[Paper Review] Norm-inflation for periodic NLS equations in negative Sobolev spaces

Rémi Carles, Thomas Kappeler|arXiv (Cornell University)|Jul 15, 2015

Advanced Mathematical Physics Problems6 references19 citations

TL;DR

This paper establishes strong ill-posedness for periodic nonlinear Schrödinger equations (NLS) with odd-order nonlinearities on the torus T^d by proving norm inflation in negative Sobolev spaces. Using a weakly nonlinear geometric optics approach with a small parameter ε, the authors construct approximate solutions via phase-amplitude expansions and demonstrate that for initial data in C^∞ with arbitrarily small H^s norm (s < 0), the solution norm blows up in finite time. The key result is that for σd ≥ 2 (or d ≥ 2 for renormalized case), and s < 0, the H^s norm of solutions can become infinite, even though initial data norm tends to zero.

ABSTRACT

In this paper we consider Schr{\"o}dinger equations with nonlinearities of odd order 2$\sigma$ + 1 on T^d. We prove that for $\sigma$d$\ge$2, they are strongly illposed in the Sobolev space H^s for any s extless{} 0, exhibiting norm-inflation with infinite loss of regularity. In the case of the one-dimensional cubic nonlinear Schr{\"o}dinger equation and its renormalized version we prove such a result for H^s with s extless{} --2/3.

Motivation & Objective

• To establish strong ill-posedness for periodic NLS equations with nonlinearities of odd order 2σ+1 on the torus T^d.
• To demonstrate norm inflation with infinite loss of regularity in H^s for s < 0, showing that solutions can grow unboundedly from arbitrarily small initial data.
• To extend the analysis to both standard and renormalized cubic NLS equations (σ=1, d=1), proving norm inflation for s < -2/3.
• To show that the renormalized version does not improve regularity or stability compared to the standard NLS in terms of norm inflation behavior.

Proposed method

• Use of a weakly nonlinear geometric optics scaling with a small parameter ε, transforming the original NLS into a form amenable to asymptotic analysis.
• Construction of first-order approximate solutions as superpositions of plane waves: u_ε^app(t,x) = Σ_j a_j(t) e^{iφ_j(t,x)/ε}, where φ_j(t,x) = j·x - |j|²t/2.
• Derivation of ODE systems for amplitudes a_j(t) based on resonance sets Res_j, defined by conservation of wave vector and frequency under nonlinear interactions.
• Identification of initial data that excite the zero-mode (a_0) via resonant interactions of nonzero modes, leading to growth in the solution norm.
• Use of scaling β < 0 to relate the approximate solution in ε to the original solution ψ, so that the zero-mode amplitude grows as ε^{-β/(2σ)} with ε → 0.
• Estimation of the error between the true solution and the approximate solution using Wiener algebra and Sobolev-type norms, showing the approximation dominates the norm inflation.

Experimental results

Research questions

• RQ1 Can norm inflation with infinite loss of regularity be proven for periodic NLS equations with odd nonlinearities in negative Sobolev spaces?
• RQ2 What is the sharp threshold for s < 0 such that norm inflation occurs in H^s(T^d) for the standard and renormalized NLS equations?
• RQ3 Does the renormalized cubic NLS equation exhibit better regularity or stability properties than the standard one in terms of norm inflation?
• RQ4 How does the scaling parameter β influence the growth of the solution norm and the convergence of initial data to zero in H^s?

Key findings

• For σd ≥ 2 (or d ≥ 2 for the renormalized case), norm inflation occurs in H^s(T^d) for all s < 0, meaning the H^s norm of the solution becomes infinite while the initial data norm tends to zero.
• In the one-dimensional cubic NLS case (σ = d = 1), norm inflation occurs for all s < -2/3, which is the sharp threshold for this case.
• The zero-mode amplitude a_0(t) grows as ε^{-β/(2σ)} under appropriate scaling, and since |a_0| bounds the H^s norm from below for s < 0, this leads to unbounded growth in any FL^{r,p} norm.
• The error between the true solution and the approximate solution is controlled and shown to be negligible compared to the approximate solution norm, ensuring that the blow-up is intrinsic to the solution.
• The renormalized NLS equation (1.2) exhibits the same norm inflation behavior as the standard NLS (1.1), indicating no improvement in regularity or stability in this regime.
• The proof relies on resonant interactions of nonzero modes that generate the zero-mode with non-vanishing initial derivative ȧ_0(0) ≠ 0, which is possible only under the specified conditions on σ and d.