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Register a new userSharp spherically averaged Strichartz estimates for the Schrodinger equation
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Authors
Zihua Guo
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We prove generalized Strichartz estimates with weaker angular integrability for the Schrodinger equation. Our estimates are sharp except some endpoints. Then we apply these new estimates to prove the scattering for the 3D Zakharov system with small data in the energy space with low angular regularity. Our results improve the results obtained recently in cite{GLNW}.
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We prove scattering for the 2D cubic derivative Schrodinger equation with small data in the critical Besov space with one degree angular regularity. The main new ingredient is that we prove a spherically averaged maximal function estimate for the 2D Schrodinger equation. We also prove a global well-posedness result for the 2D Schrodinger map in the critical Besov space with one degree angular regularity. The key ingredients for the latter results are the spherically averaged maximal function estimate, null form structure observed in cite{Bej}, as well as the generalised spherically averaged Strichartz estimates obtained in cite{Guo2} in order to exploit the null form structure.
We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schrodinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2L_x^infty$ estimates fail at the critical regularity in high dimensions by using stable Levy process in $R^d$. Moreover, we show that some spherically averaged $L_t^2L_x^infty$ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double $L_t^2$-type estimates.
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