Validity of three-dimensional axisymmetric steady Prandtl expansion

Abstract

We study steady axisymmetric Navier--Stokes flows with swirl in the circular cylinder $\mathcal C_L=(0,L)\times B_2(0,1)$ with the no-slip condition on the stationary lateral wall. For the leading Euler velocity $(u_e^0,w_e^0,θ_e^0)=(1,0,r K(r))$, where $K(r)=H(r^2)$ is a prescribed smooth radial function, we construct and justify a Prandtl expansion as $\varepsilon \to 0$ for sufficiently small fixed $L$. The expansion accounts for both axial and azimuthal boundary layers, including the case of a Blasius leading axial profile with nonzero swirl. Weighted estimates at the symmetry axis and coupled remainder estimates yield a uniform velocity error bound.