Mathematics – Analysis of PDEs
Scientific paper
2010-07-07
Mathematics
Analysis of PDEs
Scientific paper
We prove that if $u(t)$ is a log-log blow-up solution, of the type studied by Merle-Rapha\"el (2001-2005), to the $L^2$ critical focusing NLS equation $i\partial_t u +\Delta u + |u|^{4/d} u=0$ with initial data $u_0\in H^1(\mathbb{R}^d)$ 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 $i\partial_tu + \Delta u + |u|^4u=0$. This improves Rapha\"el-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.
Holmer Justin
Roudenko Svetlana
No associations
LandOfFree
Blow-up solutions on a sphere for the 3d quintic NLS in the energy space does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Blow-up solutions on a sphere for the 3d quintic NLS in the energy space, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Blow-up solutions on a sphere for the 3d quintic NLS in the energy space will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-346375