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Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on \(\mathbb R^3\). (English) Zbl 1060.35131

The authors study the following initial value problem for a cubic defocusing nonlinear Schrödinger equation \[ \begin{gathered} i\partial_t\varphi(x,t)+ \Delta_x\varphi(x,t)= |\varphi(x, t)|^2 \varphi(x,t),\quad x\in\mathbb{R}^3,\;t\geq 0,\\ \varphi(x,0)= \varphi_0(x)\in H^s(\mathbb{R}^3).\end{gathered} \] Here \(H^s(\mathbb{R}^3)\) denotes the usual inhomogeneous Sobolev space. The authors prove global existence in \(H^s(\mathbb{R}^3)\) for \(s> 4/5\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q05 Euler-Poisson-Darboux equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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