Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations
Changxing Miao Bo Zhang
Discrete & Continuous Dynamical Systems - A 2007, 17(1): 181-200 doi: 10.3934/dcds.2007.17.181
In this paper the global well-posedness in $L^2$ and $H^m$ of the Cauchy problem is proved for nonlinear Schrödinger-type equations. This we do by establishing regular Strichartz estimates for the corresponding linear equations and some nonlinear a priori estimates in the framework of Besov spaces. We further establish the regularity of the $H^m$-solution to the Cauchy problem.
keywords: Strichartz estimates Besov spaces Schrödinger-type equations Cauchy problem. Well-posedness
The energy-critical NLS with inverse-square potential
Rowan Killip Changxing Miao Monica Visan Junyong Zhang Jiqiang Zheng
Discrete & Continuous Dynamical Systems - A 2017, 37(7): 3831-3866 doi: 10.3934/dcds.2017162

We consider the defocusing energy-critical nonlinear Schrödinger equation with inverse-square potential $iu_t = -Δ u + a|x|^{-2}u + |u|^4u$ in three space dimensions. We prove global well-posedness and scattering for $a > - \frac{1}{4} + \frac{1}{{25}}$. We also carry out the variational analysis needed to treat the focusing case.

keywords: Nonlinear Schröodinger equation scattering inverse-square potential concentration compactness

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