scholarly journals Finite-time singularity formation for $C^{1,\alpha }$ solutions to the incompressible Euler equations on $\mathbb {R}^3$

2021 ◽  
Vol 194 (3) ◽  
pp. 647-727
Author(s):  
Tarek Elgindi
Keyword(s):  
Euler Equations ◽  
Finite Time ◽  
Singularity Formation ◽  
Time Singularity ◽  
Incompressible Euler Equations ◽  
Finite Time Singularity ◽  
Incompressible Euler

scholarly journals A fluid mechanic’s analysis of the teacup singularity

2020 ◽  
Vol 476 (2240) ◽  
pp. 20200348
Author(s):  
Dwight Barkley
Keyword(s):  
Euler Equations ◽  
Pressure Field ◽  
Meridional Flow ◽  
Cylinder Wall ◽  
Time Singularity ◽  
Pressure Poisson Equation ◽  
Velocity Gradients ◽  
Key Features ◽  
Finite Time Singularity ◽  
Incompressible Euler

The mechanism for singularity formation in an inviscid wall-bounded fluid flow is investigated. The incompressible Euler equations are numerically simulated in a cylindrical container. The flow is axisymmetric with the swirl. The simulations reproduce and corroborate aspects of prior studies reporting strong evidence for a finite-time singularity. The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity. The linearity of the pressure Poisson equation is exploited to decompose the pressure field into independent contributions arising from the meridional flow and from the swirl, and enforcing incompressibility and enforcing flow confinement. The key pressure field driving the blowup of velocity gradients is that confining the fluid within the cylinder walls. A model is presented based on a primitive-variables formulation of the Euler equations on the cylinder wall, with closure coming from how pressure is determined from velocity. The model captures key features in the mechanics of the blowup scenario.

scholarly journals The Singularity Formation on the Coupled Burgers–Constantin–Lax–Majda System with the Nonlocal Term

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Linrui Li ◽  
Shu Wang
Keyword(s):  
Initial Data ◽  
Finite Time ◽  
Smooth Solution ◽  
Blow Up ◽  
Nonlocal Term ◽  
Convection Term ◽  
Large Initial Data ◽  
Singularity Formation ◽  
Time Singularity ◽  
Finite Time Singularity

In this paper, we study the finite-time singularity formation on the coupled Burgers–Constantin–Lax–Majda system with the nonlocal term, which is one nonlinear nonlocal system of combining Burgers equations with Constantin–Lax–Majda equations. We discuss whether the finite-time blow-up singularity mechanism of the system depends upon the domination between the CLM type’s vortex-stretching term and the Burgers type’s convection term in some sense. We give two kinds of different finite-time blow-up results and prove the local smooth solution of the nonlocal system blows up in finite time for two classes of large initial data.

scholarly journals Finite time singularities for the free boundary incompressible Euler equations

2013 ◽  
Vol 178 (3) ◽  
pp. 1061-1134 ◽  
Author(s):  
Angel Castro ◽  
Diego Córdoba ◽  
Charles Fefferman ◽  
Francisco Gancedo ◽  
Javier Gómez-Serrano
Keyword(s):  
Free Boundary ◽  
Euler Equations ◽  
Finite Time ◽  
Incompressible Euler Equations ◽  
Time Singularities ◽  
Incompressible Euler

scholarly journals Hall MHD and Electron Inertia Effects in Current Sheet Formation at a Magnetic Neutral Line

2015 ◽  
Vol 5 (2) ◽  
pp. 109-125 ◽  
Author(s):  
Yuri E. Litvinenko ◽  
Liam C. McMahon
Keyword(s):  
Current Sheet ◽  
Finite Time ◽  
Numerical Solutions ◽  
Neutral Line ◽  
Singularity Formation ◽  
Time Singularity ◽  
Similarity Reduction ◽  
Electron Inertia ◽  
Magnetic Neutral Line ◽  
Finite Time Singularity

AbstractAn exact self-similar solution is used to investigate current sheet formation at a magnetic neutral line in incompressible Hall magnetohydrodynamics. The collapse to a current sheet is modelled as a finite-time singularity in the solution for electric current density at the neutral line. We establish that a finite-time collapse to the current sheet can occur in Hall magnetohydrodynamics, and we find a criterion for the finite-time singularity in terms of the initial conditions. We derive an asymptotic solution for the singularity formation and a formula for the singularity formation time. The analytical results are illustrated by numerical solutions, and we also investigate an alternative similarity reduction. Finally, we generalise our solution to incorporate resistive, viscous and electron inertia terms.

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