Macroscopic models for superconductivity

This paper reviews the derivation of some macroscopic models for superconductivity and also some of the mathematical challenges posed by these models. The paper begins by exploring certain analogies between phase changes in superconductors and those in solidification and melting. However, it is soon found that there are severe limitations on the range of validity of these analogies and outside this range many interesting open questions can be posed about the solutions to the macroscopic models

Scaling prediction for self-avoiding polygons revisited

We analyse new exact enumeration data for self-avoiding polygons, counted by perimeter and area on the square, triangular and hexagonal lattices. In extending earlier analyses, we focus on the perimeter moments in the vicinity of the bicritical point. We also consider the shape of the critical curve near the bicritical point, which describes the crossover to the branched polymer phase. Our recently conjectured expression for the scaling function of rooted self-avoiding polygons is further supported. For (unrooted) self-avoiding polygons, the analysis reveals the presence of an additional additive term with a new universal amplitude. We conjecture the exact value of this amplitude.Comment: 17 pages, 3 figure

Area distribution of the planar random loop boundary

We numerically investigate the area statistics of the outer boundary of planar random loops, on the square and triangular lattices. Our Monte Carlo simulations suggest that the underlying limit distribution is the Airy distribution, which was recently found to appear also as area distribution in the model of self-avoiding loops.Comment: 10 pages, 2 figures. v2: minor changes, version as publishe

Gaussian multiplicative Chaos for symmetric isotropic matrices

Motivated by isotropic fully developed turbulence, we define a theory of symmetric matrix valued isotropic Gaussian multiplicative chaos. Our construction extends the scalar theory developed by J.P. Kahane in 1985

Global generalized solutions for Maxwell-alpha and Euler-alpha equations

We study initial-boundary value problems for the Lagrangian averaged alpha models for the equations of motion for the corotational Maxwell and inviscid fluids in 2D and 3D. We show existence of (global in time) dissipative solutions to these problems. We also discuss the idea of dissipative solution in an abstract Hilbert space framework.Comment: 27 pages, to appear in Nonlinearit

Ultrametric spaces of branches on arborescent singularities

Let $S$ be a normal complex analytic surface singularity. We say that $S$ is arborescent if the dual graph of any resolution of it is a tree. Whenever $A,B$ are distinct branches on $S$, we denote by $A \cdot B$ their intersection number in the sense of Mumford. If $L$ is a fixed branch, we define $U_L(A,B)= (L \cdot A)(L \cdot B)(A \cdot B)^{-1}$ when $A \neq B$ and $U_L(A,A) =0$ otherwise. We generalize a theorem of P{\l}oski concerning smooth germs of surfaces, by proving that whenever $S$ is arborescent, then $U_L$ is an ultrametric on the set of branches of $S$ different from $L$. We compute the maximum of $U_L$, which gives an analog of a theorem of Teissier. We show that $U_L$ encodes topological information about the structure of the embedded resolutions of any finite set of branches. This generalizes a theorem of Favre and Jonsson concerning the case when both $S$ and $L$ are smooth. We generalize also from smooth germs to arbitrary arborescent ones their valuative interpretation of the dual trees of the resolutions of $S$. Our proofs are based in an essential way on a determinantal identity of Eisenbud and Neumann.Comment: 37 pages, 16 figures. Compared to the first version on Arxiv, il has a new section 4.3, accompanied by 2 new figures. Several passages were clarified and the typos discovered in the meantime were correcte

Seasonal cycle of precipitation variability in South America on intraseasonal timescales

The seasonal cycle of the intraseasonal (IS) variability of precipitation in South America is described through the analysis of bandpass filtered outgoing longwave radiation (OLR) anomalies. The analysis is discriminated between short (10--30 days) and long (30--90 days) intraseasonal timescales. The seasonal cycle of the 30--90-day IS variability can be well described by the activity of first leading pattern (EOF1) computed separately for the wet season (October--April) and the dry season (May--September). In agreement with previous works, the EOF1 spatial distribution during the wet season is that of a dipole with centers of actions in the South Atlantic Convergence Zone (SACZ) and southeastern South America (SESA), while during the dry season, only the last center is discernible. In both seasons, the pattern is highly influenced by the activity of the Madden--Julian Oscillation (MJO). Moreover, EOF1 is related with a tropical zonal-wavenumber-1 structure superposed with coherent wave trains extended along the South Pacific during the wet season, while during the dry season the wavenumber-1 structure is not observed. The 10--30-day IS variability of OLR in South America can be well represented by the activity of the EOF1 computed through considering all seasons together, a dipole but with the stronger center located over SESA. While the convection activity at the tropical band does not seem to influence its activity, there are evidences that the atmospheric variability at subtropical-extratropical regions might have a role. Subpolar wavetrains are observed in the Pacific throughout the year and less intense during DJF, while a path of wave energy dispersion along a subtropical wavetrain also characterizes the other seasons. Further work is needed to identify the sources of the 10--30-day-IS variability in South America

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final publication is available at http://www.springerlink.co

Particles and fields in fluid turbulence

The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in non-equilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.Comment: 165 pages, review article for Rev. Mod. Phy

Determining solar effects in Neptune’s atmosphere

Long-duration observations of Neptune’s brightness in two visible wavelengths provide a disk-averaged estimate of its atmospheric aerosol. Brightness variations were previously associated with the 11-year solar cycle, through solar-modulated mechanisms linked with either ultra-violet (UV) or galactic cosmic ray (GCR) effects on atmospheric particles. Here we use a recently extended brightness dataset (1972-2014), with physically realistic modelling to show that rather than alternatives, UV and GCR are likely to be modulating Neptune’s atmosphere in combination. The importance of GCR is further supported by the response of Neptune's atmosphere to an intermittent 1.5 to 1.9 year periodicity, which occurred preferentially in GCR (not UV) during the mid-1980s. This periodicity was detected both at Earth, and in GCR measured by Voyager 2, then near Neptune. A similar coincident variability in Neptune’s brightness suggests nucleation onto GCR ions. Both GCR and UV mechanisms may occur more rapidly than the subsequent atmospheric particle transport