On local well-posedness for the nonlinear Schr\"odinger equation with general power nonlinearity

Abstract

The nonlinear Schr\"odinger equation plays a fundamental role in mathematical physics, particularly in the study of quantum mechanics and Bose-Einstein condensation. This paper explores two distinct approaches to establishing the local well-posedness of solutions: the semigroup theory ansatz and the Strichartz estimates ansatz. Semigroup theory provides a general and elegant framework rooted in functional analysis, allowing for the interpretation of the time evolution of solutions as operator semigroups. Strichartz estimates, developed specifically for dispersive equations, offer an alternative technique based on refined space-time estimates and fixed-point arguments. We systematically analyze and compare both approaches and apply them to nonlinear Schr\"odinger equations where the nonlinearity is given by $F(u)=\lambda|u|^p u$ for some $\lambda \in \mathbb{R}$. So our results extend beyond the physically relevant case $p=2$.