H. J. Nussenzveig Lopes

ON THE INVISCID LIMIT FOR TWO-DIMENSIONAL INCOMPRESSIBLE FLOW WITH NAVIER FRICTION CONDITION ∗

In [Nonlinearity, 11 (1998), pp. 1625--1636], Clopeau, Mikelic, and Robert studied the inviscid limit of the two-dimensional incompressible Navier--Stokes equations in a bounded domain subject to Navier friction--type boundary conditions. They proved that the inviscid limit satisfies the incompressible Euler equations, and their result ultimately includes flows generated by bounded initial vorticities. Our purpose in this article is to adapt and, to some extent, simplify their argument in order to include pth power integrable initial vorticities, with p > 2.

Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows

We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes [10] on the vanishing viscosity limit of circularly symmetric viscous flow in a disk with rotating boundary, shown there to converge to the inviscid limit in L2-norm as long as the prescribed angular velocity α(t) of the boundary has bounded total variation. Here we establish convergence in stronger L2 and Lp-Sobolev spaces, allow for more singular angular velocities α, and address the issue of analyzing the behavior of the boundary layer. This includes an analysis of concentration of vorticity in the vanishing viscosity limit. We also consider such flows on an annulus, whose two boundary components rotate independently.

STABILITY OF TWO-DIMENSIONAL VISCOUS INCOMPRESSIBLE FLOWS UNDER THREE-DIMENSIONAL PERTURBATIONS AND INVISCID SYMMETRY BREAKING

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 Szekelyhidi, 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 ...

Existence of a weak solution for the semigeostrophic equation with integrable initial data

In this article we present a proof of existence of a weak solution for the semigeostrophic system of equations, formulated as an active scalar transport equation with Monge-Ampere coupling, with initial data in . This is an extension of a 1998 result due to Benamou and Brenier, who proved existence with initial data in .

On the large-time behavior of two-dimensional vortex dynamics

In this paper we prove two results regarding the large-time behavior of vortex dynamics in the full plane. In the first result we show that the total integral of vorticity is confined in a region of diameter growing at most like the square root of time. In the second result we show that if a dynamic rescaling of the absolute value of vorticity with spatial scale growing linearly with time converges weakly, then it must converge to a discrete sum of Dirac masses. This last result extends in scope a previous result by the authors, valid for nonnegative initial vorticity on a half-plane.

An Extension of Marchioro's Bound on the Growth of a Vortex Patch to Flows with series\itshape L

We observe that C. Marchioros cubic-root bound in time on the growth of the diameter of a patch of vorticity [Comm. Math. Phys, 164 (1994), pp. 507--524] can be extended to incompressible two-dimensional Euler flows with compactly supported initial vorticity in Lp, p > 2, and with a distinguished sign.

^\mbox\scriptsizeseries\itshapep

Abstract: In this article we study the long-time behavior of incompressible ideal flow in a half plane from the point of view of vortex scattering. Our main result is that certain asymptotic states for half-plane vortex dynamics decompose naturally into a nonlinear superposition of soliton-like states. Our approach is to combine techniques developed in the study of vortex confinement with weak convergence tools in order to study the asymptotic behavior of a self-similar rescaling of a solution of the incompressible 2D Euler equations on a half plane with compactly supported, nonnegative initial vorticity.

Vorticity

We consider approximate solution sequences of the 2D incompressible Euler equations obtained by mollifying compactly supported initial vorticities in

Large Time Behavior for Vortex Evolution in the Half-Plane

L^p

Bounds on the Dispersion of Vorticity in 2D Incompressible, Inviscid Flows with A Priori Unbounded Velocity

,