On the stability of travelling waves with vorticity obtained by minimisation

We modify the approach of Burton and Toland [Comm. Pure Appl. Math. (2011)] to show the existence of periodic surface water waves with vorticity in order that it becomes suited to a stability analysis. This is achieved by enlarging the function space to a class of stream functions that do not correspond necessarily to travelling profiles. In particular, for smooth profiles and smooth stream functions, the normal component of the velocity field at the free boundary is not required a priori to vanish in some Galilean coordinate system. Travelling periodic waves are obtained by a direct minimisation of a functional that corresponds to the total energy and that is therefore preserved by the time-dependent evolutionary problem (this minimisation appears in Burton and Toland after a first maximisation). In addition, we not only use the circulation along the upper boundary as a constraint, but also the total horizontal impulse (the velocity becoming a Lagrange multiplier). This allows us to preclude parallel flows by choosing appropriately the values of these two constraints and the sign of the vorticity. By stability, we mean conditional energetic stability of the set of minimizers as a whole, the perturbations being spatially periodic of given period.Comment: NoDEA Nonlinear Differential Equations and Applications, to appea

Nonlinear stabilitty for steady vortex pairs

In this article, we prove nonlinear orbital stability for steadily translating vortex pairs, a family of nonlinear waves that are exact solutions of the incompressible, two-dimensional Euler equations. We use an adaptation of Kelvin's variational principle, maximizing kinetic energy penalised by a multiple of momentum among mirror-symmetric isovortical rearrangements. This formulation has the advantage that the functional to be maximized and the constraint set are both invariant under the flow of the time-dependent Euler equations, and this observation is used strongly in the analysis. Previous work on existence yields a wide class of examples to which our result applies.Comment: 25 page

Regularization of point vortices for the Euler equation in dimension two

In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem [ -\ep^2 \Delta u=(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep})_+^p, \quad & x\in\Omega, u=0, \quad & x\in\partial\Omega, ] where $p>1$, $\Omega\subset\mathbb{R}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $\Omega$ is simply-connected smooth domain, then for any given non-degenerate critical point of Kirchhoff-Routh function $\mathcal{W}(x_1,...,x_m)$ with the same strength $\kappa>0$, there is a stationary classical solution approximating stationary $m$ points vortex solution of incompressible Euler equations with vorticity $m\kappa$. Existence and asymptotic behavior of single point non-vanishing vortex solutions were studied by D. Smets and J. Van Schaftingen (2010).Comment: 32page

Non-Coexistence of Infinite Clusters in Two-Dimensional Dependent Site Percolation

This paper presents three results on dependent site percolation on the square lattice. First, there exists no positively associated probability measure on {0,1}^{Z^2} with the following properties: a) a single infinite 0cluster exists almost surely, b) at most one infinite 1*cluster exists almost surely, c) some probabilities regarding 1*clusters are bounded away from zero. Second, we show that coexistence of an infinite 1*cluster and an infinite 0cluster is almost surely impossible when the underlying probability measure is ergodic with respect to translations, positively associated, and satisfies the finite energy condition. The third result analyses the typical structure of infinite clusters of both types in the absence of positive association. Namely, under a slightly sharpened finite energy condition, the existence of infinitely many disjoint infinite self-avoiding 1*paths follows from the existence of an infinite 1*cluster. The same holds with respect to 0paths and 0clusters.Comment: 17 pages, 1 figur

Stress field around arbitrarily shaped cracks in two-dimensional elastic materials

The calculation of the stress field around an arbitrarily shaped crack in an infinite two-dimensional elastic medium is a mathematically daunting problem. With the exception of few exactly soluble crack shapes the available results are based on either perturbative approaches or on combinations of analytic and numerical techniques. We present here a general solution of this problem for any arbitrary crack. Along the way we develop a method to compute the conformal map from the exterior of a circle to the exterior of a line of arbitrary shape, offering it as a superior alternative to the classical Schwartz-Cristoffel transformation. Our calculation results in an accurate estimate of the full stress field and in particular of the stress intensity factors K_I and K_{II} and the T-stress which are essential in the theory of fracture.Comment: 7 pages, 4 figures, submitted for PR

