Local well-posedness for the $H^2$-critical nonlinear Schr\"odinger equation

TL;DR

This paper establishes local well-posedness for the $H^2$-critical nonlinear Schrödinger equation in high dimensions, ensuring existence, uniqueness, and stability of solutions in the critical Sobolev space.

Contribution

It proves the local well-posedness of the nonlinear Schrödinger equation at the critical $H^2$ level for dimensions $N \\ge 5$, which was previously unresolved.

Findings

Proves local existence of solutions in $\dot H^2$ space.

Establishes unconditional uniqueness of solutions.

Demonstrates continuous dependence on initial data.

Abstract

In this paper, we consider the nonlinear Schr\"odinger equation $iu_t +\Delta u= \lambda |u|^{\frac {4} {N-4}} u$ in $\R^N $, $N\ge 5$, with $\lambda \in \C$. We prove local well-posedness (local existence, unconditional uniqueness, continuous dependence) in the critical space $\dot H^2 (\R^N) $.