Finite-Time Singularities of Prandtl's Boundary-Layer Equations: A Review

We review what is known about finite-time singularity formation ("blow-up") in the unsteady Prandtl boundary-layer equations, from the numerical discovery of the van Dommelen--Shen singularity in 1980 to the rigorous description of its self-similar structure completed in 2018--2022. The survey is organized around five questions: (i) what the singularity is physically (unsteady separation and eruption of the boundary layer); (ii) where it is known to occur (the impulsively started cylinder, vortex-induced separation, colliding wall streams); (iii) what has been proved (the E--Engquist theorem, the Kukavica--Vicol--Wang theorem, the Collot--Ghoul--Ibrahim--Masmoudi description of the stable blow-up pattern, the classification of inviscid singularities); (iv) how blow-up relates to the ill-posedness of the Prandtl equations without monotonicity and to the inviscid limit of the Navier--Stokes equations; and (v) how the reduced one-dimensional system on the symmetry axis, which carries the rigorous theory, sits inside the recent similarity transformation of Sun [Phys.\ Fluids \textbf{36}, 083616 (2024)] as its \(m=1\) member. We also record the explicit affine blow-up solution \(u=-x/(t_*-t)\), \(v=y/(t_*-t)\), which is exact for both Prandtl and Navier--Stokes, and explain why Prandtl blow-up bears no relation to the finite-energy Navier--Stokes blow-up announced in September 2026. A list of open problems closes the survey.