Explicit Solutions to the 3D Incompressible Euler Equations in Lagrangian Formulation (original) (raw)

2d incompressible Euler equations: New explicit solutions

Discrete & Continuous Dynamical Systems - A

There are not too many known explicit solutions to the 2-dimensional incompressible Euler equations in Lagrangian coordinates. Special mention must be made of the well-known ones due Gerstner and Kirchhoff, which were already discovered in the 19th century. These two classical solutions share a common characteristic, namely, the dependence of the coordinates from the initial location is determined by a harmonic map, as recognized by Abrashkin and Yakubovich, who more recently-in the 1980s-obtained new explicit solutions with a similar feature. We present a more general method for constructing new explicit solutions in Lagrangian coordinates which contain as special cases all previously known ones. This new approach shows that in fact "harmonic labelings" are special cases of a much larger family. In the classical solutions, the matrix Lie groups were essential in describing the time evolution. We see that also the geodesics in these groups are important.

On the Explicit Solutions of Separation of Variables Type for the Incompressible 2D Euler Equations

Journal of Mathematical Fluid Mechanics, 2021

We study explicit solutions to the 2 dimensional Euler equations in the Lagrangian framework. All known solutions have been of the separation of variables type, where time and space dependence are treated separately. The first such solutions were known already in the 19th century. We show that all the solutions known previously belong to two families of solutions and introduce three new families of solutions. It seems likely that these are all the solutions that are of the separation of variables type.

Euler Equations Solutions for Incompressible Fluid Flow

viXra, 2015

This paper covers the solutions of the Euler equations in 3-D and 4-D for incompressible fluid flow. The solutions are the spin-offs of the author's previous analytic solutions of the Navier-Stokes equations (vixra:1405.0251 of 2014). However, some of the solutions contained implicit terms. In this paper, the implicit terms have been expressed explicitly in terms of x, y, z and t. The author applied a new law, the law of definite ratio for fluid flow. This law states that in incompressible fluid flow, the other terms of the fluid flow equation divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. This law evolved from the author's earlier solutions of the Navier-Stokes equations. In addition to the usual approach of solving these equations, the Euler equations have also been solved by a second method in which the three equations in the system are added to produce a single equation which is ...

Euler equations for incompressible ideal fluids

Russian Mathematical Surveys, 2007

This article is a survey concerning the state-of-the-art mathematical theory of the Euler equations of incompressible homogenous ideal fluid. Emphasis is put on the different types of emerging instability, and how they may be related to the description of turbulence.

Title The 3 D incompressible euler equations with a passive scalar : A road to blow-up ?

2013

The 3D incompressible Euler equations with a passive scalar θ are considered in a smooth domain Ω ⊂ R 3 with no-normal-flow boundary conditions u •n| ∂Ω = 0. It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector B = ∇q × ∇θ, provided B has no null points initially : ω = curl u is the vorticity and q = ω • ∇θ is a potential vorticity. The presence of the passive scalar concentration θ is an essential component of this criterion in detecting the formation of a singularity. The problem is discussed in the light of a kinematic result by Graham and Henyey (2000) on the non-existence of Clebsch potentials in the neighbourhood of null points.

Geometric Properties and Nonblowup of 3D Incompressible Euler Flow

Communications in Partial Differential Equations, 2005

By exploring a local geometric property of the vorticity field along a vortex filament, we establish a sharp relationship between the geometric properties of the vorticity field and the maximum vortex stretching. This new understanding leads to an improved result of the global existence of the 3D Euler equation under mild assumptions.

Multiscale analysis in Lagrangian formulation for the 2-D incompressible Euler equation

Discrete and Continuous Dynamical Systems, 2005

We perform a systematic multiscale analysis for the 2-D incompressible Euler equation with rapidly oscillating initial data using a Lagrangian approach. The Lagrangian formulation enables us to capture the propagation of the multiscale solution in a natural way. By making an appropriate multiscale expansion in the vorticity-stream function formulation, we derive a well-posed homogenized equation for the Euler equation. Based on the multiscale analysis in the Lagrangian formulation, we also derive the corresponding multiscale analysis in the Eulerian formulation. Moreover, our multiscale analysis reveals some interesting structure for the Reynolds stress term, which provides a theoretical base for establishing systematic multiscale modeling of 2-D incompressible flow.

The three-dimensional Euler equations: Where do we stand?

The three-dimensional Euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists. While much has been discovered, the nature of solutions is still largely a mystery. This paper surveys some of the issues, such as singularity formation, that have cost so much effort in the last 25 years. In this light we review the Beale-Kato-Majda theorem and its consequences and then list some of the results of numerical experiments that have been attempted. A different line of endeavour focuses on work concerning the pressure Hessian and how it may be used and modelled. The Euler equations are finally discussed in terms of their membership of a class of general Lagrangian evolution equations. Using Hamilton's quaternions, these are reformulated in an elegant manner to describe the motion and rotation of fluid particles.

The 3D incompressible Euler equations with a passive scalar : a road to blow-up?

The 3D incompressible Euler equations with a passive scalar θ are considered in a smooth domain Ω ⊂ R 3 with no-normal-flow boundary conditions u ·n| ∂Ω = 0. It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector B = ∇q × ∇θ, provided B has no null points initially : ω = curl u is the vorticity and q = ω · ∇θ is a potential vorticity. The presence of the passive scalar concentration θ is an essential component of this criterion in detecting the formation of a singularity. * Electronic address: j.d.gibbon@ic.ac.uk

On the free boundary problem of three-dimensional incompressible Euler equations

Communications on Pure and Applied Mathematics, 2008

In this paper, we consider in three dimensions the motion of a general inviscid, incompressible fluid with a free interface that separates the fluid region from the vacuum. We assume that the fluid region is below the vacuum and that there is no surface tension on the free surface. Then we prove the local well-posedness of the free boundary problem in Sobolev space provided that there is no selfintersection point on the initial surface and under the stability assumption that @p @n . /j tD0 Ä 2c 0 < 0 with being restricted to the initial surface.