A note on the nonlinear Schrödinger equation in a general domain

TL;DR

This paper develops a constructive approach to the Cauchy problem for nonlinear Schrödinger equations on general domains $Ω \subset \mathbb{R}^N$ by proving that approximate solutions form a Cauchy sequence in a Banach space, avoiding traditional compactness arguments. It treats three nonlinearities: power-type in 2D ($g(u)=\lambda|u|^\alpha u$, with $\alpha \le 2$), the logarithmic nonlinearity $g(u)=u\log|u|^2$, and a damping nonlinearity $g(u)=i\,\frac{u}{|u|^{\alpha}}$ with $0<\alpha<1$. Two approximation schemes are developed—truncated nonlinearity and Yosida-type regularization—and used to establish local well-posedness in the 2D setting, as well as global well-posedness for the logarithmic and damped cases in appropriate spaces, with mass and energy conservation where applicable. The paper also discusses extensions to fractional/logarithmic variants and partial results on unbounded domains, highlighting a constructive framework complementary to compactness-based methods.

Abstract

We consider the Cauchy problem for nonlinear Schrödinger equations in a general domain $Ω\subset\mathbb{R}^N$. Construction of solutions has been only done by classical compactness method in previous results. Here, we construct solutions by a simple alternative approach. More precisely, solutions are constructed by proving that approximate solutions form a Cauchy sequence in some Banach space. We discuss three different types of nonlinearities: power type nonlinearities, logarithmic nonlinearities and damping nonlinearities.

Theorems & Definitions (23)

• Theorem 2.1: Local well-posedness V84Og90C03
• Remark 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 3.1: C83
• Lemma 3.2: C83
• Theorem 3.3: CH80C03
• Lemma 3.4: CH80