The inviscid Boussinesq system is a system of nonlinear partial differential equations for a temperature anomaly 
, an incompressible vector
field 
,
and pressure 
.
With external forces 
and 
in two dimensions, it can be written
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is the upward coordinate vector. The corresponding vorticity 
satisfies 
.
Alpöge and Buckmaster (2026a) claim a solution on 
from smooth, compactly supported initial temperature and
zero initial velocity, with forces that are smooth in space and time and have compact support in one fixed ball.
The temperature remains bounded, while the norm of its gradient
tends to infinity and the vorticity norm has infinite
limit superior at a finite terminal time. The solution remains smooth and unique
on every closed preterminal interval.
The authors report that Claude and Codex helped develop and simplify the construction. Two accompanying Lean 4 developments formalize the claimed result, but independent human verification had not been reported as of Sep. 9, 2026 (Alpöge and Buckmaster 2026b). This system is distinct from the scalar Boussinesq equation.
