Math Conjectures

Global regularity for the two-dimensional inviscid Boussinesq system

OPENMajorOpen problemProposed c. 1980 · Canonical special case

Canonical statement

For every u0∈S(R2;R2)u_0\in\mathcal S(\mathbb R^2;\mathbb R^2) and θ0∈S(R2)\theta_0\in\mathcal S(\mathbb R^2) with ∇⋅u0=0\nabla\cdot u_0=0, the maximal classical solution of
∂tu+(u⋅∇)u+∇p=θe2,∂tθ+u⋅∇θ=0,∇⋅u=0,(u,θ)∣t=0=(u0,θ0) \partial_tu+(u\cdot\nabla)u+\nabla p=\theta e_2,\qquad \partial_t\theta+u\cdot\nabla\theta=0,\qquad \nabla\cdot u=0,\qquad (u,\theta)|_{t=0}=(u_0,\theta_0)
exists for all t≥0t\ge0 and is smooth.
View source LaTeX
For every \(u_0\in\mathcal S(\mathbb R^2;\mathbb R^2)\) and \(\theta_0\in\mathcal S(\mathbb R^2)\) with \(\nabla\cdot u_0=0\), the maximal classical solution of \[ \partial_tu+(u\cdot\nabla)u+\nabla p=\theta e_2,\qquad \partial_t\theta+u\cdot\nabla\theta=0,\qquad \nabla\cdot u=0,\qquad (u,\theta)|_{t=0}=(u_0,\theta_0) \] exists for all \(t\ge0\) and is smooth.
PDE-005:[Chae2006Boussinesq][ChaeMiaoXue2022BoussinesqFronts][Wu2025BoussinesqSurvey]
Partial differential equationsMathematical physicsDynamical systems

Notes

The problem asks whether the two-dimensional Boussinesq system with neither viscosity nor thermal diffusion is globally regular: a velocity field uu is driven by the buoyancy force θe2\theta e_2 while the temperature θ\theta is passively transported, and one asks whether smooth, rapidly decaying data always produce solutions smooth for all t≥0t\ge 0. The question crystallized around 1980 and is regarded as a two-dimensional relative of the three-dimensional Euler problem, since the buoyancy term plays the role of a vortex-stretching mechanism in the vorticity equation.

By contrast, variants with even partial dissipation are much better understood: Chae proved global regularity when either viscosity or thermal diffusion alone is present [Chae2006Boussinesq], global results are known for nondiffusive temperature fronts in the viscous system [ChaeMiaoXue2022BoussinesqFronts], and the extensive literature on partial and fractional dissipation is surveyed in [Wu2025BoussinesqSurvey].

For the fully inviscid system no general global a priori bound is known, and no smooth finite-time blow-up example has been constructed; the problem is open in both directions.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.