Finite time blowup for a supercritical defocusing nonlinear Schr\"odinger system

We consider the global regularity problem for defocusing nonlinear Schr"odinger systems $$i{\mathrm{\partial }}_t+\mathrm{\Delta }u=({\mathrm{\nabla }}_{{\mathbf{R}}^m}F)(u)+G$$ on Galilean spacetime $\mathbf{R}\times {\mathbf{R}}^d$, where the field $u\colon {\bf R}^{1+d} \to {\bf C}^m$isvector-valued,$F\colon {\bf C}^m \to {\bf R}$ is a smooth potential which is positive, phase-rotation-invariant, and homogeneous of order $p+1$ outside of the unit ball for some exponent $p>1$, and $G:\mathbf{R}\times {\mathbf{R}}^d\to {\mathbf{C}}^m$ is a smooth, compactly supported forcing term. This generalises the scalar defocusing nonlinear Schr"odinger (NLS) equation, in which $m=1$ and $F(v) = \frac{1}{p+1} |v|^{p+1}$\mathrm{.}Inthispaperwestudythesupercriticalcasewhere$d \geq 3$and$p > 1 + \frac{4}{d-2}$. We show that in this case, there exists a smooth potential $F$ for some sufficiently large $m$, positive and homogeneous of order $p+1$ outside of the unit ball, and a smooth compactly choice of initial data $u(0)$ and forcing term $G$ for which the solution develops a finite time singularity. In fact the solution is locally discretely self-similar with respect to parabolic rescaling of spacetime. This demonstrates that one cannot hope to establish a global regularity result for the scalar defocusing NLS unless one uses some special property of that equation that is not shared by these defocusing nonlinear Schr"odinger systems. As in a previous paper of the author considering the analogous problem for the nonlinear wave equation, the basic strategy is to first select the mass, momentum, and energy densities of $u$, then $u$ itself, and then finally design the potential $F$ in order to solve the required equation.