/2.
• Employs Duhamel's formula to express the nonlinear solution as a perturbation of the linear solution.
• Controls the nonlinear term via high-order Sobolev norms (Hᵏ) to bound the integral term in Duhamel's formula.
• Uses a fixed-point argument in a time-weighted norm space to control the solution globally in time.
• Applies Gronwall's inequality and compact Sobolev embeddings to pass to the limit in approximating sequences.
• Relies on the distorted Fourier transform and oscillatory integral analysis for technical estimates, particularly in the nonlinear term.

Experimental results

Research questions

• RQ1 Can optimal L∞ decay estimates be established for the Hartree-type NLS with external potentials in dimensions d ≥ 3?
• RQ2 Does the presence of a potential V ≠ 0 affect the decay rate of solutions, compared to the free case?
• RQ3 Can small initial data in Hᵏ(ℝᵈ) lead to the same decay rate as in the free case, ||u(t)||∞ ≤ C(1 + |t|)⁻ᵈᐟ²?
• RQ4 Is the decay rate preserved for the time derivative ∂ₜu under similar assumptions?
• RQ5 Can the proof strategy used for V = 0 be adapted to the case of non-zero potentials with sufficient regularity and decay?

Key findings

• The solution satisfies the optimal decay estimate ||u(t)||∞ ≤ C₀(1 + |t|)⁻ᵈᐟ² for all t ≥ 0, with C₀ depending only on d, V, and ||w||₁.
• The decay rate matches that of the free Schrödinger equation, confirming dispersive behavior persists under small external potentials.
• The result holds for small initial data in Hᵏ(ℝᵈ) with k the smallest even integer greater than d/2.
• A similar decay estimate is established for the time derivative: ||∂ₜu(t)||∞ ≤ ˜C₀(1 + |t|)⁻ᵈᐟ² under additional smallness assumptions on initial data in Hᵏ and the propagator.
• The proof relies on high Sobolev norm control and a fixed-point argument in a time-weighted norm space, ensuring global existence and decay.
• The result is new even in dimension d = 3, extending prior work by Grillakis and Machedon which required V = 0 and allowed large initial data.