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Uniqueness of solutions for the logarithmic Schrödinger equation

Masayuki Hayashi

TL;DR

This work addresses the longstanding question of uniqueness for the logarithmic Schrödinger equation $i\partial_t u+\Delta u+\lambda u\log(|u|^2)=0$ in low-regularity Sobolev spaces. It develops an unconditional uniqueness framework in $H^s(\mathbb{R}^d)$ for $s\in(0,1)$ by coupling integral-equation formulations with a localization argument and local smoothing, enabled by a quantitative bound on the sublinear logarithmic nonlinearity. The results extend to the torus and to equations perturbed by pure power nonlinearities, using Zygmund-type bounds and Strichartz estimates to control nonlinear effects. Collectively, the paper provides a robust, dispersive-tools-based approach to well-posedness for logarithmic nonlinearities, resolving a gap in low-regularity uniqueness and offering a template for related logarithmic evolution equations.

Abstract

We consider the Cauchy problem for the logarithmic Schrödinger equation and prove uniqueness of weak $H^s(\mathbb{R}^d)$ solutions for $s\in(0,1)$, which improves on the previous uniqueness result in $H^1(\mathbb{R}^d)$. The proof is achieved by combining a nontrivial use of integral equations, local smoothing estimates, and quantitative estimates of the sublinear effect of the nonlinearity, based on the localization argument. We also study uniqueness on the torus and uniqueness of the equation perturbed by pure power nonlinearities.

Uniqueness of solutions for the logarithmic Schrödinger equation

TL;DR

This work addresses the longstanding question of uniqueness for the logarithmic Schrödinger equation in low-regularity Sobolev spaces. It develops an unconditional uniqueness framework in for by coupling integral-equation formulations with a localization argument and local smoothing, enabled by a quantitative bound on the sublinear logarithmic nonlinearity. The results extend to the torus and to equations perturbed by pure power nonlinearities, using Zygmund-type bounds and Strichartz estimates to control nonlinear effects. Collectively, the paper provides a robust, dispersive-tools-based approach to well-posedness for logarithmic nonlinearities, resolving a gap in low-regularity uniqueness and offering a template for related logarithmic evolution equations.

Abstract

We consider the Cauchy problem for the logarithmic Schrödinger equation and prove uniqueness of weak solutions for , which improves on the previous uniqueness result in . The proof is achieved by combining a nontrivial use of integral equations, local smoothing estimates, and quantitative estimates of the sublinear effect of the nonlinearity, based on the localization argument. We also study uniqueness on the torus and uniqueness of the equation perturbed by pure power nonlinearities.

Paper Structure

This paper contains 12 sections, 14 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Main results
  4. Organization of the paper
  5. Uniqueness in $H^s({\mathbb R}^d)$
  6. Review on the previous result
  7. Localization argument
  8. Proof of unconditional uniqueness
  9. Further results
  10. Preliminaries
  11. Uniqueness on the torus
  12. Perturbation of pure power nonlinearities

Key Result

Theorem 1.2

The flow map $\Phi:u_0\mapsto u$ for eq:1.1 is uniquely extended from $H^1$ to $L^2$. If $u_0\in L^2({\mathbb R}^d)$, $\Phi(u_0)\in C({\mathbb R},L^2({\mathbb R}^d))$ is a strong solution to eq:1.1, and $\Phi$ is Lipschitz continuous: for any $u_0,v_0\in L^2({\mathbb R}^d)$ and all $t\in{\mathbb R}$. If in addition $u_0\in H^s({\mathbb R}^d)$ for $s\in(0,1)$, then $\Phi(u_0)\in C({\mathbb R}, H^s

Theorems & Definitions (28)

  • Definition 1.1
  • Theorem 1.2: CHO24
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: Local smoothing estimates
  • ...and 18 more