Regularity criteria: Ladyzhenskaya-Prodi-Serrin and Beale-Kato-Majda

A conditional regularity result says: if a solution has some extra property, then it is smooth. These criteria are both tools and constraints, because any singularity must violate all of them.

The Ladyzhenskaya-Prodi-Serrin criterion (Prodi 1959, Serrin 1962, Ladyzhenskaya 1967). Let \(\mathbf{u}\) be a Leray-Hopf weak solution on \(\mathbb{R}^3\times(0,T)\). If \[ \mathbf{u} \in L^p(0,T;L^q(\mathbb{R}^3)), \qquad \frac{2}{p} + \frac{3}{q} \le 1, \qquad 3 \lt q \le \infty, \] then \(\mathbf{u}\) is smooth on \(\mathbb{R}^3\times(0,T]\) and unique. For \(q = 4\) the condition is \(p \ge 8\); for \(q = \infty\) it is \(p \ge 2\). The endpoint \(q = 3\), \(p = \infty\) is much harder and was proved by Escauriaza, Seregin and Šverák (2003) using backward uniqueness for parabolic equations; Seregin (2012) went further and showed that at a first blow-up time \(\|\mathbf{u}(t)\|_{L^3} \to \infty\).

Why these exponents? Under the scaling \(\mathbf{u}_\lambda = \lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2t)\), \[ \|\mathbf{u}_\lambda\|_{L^p_tL^q_x} = \lambda^{1 - 2/p - 3/q}\,\|\mathbf{u}\|_{L^p_tL^q_x}, \] so equality in \(\frac2p + \frac3q = 1\) is exactly scale invariance. The proof shows where it enters. Testing the equation with \(-\Delta\mathbf{u}\) gives \(\frac12\frac{d}{dt}\|\nabla\mathbf{u}\|^2 + \nu\|\Delta\mathbf{u}\|^2 \le \int|\mathbf{u}||\nabla\mathbf{u}||\Delta\mathbf{u}|\). Hölder's inequality bounds the right side by \(\|\mathbf{u}\|_{L^q}\|\nabla\mathbf{u}\|_{L^{2q/(q-2)}}\|\Delta\mathbf{u}\|_{L^2}\), interpolation gives \(\|\nabla\mathbf{u}\|_{L^{2q/(q-2)}} \le C\|\nabla\mathbf{u}\|^{1-3/q}\|\Delta\mathbf{u}\|^{3/q}\), and Young's inequality absorbs \(\|\Delta\mathbf{u}\|\) into the left side, leaving \[ \frac{d}{dt}\|\nabla\mathbf{u}\|^2_{L^2} \le C_\nu\,\|\mathbf{u}\|^p_{L^q}\,\|\nabla\mathbf{u}\|^2_{L^2}, \qquad p = \frac{2q}{q-3}. \] Grönwall's inequality then bounds \(\|\nabla\mathbf{u}(t)\|^2\) by \(\|\nabla\mathbf{u}_0\|^2\exp\bigl(C_\nu\int_0^t\|\mathbf{u}\|^p_{L^q}\bigr)\), finite under the hypothesis, and higher derivatives follow.

The Beale-Kato-Majda criterion (1984) controls blow-up through vorticity. If a smooth solution of the 3D Euler equations exists on \([0,T^*)\) and cannot be continued past \(T^*\), then \[ \int_0^{T^*}\|\boldsymbol{\omega}(t)\|_{L^\infty}\,dt = \infty, \] and the same holds for Navier-Stokes. So vorticity must blow up, and fast: a rate \(\|\boldsymbol{\omega}(t)\|_\infty \sim (T^*-t)^{-\alpha}\) is only possible with \(\alpha \ge 1\), the rate the scaling predicts, since \(\boldsymbol{\omega}_\lambda\) carries a factor \(\lambda^2\) and time a factor \(\lambda^{-2}\). The proof uses a logarithmic Sobolev inequality to bound \(\|\nabla\mathbf{u}\|_\infty\) by \(\|\boldsymbol{\omega}\|_\infty\) up to a log of a higher norm. Combined with the 2D fact that \(\|\omega\|_\infty\) never grows, BKM gives another proof that 2D solutions stay smooth. Constantin and Fefferman (1993) showed for Navier-Stokes that the direction of vorticity matters too: if it varies in a Lipschitz way where vorticity is large, the solution stays smooth.

Picture it: a list of alarms, each wired to a scale-invariant quantity; a singularity, if one exists, must set off every alarm at once. Think it: the sharpest criteria sit at critical scaling, and every known a priori bound for general data sits strictly below it. A proof of global regularity would need a bound at the critical level; a construction of blow-up would need to defeat all the criteria simultaneously.