arXiv research

DGNet solves complex dynamical systems with neural networks and constraints.

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.

We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, \fw) \in H^s\times H^s\times H^{s'}$, $2<s'<s$. The classical local well-posedness result for the compressible Euler equations in $3$-D holds f…

This research smooths out fluid equations to avoid sudden shocks.

New SGD variant makes neural networks compressible without assumptions.

Recent work has introduced a simple numerical method for solving partial differential equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the method while applying it to analyze one of the most fundamental features in numerical PDEs and nonlinear analysis: irregular solutions. First, the S…

Extends Zeitlin's model to 3-D axisymmetric Euler equations.

We study the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed eigencurves, i.e. the eigenvector fields of the Jacobian of the flux are given. We formulate this as a typically overdetermined system of equations for the eigenvalues-to-be. Equivalent formulations in te…

Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa--Holm equations are well-studied examples.A beautiful approach to well-posedness is to go from the Eulerian to a Lagrangian description.Geometrically it corresponds …

Euler derived elastica equation using modern mathematical concepts.

Schwarzian derivative connects to Euler-Lagrange equations in variational calculus.

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 stability of multi-dimensional rarefaction waves in gas dynamics.

Study on stability in discretized hydrodynamics model.

In this paper, we will generalize the Bott-Virasoro group, applying the concept of the connection cochain, and derive the Euler equations corresponding to the generalized Bott-Virasoro group. We will show the relationships between the new Euler equations and the old ones. Moreover, we will study the geodesic equation c…

New equations for Cosserat media motions derived from bundle automorphisms.

We consider stochastic versions of Euler--Arnold equations using the infinite-dimensional geometric approach as pioneered by Ebin and Marsden. For the Euler equation on a compact manifold (possibly with smooth boundary) we establish local existence and uniqueness of a strong solution (in the stochastic sense) in spaces…

Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.

This paper extends invariant Euler-Lagrange equations to higher dimensions and groups.

We develop an integral geometry of stationary Euler equations defining some function $w$ on the Grassmannian of affine lines in the space. This function depends on a putative compactly supported solution $v$ of the system, and we deduce a linear differential equation for $w$. We prove also that the purported annulation…

We present a contact transformation of the generalized Hunter--Saxton equation to the Euler--Poisson equation with special values of the Ovsiannikov invariants. We also find the general solution for the generalized Hunter--Saxton equation.

Geometric framework for dissipative systems on Lie algebroids.

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

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

Paper derives invariantised Euler-Lagrange equations for Herglotz problems.

Geometric mechanics approach to constrained and floating multibody systems using Hamel's equations.

In this paper we show that there are applications that transform the movement of a pendulum into movements in $\mathbb{R}^3$. This can be done using Euler top system of differential equations. On the constant level surfaces, Euler top system reduces to the equation of a pendulum. Those properties are also considered in…

Toolbox for stochastic Euler equations using Ebin-Marsden theory.

Develops higher-order Euler-Poincaré field equations for principal G-bundles.

Optical interpretation of Euler's angle problem for caustics of light rays.

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

In this article we write the equations of barotropic compressible fluid mechanics as a geodesic equation on an infinite-dimensional manifold. The equations are given by \begin{align} u_t + \nabla_uu = -\frac{1}ρ \grad p \\ ρ_t + \diver{(ρu)} = 0, \end{align} where the fluid fills up a compact manifold $M$, $u$ is a tim…

We solve Euler equations on graph manifolds, classifying steady flows with Morse-Bott Bernoulli functions.

We prove a version of the variational Euler-Lagrange equations valid for functionals defined on Fréchet manifolds, such as the spaces of sections of differentiable vector bundles appearing in various physical theories.

Study of closed real plane curves with hyperelliptic genus three solutions.

In this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh-Taylor sign condition holds. We prove that as the surface tension tends to zero, when the Rayleigh-Taylor condition is satisfied, solutions converge to the Euler …

In this paper we consider the log-aesthetic curves and their generalization which are used in CAGD. We consider those curves under similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by the Burgers equation. We propose a variational formulation of those curves whose…

Lowered regularity assumption for a phase-dependent Helfrich energy equation.

Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for ana…

Paper studies critical points of curvature energies in 4D.

In this paper we provide a variational derivation of the Euler-Poincaré equations for systems subjected to external forces using an adaptation of the techniques introduced by Galley and others. Moreover, we study in detail the underlying geometry which is related to the notion of Poisson groupoid. Finally, we apply the…

In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on Lie groups, deriving a discrete version of the second-order Euler-Lagrange equatio…

Develops multifactor approximations for SVEs with completely monotone kernels.

The introduction of a covariant derivative on the velocity phase space is needed for a global expression of Euler-Lagrange equations. The aim of this paper is to show how its torsion tensor turns out to be involved in such a version.

Researchers study surface area functionals in CR manifolds, deducing equations for various cases.

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…

We prove that under certain assumptions a partial differential equation can be derived from a variational principle. It is well-known from Noether's theorem that symmetries of a variational functional lead to conservation laws of the corresponding Euler-Lagrange equation. We reverse this statement and prove that a diff…

If a Lagrangian defining a variational problem has order $k$ then its Euler-Lagrange equations generically have order $2k$. This paper considers the case where the Euler-Lagrange equations have order strictly less than $2k$, and shows that in such a case the Lagrangian must be a polynomial in the highest-order derivati…