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

Abstract: 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 $\mathbb T<sup>2$ with a uniformly spacetime smooth force. In particular, we show there exist a smooth odd initial density, a smooth odd force $$F\in C^{\infty([0,1]\times\mathbb} T<sup>2)$$, and a classical solution $ρ$ on $[0,1)$ whose density gradient and spatial velocity gradient diverge in $L<sup>\infty$ as $t\uparrow 1$. Nevertheless, $ρ(t)$ converges in $C<sup>η$ for every $0\leqη&lt;1$.