[Paper Review] On the ill-posedness of the cubic nonlinear Schrödinger equation on the circle

Tadahiro Oh, Yuzhao Wang|Edinburgh Research Explorer|Aug 4, 2015

Advanced Mathematical Physics Problems18 references8 citations

TL;DR

This paper establishes norm inflation for the cubic nonlinear Schrödinger equation (NLS) on the torus in negative Sobolev spaces below the critical regularity $ s_{\text{crit}} = -\frac{1}{2} $, demonstrating ill-posedness via a high-to-low frequency cascade mechanism. The authors adapt Christ-Colliander-Tao's argument to the periodic setting, proving that the solution map fails to be uniformly continuous for $ s \leq -\frac{1}{2} $, both for the standard and Wick-ordered NLS.

ABSTRACT

In this note, we consider the ill-posedness issue for the cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, adapting the argument by Christ-Colliander-Tao [14] to the periodic setting, we exhibit a norm inflation phenomenon for both the usual cubic NLS and the Wick ordered cubic NLS for $s \leq s_ ext{crit} :=- \frac 12$. We also discuss norm inflation phenomena for general cubic fractional NLS on the circle.

Motivation & Objective

• To establish the ill-posedness of the cubic nonlinear Schrödinger equation (NLS) on the torus $ \mathbb{T} $ in negative Sobolev spaces below the scaling-critical regularity $ s_{\text{crit}} = -\frac{1}{2} $.
• To extend the norm inflation argument of Christ-Colliander-Tao from the non-periodic to the periodic setting, specifically for the cubic NLS on $ \mathbb{T} $.
• To analyze both the standard cubic NLS and the Wick-ordered cubic NLS, showing that ill-posedness persists in the same regularity regime.
• To investigate the behavior of general cubic fractional NLS on the circle, extending the norm inflation phenomenon to broader classes of nonlinearities.
• To clarify the distinction between the standard and Wick-ordered NLS in low regularity, particularly in the absence of $ L^2 $-conservation below $ s = 0 $.

Proposed method

• Adapts the high-to-low frequency cascade argument from Christ-Colliander-Tao (2014) to the periodic setting on $ \mathbb{T} $, using a randomized initial data construction.
• Employs a gauge transformation to define the Wick-ordered NLS, which renormalizes the cubic nonlinearity by subtracting the spatial average, improving regularity properties.
• Constructs a sequence of approximate solutions via a Neumann series expansion in frequency annuli, analyzing the $ k $-th order term $ \Xi_k(t) $ in the solution expansion.
• Applies Hölder's and Young's inequalities to control the $ H^s $-norm of the solution components, with $ s < 0 $, using the decay of Fourier coefficients.
• Uses the convolution estimate $ \mathbf{1}_{a+Q_A} * \mathbf{1}_{b+Q_A} \gtrsim A \cdot \mathbf{1}_{a+b+Q_A} $ to localize the frequency support and estimate the low-frequency projection of $ \Xi_1(t) $.
• Derives sharp lower and upper bounds on $ \|\Xi_k(t)\|_{H^s} $, showing that $ \|\Xi_1(t)\|_{H^s} \gtrsim t R^3 A^2 f(A) $ for $ s < 0 $, with $ f(A) \sim \|\langle \xi \rangle^s\|_{L^2(Q_A)} $.

Experimental results

Research questions

• RQ1 Does the cubic NLS on $ \mathbb{T} $ exhibit norm inflation for regularities $ s \leq -\frac{1}{2} $, the scaling-critical threshold?
• RQ2 Is the solution map for the standard cubic NLS on $ \mathbb{T} $ discontinuous in $ H^s $ for $ s < 0 $, and does this extend to the Wick-ordered version?
• RQ3 Can the norm inflation mechanism be extended to general cubic fractional NLS on the circle, and what regularity thresholds are involved?
• RQ4 What is the role of the Galilean and scaling symmetries in determining the critical regularity for ill-posedness in the periodic setting?
• RQ5 Is the critical case $ s = -\frac{1}{2} $ amenable to the same method, or does it require a fundamentally new approach due to conflicting scaling constraints?

Key findings

• Norm inflation occurs for the standard cubic NLS on $ \mathbb{T} $ in $ H^s $ for all $ s \leq -\frac{1}{2} $, implying ill-posedness below the scaling-critical regularity.
• The same norm inflation is established for the Wick-ordered cubic NLS on $ \mathbb{T} $, showing that the renormalization does not restore well-posedness below $ s = -\frac{1}{2} $.
• For $ s < 0 $, the solution map $ \Phi(t): u_0 \mapsto u(t) $ fails to be uniformly continuous in $ H^s(\mathbb{T}) $, confirming mild ill-posedness.
• The authors prove that $ \|\Xi_1(T_N)\|_{H^s} \gtrsim (\log N)^{1/4} $ in case (i), $ \|\Xi_1(T_N)\|_{H^s} \gtrsim N^{-s} (\log N)^{-2} g(N) $ in case (ii), and $ \|\Xi_1(T_N)\|_{H^s} \gtrsim N^{-1/2 - s - 3\theta} $ in case (iii), with $ T_N \to 0 $.
• The method fails in the critical case $ s = -\frac{1}{2} $ due to conflicting scaling constraints: $ D^{-3} E^{-1-s} \ll F \lesssim 1 $ cannot hold as $ D \to 0 $, $ E \ll 1 $, and $ s \in (-\frac{1}{2}, 0) $, indicating a need for new techniques.
• The analysis confirms that the standard and Wick-ordered NLS are dynamically equivalent only in $ L^2(\mathbb{T}) $, and the gauge transformation is ill-defined below $ L^2 $, justifying the need for separate analysis in negative Sobolev spaces.