STABILITY OF TWO-DIMENSIONAL VISCOUS INCOMPRESSIBLE FLOWS UNDER THREE-DIMENSIONAL PERTURBATIONS AND INVISCID SYMMETRY BREAKING Academic Article uri icon

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 Szkelyhidi, 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. 2013 Society for Industrial and Applied Mathematics.

published proceedings

  • SIAM JOURNAL ON MATHEMATICAL ANALYSIS

author list (cited authors)

  • Bardos, C., Lopes Filho, M. C., Niu, D., Nussenzveig Lopes, H. J., & Titi, E. S.

citation count

  • 29

complete list of authors

  • Bardos, C||Lopes Filho, MC||Niu, Dongjuan||Nussenzveig Lopes, HJ||Titi, ES

publication date

  • January 2013