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The Cauchy problem for the logarithmic Schrödinger equation revisited

Masayuki Hayashi, Tohru Ozawa

TL;DR

The paper addresses the Cauchy problem for the logarithmic Schrödinger equation $i\\partial_t u+\\Delta u+\\lambda u\\log(|u|^2)=0$, tackling the non-Lipschitz, singular nonlinear term by working in energy spaces $W_1$ and $W_2$. A constructive approximation scheme with $g_\\varepsilon(u)=2u\\log(|u|+\\varepsilon)$ yields strong solutions in $H^1$ and in the $H^2$-energy space, handling both signs of $\\lambda$ and extending to general domains. The authors derive uniform a priori estimates, prove convergence of approximate solutions without compactness arguments, and establish a priori identities (including an $H^2$-identity) that control time derivatives, Laplacians, and nonlinear logs. They obtain $u\\in C(\\mathbb{R},W_1)$ (and $W_2$ when applicable), energy conservation for $\\lambda>0$, and strong continuity results at $t=0$, providing a robust well-posedness theory for the logarithmic nonlinearity. The results advance the understanding of non-Lipschitz dispersive PDEs and offer a framework applicable to general domains with Dirichlet boundary conditions.

Abstract

We revisit the Cauchy problem for the logarithmic Schrödinger equation and construct strong solutions in $H^1$, the energy space, and the $H^2$-energy space. The solutions are provided in a constructive way, which does not rely on compactness arguments, that a sequence of approximate solutions forms a Cauchy sequence in a complete function space and then actual convergence is shown to be in a strong sense.

The Cauchy problem for the logarithmic Schrödinger equation revisited

TL;DR

The paper addresses the Cauchy problem for the logarithmic Schrödinger equation , tackling the non-Lipschitz, singular nonlinear term by working in energy spaces and . A constructive approximation scheme with yields strong solutions in and in the -energy space, handling both signs of and extending to general domains. The authors derive uniform a priori estimates, prove convergence of approximate solutions without compactness arguments, and establish a priori identities (including an -identity) that control time derivatives, Laplacians, and nonlinear logs. They obtain (and when applicable), energy conservation for , and strong continuity results at , providing a robust well-posedness theory for the logarithmic nonlinearity. The results advance the understanding of non-Lipschitz dispersive PDEs and offer a framework applicable to general domains with Dirichlet boundary conditions.

Abstract

We revisit the Cauchy problem for the logarithmic Schrödinger equation and construct strong solutions in , the energy space, and the -energy space. The solutions are provided in a constructive way, which does not rely on compactness arguments, that a sequence of approximate solutions forms a Cauchy sequence in a complete function space and then actual convergence is shown to be in a strong sense.
Paper Structure (15 sections, 27 theorems, 196 equations)

This paper contains 15 sections, 27 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. The Cauchy problem in $H^1$ and the energy space
  3. Approximate problems
  4. Construction of weak $H^1$-solutions
  5. Uniqueness and regularity
  6. Construction of solutions in $W_1$
  7. The Cauchy problem in the $H^2$-energy space
  8. Construction of weak $H^2$-solutions
  9. A priori estimates in $W_2$
  10. Further regularity
  11. The Cauchy problem in a general domain
  12. Logarithmic inequalities
  13. The energy spaces $W_1$ and $W_2$
  14. Characterization in terms of Orlicz spaces
  15. Inclusion relation in function spaces

Key Result

Theorem 1.1

Let $\lambda\in{\mathbb R}\setminus\{0\}$. For any $\varphi\in H^1({\mathbb R}^d)$, there exists a unique solution $C({\mathbb R}, H^1({\mathbb R}^d))$ to eq:1.1 in the sense of for all bounded open sets $\Omega\subset{\mathbb R}^d$ and all $t\in{\mathbb R}$, and with $u(0)=\varphi$. If in addition we assume $\varphi\in W_1$, the $H^1$-solution above satisfies $u\in (C\cap L^\infty)({\mathbb R},W

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 52 more