Onsager's Conjecture for the Incompressible Euler Equations in Bounded Domains

The goal of this note is to show that, also in a bounded domain $\Omega \subset \mathbb{R}^n$, with $\partial \Omega\in C^2$, any weak solution, $(u(x,t),p(x,t))$, of the Euler equations of ideal incompressible fluid in $\Omega\times (0,T) \subset \mathbb{R}^n\times\mathbb{R}_t$, with the impermeability boundary condition: $u\cdot \vec n =0$ on $\partial\Omega\times(0,T)$, is of constant energy on the interval $(0,T)$ provided the velocity field $u \in L^3((0,T); C^{0,\alpha}(\overline{\Omega}))$, with $\alpha>\frac13\,.

Interaction of two systems with saddle-node bifurcations on invariant circles. I. Foundations and the mutualistic case

The saddle-node bifurcation on an invariant circle (SNIC) is one of the codimension-one routes to creation or destruction of a periodic orbit in a continuous-time dynamical system. It governs the transition from resting behaviour to periodic spiking in many class I neurons, for example. Here, as a first step towards theory of networks of such units the effect of weak coupling between two systems with a SNIC is analysed. Two crucial parameters of the coupling are identified, which we call \delta_1 and \delta_2. Global bifurcation diagrams are obtained here for the "mutualistic" case \delta_1 \delta_2 > 0. According to the parameter regime, there may coexist resting and periodic attractors, and there can be quasiperiodic attractors of torus or cantorus type, making the behaviour of even such a simple system quite non-trivial. In a second paper we will analyse the mixed case \delta_1 \delta_2 < 0 and summarise the conclusions of this study.Comment: 37 pages, 27 figure

Abrupt bifurcations in chaotic scattering : view from the anti-integrable limit

Bleher, Ott and Grebogi found numerically an interesting chaotic phenomenon in 1989 for the scattering of a particle in a plane from a potential field with several peaks of equal height. They claimed that when the energy E of the particle is slightly less than the peak height Ec there is a hyperbolic suspension of a topological Markov chain from which chaotic scattering occurs, whereas for E > Ec there are no bounded orbits. They called the bifurcation at E = Ec an abrupt bifurcation to chaotic scattering. The aim of this paper is to establish a rigorous mathematical explanation for how chaotic orbits occur via the bifurcation, from the viewpoint of the anti-integrable limit, and to do so for a general range of chaotic scattering problems

Methods for the Study of Transverse Momentum Differential Correlations

We introduce and compare three differential correlation functions for the study of transverse momentum correlation in $p+p$ and $A+A$ collisions. These consist of {\it inclusive}, {\it event-wise} and a differential version of the correlation measure $\tilde C$ introduced by Gavin \cite{Gavin} for experimental study of the viscosity per unit entropy of the matter produced in $A+A$ collisions. We study the quantitative difference between the three observables on the basis of PYTHIA simulations of $p+p$ collisions and $A+A$ collisions consisting of an arbitrary superposition of $p+p$ collision events at $\sqrt{s} =$200 GeV. We observe that {\it inclusive} and {\it event-wise} correlation functions are remarkably identical to each other where as the observable $\tilde C$ differs from the two. We study the robustness and efficiency dependencies of these observables based on truncated Taylor expansions in efficiency in $p+p$ collisions and on the basis of Monte Carlo simulation using an adhoc detector efficiency parameterization. We find that all the three observables are essentially independent of detector efficiency. We additionally study the scaling of the correlation measures and find all the observables exhibit an approximate $1/N$ dependence of the number of participants ({\it N}) in $A+A$ collisions. Finally, we study the impact of flow-like anisotropy on the {\it inclusive} correlation function and find flow imparts azimuthal modulations similar to those observed with two-particle densities.Comment: 19 pages, 8 figure

Observations on staggered fermions at non-zero lattice spacing

We show that the use of the fourth-root trick in lattice QCD with staggered fermions corresponds to a non-local theory at non-zero lattice spacing, but argue that the non-local behavior is likely to go away in the continuum limit. We give examples of this non-local behavior in the free theory, and for the case of a fixed topologically non-trivial background gauge field. In both special cases, the non-local behavior indeed disappears in the continuum limit. Our results invalidate a recent claim that at non-zero lattice spacing an additive mass renormalization is needed because of taste-symmetry breaking.Comment: 17 pages, two refs. and a note adde

The Hydraulic Jump in Liquid Helium

We present the results of some experiments on the circular hydraulic jump in normal and superfluid liquid helium. The radius of the jump and the depth of the liquid outside the jump are measured through optical means. Although the scale of the apparatus is rather small, the location of the jump is found to be consistent with the assumption that the jump can be treated as a shock, if the surface tension is taken into account. The radius of the jump does not change when going down in temperature through the lambda point; we think that the flow is supercritical. A remarkable feature of the experiment is the observation of stationary ripples within the jump when the liquid is superfluid.Comment: Submitted to the proceedings of the 24th International Conference on Low Temperature Physics. 2 figure

On the effect of rotation on magnetohydrodynamic turbulence at high magnetic Reynolds number

This article is focused on the dynamics of a rotating electrically conducting fluid in a turbulent state. As inside the Earth's core or in various industrial processes, a flow is altered by the presence of both background rotation and a large scale magnetic field. In this context, we present a set of 3D direct numerical simulations of incompressible decaying turbulence. We focus on parameters similar to the ones encountered in geophysical and astrophysical flows, so that the Rossby number is small, the interaction parameter is large, but the Elsasser number, defining the ratio between Coriolis and Lorentz forces, is about unity. These simulations allow to quantify the effect of rotation and thus inertial waves on the growth of magnetic fluctuations due to Alfv\'en waves. Rotation prevents the occurrence of equipartition between kinetic and magnetic energies, with a reduction of magnetic energy at decreasing Elsasser number {\Lambda}. It also causes a decrease of energy transfer mediated by cubic correlations. In terms of flow structure, a decrease of {\Lambda} corresponds to an increase in the misalignment of velocity and magnetic field.Comment: 18 pages, 12 figure