Asymptotic behavior of 2D incompressible ideal flow around small disks

In this article, we study the homogenization limit of a family of solutions to the incompressible 2D Euler equations in the exterior of a family of $n_k$ disjoint disks with centers ${z^k_i}$ and radii $varepsilon_k$. We assume that the initial velocities $u_0^k$ are smooth, divergence-free, tangent to the boundary and that they vanish at infinity. We allow, but we do not require, $n_k to infty$, and we assume $varepsilon_k to 0$ as $kto infty$. Let $gamma^k_i$ be the circulation of $u_0^k$ around the circle ${|x-z^k_i|=varepsilon_k}$. We prove that the homogenization limit retains information on the circulations as a time-independent coefficient. More precisely, we assume that: (1) $omega_0^k = mbox{ curl }u_0^k$ has a uniform compact support and converges weakly in $L^{p_0}$, for some $p_0>2$, to $omega_0 in L^{p_0}_{c}(mathbb{R}^2)$, (2) $sum_{i=1}^{n_k} gamma^k_i delta_{z^k_i} rightharpoonup mu$ weak-$ast$ in $mathcal{BM}(mathbb{R}^2)$ for some bounded Radon measure $mu$, and (3) the radii $varepsilon_k$ are sufficiently small. Then the corresponding solutions $u^k$ converge strongly to a weak solution $u$ of a modified Euler system in the full plane. This modified Euler system is given, in vorticity formulation, by an active scalar transport equation for the quantity $omega=mbox{ curl } u$, with initial data $omega_0$, where the transporting velocity field is generated from $omega$ so that its curl is $omega + mu$. As a byproduct, we obtain a new existence result for this modified Euler system.

On the Onsager conjecture in two dimensions

This note addresses the question of energy conservation for the 2D Euler system with an $L^p$-control on vorticity. We provide a direct argument, based on a mollification in physical space, to show that the energy of a weak solution is conserved if $omega = abla times u in L^{frac32}$. An example of a 2D field in the class $omega in L^{frac32 - epsilon}$ for any $epsilon>0$, and $uin B^{1/3}_{3,infty}$ (Onsager critical space) is constructed with non-vanishing energy flux. This demonstrates sharpness of the kinematic argument. Finally we prove that any solution to the Euler equation produced via a vanishing viscosity limit from Navier-Stokes, with $omega in L^p$, for $p>1$, conserves energy. This is an Onsager-supercritical condition under which the energy is still conserved, pointing to a new mechanism of energy balance restoration.

Vanishing viscosity limit for an expanding domain in space

We study the limiting behavior of viscous incompressible flows when the fluid domain is allowed to expand as the viscosity vanishes. We describe precise conditions under which the limiting flow satisfies the full space Euler equations. The argument is based on truncation and on energy estimates, following the structure of the proof of Katos criterion for the vanishing viscosity limit. This work complements previous work by the authors, see [Kelliher, Comm. Math. Phys. 278 (2008), 753-773] and [arXiv:0801.4935v1].

Incompressible flow around a small obstacle and the vanishing viscosity limit

In this article we consider viscous flow in the exterior of an obstacle satisfying the standard no-slip boundary condition at the surface of the obstacle. We seek conditions under which solutions of the Navier-Stokes system in the exterior domain converge to solutions of the Euler system in the full space when both viscosity and the size of the obstacle vanish. We prove that this convergence is true assuming two hypothesis: first, that the initial exterior domain velocity converges strongly in $L^2$ to the full-space initial velocity and second, that the diameter of the obstacle is smaller than a suitable constant times viscosity, or, in other words, that the obstacle is sufficiently small. The convergence holds as long as the solution to the limit problem is known to exist and stays sufficiently smooth. This work complements the study of incompressible flow around small obstacles, which has been carried out in [1,2,3] [1] D. Iftimie and J. Kelliher, {it Remarks on the vanishing obstacle limit for a 3D viscous incompressible fluid.} Preprint available at http://math.univ-lyon1.fr/~iftimie/ARTICLES/viscoushrink3d.pdf . [2] D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes. {it Two dimensional incompressible ideal flow around a small obstacle.} Comm. Partial Differential Equations {bf 28} (2003), no. 1-2, 349--379. [3] D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes. {it Two dimensional incompressible viscous flow around a small obstacle.} Math. Ann. {bf 336} (2006), no. 2, 449--489.

Vanishing Viscosity Limits and Boundary Layers for Circularly Symmetric 2D Flows

We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes [LMN], 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 $L^2$-norm as long as the prescribed angular velocity $alpha(t)$ of the boundary has bounded total variation. Here we establish convergence in stronger $L^2$ and $L^p$-Sobolev spaces, allow for more singular angular velocities $alpha$, 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. [LMN] Lopes Filho, M. C., Mazzucato, A. L. and Nussenzveig Lopes, H. J., Vanishing viscosity limit for incompressible flow inside a rotating circle, preprint 2006.