scholarly journals On the approximation of vorticity fronts by the Burgers–Hilbert equation

2021 ◽  
pp. 1-37
Author(s):  
John K. Hunter ◽  
Ryan C. Moreno-Vasquez ◽  
Jingyang Shu ◽  
Qingtian Zhang
Keyword(s):  
Euler Equations ◽  
Asymptotic Expansions ◽  
Energy Method ◽  
Two Dimensional ◽  
Asymptotic Equation ◽  
Contour Dynamics ◽  
Incompressible Euler Equations ◽  
Incompressible Euler

This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

scholarly journals A Characteristic Mapping method for the two-dimensional incompressible Euler equations

2021 ◽  
Vol 424 ◽  
pp. 109781
Author(s):  
Xi-Yuan Yin ◽  
Olivier Mercier ◽  
Badal Yadav ◽  
Kai Schneider ◽  
Jean-Christophe Nave
Keyword(s):  
Euler Equations ◽  
Mapping Method ◽  
Two Dimensional ◽  
Incompressible Euler Equations ◽  
Characteristic Mapping ◽  
Incompressible Euler

Variety of unsymmetric multibranched logarithmic vortex spirals

2017 ◽  
Vol 30 (1) ◽  
pp. 23-38 ◽  
Author(s):  
V. ELLING ◽  
M. V. GNANN
Keyword(s):  
Euler Equations ◽  
Two Dimensional ◽  
Incompressible Euler Equations ◽  
Incompressible Euler

Building on work of Prandtl and Alexander, we study logarithmic vortex spiral solutions of the two-dimensional incompressible Euler equations. We consider multi-branched spirals that are not symmetric, including mixtures of sheets and continuum vorticity. We find that non-trivial solutions allow only sheets, that there is a large variety of such solutions, but that only the Alexander spirals with three or more symmetric branches appear to yield convergent Biot–Savart integral.

ASYMPTOTIC EXPANSIONS IN A STEADY STATE EULER–POISSON SYSTEM AND CONVERGENCE TO INCOMPRESSIBLE EULER EQUATIONS

2005 ◽  
Vol 15 (05) ◽  
pp. 717-736 ◽  
Author(s):  
YUE-JUN PENG ◽  
INGRID VIOLET
Keyword(s):  
Steady State ◽  
Euler Equations ◽  
Asymptotic Expansions ◽  
Electron Mass ◽  
Poisson System ◽  
Drift Diffusion Model ◽  
Drift Diffusion ◽  
Incompressible Euler Equations ◽  
Potential Flows ◽  
Incompressible Euler

This work is concerned with a steady state Euler–Poisson system for potential flows arising in mathematical modeling for plasmas and semiconductors. We study the zero electron mass limit and zero relaxation time limit of the system by using the method of asymptotic expansions. These two limits are expressed by the Maxwell–Boltzmann relation and the classical drift-diffusion model, respectively. For each limit, we show the existence and uniqueness of profiles and justify the asymptotic expansions up to any order. These results also give new approaches for the convergence of the Euler–Poisson system to incompressible Euler equations, which has already been obtained via the quasi-neutral limit.

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