17 research outputs found
Mathematics for 2d Interfaces
We present here a survey of recent results concerning the mathematical
analysis of instabilities of the interface between two incompressible, non
viscous, fluids of constant density and vorticity concentrated on the
interface. This configuration includes the so-called Kelvin-Helmholtz (the two
densities are equal), Rayleigh-Taylor (two different, nonzero, densities) and
the water waves (one of the densities is zero) problems. After a brief review
of results concerning strong and weak solutions of the Euler equation, we
derive interface equations (such as the Birkhoff-Rott equation) that describe
the motion of the interface. A linear analysis allows us to exhibit the main
features of these equations (such as ellipticity properties); the consequences
for the full, non linear, equations are then described. In particular, the
solutions of the Kelvin-Helmholtz and Rayleigh-Taylor problems are necessarily
analytic if they are above a certain threshold of regularity (a consequence is
the illposedness of the initial value problem in a non analytic framework). We
also say a few words on the phenomena that may occur below this regularity
threshold. Finally, special attention is given to the water waves problem,
which is much more stable than the Kelvin-Helmholtz and Rayleigh-Taylor
configurations. Most of the results presented here are in 2d (the interface has
dimension one), but we give a brief description of similarities and differences
in the 3d case.Comment: Survey. To appear in Panorama et Synth\`ese
Geodesic boundary value problems with symmetry
This paper shows how left and right actions of Lie groups on a manifold may
be used to complement one another in a variational reformulation of optimal
control problems equivalently as geodesic boundary value problems with
symmetry. We prove an equivalence theorem to this effect and illustrate it with
several examples. In finite-dimensions, we discuss geodesic flows on the Lie
groups SO(3) and SE(3) under the left and right actions of their respective Lie
algebras. In an infinite-dimensional example, we discuss optimal
large-deformation matching of one closed curve to another embedded in the same
plane. In the curve-matching example, the manifold \Emb(S^1, \mathbb{R}^2)
comprises the space of closed curves embedded in the plane
. The diffeomorphic left action \Diff(\mathbb{R}^2) deforms the
curve by a smooth invertible time-dependent transformation of the coordinate
system in which it is embedded, while leaving the parameterisation of the curve
invariant. The diffeomorphic right action \Diff(S^1) corresponds to a smooth
invertible reparameterisation of the domain coordinates of the curve. As
we show, this right action unlocks an important degree of freedom for
geodesically matching the curve shapes using an equivalent fixed boundary value
problem, without being constrained to match corresponding points along the
template and target curves at the endpoint in time.Comment: First version -- comments welcome
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
The variational particle-mesh method for matching curves
Diffeomorphic matching (only one of several names for this technique) is a
technique for non-rigid registration of curves and surfaces in which the curve
or surface is embedded in the flow of a time-series of vector fields. One seeks
the flow between two topologically-equivalent curves or surfaces which
minimises some metric defined on the vector fields, \emph{i.e.} the flow
closest to the identity in some sense.
In this paper, we describe a new particle-mesh discretisation for the
evolution of the geodesic flow and the embedded shape. Particle-mesh algorithms
are very natural for this problem because Lagrangian particles (particles
moving with the flow) can represent the movement of the shape whereas the
vector field is Eulerian and hence best represented on a static mesh. We
explain the derivation of the method, and prove conservation properties: the
discrete method has a set of conserved momenta corresponding to the
particle-relabelling symmetry which converge to conserved quantities in the
continuous problem. We also introduce a new discretisation for the geometric
current matching condition of (Vaillant and Glaunes, 2005). We illustrate the
method and the derived properties with numerical examples.Comment: I uploaded the wrong paper before! Here is the correct on
Statistical Equilibrium of Circulating Fluids
We are investigating the inviscid limit of the Navier-Stokes equation, and we
find previously unknown anomalous terms in Hamiltonian, Dissipation, and
Helicity, which survive this limit and define the turbulent statistics.
We find various topologically nontrivial configurations of the confined
Clebsch field responsible for vortex sheets and lines. In particular, a stable
vortex sheet family is discovered, but its anomalous dissipation vanishes as
.
Topologically stable stationary singular flows, which we call Kelvinons, are
introduced. They have a conserved velocity circulation around
the loop and another one for an infinitesimal closed loop
encircling , leading to a finite helicity. The anomalous
dissipation has a finite limit, which we computed analytically.
The Kelvinon is responsible for asymptotic PDF tails of velocity circulation,
\textbf{perfectly matching numerical simulations}.
The loop equation for circulation PDF as functional of the loop shape is
derived and studied. This equation is \textbf{exactly} equivalent to the
Schr\"odinger equation in loop space, with viscosity playing the role of
Planck's constant.
Kelvinons are fixed points of the loop equation at WKB limit . The anomalous Hamiltonian for the Kelvinons contains a large parameter
. The leading powers of this parameter can be
summed up, leading to familiar asymptotic freedom, like in QCD. In particular,
the so-called multifractal scaling laws are, as in QCD, modified by the powers
of the logarithm.Comment: 246 pages, 96 figures, and six appendixes. Submitted to Physics
Reports. Revised the energy balance analysis and discovered asymptotic
freedom leading to powers of logarithm of scale modifying K41 scaling law
Proceedings of the First International Workshop on Mathematical Foundations of Computational Anatomy (MFCA'06) - Geometrical and Statistical Methods for Modelling Biological Shape Variability
International audienceNon-linear registration and shape analysis are well developed research topic in the medical image analysis community. There is nowadays a growing number of methods that can faithfully deal with the underlying biomechanical behaviour of intra-subject shape deformations. However, it is more difficult to relate the anatomical shape of different subjects. The goal of computational anatomy is to analyse and to statistically model this specific type of geometrical information. In the absence of any justified physical model, a natural attitude is to explore very general mathematical methods, for instance diffeomorphisms. However, working with such infinite dimensional space raises some deep computational and mathematical problems. In particular, one of the key problem is to do statistics. Likewise, modelling the variability of surfaces leads to rely on shape spaces that are much more complex than for curves. To cope with these, different methodological and computational frameworks have been proposed. The goal of the workshop was to foster interactions between researchers investigating the combination of geometry and statistics for modelling biological shape variability from image and surfaces. A special emphasis was put on theoretical developments, applications and results being welcomed as illustrations. Contributions were solicited in the following areas: * Riemannian and group theoretical methods on non-linear transformation spaces * Advanced statistics on deformations and shapes * Metrics for computational anatomy * Geometry and statistics of surfaces 26 submissions of very high quality were recieved and were reviewed by two members of the programm committee. 12 papers were finally selected for oral presentations and 8 for poster presentations. 16 of these papers are published in these proceedings, and 4 papers are published in the proceedings of MICCAI'06 (for copyright reasons, only extended abstracts are provided here)