A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
The study identifies unique fluid flow patterns.
Incompressible fluid dynamics on special manifolds.
There is a remarkable and canonical problem in 3D geometry and topology: To understand existing models of 3D fluid motion or to create new ones that may be useful. We discuss from an algebraic viewpoint the PDE called Euler's equation for incompressible frictionless fluid motion. In part I we define a "finite dimension…
Counterexamples show no simple generalization of Hasimoto transform for higher-dimensional Euler fluids.
We prove short-time existence for the Einstein-Euler-Entropy system for non-isentropic fluids with data in uniformly local Sobolev spaces. The cases of compact as well as non-compact Cauchy surfaces are covered. The method employed uses a Lagrangian description of the fluid flow which is based on techniques developed b…
We study the free boundary Euler equations in two spatial dimensions. We prove that if the boundary is sufficiently regular, then solutions of the free boundary fluid motion converge to solutions of the Euler equations in a fixed domain when the coefficient of surface tension tends to infinity.
The paper explores the geometric properties of fluid flows and their symmetries.
Arnold discovered geodesics in fluid dynamics.
Paper proves existence of conjugate points on ellipsoids but not on spheres.
Study axisymmetric ideal fluids on 3-manifolds, proving Fredholm properties.
New techniques analyze steady fluid flows on non-compact manifolds.
Toolbox for stochastic Euler equations using Ebin-Marsden theory.
Study shows singular sets for certain fluid equations are negligible.
Novel discretization of Euler equations for incompressible fluids.
Study the exponential map on surfaces using fluid dynamics.
We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we develop such connections for the case of two spatial dimensions, and demonstrate tha…
Study calculates curvature for fluid dynamics group, proving positivity.
In this note we survey some recent results for the Euler equations in compressible and incompressible fluid dynamics. The main point of all these theorems is the surprising fact that a suitable variant of Gromov's -principle holds in several cases.
The paper constructs chaotic solutions to the Euler equations on high-dimensional manifolds.
This research smooths out fluid equations to avoid sudden shocks.
Paper finds new criteria for conjugate points in fluid flows.
Study shows instability of naked singularities in perfect fluid models.
Geometric Hydrodynamics tackles open problems in fluid dynamics.
We present a geometric analysis of the incompressible averaged Euler equations for an ideal inviscid fluid. We show that solutions of these equations are geodesics on the volume-preserving diffeomorphism group of a new weak right invariant pseudo metric. We prove that for precompact open subsets of , thi…
In this article, we show how to embed the so-called CH2 equations into the geodesic flow of the Hdiv metric in 2D, which, itself, can be embedded in the incompressible Euler equation of a non compact Riemannian manifold. The method consists in embedding the incompressible Euler equation with a potential term coming fro…
The study identifies conjugate and cut points in ideal fluid motion configurations.
Study on stability in discretized hydrodynamics model.
The Lagrangian and Hamiltonian structures for an ideal gauge-charged fluid are determined. Using a Kaluza-Klein point of view, the equations of motion are obtained by Lagrangian and Poisson reductions associated to the automorphism group of a principal bundle. As a consequence of the Lagrangian approach, a Kelvin-Noeth…
This paper provides a precise sense in which the time t map for the Euler equations of an ideal fluid in a region in R^n (or a smooth compact n-manifold with boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphism group. This is interesting and nontri…
We consider solutions to the complex Trkalian equation,~$ \vec{\nabla} \times \vc = \vc ,$ where~$\vc$ is a 3 component vector function with each component in the complex field, and may be expressed in the form~$ \vc = e^{ig} \vec{\nabla} F, $ with~ real and~ complex. We find, there are precisely two classes of s…
We explore a computational model of an incompressible fluid with a multi-phase field in three-dimensional Euclidean space. By investigating an incompressible fluid with a two-phase field geometrically, we reformulate the expression of the surface tension for the two-phase field found by Lafaurie, Nardone, Scardovelli, …
We solve Euler equations on graph manifolds, classifying steady flows with Morse-Bott Bernoulli functions.
Euler's equations for a two-dimensional system can be written in Hamiltonian form, where the Poisson bracket is the Lie-Poisson bracket associated to the Lie algebra of divergence free vector fields. We show how to derive the Poisson brackets of 2d hydrodynamics of ideal fluids as a reduction from the one associated to…
This paper is devoted to the geometric analysis of the incompressible averaged Euler equations on compact Riemannian manifolds with boundary. The equation also coincides with the model for a second-grade non-Newtonian fluid. We study the analytical and geometrical properties of the Lagrangian flow map. We prove existen…
Madelung transform connects quantum and fluid dynamics via symplectic geometry.
Non-vanishing steady Euler flows and Beltrami fields found in high dimensions.
Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite extent (stars with a vacuum exterior) or infinite extent. In the l…
New approach links 2D fluid dynamics to matrix theory.
Given a principal bundle G \rightarrow P \rightarrow B (each being compact, connected and oriented) and a G-invariant metric h^{P} on P which induces a volume form μ^{P}, we consider the group of all unimodular automorphisms SAut(P,μ^{P}):={\varphi\in Diff(P) | \varphi^{*}μ^{P}=μ^{P} and \varphi is G-equivariant} of P …
We investigate the initial value problem for the Einstein-Euler equations of general relativity under the assumption of Gowdy symmetry on T3, and we construct matter spacetimes with low regularity. These spacetimes admit, both, impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularitie…
We describe first integrals of geostrophic equations, which are similar to the enstrophy invariants of the Euler equation for an ideal incompressible fluid. We explain the geometry behind this similarity, give several equivalent definitions of the Poisson structure on the space of smooth densities on a symplectic manif…
Constraint-aware neural networks improve accuracy in fluid flow simulations.
In this note we first set up an analogy between spin and vorticity of a perfect 2d-fluid flow, based on the Borel-Weil contruction of the irreducible unitary representations of SU(2), and looking at the Madelung-Bohm velocity attached to the ensuing spin wave functions. We also show that, in the framework of finite dim…
The paper proves stability of certain cosmological models with negative spatial curvature.
Study semi-invariant metrics in hydrodynamics for better understanding of fluid motion.
Study on half spheres solves Nirenberg problem with complex blow-up analysis.
Dirac structures on tangent bundles provide a unified framework for Lagrange--Dirac dynamical systems.
Let (N,g) be a Riemannian manifold. For a compact, connected and oriented submanifold M of N. we define the space of volume preserving embeddings Emb_μ(M,N) as the set of smooth embeddings f:M \rightarrow N such that f*μ^{f}=μ, where μ^{f} (resp. μ) is the Riemannian volume form on f(M) (resp. M) induced by the ambient…