Control of blow-up profiles for the mass-critical focusing nonlinear Schr\"odinger equation on bounded domains

In this paper, we consider the mass-critical focusing nonlinear Schr\"odinger on bounded two-dimensional domains with Dirichlet boundary conditions. In the absence of control, it is well-known that free solutions starting from initial data sufficiently large can blow-up. More precisely, given a finite number of points, there exists particular profiles blowing up exactly at these points at the blow-up time. For pertubations of these profiles, we show that, with the help of an appropriate nonlinear feedback law located in an open set containing the blow-up points, the blow-up can be prevented from happening. More specifically, we construct a small-time control acting just before the blow-up time. The solution may then be extended globally in time. This is the first result of control for blow-up profiles for nonlinear Schr\"odinger type equations. Assuming further a geometrical control condition on the support of the control, we are able to prove a null-controllability result for such blow-up profiles. Finally, we discuss possible extensions to three-dimensional domains.