The Vortex-Wave equation with a single vortex as the limit of the Euler equation

In this article we consider the physical justification of the Vortex-Wave equation introduced by Marchioro and Pulvirenti in the case of a single point vortex moving in an ambient vorticity. We consider a sequence of solutions for the Euler equation in the plane corresponding to initial data consisting of an ambient vorticity in $L^1\cap L^\infty$ and a sequence of concentrated blobs which approach the Dirac distribution. We introduce a notion of a weak solution of the Vortex-Wave equation in terms of velocity (or primitive variables) and then show, for a subsequence of the blobs, the solutions of the Euler equation converge in velocity to a weak solution of the Vortex-Wave equation.Comment: 24 pages, to appea

Weak in Space, Log in Time Improvement of the Lady{\v{z}}enskaja-Prodi-Serrin Criteria

In this article we present a Lady{\v{z}}enskaja-Prodi-Serrin Criteria for regularity of solutions for the Navier-Stokes equation in three dimensions which incorporates weak $L^p$ norms in the space variables and log improvement in the time variable.Comment: 14 pages, to appea

Mathematical results for some α\displaystyle{\alpha} models of turbulence with critical and subcritical regularizations

In this paper, we establish the existence of a unique "regular" weak solution to turbulent flows governed by a general family of $\alpha$ models with critical regularizations. In particular this family contains the simplified Bardina model and the modified Leray-$\alpha$ model. When the regularizations are subcritical, we prove the existence of weak solutions and we establish an upper bound on the Hausdorff dimension of the time singular set of those weak solutions. The result is an interpolation between the bound proved by Scheffer for the Navier-Stokes equations and the regularity result in the critical case

On the motion of a small light body immersed in a two dimensional incompressible perfect fluid with vorticity

In this paper we consider the motion of a rigid body immersed in a two dimensional unbounded incom-pressible perfect fluid with vorticity. We prove that when the body shrinks to a massless pointwise particle with fixed circulation, the "fluid+rigid body" system converges to the vortex-wave system introduced by Marchioro and Pulvirenti in [11]. This extends both the paper [2] where the case of a solid tending to a massive pointwise particle was tackled and the paper [3] where the massless case was considered but in a bounded cavity filled with an irrotational fluid.DYnamique des Fluides, Couches Limites, Tourbillons et Interface