Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation

Details Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation

2010-07-25

arxiv.org/abs/1007.4299v3

Mathematics

Analysis of PDEs

30 pages, 4 figures; We obtain up to some endpoints the full range of the radial Strichartz estimates for the Schrodinger equa

Scientific paper

We prove some new Strichartz estimates for a class of dispersive equations with radial initial data. In particular, we obtain up to some endpoints the full radial Strichartz estimates for the Schr\"odinger equation. The ideas of proof are based on Shao's ideas \cite{Shao} and some ideas in \cite{GPW} to treat the non-homogeneous case, while at the endpoint we need to use subtle tools to overcome some logarithmic divergence. We also apply the improved Strichartz estimates to the nonlinear problems. First, we prove the small data scattering and large data LWP for the nonlinear Schr\"odinger equation with radial critical $\dot{H}^s$ initial data below $L^2$; Second, for radial data we improve the results of the $\dot{H}^s\times \dot{H}^{s-1}$ well-posedness for the nonlinear wave equation in \cite{SmithSogge}; Finally, we obtain the well-posedness theory for the fractional order Schr\"odinger equation in the radial case.

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