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On the propagation of high regularity for the logarithmic Schr{ö}dinger equation

Quentin Chauleur, Guillaume Ferriere

TL;DR

We address the propagation of high Sobolev regularity for the one-dimensional logarithmic Schrödinger equation $i∂_t u + Δu = λ u log|u|^2$ on symmetric domains. Our approach combines a linear toy model with a detailed fixed-point analysis to study how cancellation points affect regularity, using the integrated quantity $\mathcal{V}[f](x)=∫_0^1 f(σx) dσ$ to capture near-origin dynamics. We prove that, for odd initial data in a broad class, the $H^s$-norm cannot remain finite for $s>7/2$ (instantaneous blow-up), while under Neumann boundary conditions and first-order cancellation with non-vanishing away from the origin, $H^3$-regularity is propagated for short times. Numerical experiments support the theoretical findings and illustrate the interplay between cancellation and dispersion, informing both analysis and high-order numerics for logNLS.

Abstract

We investigate both the instantaneous loss and the persistence of high regularity for the one-dimensional logarithmic Schr{ö}dinger equation in symmetric domains under various boundary conditions. We show that for a broad class of odd initial data, the $H^s$-norm of solutions exhibits instantaneous blow-up for all $s > 7/2 $. Conversely, we establish that $H^3$-regularity is preserved for solutions that are odd with first-order cancellation, non-vanishing behavior away from the origin and Neumann boundary conditions on symmetric bounded domains. These theoretical results are further supported and illustrated by numerical simulations.

On the propagation of high regularity for the logarithmic Schr{ö}dinger equation

TL;DR

We address the propagation of high Sobolev regularity for the one-dimensional logarithmic Schrödinger equation on symmetric domains. Our approach combines a linear toy model with a detailed fixed-point analysis to study how cancellation points affect regularity, using the integrated quantity to capture near-origin dynamics. We prove that, for odd initial data in a broad class, the -norm cannot remain finite for (instantaneous blow-up), while under Neumann boundary conditions and first-order cancellation with non-vanishing away from the origin, -regularity is propagated for short times. Numerical experiments support the theoretical findings and illustrate the interplay between cancellation and dispersion, informing both analysis and high-order numerics for logNLS.

Abstract

We investigate both the instantaneous loss and the persistence of high regularity for the one-dimensional logarithmic Schr{ö}dinger equation in symmetric domains under various boundary conditions. We show that for a broad class of odd initial data, the -norm of solutions exhibits instantaneous blow-up for all . Conversely, we establish that -regularity is preserved for solutions that are odd with first-order cancellation, non-vanishing behavior away from the origin and Neumann boundary conditions on symmetric bounded domains. These theoretical results are further supported and illustrated by numerical simulations.
Paper Structure (21 sections, 41 theorems, 146 equations, 4 figures)

This paper contains 21 sections, 41 theorems, 146 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Functional setting and main results
  3. Notations
  4. Background on logarithmic Schrödinger equations
  5. Main results
  6. Non-propagation of the $H^s$ regularity
  7. Behavior and integrability near the origin
  8. Integrated equation on the derivative and conclusion
  9. Linear toy model
  10. Preliminary estimates
  11. Sobolev embeddings and inequalities
  12. A decomposition through Taylor formula
  13. Fixed-point argument
  14. Properties of the $V_i$
  15. Duhamel formulation
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $\lambda \in \mathbb{R} \setminus \{ 0 \}$ and let $\Omega \subset \mathbb{R}$ be either: Then for any $\varphi \in W_2 (\Omega)$ (where $W_2 (\Omega)$ is defined in eq:W_2_def), there exists a unique solution to logNLS in the sense that for all open bounded subset $\omega \subset \Omega$ and $a.e.$$t \in \mathbb{R}$, with $u (0) = \varphi$. Moreover, if $\Omega$ is a bounded domain or if $

Figures (4)

  • Figure 1: The function $\chi$ on $\left[-3 , 3\right]$.
  • Figure 2: Time evolution of the $H^s$-norms for the numerical solution $(u^j)_{0 \leq j \leq J}$ as the spatial discretization step $h$ tends to 0.
  • Figure 3: Plots of the real part (left pannel) and imaginary part (right pannel) of solution of equation \ref{['logNLS']} with initial condition $\varphi(x)=\tanh(x)$ at time $t=0$, $1$ and $2$.
  • Figure 4: Plots of the absolute value $|u|$ of solution of equation \ref{['logNLS']} with initial condition $\varphi(x)=1-\cos\left(\frac{\pi x}{16}\right)$ at time $t=0$, $0.05$ and $0.1$(left panel), as well as the evolution over time of the minimum of its absolute value (right panel).

Theorems & Definitions (77)

  • Lemma 2.1
  • Theorem 2.2: Carles_Ferriere_logGP
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Lemma 3.1
  • ...and 67 more