Scattering in the energy space and below for 3D NLS
The author studies uniform boundedness problems arising in scattering in \(H^s(\mathbb{R}^3)\) for the defocusing 3D nonlinear Schrödinger equation (NLS) \[ iu_t+\Delta u-u|u|^{p-2}=0, \quad p<6, \qquad u(0)=\varphi. \] First, when \(\frac{10}{3}<p\) and \(\varphi\in H^1\), he obtains \(L_t^qL_x^p\)-bounds on the solutions \(u\) for all admissible pairs \((p,q)\) in terms of the \(H^1\)-norm of the Cauchy data \(\varphi\), thus answering an issue left open by \textit{J. Ginibre} and \textit{G. Velo} [J. Math. Pures Appl., IX. Sér. 64, 363-401 (1985; Zbl 0582.35090)]. This makes use of uniform decay estimates without decay assumption on \(\varphi\), which is a refinement of those obtained by \textit{J.-E. Lin} and \textit{W. A. Strauss} [J. Funct. Anal. 30, 245-263 (1978; Zbl 0395.35070)] and Ginibre and Velo [ibid], and is based on the Morawetz' inequality. For the case \(\varphi\in H^s\) where \(s\geq 1\), and in particular, when \(p=4\) where it is known that the Cauchy problem is well posed, the author obtains \(H^s\)-bounds on \(u(t)\) in terms of the \(H^s\)-norm of \(\varphi\) for all \(t\). The \(s>1\) case differs from the \(s=1\) case in that the bound in the former case is not implied by a conservation law. Estimates for the wave maps are also obtained. Further, the author considers the case below the energy norm, i.e., when \(s<1\), again using the case \(p=4\) as an example. Adapting his own arguments for the 2D case [Int. Math. Res. Not. 5, 253-283 (1998; Zbl 0917.35126)] and using an improved Strichartz inequality, the author establishes global well-posedness for all Cauchy data with \(s>\frac{11}{13}\) and scattering results. Results for the case when \(\varphi\) is radial are also given.
scientific article; zbMATH DE number 1221960
scientific article