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| Format: | Preprint |
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2018
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| Online Access: | https://arxiv.org/abs/1812.00957 |
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| _version_ | 1866914688703397888 |
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| author | Lam, F. |
| author_facet | Lam, F. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1812_00957 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Leray self-similarity equations in fluid dynamics Lam, F. Fluid Dynamics 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. |
| title | Leray self-similarity equations in fluid dynamics |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/1812.00957 |