Finite-time blowup for the inviscid Boussinesq system with smooth forcing

Yes. Blowup for the inviscid Boussinesq system on $\mathbb R^2$ with forces in $C^\infty(\mathbb R^2\times[0,T_*])$ in both equations, supported in one fixed spatial ball, from smooth compactly supported initial temperature and zero initial velocity. The temperature stays bounded while $\|\nabla\theta(t)\|_\infty\to\infty$ and the vorticity norm has infinite limsup as $t\uparrow T_*$. The solution is smooth on every closed interval before blowup and unique in a finite-energy Lipschitz class. This lifts the force from barely $C^1$ to fully smooth, and is the construction the Euler paper then builds on. 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. Blowup for the inviscid Boussinesq system on $\mathbb R^2$ with forces in $C^\infty(\mathbb R^2\times[0,T_*])$ in both equations, supported in one fixed spatial ball, from smooth compactly supported initial temperature and zero initial velocity. The temperature stays bounded while $\|\nabla\theta(t)\|_\infty\to\infty$ and the vorticity norm has infinite limsup as $t\uparrow T_*$. The solution is smooth on every closed interval before blowup and unique in a finite-energy Lipschitz class. This lifts the force from barely $C^1$ to fully smooth, and is the construction the Euler paper then builds on. 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".