Finite-time blowup for the 3D incompressible Euler equations with smooth forcing

Yes. Theorem 1.1: for every $r_0>0$ and $z_0$ there are a time $T_*>0$, a divergence-free axisymmetric $u_0\in C_c^\infty$ supported in a fixed solid torus with nonzero swirl and zero meridional velocity, and an axisymmetric force $f\in C^\infty(\mathbb R^3\times[0,T_*])$ supported in that torus, with a solution smooth on $[0,T_*)$ for which circulation and meridional velocity stay bounded while $\|\nabla\Gamma(t)\|_\infty$ and $\|\omega(t)\|_\infty$ tend to infinity and $\int_0^{T_*}\|\omega(t)\|_\infty\,dt=\infty$, so the blowup is genuine by Beale-Kato-Majda. It is unique among divergence-free locally space-time Lipschitz solutions with the same data, and competitors need not be axisymmetric. 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 1.1: for every $r_0>0$ and $z_0$ there are a time $T_*>0$, a divergence-free axisymmetric $u_0\in C_c^\infty$ supported in a fixed solid torus with nonzero swirl and zero meridional velocity, and an axisymmetric force $f\in C^\infty(\mathbb R^3\times[0,T_*])$ supported in that torus, with a solution smooth on $[0,T_*)$ for which circulation and meridional velocity stay bounded while $\|\nabla\Gamma(t)\|_\infty$ and $\|\omega(t)\|_\infty$ tend to infinity and $\int_0^{T_*}\|\omega(t)\|_\infty\,dt=\infty$, so the blowup is genuine by Beale-Kato-Majda. It is unique among divergence-free locally space-time Lipschitz solutions with the same data, and competitors need not be axisymmetric. 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".