We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior. Contents 1 The porous medium equation as a gradient flow 2 1.1 The porous medium equation . . . . . . . . . . . . . . . . . . 2 1.2 Abstract gradient flow . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Two interpretations of the porous medium equation as gradient flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 A physical argument in favor of the new gradient flow 6 3 A mathematical argument in favor of new gradient flow 9 3.1 Self similar solutions and asymptotic behaviour . . . . . . . . 9 3.2 A new asymptotic result . . . . . . . . . . . . . . . . . . . . . 10 3.3 The asymptotic result express...

We study Euler–Poincaré systems (i.e., the Lagrangian analogue of Lie-Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d’Alembert type. Then we derive an abstract Kelvin-Noether theorem for these equations. We also explore their relation with the theory of Lie-Poisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; thus, it does not produce a corresponding Euler–Poincaré system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler–Poincaré systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional Camassa-Holm equations, which have many potentially interesting analytical properties. These

Reduction in the category of Poisson manifolds is defined and some basic properties are derived. The context is chosen to include the usual theorems on reduction of symplectic manifolds, as well as results such as the Dirac bracket and the reduction to the Lie-Poisson bracket. 1

Based on recent advances in the theory of Euler--Poincare (EP) equations with advected parameters and using the methods of Hamilton's principle asymptotics and averaged Lagrangians, we propose a new class of models for ideal incompressible fluids in three dimensions, including stratification and rotation for GFD applications. In these models, the amplitude of the rapid fluctuations introduces a length scale, #, below which wave activity is filtered by both linear and nonlinear dispersion. This filtering enhances the stability and regularity of the new fluid models without compromising either their large scale behavior, or their conservation laws. These models also describe geodesic motion on the volume-preserving di#eomorphism group for a metric containing the H 1 norm of the fluid velocity. PACS Numbers: 03.40.Gc, 47.10.+g, 03.40.-t, 03.40.-z 1 Euler--Poincare models of ideal fluids 2 Linear dispersion is well known to have profound e#ects on wave mean flow interaction in fluids [...

We review the properties of the nonlinearly dispersive Navier–Stokes-alpha (NS-α) model of incompressible fluid turbulence — also called the viscous Camassa–Holm equations in the literature. We first re-derive the NS-α model by filtering the velocity of the fluid loop in Kelvin’s circulation theorem for the Navier–Stokes equations. Then we show that this filtering causes the wavenumber spectrum of the translational kinetic energy for the NS-α model to roll off as k −3 for kα> 1 in three dimensions, instead of continuing along the slower Kolmogorov scaling law, k −5/3, that it follows for kα < 1. This roll off at higher wavenumbers shortens the inertial range for the NS-α model and thereby makes it more computable. We also explain how the NS-α model is related to large eddy simulation (LES) turbulence modeling and to the stress tensor for second-grade fluids. We close by surveying recent results in the literature for the NS-α model and its inviscid limit (the Euler-α model).

Abstract. In [HMR1], Holm, Marsden, and Ratiu derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equation ˙V (t)+∇U(t)V (t)−α2 [∇U(t)] t ·△U(t) = −grad p(t) where divU = 0, and V = (1 − α2△)U. In this model, the momentum V is transported by the velocity U, with the effect that nonlinear interaction between modes corresponding to length scales smaller than α is negligible. We generalize this equation to the setting of an n dimensional compact Riemannian manifold. The resulting equation is the Euler-Poincaré equation associated with the geodesic flow of the H1 right invariant metric on Ds µ, the group of volume preserving Hilbert diffeomorphisms of class Hs. We prove that the geodesic spray is continuously differentiable from T Ds µ(M) into TTD s µ(M) so that a standard Picard iteration argument proves existence and uniqueness on a finite time interval. Our goal in this paper is to establish the foundations for Lagrangian stability analysis following Arnold [A]. To do so, we use submanifold geometry, and prove that the weak curvature tensor of the right invariant H1 metric on Ds µ is a bounded trilinear map in the Hs topology, from which it follows that solutions to Jacobi’s equation exist. Using such solutions, we are able to study the infinitesimal stability behavior of geodesics. 1.

This paper studies the stability and bifurcations of the relative equilibria of the double spherical pendulum, which has the circle as its symmetry group. This example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian reduction. In particular, we establish a satisfactory global theory of Lagrangian reduction that is consistent with the classical local Routh theory for systems with an abelian symmetry group. 1

The three-dimensional motion of an incompressible inviscid fluid is classically described by the Euler equations, but can also be seen, following Arnold [1], as a geodesic on a group of volume-preserving maps. Local existence and uniqueness of minimal geodesics have been established by Ebin and Marsden [16]. In the large, for a large class of data, the existence of minimal geodesics may fail, as shown by Shnirelman [26]. For such data, we show that the limits of approximate solutions are solutions of a suitable extension of the Euler equations or, equivalently, as sharp measure-valued solutions to the Euler equations in the sense of DiPerna and Majda [14].

this paper's content. In the first section we introduce, following [19], the relevant energy E. E is a functional of the phase configuration \Omega\Gamma Throughout the sequel, will denote the characteristic function of\Omega ((x) = 1 if x

Computational anatomy (CA) is the mathematical study of anatomy I a I = I a BG, an orbit under groups of diffeomorphisms (i.e., smooth invertible mappings) g a G of anatomical exemplars Iaa I. The observable images are the output of medical imaging devices. There are three components that CA examines: (i) constructions of the anatomical submanifolds, (ii) comparison of the anatomical manifolds via estimation of the underlying diffeomorphisms g a G defining the shape or geometry of the anatomical manifolds, and (iii) generation of probability laws of anatomical variation P(d) on the images I for inference and disease testing within anatomical models. This paper reviews recent advances in these three areas applied to shape, growth, and atrophy.