Desingularization of vortices for the Euler equation

We study the existence of stationary classical solutions of the incompressible Euler equation in the plane that approximate singular stationnary solutions of this equation. The construction is performed by studying the asymptotics of equation -\eps^2 \Delta u^\eps=(u^\eps-q-\frac{\kappa}{2\pi} \log \frac{1}{\eps})_+^p with Dirichlet boundary conditions and $q$ a given function. We also study the desingularization of pairs of vortices by minimal energy nodal solutions and the desingularization of rotating vortices.Comment: 40 page

The phase transition of the quantum Ising model is sharp

An analysis is presented of the phase transition of the quantum Ising model with transverse field on the d-dimensional hypercubic lattice. It is shown that there is a unique sharp transition. The value of the critical point is calculated rigorously in one dimension. The first step is to express the quantum Ising model in terms of a (continuous) classical Ising model in d+1 dimensions. A so-called `random-parity' representation is developed for the latter model, similar to the random-current representation for the classical Ising model on a discrete lattice. Certain differential inequalities are proved. Integration of these inequalities yields the sharpness of the phase transition, and also a number of other facts concerning the critical and near-critical behaviour of the model under study.Comment: Small changes. To appear in the Journal of Statistical Physic

The use of synchrotron edge topography to study polytype nearest neighbour relationships in SiC

A brief review of the phenomenon of polytypism is presented and its prolific abundance in Silicon Carbide discussed. An attempt has been made to emphasise modern developments in understanding this unique behaviour. The properties of Synchrotron Radiation are shown to be ideally suited to studies of polytypes in various materials and in particular the coalescence of polytypes in SiC. It is shown that with complex multipolytypic crystals the technique of edge topography allows the spatial extent of disorder to be determined and, from the superposition of Laue type reflections, neighbourhood relationships between polytypes can be deduced. Finer features have now been observed with the advent of second generation synchrotrons, the resolution available enabling the regions between adjoining polytypes to be examined more closely. It is shown that Long Period Polytypes and One Dimensionally Disordered layers often found in association with regions of high defect density are common features at polytype boundaries. An idealised configuration termed a "polytype sandwich" is presented as a model for the structure of SiC grown by the modified Lely technique. The frequency of common sandwich edge profiles are classified and some general trends of polytype neighbourism are summarised

Fish Spawning Aggregations: Where Well-Placed Management Actions Can Yield Big Benefits for Fisheries and Conservation

Marine ecosystem management has traditionally been divided between fisheries management and biodiversity conservation approaches, and the merging of these disparate agendas has proven difficult. Here, we offer a pathway that can unite fishers, scientists, resource managers and conservationists towards a single vision for some areas of the ocean where small investments in management can offer disproportionately large benefits to fisheries and biodiversity conservation. Specifically, we provide a series of evidenced-based arguments that support an urgent need to recognize fish spawning aggregations (FSAs) as a focal point for fisheries management and conservation on a global scale, with a particular emphasis placed on the protection of multispecies FSA sites. We illustrate that these sites serve as productivity hotspots - small areas of the ocean that are dictated by the interactions between physical forces and geomorphology, attract multiple species to reproduce in large numbers and support food web dynamics, ecosystem health and robust fisheries. FSAs are comparable in vulnerability, importance and magnificence to breeding aggregations of seabirds, sea turtles and whales yet they receive insufficient attention and are declining worldwide. Numerous case-studies confirm that protected aggregations do recover to benefit fisheries through increases in fish biomass, catch rates and larval recruitment at fished sites. The small size and spatio-temporal predictability of FSAs allow monitoring, assessment and enforcement to be scaled down while benefits of protection scale up to entire populations. Fishers intuitively understand the linkages between protecting FSAs and healthy fisheries and thus tend to support their protection