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- W2951787114 abstract "We prove that if $u(t)$ is a log-log blow-up solution, of the type studied by Merle-Raphael (2001-2005), to the $L^2$ critical focusing NLS equation $ipartial_t u +Delta u + |u|^{4/d} u=0$ with initial data $u_0in H^1(mathbb{R}^d)$ in the cases $d=1, 2$, then $u(t)$ remains bounded in $H^1$ away from the blow-up point. This is obtained without assuming that the initial data $u_0$ has any regularity beyond $H^1(mathbb{R}^d)$. As an application of the $d=1$ result, we construct an open subset of initial data in the radial energy space $H^1_{rad}(mathbb{R}^3)$ with corresponding solutions that blow-up on a sphere at positive radius for the 3d quintic ($dot H^1$-critical) focusing NLS equation $ipartial_tu + Delta u + |u|^4u=0$. This improves Raphael-Szeftel (2009), where an open subset in $H^3_{rad}(mathbb{R}^3)$ is obtained. The method of proof can be summarized as follows: on the whole space, high frequencies above the blow-up scale are controlled by the bilinear Strichartz estimates. On the other hand, outside the blow-up core, low frequencies are controlled by finite speed of propagation." @default.
- W2951787114 created "2019-06-27" @default.
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- W2951787114 date "2010-07-07" @default.
- W2951787114 modified "2023-09-27" @default.
- W2951787114 title "Blow-up solutions on a sphere for the 3d quintic NLS in the energy space" @default.
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