In this article we study the asymptotic behavior of incompressible, ideal, timedependent two dimensional flow in the exterior of a single smooth obstacle when the size of the obstacle becomes very small. Our main purpose is to identify the equation satisfied by the limit flow. We will see that the asymptotic behavior depends on γ, the circulation around the obstacle. For smooth flow around a single obstacle, γ is a conserved quantity which is determined by the initial data. We will show that if γ = 0, the limit flow satisfies the standard incompressible Euler equations in the full plane but, if γ = 0, the limit equation acquires an additional forcing term. We treat this problem by first constructing a sequence of approximate solutions to the incompressible 2D Euler equation in the full plane from the exact solutions obtained when solving the equation on the exterior of each obstacle and then passing to the limit on the weak formulation of the equation. We use an explicit treatment of the Green's function of the exterior domain based on conformal maps, a priori estimates obtained by carefully examining the limiting process and the Div-Curl Lemma, together with a standard weak convergence treatment of the nonlinearity for the passage to the limit.

Motivated by the presence of a finite number of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages for dissipative dynamical systems, we present a continuous data assimilation algorithm for the three-dimensional Navier-Stokes-α model. This algorithm consists of introducing a nudging process through general type of approximation interpolation operator (that is constructed from observational measurements) that synchronizes the large spatial scales of the approximate solutions with those of the unknown solutions the Navier-Stokes-α equations that corresponds to these measurements. Our main result provides conditions on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, that is obtained from these collected data, converges to the unkown reference solution (physical reality) over time. These conditions are given in terms of some physical parameters, such as kinematic viscosity, the size of the domain and the forcing term.

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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.

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