The conjecture that slower-than-$C\nu^{1/4}$ convergence invalidates Prandtl theory

Let $u$ solve the two-dimensional Navier–Stokes equations with viscosity $u$, and let $\overline{u}$ denote the corresponding Euler solution. Suppose the vanishing viscosity limit holds in ($VV$), with convergence rate described by $F(\nu)$, and interpret “slower than $C\nu^{1/4}$” as a rate asymptotically worse than $C\nu^{1/4}$. Conjecture. If the vanishing viscosity limit in ($VV$) holds at a rate slower than $C\nu^{\frac{1}{4}}$ in two dimensions, then Prandtl theory fails. The conjecture concerns the relationship between the convergence rate in the vanishing viscosity limit and the validity of Prandtl boundary-layer theory; the surrounding discussion argues that slower convergence places the matching between the Navier–Stokes and Euler solutions outside the Prandtl layer.