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We discuss the vacuum energy of a quantized scalar field in the presence of classical surfaces, defining bounded domains $\Omega \subset {\mathbb{R}}^{d}$, where the field satisfies ideal or non-ideal boundary conditions. For the electromagnetic case, this situation describes the conductivity correction to the zero-point energy. Using an analytic regularization procedure, we obtain the vacuum energy for a massless scalar field at zero temperature in the presence of a slab geometry $\Omega=\mathbb R^{d-1}\times[0, L]$ with Dirichlet boundary conditions. To discuss the case of non-ideal boundary conditions, we employ an asymptotic expansion, based on an approximate functional equation for the Riemann zeta-function, where finite sums outside their original domain of convergence are defined. Finally, to obtain the Casimir energy for a massless scalar field in the presence of a rectangular box, with lengths $L_{1}$ and $L_{2}$, i.e., $\Omega=[0,L_{1}]\times[0,L_{2}]$ with non-ideal boundary conditions, we employ an approximate functional equation of the Epstein zeta-function., Comment: 10 pages

Lei and Lin have recently given a proof of a global mild solution of the three-dimensional Navier-Stokes equations in function spaces based on the Wiener algebra. An alternative proof of existence of these solutions was then developed by Bae, and this new proof allowed for an estimate of the radius of analyticity of the solutions at positive times. We adapt the Bae proof to prove existence of the Lei-Lin solution in the spatially periodic setting, finding an improved bound for the radius of analyticity in this case., Comment: 15 pages

The Monte Carlo (MC) method is the most common technique used for uncertainty quantification, due to its simplicity and good statistical results. However, its computational cost is extremely high, and, in many cases, prohibitive. Fortunately, the MC algorithm is easily parallelizable, which allows its use in simulations where the computation of a single realization is very costly. This work presents a methodology for the parallelization of the MC method, in the context of cloud computing. This strategy is based on the MapReduce paradigm, and allows an efficient distribution of tasks in the cloud. This methodology is illustrated on a problem of structural dynamics that is subject to uncertainties. The results show that the technique is capable of producing good results concerning statistical moments of low order. It is shown that even a simple problem may require many realizations for convergence of histograms, which makes the cloud computing strategy very attractive (due to its high scalability capacity and low-cost). Additionally, the results regarding the time of processing and storage space usage allow one to qualify this new methodology as a solution for simulations that require a number of MC realizations beyond the standard.

In this article we study the limit when $\alpha \to 0$ of solutions to the $\alpha$-Euler system in the half-plane, with no-slip boundary conditions, to weak solutions of the 2D incompressible Euler equations with non-negative initial vorticity in the space of bounded Radon measures in $H^{-1}$. This result extends the analysis done in arXiv:1611.05300 and arXiv:1403.5682. It requires a substantially distinct approach, analogous to that used for Delort's Theorem, and a new detailed investigation of the relation between (no-slip) filtered velocity and potential vorticity in the half-plane.

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 $k\to \infty$. 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.

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 $\omega = \nabla \times u \in L^{\frac32}$. An example of a 2D field in the class $\omega \in L^{\frac32 - \epsilon}$ for any $\epsilon>0$, and $u\in B^{1/3}_{3,\infty}$ (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., Comment: 14 pages

We consider the $\alpha$-Euler equations on a bounded three-dimensional domain with frictionless Navier boundary conditions. Our main result is the existence of a strong solution on a positive time interval, uniform in $\alpha$, for $\alpha$ sufficiently small. Combined with the convergence result in a previous article by the same authors, this implies convergence of solutions of the $\alpha$-Euler equations to solutions of the incompressible Euler equations when $\alpha \to 0$. In addition, we obtain a new result on local existence of strong solutions for the incompressible Euler equations on bounded three-dimensional domains. The proofs are based on new {\it a priori} estimates in conormal spaces.

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 Kato's 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]., Comment: 23 pages, submitted for publication

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., Comment: Submitted for publication

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.

In [Math. Meth. Appl. Sci. 19 (1996) 53-62], C. Marchioro examined the problem of vorticity confinement in the exterior of a smooth bounded domain. The main result in Marchioro's paper is that solutions of the incompressible 2D Euler equations with compactly supported nonnegative initial vorticity in the exterior of a connected bounded region have vorticity support with diameter growing at most like $\mathcal{O}(t^{(1/2)+\vare})$, for any $\vare>0$. In addition, if the domain is the exterior of a disk, then the vorticity support is contained in a disk of radius $\mathcal{O}(t^{1/3})$. The purpose of the present article is to refine Marchioro's results. We will prove that, if the initial vorticity is even with respect to the origin, then the exponent for the exterior of the disk may be improved to 1/4. For flows in the exterior of a smooth, connected, bounded domain we prove a confinement estimate with exponent 1/2 (i.e. we remove the $\vare$) and in certain cases, depending on the harmonic part of the flow, we establish a logarithmic improvement over the exponent 1/2. The main new ingredients in our approach are: (1) a detailed asymptotic description of solutions to the exterior Poisson problem near infinity, obtained by the use of Riemann mappings; (2) renormalized energy estimates and bounds on logarithmic moments of vorticity and (3) a new {\it a priori} estimate on time derivatives of logarithmic perturbations of the moment of inertia., Comment: Submitted for publication

In this work we study the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity $\omega_0$ and the circulation $\gamma$ of the initial flow around the obstacle. We prove that, if $\gamma$ is sufficiently small, the limit flow satisfies the full-plane Navier-Stokes system, with initial vorticity $\omega_0 + \gamma \delta$, where $\delta$ is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors in [Comm P.D.E. 28 (2003) 349-379], where the effect of the small obstacle appears in the coefficients of the PDE and not only on the initial data. The main ingredients of the proof are $L^p-L^q$ estimates for the Stokes operator in an exterior domain, a priori estimates inspired on Kato's fixed point method, energy estimates, renormalization and interpolation.

In this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible Euler equations and as a (weak) solution of the Birkhoff-Rott equations. We begin by discussing the classical Birkhoff-Rott equations with respect to arbitrary parametrizations of the sheet. We introduce a notion of weak solution to the Birkhoff-Rott system and we prove consistency of this notion with the classical formulation of the equations. Our main purpose in this paper is to present a sharp criterion for the equivalence of the weak Euler and weak Birkhoff-Rott descriptions of vortex sheet dynamics., Comment: 22 pages

In [1], T. Clopeau, A. Mikeli\'c, and R. Robert studied the inviscid limit of the 2D 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 $p$-th power integrable initial vorticities, with $p>2$. [1] Clopeau, T., Mikeli\'c, A., Robert, R., {\it On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions}, Nonlinearity {\bf 11} (1998) 1625--1636., Comment: 18 pages

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., Comment: 30 pages, no figures

In this article we study the asymptotic behavior of incompressible, ideal, time-dependent 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 $\gamma$, the circulation around the obstacle. For smooth flow around a single obstacle, $\gamma$ is a conserved quantity which is determined by the initial data. We will show that if $\gamma = 0$, the limit flow satisfies the standard incompressible Euler equations in the full plane but, if $\gamma \neq 0$, 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, {\it 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., Comment: 30 pages, 1 figure, submitted for publication, AMS classification: Primary: 35Q35, Secondary: 76B03,76B47