Consider a three-dimensional vortex sheet in inviscid, incompressible flow which is irrotational away from the sheet. We derive an equation for the evolution of a vortex sheet in Lagrangian coordinates, i.e. an equation that is restricted to the sheet itself and is analogous to the Birkhoff-Rott equation for a two-dimensional (planar) sheet. This general equation is specialized to sheets with a...

We show a superconvergence property for the Semi-Lagrangian Discontinuous Galerkin scheme of arbitrary degree in the case of constant linear advection equation with periodic boundary conditions.

The classical quantization of a family of a quadratic Liénard-type equation (Liénard II equation) is achieved by a quantization scheme (M. C. Nucci. Theor. Math. Phys., 168:994– 1001, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. This method straightforwardly yields the Schrödinger equation as given in (A. Ghose Ch...

4. Series of lectures on statistical turbulence written for amateurs but not experts. Elementary aspects and problems of turbulence in two and three dimensional Navier-Stokes equation are introduced. A few properties of scalar turbulence and transport phenomena in turbulent flows are described. Kraichnan’s model of passive advection is discussed a bit more precisely. Part 1: Approaching turbule...

In this paper we consider the convection-diffusion problem of a passive scalar in Lagrangian coordinates, i.e., in a coordinate system fixed on fluid particles. Both the convection-diffusion partial differential equation and the Langevin equation are expressed in Lagrangian coordinates and are shown to be equivalent for uniform, isotropic diffusion. The Lagrangian diffusivity is proportional to...

Abstract The Lagrangian average (LA) of the ideal fluid equations preserves their fundamental transport structure. This transport structure is responsible for the Kelvin circulation theorem of the LA flow and, hence, for its potential vorticity convection and helicity conservation. We show that Lagrangian averaging also preserves the Euler–Poincaré variational framework that implies the exact i...

We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve in the 3-sphere or a Legendrian curve in the anti-de Sitter 3-space. We describe ruled Lagrangian surfaces and characterize the cyclic and ru...

The computation of Lagrangian invariant subspaces of a Hamiltonian matrix, or the closely related task of solving algebraic Riccati equations, is an important issue in linear optimal control, stochastic control and H∞-design. We propose a new class of Riemannian Newton methods that allows to compute isolated Lagrangian invariant subspaces of a Hamiltonian matrix. The algorithm implements a vari...

We describe the remapped particle-mesh method, a new mass-conserving method for solving the density equation which is suitable for combining with semi-Lagrangian methods for compressible flow applied to numerical weather prediction. In addition to the conservation property, the remapped particle-mesh method is computationally efficient and at least as accurate as current semi-Lagrangian methods...

We associate a periodic two-dimensional Schrödinger operator to every Lagrangian torus in CP 2 and define the spectral curve of a torus as the Floquet spectrum of this operator on the zero energy level. In this event minimal Lagrangian tori correspond to potential operators. We show that Novikov–Veselov hierarchy of equations induces integrable deformations of minimal Lagrangian torus in CP 2 p...