  In the present note, we show that, as a priori bounds, the vorticity dynamics
derived from Leray's backward self-similarity hypothesis admits only trivial
solution in viscous as well as inviscid flows. By analogy, there is no non-zero
solution in the forward self-similar equation. Since the Navier-Stokes or Euler
equations are invariant under space translation in the whole space, our
analysis establishes that technically flawed arguments have been exploited in a
number of past papers, notably in Necas, Ruzicka & Sverak (1996); Tsai (1998);
and Pomeau (2016), where the presumed decays or bounds at infinity are
ill-defined and non-existent. Furthermore, an effort has been made to exemplify
an inappropriate application of the familiar extremum principles in the theory
of linear elliptic equation.
  In the appendix, we give a counterexample to the Sobolev inequality and,
hence illustrate the nature of self contradiction. In the totality comparison
of Lp norms, its scope of application is not significant.
