arXiv research

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…

Paper proves existence of conjugate points on ellipsoids but not on spheres.

The study identifies unique fluid flow patterns.

A nontrivial smooth steady incompressible Euler flow in three dimensions with compact support is constructed. Another uncommon property of this solution is the dependence between the Bernoulli function and the pressure.

We point out a duality between steady incompressible Euler flows and solutions of the strongly coupled Faddeev-Skyrme sigma model with potential (mass) term. We supplement this result with various applications and several explicit examples.

Existence of a conjugate point proven on 3D ellipsoid.

Study how large-scale flows align small-scale vortices in 3D Euler equations.

Incompressible fluid dynamics on special manifolds.

The paper explores the geometric properties of fluid flows and their symmetries.

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…

These notes briefly summarize the lectures for the Summer School "Optimal transportation: Theory and applications" held by the second author in Grenoble during the week of June 22-26, 2009. Their goal is to describe some recent results on Brenier's variational models for incompressible Euler equation.

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, …

Study calculates curvature for fluid dynamics group, proving positivity.

The study develops a word mechanism for knot and link diagrams.

New obstruction found for embedding Riemannian manifolds into Euclidean spaces.

The binormal (or vortex filament) equation provides the localized induction approximation of the 3D incompressible Euler equation. We present explicit solutions of the binormal equation in higher-dimensions that collapse in finite time. The local nature of this phenomenon suggests the appearance of singularity in nearb…

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~$g$ real and~$F$ complex. We find, there are precisely two classes of s…

This paper explores vortices and harmonic flows on compact surfaces, using Hodge decomposition.

Geometric approach solves Euler equations with random forces.

A triangulation of a compact 3-manifold is annular-efficient if it is 0-efficient and the only normal, incompressible annuli are thin edge-linking. If a compact 3-manifold has an annular-efficient triangulation, then it is irreducible, boundary-irreducible, and an-annular. Conversely, it is shown that for a compact, ir…

We draw connections between the field of contact topology and the study of Beltrami fields in hydrodynamics on Riemannian manifolds in dimension three. We demonstrate an equivalence between Reeb fields (vector fields which preserve a transverse nowhere-integrable plane field) up to scaling and rotational Beltrami field…

The paper studies minimal resistance dynamics in radial fields, finding unique solutions for incompressible flows.

The group of diffeomorphisms of a compact manifold endowed with the L^2 metric acting on the space of probability densities gives a unifying framework for the incompressible Euler equation and the theory of optimal mass transport. Recently, several authors have extended optimal transport to the space of positive Radon …

Paper extends port-Hamiltonian model to include internal energy for compressible and incompressible flow.

Novel discretization of Euler equations for incompressible fluids.

The paper finds a short graph incompressible in a complex with specific properties.

Generative models tackle incompressible fluid flows by enforcing divergence-free constraints.

Toolbox for stochastic Euler equations using Ebin-Marsden theory.

Neural Networks improve incompressible flow simulations without complex kernels.

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 $h$-principle holds in several cases.

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…

Let $M$ be a smooth connected compact surface and $P$ be either a real line or a circle. This paper proceeds the study of the stabilizers and orbits of smooth functions on $M$ with respect to the right action of the group of diffeomorphisms of $M$. A large class of smooth maps $f:M\to P$ with isolated singularities is …

In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of Hamilton's classification theorem on four-manifolds with positive isotropic curvature and with no essential incom…

Develops methods for constructing exact, non-stationary solutions to Euler equations.

A theory for the evolution of a metric $g$ driven by the equations of three-dimensional continuum mechanics is developed. This metric in turn allows for the local existence of an evolving three-dimensional Riemannian manifold immersed in the six-dimensional Euclidean space. The Nash-Kuiper theorem is then applied to th…

Arnold discovered geodesics in fluid dynamics.

The paper constructs chaotic solutions to the Euler equations on high-dimensional manifolds.

Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three-dimensional (3D) ideal flow (Frisch & Zheligovsky, Commun. Math. Phys., vol. 326, 2014, pp. 499-505, Podvigina et al., J. Comput. Phys., vol. 306, 2016, pp. 320-342). Looking at such invariants with the modern tools …

Smooth flow around a helix, extending previous circle solution.

We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative …

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 ${\mathbb R}^n$, thi…

New probabilistic constructions for Kähler-Einstein metrics.

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 …

Paper develops physics-informed, boundary-constrained Gaussian process for fluid flow field reconstruction.

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…

Forward-Euler fails for simulating Wasserstein gradient flows with KL divergence.

In this paper, we characterize all links in the 3-sphere with bridge number at least three that have a bridge sphere of distance two. We show that a link L has a bridge sphere of distance at most two then it falls into at least one of three categories: (1) The exterior of L contains an essential meridional sphere. (2) …

We show that if M is a complete, finite-volume, hyperbolic 3-manifold having exactly one cusp, and if H_1(M;Z_2) has dimension at least 6, then M has volume greater than 5.06. We also show that if M is a closed, orientable hyperbolic 3-manifold such that H_1(M;Z_2) has dimension at least 4, and if the image of the cup …