# Leray self-similarity equations in fluid dynamics

## TL;DR

This paper demonstrates that Leray's self-similarity equations in fluid dynamics admit only trivial solutions, highlighting flaws in previous arguments and clarifying the mathematical limitations of self-similar solutions in viscous and inviscid flows.

## Contribution

It provides a rigorous analysis showing the non-existence of non-trivial self-similar solutions, criticizing past flawed assumptions and illustrating mathematical inconsistencies.

## Key findings

- Leray's equations admit only trivial solutions in viscous and inviscid flows.
- Past papers used ill-defined decay assumptions at infinity.
- Counterexample to Sobolev inequality illustrates self-contradiction.

## Abstract

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.