Steady-State Solutions of the Euler Equations in Two Dimensions II. Local Analysis of Limiting V-States

1986 ◽  
Vol 46 (5) ◽  
pp. 765-800 ◽  
Author(s):  
Edward A. Overman II
Keyword(s):  
Steady State ◽  
Euler Equations ◽  
Two Dimensions ◽  
Local Analysis ◽  
Steady State Solutions

scholarly journals Stability and instability results for the 2D [alpha]-Euler equations

2017 ◽  
Author(s):  
◽  
Shibi Kapisthalam Vasudevan
Keyword(s):  
Steady State ◽  
Euler Equations ◽  
Nonlinear Term ◽  
Zero Mode ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Fourier Decomposition ◽  
Inviscid Incompressible Fluid ◽  
Stability And Instability ◽  
Steady State Solutions

We study stability and instability of time independent solutions of the two dimensional a-Euler equations and Euler equations; the a-Euler equations are obtained by replacing the nonlinear term (u [times] [del.])u in the classical Euler equations of inviscid incompressible fluid by the term (v [times] [del.])u, where v is the regularized velocity satisfying (1 -[alpha]2[delta])v [equals] u, and [alpha] [greater than] 0. In the first part of the thesis, for the a-model, we develop analogues of the classical Arnol'd type stability criteria based on the energy-Casimir method for several settings including multi connected domains, periodic channels, and others. In the second part of the thesis, we study stability of a particular steady state, the unidirectional solution of the a-Euler equation on the two dimensional torus, having only one non zero mode in its Fourier decomposition. Using continued fractions, we give a proof of instability of the steady state under fairly general conditions. In the third part of the thesis we study various properties of a family of elliptic operators introduced by Zhiwu Lin in his work on instability of steady state solutions of the two dimensional Euler equations. This involves Birman-Schwinger type operators associated with the linearization of the Euler equations about the steady state and certain perturbation determinants.

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