Global well-posedness and scattering for mass-critical inhomogeneous NLS when $d\ge3$
Xuan Liu, Changxing Miao, Jiqiang Zheng
TL;DR
The paper establishes global well-posedness and scattering for the mass-critical inhomogeneous NLS $i\partial_t u+\Delta u=\mu |x|^{-b}|u|^{(4-2b)/d}u$ in dimensions $d\ge3$ with $0<b<\min\{2,d/2\}$, including the focusing case under a mass constraint relative to the ground state. It extends the profile/rigidity framework to the inhomogeneous setting by working in Lorentz spaces, using long-time Strichartz estimates and a frequency-localized Morawetz approach to rule out almost periodic counterexamples. The decay of the inhomogeneity plays a crucial role in preventing concentration at infinity and enables Morawetz rather than interaction Morawetz techniques. The results provide a robust methodology for handling mass-critical INLS with singular inhomogeneities and broken symmetries, yielding global dynamics for large data and clarifying the role of the ground-state threshold in the focusing case.
Abstract
We prove global well-posedness and scattering for solutions to the mass-critical inhomogeneous nonlinear Schrödinger equation $i\partial_{t}u+Δu=\pm |x|^{-b}|u|^{\frac{4-2b}{d}}u$ for large $L^2(\mathbb{R} ^d)$ initial data with $d\ge3,0<b<\min \left\{ 2,\frac{d}{2} \right\}$; in the focusing case, we require that the mass is strictly less than that of the ground state. Compared with the classical Schrödinger case ($b=0$, Dodson, J. Amer. Math. Soc. (2012), Adv. Math. (2015)), the main differences for the inhomogeneous case ($b>0$) are that the presence of the inhomogeneity $|x|^{-b}$ creates a nontrivial singularity at the origin, and breaks the translation symmetry as well as the Galilean invariance of the equation, which makes the establishment of the profile decomposition and long time Strichartz estimates more difficult. To overcome these difficulties, we perform the concentration compactness/rigidity methods of [Kenig and Merle, Invent. Math. (2006)] in the Lorentz space framework, and reduces the problem to the exclusion of almost periodic solutions. The exclusion of these solutions will utilize fractional estimates and long time Strichartz estimates in Lorentz spaces. In our study, we obseve that the decay of the inhomogeneity $|x|^{-b}$ at infinity prevents the concentration of the almost periodic solution at infinity in either physical or frequency space. Therefore, we can use classical Morawetz estimates, rather than interaction Morawetz estimates, to exclude the existence of the quasi-soliton.