Mathematics – Analysis of PDEs
Scientific paper
2004-02-09
Mathematics
Analysis of PDEs
23 pages, no figures, to appear, New York J. Math. This is the final version
Scientific paper
In any dimension $n \geq 3$, we show that spherically symmetric bounded energy solutions of the defocusing energy-critical non-linear Schr\"odinger equation $i u_t + \Delta u = |u|^{\frac{4}{n-2}} u$ in $\R \times \R^n$ exist globally and scatter to free solutions; this generalizes the three and four dimensional results of Bourgain and Grillakis. Furthermore we have bounds on various spacetime norms of the solution which are of exponential type in the energy, which improves on the tower-type bounds of Bourgain. In higher dimensions $n \geq 6$ some new technical difficulties arise because of the very low power of the non-linearity.
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