Extending the Córdoba-Martínez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing

By adapting the techniques used in the IPM blowup result of Córdoba-Martínez-Zoroa with spatially smooth force, we prove finite-time blow-up for the IPM equation on ${\mathbb{T}}^2$ with a uniformly spacetime smooth force. In particular, we show there exist a smooth odd initial density, a smooth odd force $F\in C^{\mathrm{\infty }}([0,1]\times {\mathbb{T}}^2)$, and a classical solution $\rho$ on $[0,1)$ whose density gradient and spatial velocity gradient diverge in $L^{\mathrm{\infty }}$ as $t\uparrow 1$. Nevertheless, $\rho (t)$ converges in $C^{\eta }$ for every $0\le \eta <1$.