[Paper Review] A remark on norm inflation for nonlinear Schrödinger equations

Nobu Kishimoto|arXiv (Cornell University)|Jun 26, 2018

Advanced Mathematical Physics Problems40 references8 citations

TL;DR

This paper establishes norm inflation for semilinear Schrödinger equations with polynomial nonlinearities in negative Sobolev spaces on both non-periodic and periodic domains. By extending the Iwabuchi-Ogawa method to general nonlinearities—including non-gauge-invariant and complex-coefficient cases—it proves ill-posedness in $ H^s $ for $ s < \min\{s_c(d,p), 0\} $, including critical and subcritical regimes, demonstrating strong instability via high-to-low frequency energy cascade.

ABSTRACT

We consider semilinear Schrödinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on $\mathbb{R}^d$ or on the torus. Norm inflation (ill-posedness) of the associated initial value problem is proved in Sobolev spaces of negative indices. To this end, we apply the argument of Iwabuchi and Ogawa (2012), who treated quadratic nonlinearities. This method can be applied whether the spatial domain is non-periodic or periodic and whether the nonlinearity is gauge/scale-invariant or not.

Motivation & Objective

• To establish norm inflation (NI s) for semilinear Schrödinger equations with polynomial nonlinearities in negative Sobolev spaces $ H^s $, indicating strong ill-posedness.
• To extend the Iwabuchi-Ogawa method beyond quadratic nonlinearities and gauge-invariant cases to general polynomial nonlinearities with complex coefficients.
• To analyze the behavior of the solution map in $ H^s $ for $ s < 0 $, particularly in regimes where scaling criticality and periodicity complicate standard approaches.
• To demonstrate that norm inflation occurs not only below the scaling-critical regularity but also at the critical index $ s = s_c(d,p) $ for specific cases like $ (d,p) = (1,3), (2,2) $.

Proposed method

• Applies the Iwabuchi-Ogawa argument to analyze the power series expansion of the solution, focusing on the growth of individual terms in frequency-localized norms.
• Uses frequency localization via dyadic projections $ U_k[\phi] $ to track energy transfer from high to low frequencies.
• Employs $ L^p $-based modulation spaces $ D^{[\alpha]}_{p,q} $ and estimates involving the Fourier transform and Young's inequality to control the size of solution components.
• Implements a recursive estimate for the $ k $-th order term in the solution expansion, showing that higher-order terms dominate in negative Sobolev norms under specific parameter choices.
• Constructs initial data $ \phi $ with controlled $ H^s $-norm but large high-frequency content, tuned to trigger norm inflation at time $ T \sim N^{-2} $.
• Validates the norm inflation condition by choosing parameters $ r, A, T $ such that $ \|\phi\|_{H^s} \ll \delta $, $ T \ll \delta $, and $ \|u(T)\|_{H^s} \gg \delta^{-1} $.

Experimental results

Research questions

• RQ1 Does norm inflation occur for general polynomial nonlinearities in negative Sobolev spaces on both $ \mathbb{R}^d $ and $ \mathbb{T}^d $, regardless of gauge invariance or coefficient reality?
• RQ2 Can the Iwabuchi-Ogawa method be extended to non-quadratic nonlinearities and non-gauge-invariant cases?
• RQ3 What is the sharp range of regularity $ s $ for which norm inflation holds, particularly at or below the scaling-critical index $ s_c(d,p) $?
• RQ4 Is norm inflation possible in the periodic setting for $ (p,q) = (2,1) $, and what regularity thresholds apply?
• RQ5 Can the method detect infinite loss of regularity in the solution map, and under what conditions?

Key findings

• Norm inflation occurs in $ H^s $ for all $ s < \min\{s_c(d,p), 0\} $, regardless of the spatial domain ($ \mathbb{R}^{d_1} \times \mathbb{T}^{d_2} $) or nonlinearity type.
• For $ (d,p) = (1,3) $ and $ (2,2) $, norm inflation holds at the scaling-critical index $ s = s_c(d,p) = -\frac{1}{2}, -1 $, respectively.
• In the $ (p,q) = (2,1) $ case, norm inflation is proven for $ s < -\frac{1}{4} $ on $ \mathbb{R}^d $, $ d \leq 3 $, and for $ s < 0 $ on $ \mathbb{R}^{d_1} \times \mathbb{T}^{d_2} $ with $ d_1 + d_2 \leq 3 $, $ d_2 \geq 1 $.
• For $ \mathbb{T} $, norm inflation holds for $ (p,q) = (4,1), (4,2), (4,3) $ at any $ s < 0 $, extending previous results.
• The method provides a new proof of infinite loss of regularity for smooth nonlinearities, recovering and extending results from Carles, Dumas, and Sparber (2012) and Carles and Kappeler (2014).
• The construction achieves $ \|u(T)\|_{H^s} \gg \delta^{-1} $ while $ \|\phi\|_{H^s} < \delta $, $ T < \delta $, confirming strong instability beyond mere discontinuity of the solution map.