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

Abstract

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 threedimensional 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-halfdimensional, helical and axi-symmetric 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 twodimensional 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. MSC Subject Classifications:35Q35, 65M70.

The primitive equations as the small aspect ratio limit of the Navier–Stokes equations: Rigorous justification of the hydrostatic approximation

“…Proof. We will follow the argument of Serrin [33] (see also [2] and the references therein). Recalling the definition of weak solutions to (SNS), the following integral identity holds…”

“…Instead, one can only perform the energy estimates in the framework of the weak solutions. To this end, we adopt the idea, which was introduced in Serrin [33] (see also Bardos et al [2] and the reference therein) to prove the weak-strong uniqueness of the Navier-Stokes equations; however, the difference in our case is that, the role of "strong solutions" is now played by the solutions of the (PEs), while the role of "weak solutions" is now played by those of (SNS), or intuitively, we are somehow doing the weak-strong uniqueness between two different systems. Precisely, we will: (i) use (v, w) as the testing functions for (SNS); (ii) test (PEs) by v ε ; (iii) perform the basic energy identity of (PEs); (iv) use the energy inequality for (SNS).…”

Planar Limits of Three-Dimensional Incompressible Flows with Helical Symmetry

“…It was shown in [12,Theorem 3.4] that weak solutions of (2.7) with helical symmetry are unique, global in time, and agree with a strong solution, if the initial data belongs to H 1 0,per (Ω σ ) and the associated pressure p is also a helical function. (See also [1] for more elaborate discussion regarding this matter. )…”

“…While there is an existence theory for weak solutions of the Euler equation in three dimensions [4,17], we will give here a definition of weak solution to (5.1) adapted to the geometry of the problem and amenable to the analysis of the limit σ → ∞ (for further discussion on the uniqueness of helical weak solutions, we refer the reader to [1].) Let ψ be the unique weak solution of (5.11) with initial condition…”

“…In fact, for the case of a circular pipe they established both global existence of a weak helical solution with initial data in L 2 , and global existence and uniqueness of a strong solution with initial data in the Sobolev space H 1 . (For a discussion about uniqueness of weak solutions, , within the class of all LerayHopf weak solutions of the three-dimensional Navier-Stokes with helical initial data, see [1].) The situation is different, and rather interesting, in the case of ideal fluid governed by the Euler equations, see [5,6].…”