scholarly journals On the Convergence of the Spectral Viscosity Method for the Two-Dimensional Incompressible Euler Equations with Rough Initial Data

2019 ◽  
Vol 20 (5) ◽  
pp. 1309-1362
Author(s):  
Samuel Lanthaler ◽  
Siddhartha Mishra
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Two Dimensional ◽  
Viscosity Method ◽  
Incompressible Euler Equations ◽  
Rough Initial Data ◽  
Incompressible Euler

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.

scholarly journals Computation of measure-valued solutions for the incompressible Euler equations

2015 ◽  
Vol 25 (11) ◽  
pp. 2043-2088 ◽  
Author(s):  
Samuel Lanthaler ◽  
Siddhartha Mishra
Keyword(s):  
Euler Equations ◽  
Computational Study ◽  
Vortex Sheet ◽  
Young Measures ◽  
Viscosity Method ◽  
Ensemble Averaging ◽  
Incompressible Euler Equations ◽  
Numerical Resolution ◽  
Admissible Measure ◽  
Incompressible Euler

We combine the spectral (viscosity) method and ensemble averaging to propose an algorithm that computes admissible measure-valued solutions of the incompressible Euler equations. The resulting approximate young measures are proved to converge (with increasing numerical resolution) to a measure-valued solution. We present numerical experiments demonstrating the robustness and efficiency of the proposed algorithm, as well as the appropriateness of measure-valued solutions as a solution framework for the Euler equations. Furthermore, we report an extensive computational study of the two-dimensional vortex sheet, which indicates that the computed measure-valued solution is non-atomic and implies possible non-uniqueness of weak solutions constructed by Delort.

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