Math Conjectures

Global regularity for three-dimensional incompressible MHD

OPENMajorOpen problemProposed c. 1950 · Canonical special case

Canonical statement

For every pair of smooth divergence-free vector fields u0,b0:T3→R3u_0,b_0:\mathbb T^3\to\mathbb R^3, the unique local smooth solution of
∂tu+(u⋅∇)u−(b⋅∇)b+∇p=Δu,∂tb+(u⋅∇)b−(b⋅∇)u=Δb,∇⋅u=∇⋅b=0,(u,b)∣t=0=(u0,b0) \begin{aligned} \partial_tu+(u\cdot\nabla)u-(b\cdot\nabla)b+\nabla p&=\Delta u,\\ \partial_tb+(u\cdot\nabla)b-(b\cdot\nabla)u&=\Delta b,\\ \nabla\cdot u=\nabla\cdot b&=0,\\ (u,b)|_{t=0}&=(u_0,b_0) \end{aligned}
extends smoothly for all t≥0t\ge0.
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For every pair of smooth divergence-free vector fields \(u_0,b_0:\mathbb T^3\to\mathbb R^3\), the unique local smooth solution of \[ \begin{aligned} \partial_tu+(u\cdot\nabla)u-(b\cdot\nabla)b+\nabla p&=\Delta u,\\ \partial_tb+(u\cdot\nabla)b-(b\cdot\nabla)u&=\Delta b,\\ \nabla\cdot u=\nabla\cdot b&=0,\\ (u,b)|_{t=0}&=(u_0,b_0) \end{aligned} \] extends smoothly for all \(t\ge0\).
PDE-004:[SermangeTemam1983MHD][CaflischKlapperSteele1997IdealMHD][ChenMiaoZhang2008MHDRegularity][JiaZhou2026MHDCriteria]
Partial differential equationsMathematical physics

Notes

The problem asks for global smooth solutions of the three-dimensional incompressible magnetohydrodynamics (MHD) system, which couples a Navier–Stokes-type equation for the velocity uu to an induction equation for the magnetic field bb, both with full dissipation. Given smooth divergence-free data on the torus, does the unique local solution extend smoothly for all t≥0t\ge 0? The question emerged in the mid-twentieth century with the mathematical study of conducting fluids; a standard mathematical formulation and foundational results appear in Sermange and Temam [SermangeTemam1983MHD].

Much of the Navier–Stokes theory carries over: global weak solutions exist, strong solutions are global for small data, and numerous conditional regularity criteria are available, imposing integrability conditions on the solution that preclude blow-up [SermangeTemam1983MHD] [ChenMiaoZhang2008MHDRegularity] [JiaZhou2026MHDCriteria]. Questions of singularities and energy dissipation for the ideal (non-dissipative) counterpart were examined in [CaflischKlapperSteele1997IdealMHD].

Since taking b0=0b_0=0 reduces the system to the Navier–Stokes equations, the MHD problem contains the Navier–Stokes regularity problem and is at least as hard; large-data smooth global regularity remains open.

References (4)

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.