American Institute of Mathematical Sciences

We present a survey of the relations between infinite dimensional integrals, both of the probabilistic type (e.g. Wiener path integrals) and of oscillatory type (e.g. Feynman path integrals).

Besides their mutual relations (analogies and differences) we also discuss their relations with certain types of partial differential equations (parabolic resp. hyperbolic), describing time evolution with or without stochastic terms.

The connection of these worlds of deterministic and stochastic evolutions with the world of quantum phenomena is also briefly illustrated. The survey spans a bridge from basic concepts and methods in these areas to recent developments concerning their relations.

We consider the variational formulation for vertical slice models introduced in Cotter and Holm (Proc Roy Soc, 2013). These models have a Kelvin circulation theorem that holds on all materially-transported closed loops, not just those loops on isosurfaces of potential temperature. Potential vorticity conservation can be derived directly from this circulation theorem. In this paper, we show that this property is due to these models having a relabelling symmetry for every single diffeomorphism of the vertical slice that preserves the density, not just those diffeomorphisms that preserve the potential temperature. This is developed using the methodology of Cotter and Holm (Foundations of Computational Mathematics, 2012).

3D stochastic Euler equations with a special form of multiplicative noise are considered. A Constantin-Iyer type representation in Euler-Lagrangian form is given, based on stochastic characteristics. Local existence and uniqueness of solutions in suitable Hölder spaces is proved from the Euler-Lagrangian formulation.

The present paper is dedicated to integrable models with Mikhailov reduction groups \begin{document}$G_R \simeq \mathbb{D}_h.$\end{document} Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on the realization of the \begin{document}$G_R$\end{document}-action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with \begin{document}$\mathbb{D}_h$\end{document} symmetries are presented.

The small length scales of the dissipative processes of physical viscosity and heat conduction are typically not resolved in the numerical simulation of high Reynolds number flows in the discrete geometry of computational grids. Historically, the simulations of flows with shocks and/or turbulence have relied on solving the Euler equations with dissipative regularization. In this paper, we begin by reviewing the regularization strategies used in shock wave calculations in both a Lagrangian and an Eulerian framework. We exhibit the essential similarities with Large Eddy Simulation models of turbulence, namely that almost all of these depend on the square of the size of the computational cell. In our principal result, we justify that dependence by deriving the evolution equations for a finite-sized volume of fluid. Those evolution equations, termed finite scale Navier-Stokes (FSNS), contain dissipative terms similar to the artificial viscosity first proposed by von Neumann and Richtmyer. We describe the properties of FSNS, provide a physical interpretation of the dissipative terms and show the connection to recent concepts in fluid dynamics, including inviscid dissipation and bi-velocity hydrodynamics.

Recently, Gallouët and Vialard [11] showed that the CH equation can be embedded in the incompressible Euler equation on a non compact Riemannian manifold. After surveying this result from a geometric point of view, we extend it to a broader class of PDEs, namely the so-called CH2 equations and the Holm-Staley \begin{document}$b$\end{document}-family of equations. A salient feature of these embeddings is the cone singularity of the Riemannian manifold on which the incompressible Euler equation is considered.

We present an account of dual pairs and the Kummer shapes for \begin{document}$ n:m $\end{document} resonances that provides an alternative to Holm and Vizman's work. The advantages of our point of view are that the associated Poisson structure on \begin{document}$ \mathfrak{su}(2)^{*} $\end{document} is the standard \begin{document}$ (+) $\end{document}-Lie-Poisson bracket independent of the values of \begin{document}$ (n,m) $\end{document} as well as that the Kummer shape is regularized to become a sphere without any pinches regardless of the values of \begin{document}$ (n,m) $\end{document}. A similar result holds for \begin{document}$ n:-m $\end{document} resonance with a paraboloid and \begin{document}$ \mathfrak{su}(1,1)^{*} $\end{document}. The result also has a straightforward generalization to multidimensional resonances as well.

Under periodic boundary conditions, a one-dimensional dispersive medium driven by a Lamb oscillator exhibits a smooth response when the dispersion relation is asymptotically linear or superlinear at large wave numbers, but unusual fractal solution profiles emerge when the dispersion relation is asymptotically sublinear. Strikingly, this is exactly the opposite of the superlinear asymptotic regime required for fractalization and dispersive quantization, also known as the Talbot effect, of the unforced medium induced by discontinuous initial conditions.

In this paper we present two dual pairs that can be seen as the linear analogues of the following two dual pairs related to fluids: the EPDiff dual pair due to Holm and Marsden, and the ideal fluid dual pair due to Marsden and Weinstein.

Left Lie reduction is a technique used in the study of curves in bi-invariant Lie groups [32, 33, 40]. Although the manifold \begin{document}$ \operatorname{SPD}(n) $\end{document} of all \begin{document}$ n\times n $\end{document} symmetric positive-definite matrices is not a Lie group with respect to the standard matrix multiplication, it is a symmetric space with a left action of \begin{document}$ GL(n) $\end{document} and an isotropy group \begin{document}$ SO(n) $\end{document} leaving the identity matrix fixed. The main purpose of this paper is to extend the method of left Lie reduction to \begin{document}$ \operatorname{SPD}(n) $\end{document} and use it to study two second order variational curves: Riemannian cubics and elastica. Riemannian cubics in \begin{document}$ \operatorname{SPD}(n) $\end{document} are reduced to so-called Lie quadratics in the Lie algebra \begin{document}$ \mathfrak{gl}(n) $\end{document} and geometric analyses are presented. Besides, by using the Frenet-Serret frames and the extended left Lie reduction separately, we investigate elastica in the manifold \begin{document}$ \operatorname{SPD}(n) $\end{document}. The latter presents a comparatively simple form of the equations for elastica in \begin{document}$ \operatorname{SPD}(n) $\end{document}.