Euler Equations of Incompressible Ideal Fluids

Abstract

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.

References in corpus (2)

• Dynamic Stability of the 3D Axi-symmetric Navier-Stokes Equations with Swirl
• Recent progress in mathematical analysis of vortex sheets

Cited by in corpus (26)

• Potentially Singular Solutions of the 3D Incompressible Euler Equations
• Finite time blowup of 2D Boussinesq and 3D Euler equations with $C^{1,\alpha }$ velocity and boundary
• Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem
• On the analyticity and Gevrey class regularity up to the boundary for the Euler Equations
• On the convergence rate of the Euler-$\alpha$, an inviscid second-grade complex fluid, model to the Euler equations
• Ideal evolution of MHD turbulence when imposing Taylor-Green symmetries
• A Kato type Theorem for the inviscid limit of the Navier-Stokes equations with a moving rigid body
• Global regularity for a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations
• Vortices, Maximum Growth and the Problem of Finite-Time Singularity Formation
• Non-uniqueness for the Euler equations: the effect of the boundary
• From the Boltzmann Equation to the Euler Equations in the Presence of Boundaries
• Entire solutions of hydrodynamical equations with exponential dissipation
• The 3D incompressible Euler equations with a passive scalar: a road to blow-up?
• A regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations
• Infinite energy solutions for Dissipative Euler equations in R^2
• The role of BKM-type theorems in $3D$ Euler, Navier-Stokes and Cahn-Hilliard-Navier-Stokes analysis
• On complex singularities of the 2D Euler equation at short times
• Atypical late-time singular regimes accurately diagnosed in stagnation-point-type solutions of 3D Euler flows
• Global regularity and convergence of a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations
• Symmetry-plane model of 3D Euler flows and mapping to regular systems to improve blowup assessment using numerical and analytical solutions
• Axisymmetric Euler-$\alpha$ Equations without Swirl: Existence, Uniqueness, and Radon Measure Valued Solutions
• On power series solutions for the Euler equation, and the Behr-Necas-Wu initial datum
• Global Existence and Uniqueness of Weak Solutions of 3-D Euler Equations with Helical Symmetry in the Absence of Vorticity Stretching
• Analytic structure of solutions of the one-dimensional Burgers equation with modified dissipation
• Hölder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains
• Finite Time Blow-up of a 3D Model for Incompressible Euler Equations