Marcel Oliver

The vortex blob method as a second-grade non-Newtonian fluid

We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity and initial data in the space of Radon measures M (R 2). The solutions of this regularized PDE, also known as the isotropic Lagrangian averaged Euler or Euler-α equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in M (R 2) such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes as are common in practical vortex computations.

Approximate momentum conservation for spatial semidiscretizations of semilinear wave equations

Summary.We prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions, analytic nonlinearity, and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize. Our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic. The method of proof is that of backward error analysis, whereby we construct a modified equation which is itself Lagrangian and translation invariant, and therefore also conserves momentum. This modified equation interpolates the semidiscrete system for all time, and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system. These properties directly imply approximate momentum conservation for the semidiscrete system. We also consider discretizations that are not variational as well as discretizations on non-uniform grids. Through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum.

Length scales in solutions of the complex Ginzburg-Landau equation

Abstract We generalise and in certain cases improve on previous a priori estimates of Sobolev norms of solutions to the generalised complex Ginzburg-Landau equation. A set of dynamic length scales based on ratios of these norms is defined. We are able to derive lower bounds for time averages and long-time limits of these length scales. The bounds scale like the inverses of our L∞ bounds.

Global well-posedness for the lake equations

Abstract We prove global well-posedness for the lake equations. These equations arise to leading order in a low aspect ratio, low Froude number (i.e. low wave speed) and very small wave amplitude expansion of the three-dimensional incompressible Euler equations in a basin with a free upper surface and a spatially varying bottom topography. Well-posedness means that there exists a solution, that it is unique and that it depends continuously on the data, i.e. the initial condition and the bottom topography. Our approach follows the works of Yudovitch and Bardos in constructing the solutions as the inviscid limit of solutions to a system with artificial viscosity which is the analog of the Navier-Stokes with respect to the Euler equations. One of the main assumptions in the present work is a nondegenerate bottom topography.

Variational asymptotics for rotating shallow water near geostrophy: a transformational approach

We introduce a unified variational framework in which the classical balance models for nearly geostrophic shallow water as well as several new models can be derived. Our approach is based on consistently truncating an asymptotic expansion of a near identity transformation of the rotating shallow water Lagrangian. Model reduction is achieved by imposing either degeneracy (for models in a semigeostrophic scaling) or incompressibility (for models in a quasigeostrophic scaling) with respect to the new coordinates. At first order, we recover the classical semigeostrophic and quasigeostrophic equations, Salmon’s L1 and large-scale semigeostrophic equations, as well as a one-parameter family of models that interpolate between the two. We identify one member of this family, dierent from previously known models, that promises better regularity—hence consistency with large-scale vortical motion—than all other first order models. Moreover, we explicitly derive second order models for all cases considered. While these second order models involve nonlinear potential vorticity inversion and do not obviously share the good properties or their first order counterparts, we oer an explicit survey of second order models and point out several avenues for exploration.

Parcel Eulerian-Lagrangian fluid dynamics for rotating geophysical flows

Parcel Eulerian–Lagrangian Hamiltonian formulations have recently been used in structure-preserving numerical schemes, asymptotic calculations and in alternative explanations of fluid parcel (in)stabilities. A parcel formulation describes the dynamics of one fluid parcel with a Lagrangian kinetic energy but an Eulerian potential evaluated at the parcels position. In this paper, we derive the geometric link between the parcel Eulerian–Lagrangian formulation and well-known variational and Hamiltonian formulations for three models of ideal and geophysical fluid flow: generalized two-dimensional vorticity–stream function dynamics, the rotating two-dimensional shallow-water equations and the rotating three-dimensional compressible Euler equations.

Boltzmann's Dilemma: An Introduction to Statistical Mechanics via the Kac Ring

The process of coarse-graining—here, in particular, of passing from a deterministic, simple, and time-reversible dynamics at the microscale to a typically irreversible description in terms of averaged quantities at the macroscale—is of fundamental importance in science and engineering. At the same time, it is often difficult to grasp and, if not interpreted correctly, implies seemingly paradoxical results. The kinetic theory of gases, historically the first and arguably most significant example, occupied physicists for the better part of the 19th century and continues to pose mathematical challenges to this day. In this paper, we describe the so-called Kac ring model, suggested by Mark Kac in 1956, which illustrates coarse-graining in a setting so simple that all aspects can be exposed both through elementary, explicit computation and through easy numerical simulation. In this setting, we explain a Boltzmannian “Stoszahlansatz,” ensemble averages, the difference between ensemble averaged and “typical” system behavior, and the notion of entropy.

Distribution-valued initial data for the complex Ginzburg-Landau equation

The generalized complex Ginzburg-Landau (CGL) equation with a nonlinearity of order 2{sigma} + 1 in d spatial dimensions has a unique local classical solution for distributional initial data in the Sobolev space H{sup q} provided that q > d/2 - 1/{sigma}. This result directly corresponds to a theorem for the nonlinear Schroedinger (NLS) equation which has been proved by Cazenave and Weissler in 1990. While the proof in the NLS case relies on Besov space techniques, it is shown here that for the CGL equation, the smoothing properties of the linear semigroup can be eased to obtain an almost optimal result by elementary means. 1 fig.

Slow dynamics via degenerate variational asymptotics

We introduce the method of degenerate variational asymptotics for a class of singularly perturbed ordinary differential equations in the limit of strong gyroscopic forces. Such systems exhibit dynamics on two separate time scales. We derive approximate equations for the slow motion to arbitrary order through an asymptotic expansion of the Lagrangian in suitably transformed coordinates. We prove that the necessary near-identity change of variables can always be constructed and that solutions of the slow limit equations shadow solutions of the full parent model at the expected order over a finite interval of time.

Long-Time Accuracy for Approximate Slow Manifolds in a Finite-Dimensional Model of Balance

We study the slow singular limit for planar anharmonic oscillatory motion of a charged particle under the influence of a perpendicular magnetic field when the mass of the particle goes to zero. This model has been used by the authors as a toy model for exploring variational high-order approximations to the slow dynamics in rotating fluids. In this paper, we address the long time validity of the slow limit equations in the simplest nontrivial case. We show that the first-order reduced model remains O(ε) accurate over a long 1/ε timescale. The proof is elementary, but involves subtle estimates on the nonautonomous linearized dynamics.