  We show that the 1D Hou-Luo model on the real line admits exact self-similar
finite-time blowup solutions with smooth self-similar profiles. The existence
of these profiles is established via a fixed-point method that is purely
analytic. We also prove that the profiles satisfy some monotonicity and
convexity properties that are unknown before, and we give rigorous estimates on
the algebraic decay rates of the profiles in the far field. Our result
supplements the previous computer-assisted proof of self-similar finite-time
blowup for the Hou-Luo model with finer characterizations of the profiles.
