Continuous dependence for $H^{2}$ critical nonlinear Schr\"{o}dinger equations in high dimensions

TL;DR

This paper proves that solutions to high-dimensional $H^{2}$ critical nonlinear Schr"{o}dinger equations depend continuously on initial data, overcoming previous difficulties caused by non-Lipschitz nonlinearities using exotic Strichartz spaces.

Contribution

It establishes continuous dependence in $H^{2}$ for solutions in high dimensions, addressing a gap in the well-posedness theory for these equations.

Findings

Established continuous dependence of solutions on initial data in $H^{2}$.

Introduced exotic Strichartz spaces to handle non-Lipschitz nonlinearities.

Resolved a key difficulty in high-dimensional nonlinear Schr"{o}dinger equations.

Abstract

The global existence of solutions in $H^{2}$ is well known for $H^{2}$ critical nonlinear Schr\"{o}dinger equations with small initial data in high dimensions $d\geq8$. However, even though the solution is constructed by a fixed-point technique, continuous dependence in $H^{2}$ does not follow from the contraction mapping argument. Comparing with the low dimension cases $4<d<8$, there is an obstruction to this approach because of the sub-quadratic nature of the nonlinearity(which makes the derivative of the nonlinearity non-Lipschitz). In this paper, we resolve this difficulty by applying exotic Strichartz spaces of lower order instead and show that the solution depends continuously on the initial value in the sense that the local flow is continuous $H^{2}\rightarrow H^{2}$.