Finite-time blowup for the IPM equation with a uniformly space-time smooth force

Yes. Theorem 2.1: there are a smooth odd initial density $\rho_{in}\in C^\infty(\mathbb T^2)$ of zero spatial mean, an odd force $F\in C^\infty([0,1]\times\mathbb T^2)$, and a solution smooth on $[0,T]$ for every $T<1$ with $\rho(t)\to\rho_*$ in $C^\eta$ for every $0\le\eta<1$, yet $\|\nabla\rho(t)\|_\infty$ and $\|D_xu_{\mathbb T}(\rho(t))\|_\infty$ both diverging as $t\uparrow1$. The advance over Córdoba and Martínez-Zoroa is precisely the force class, from $L^\infty_t C^\infty_x$ to uniformly space-time smooth, on the torus rather than the plane. Not the Clay Millennium problem: that problem is Navier-Stokes with viscosity, and Fefferman's official description states that the Euler equation "is not on the Clay Institute's list of prize problems".

Yes. Theorem 2.1: there are a smooth odd initial density $\rho_{in}\in C^\infty(\mathbb T^2)$ of zero spatial mean, an odd force $F\in C^\infty([0,1]\times\mathbb T^2)$, and a solution smooth on $[0,T]$ for every $T<1$ with $\rho(t)\to\rho_*$ in $C^\eta$ for every $0\le\eta<1$, yet $\|\nabla\rho(t)\|_\infty$ and $\|D_xu_{\mathbb T}(\rho(t))\|_\infty$ both diverging as $t\uparrow1$. The advance over Córdoba and Martínez-Zoroa is precisely the force class, from $L^\infty_t C^\infty_x$ to uniformly space-time smooth, on the torus rather than the plane. Not the Clay Millennium problem: that problem is Navier-Stokes with viscosity, and Fefferman's official description states that the Euler equation "is not on the Clay Institute's list of prize problems".