Path 2: Regularity Criteria and Blow-up Conditions

Base Camp 2.1 – Ladyzhenskaya–Prodi–Serrin Conditions

Studies integrability criteria that guarantee a weak solution is actually smooth (preventing blowup). A classical result by Ladyzhenskaya (1958) and Prodi–Serrin (1962) asserts that if a weak solution $u(x,t)$ lies in certain $L^p_tL^q_x$ spaces (with $2/p + 3/q \le 1$), then no singularities occur. In particular, Serrin's 1962 paper showed that if $u\in L^r(0,T;L^s(\mathbb{R}^3))$ with $2/r+3/s\le 1$ (e.g. $L^\infty_tL^3_x$ or $L^4_tL^{12}_x$), then $u$ is smooth on $(0,T)$. These criteria, proved via interpolation and energy estimates, essentially require that the solution is not too large on average. (Intermediate level: conditional regularity theorems.)

Stepping Stones

• Stepping Stone: Serrin's Criterion – If $u$'s time-space integrability satisfies $2/p+3/q<1$ (or equals 1 under mild endpoint conditions), then $u$ remains regular.
• Stepping Stone: Examples – Notably, $u\in L^\infty(0,T;L^3_x)$ or $u\in L^5(0,T;L^{10}_x)$ are sufficient to preclude blowup (these meet Serrin's condition).
• Stepping Stone: Contrast with Weak Solutions – Leray–Hopf solutions are known to belong to $L^2_tL^6_x$ (for incompressible flow), which falls outside the Serrin range (since $2/2+3/6=3/2>1$), highlighting the gap between known existence and known regularity regimes.

Base Camp 2.2 – Vorticity & Critical Norm Criteria

Explores blow-up criteria in critical spaces, often involving the vorticity $\omega = \nabla \times u$. A celebrated criterion by Beale–Kato–Majda (1984) states that a smooth solution of the 3D Euler equations can blow up only if $\int_0^T |\omega(t)|_{L^\infty}dt = \infty$. For the viscous Navier–Stokes case, this implies that as long as the vorticity remains bounded in sup-norm (no infinite swirl), the solution cannot develop a singularity. Another milestone result (Escauriaza–Seregin–Šverák 2003) proved that if $u$ is bounded in critical Lebesgue space $L^\infty(0,T;L^3_x)$, then $u$ is actually smooth up to time $T$. In other words, the borderline Serrin case $(p,q)=(\infty,3)$ still precludes blowup. Researchers have also found regularity criteria based on one component of velocity or pressure – for example, requiring just one velocity component to satisfy a Serrin-type condition can ensure smoothness. (Advanced level: sharpening conditions for no-blowup.)

Stepping Stones

• Stepping Stone: BKM Criterion – (Euler analog) Blow-up can occur only if vorticity norms blow up: $\sup_{t<T}|\omega(\cdot,t)|_{\infty} = \infty$ or equivalently $\int_0^T\!|\omega|_{\infty}dt=\infty$. No infinite vorticity accumulation ⇒ extend solution beyond $T$.
• Stepping Stone: Escauriaza–Seregin–Šverák Theorem – If $u(t)\in L^\infty(0,T;L^3(\mathbb{R}^3))$, then $u$ does not blow up at $T$ (in fact $u$ is Hőlder-continuous near $T$). This solved the last open endpoint of Serrin's condition.
• Stepping Stone: Further Regularity Criteria – E.g., if one component of the velocity $u_3$ remains in $L^2_tL^\infty_x$ (a “one-component Serrin condition”), then no blowup occurs. Similarly, conditions on the pressure (e.g. $p \in L^{5/4}_{t,x}$) or the direction of vorticity can prevent singularities – though these are more technical refinements.

What to Upload Next

To dive deeper into each Base Camp, the following books and papers are top priorities:

Base Camp 2.1 – Ladyzhenskaya–Prodi–Serrin Conditions

• James Serrin (1962), “On the interior regularity of weak solutions of the Navier–Stokes equations,” Arch. Ration. Mech. Anal. 9:187–195. – Original paper establishing the $L^r_tL^s_x$ regularity criterion. (Classical, rigorous.)
• O. A. Ladyzhenskaya (1963), The Mathematical Theory of Viscous Incompressible Flow – Contains Ladyzhenskaya's early regularity conditions (e.g. $u\in L^4_{t,x}$) and proofs. Provides a very clear classical PDE approach (intermediate level, monograph).
• J. G. Heywood (1980), “The interior regularity of the weak solutions of the Navier–Stokes equations,” Acta Math. 142:159–178. – A detailed exposition of Serrin's condition and related results, with full proofs and slight extensions. (Rigorous.)
• Giovanni Galdi (2011), An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Vol. I. – Sections on Leray–Hopf solutions and Criteria à la Serrin. A comprehensive modern textbook, providing context and detailed proofs (graduate level).

Base Camp 2.2 – Vorticity & Critical Norm Criteria

• J. T. Beale, T. Kato, A. Majda (1984), “Remarks on the breakdown of smooth solutions for the 3-D Euler equations,” Comm. Math. Phys. 94:61–66. – The BKM paper. Short and incisive, it proves the vorticity blow-up criterion. A must-read for understanding blowup mechanisms in ideal fluids.
• Luis Escauriaza, G. Seregin, V. Šverák (2003), “$L_{3,\infty}$-solutions of Navier–Stokes and backward uniqueness,” Russ. Math. Surveys 58(2):211–250. – Proves the critical $L^\infty_tL^3_x$ regularity criterion. Also introduces techniques of backward uniqueness. (Advanced, but crucial result.)
• M. R. Ukhovskii, V. I. Yudovich (1968), “Axially symmetric flows of ideal and viscous fluids,” J. Appl. Math. Mech. 32:52–61. – Not directly a criterion, but provides the context for how special structure (no swirl) prevents blowup. Reading this alongside criteria papers shows how geometry can enforce conditions akin to Serrin's.
• Alexander Majda & Andrea Bertozzi (2002), Vorticity and Incompressible Flow – Chapter 3 covers blow-up criteria and vorticity estimates in both Euler and Navier–Stokes. Offers intuition linking the BKM criterion with physical vortex stretching, at an accessible level.