NDLI: Stability of Two-Dimensional Viscous Incompressible Flows under Three-Dimensional Perturbations and Inviscid Symmetry Breaking

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Bardos, C. Titi, E. S. Lopes Filho, M. C. Nussenzveig Lopes, H. J. Niu, Dongjuan
Copyright Year	2013
Abstract	In this article we consider weak solutions of the three-dimensional incompressible fluid flow equations with initial data admitting a one-dimensional symmetry group. We examine both the viscous and inviscid cases. For the case of viscous flows, we prove that Leray--Hopf weak solutions of the three-dimensional Navier--Stokes equations preserve initially imposed symmetry and that such symmetric flows are stable under general three-dimensional perturbations, globally in time. We work in three different contexts: two-and-a-half-dimensional, helical, and axisymmetric flows. In the inviscid case, we observe that as a consequence of recent work by De Lellis and Székelyhidi, there are genuinely three-dimensional weak solutions of the Euler equations with two-dimensional initial data. We also present two partial results where restrictions on the set of initial data and on the set of admissible solutions rule out spontaneous symmetry breaking; one is due to P.-L. Lions and the other is a consequence of our viscous stability result.
Starting Page	1871
Ending Page	1885
Page Count	15
File Format	PDF
ISSN	00361410
DOI	10.1137/120862569
e-ISSN	10957154
Journal	SIAM Journal on Mathematical Analysis (SJMAAH)
Issue Number	3
Volume Number	45
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-06-19
Access Restriction	Subscribed
Subject Keyword	uniqueness of weak solutions Spectral, collocation and related methods PDEs in connection with fluid mechanics helical symmetry axisymmetric flow Navier--Stokes equations Leray--Hopf weak solutions Euler equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Mathematics

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