BealeKatoMajda

Beale–Kato–Majda (BKM) theorem is a foundational result in the mathematical theory of the three-dimensional incompressible Euler equations. It concerns the evolution of the velocity field u(x,t) and its vorticity ω = ∇×u, under the equations ∂t u + (u·∇)u + ∇p = 0 and ∇·u = 0. Consider a smooth solution on a time interval [0,T). The theorem states that u remains smooth on [0,T) if and only if the time integral of the L-infinity norm of the vorticity is finite, i.e. ∫0^T ||ω(·, t)||∞ dt < ∞. Equivalently, if a finite-time breakdown occurs at time T, then ∫0^{T} ||ω(·, t)||∞ dt diverges.

The proof relies on the vorticity formulation ∂t ω + (u·∇)ω = (ω·∇)u and Grönwall-type estimates to show that higher norms of the velocity (and hence smoothness) can be controlled by the accumulation of the maximum vorticity over time. A key consequence is that the growth of the gradient of u is governed by the integral of ||ω||∞, linking potential singularities to the behavior of vorticity.

Beale–Kato–Majda provides a sharp criterion for breakdown in 3D Euler and clarifies what would be required for a finite-time blow-up. It does not resolve global regularity, but it has influenced further work on Euler and Navier–Stokes equations, including related refinements and extensions by other researchers.