Surface waves on steady perfect‐fluid flows with vorticity

This is a theory of two‐dimensional steady periodic surface waves on flows under gravity in which the given data are three quantities that are independent of time in the corresponding evolution problem: the volume of fluid per period, the circulation per period on the free stream line, and the rearrangement class (equivalently, the distribution function) of the vorticity field. A minimizer of the total energy per period among flows satisfying these three constraints is shown to be a weak solution of the surface wave problem for which the vorticity is a decreasing function of the stream function. This decreasing function can be thought of as an infinite‐dimensional Lagrange multiplier corresponding to the vorticity rearrangement class being specified in the minimization problem. (Note that functional dependence of vorticity on the stream function was not specified a priori but is part of the solution to the problem and ensures the flow is steady.) To illustrate the idea with a minimum of technical difficulties, the existence of nontrivial waves on the surface of a fluid flowing with a prescribed distribution of vorticity and confined beneath an elastic sheet is proved. The theory applies equally to irrotational flows and to flows with locally square‐integrable vorticity. © 2011 Wiley Periodicals, Inc.     

Galerkin’s method, monotonicity and linking for indefinite Hamiltonian systems with bounded potential energy

A combination of Galerkins method and linking theory with monotonicity in the calculus of variations is used to study Hamiltonian systems in which the kinetic-energy functional is a (not necessarily definite) quadratic form and the potential-energy functional may be bounded. The existence of non-constant brake periodic orbits for almost all prescribed energies is established. An example of a Hamiltonian system which satisfies our hypotheses but has no non-constant brake periodic orbits with energy in an uncountable set of measure zero is given. Additional hypotheses, sufficient to ensure the existence of non-constant brake periodic orbits of all energies, are found.Received: 28 November 2003, Accepted: 2 June 2004, Published online: 3 September 2004Mathematics Subject Classification (2000): 37J45     

Self-Adjoint Operators and Cones

Suppose that K is a cone in a real Hilbert space with K = {0},and that A: is a self-adjoint operator which maps K intoitself. If ||A|| is an eigenvalue of A, it is shown that ithas an eigenvector in the cone. As a corollary, it follows thatif ||A||ⁿ is an eigenvalue of Aⁿ, then ||A|| is an eigenvalueof A which has an eigenvector in K. The role of the support-boundaryof K in the simplicity of the principal eigenvalue ||A|| isinvestigated. If H is a separable Hilbert space, it is shownthat ||A|| (A); that is, the spectral radius of A lies in thespectrum of A. When A is compact, we obtain a very elementaryproof of the Krein-Rutman Theorem in the self-adjoint case withoutassuming that K = {0}.     

Electron Sagnac gyroscope in an array of mesoscopic quantum rings

The Sagnac effect is an important phase coherent effect in optical and atom interferometers where rotations with respect to an inertial frame are measured in the interference pattern. We analyze the Sagnac effect in a serial array of mesoscopic ring shaped electron interferometers comprised of rings with half-circumferences comparable to the mean free path. The entire array is, however, much larger than the phase coherence length. Phase coherent transport at the level of individual rings leads to a measurable Sagnac effect in the conductance of the chain. We use the signal to noise ratio (SNR) to determine the number of rings needed to measure a desired rotation rate.     

Standing waves on infinite depth

The two-dimensional standing wave problem, for an infinitely deep layer, is considered, based on the formulation of the problem as a second order non local PDE. Despite the presence of infinitely many resonances in the linearized problem, we use the Nash–Moser implicit function theorem to prove the existence of standing waves corresponding to values of the amplitude ε having 0 as a Lebesgue point. To cite this article: G. Iooss et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

The Sub-Harmonic Bifurcation of Stokes Waves

Steady periodic water waves on the free surface of an infinitely deep irrotational flow under gravity without surface tension (Stokes waves) can be described in terms of solutions of a quasi-linear equation which involves the Hilbert transform and which is the Euler-Lagrange equation of a simple functional. The unknowns are a 2π-periodic function w which gives the wave profile and the Froude number, a dimensionless parameter reflecting the wavelength when the wave speed is fixed (and vice versa). Although this equation is exact, it is quadratic (with no higher order terms) and the global structure of its solution set can be studied using elements of the theory of real analytic varieties and variational techniques. In this paper it is shown that there bifurcates from the first eigenvalue of the linearised problem a uniquely defined arc-wise connected set of solutions with prescribed minimal period which, although it is not necessarily maximal as a connected set of solutions and may possibly self-intersect, has a local real analytic parametrisation and contains a wave of greatest height in its closure (suitably defined). Moreover it contains infinitely many points which are either turning points or points where solutions with the prescribed minimal period bifurcate. (The numerical evidence is that only the former occurs, and this remains an open question.) It is also shown that there are infinitely many values of the Froude number at which Stokes waves, having a minimal wavelength that is an arbitrarily large integer multiple of the basic wavelength, bifurcate from the primary branch. These are the sub-harmonic bifurcations in the paper's title. (In 1925 Levi-Civita speculated that the minimal wavelength of a Stokes wave propagating with speed c did not exceed 2πc ²/g. This is disproved by our result on sub-harmonic bifurcation, since it shows that there are Stokes waves with bounded propagation speeds but arbitrarily large minimal wavelengths.) Although the work of Benjamin & Feir} and others [9, 10] has shown Stokes waves on deep water to be unstable, they retain a central place in theoretical hydrodynamics. The mathematical tools used to study them here are real analytic-function theory, spectral theory of periodic linear pseudo-differential operators and Morse theory, all combined with the deep influence of a paper by Plotnikov [36]. Accepted: December 6, 1999