scholarly journals Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations

2011 ◽  
Vol 121 (7) ◽  
pp. 1445-1463 ◽  
Author(s):  
F. Flandoli ◽  
M. Gubinelli ◽  
E. Priola
Keyword(s):  
Euler Equations ◽  
Vortex Dynamics ◽  
Point Vortex ◽  
Well Posedness ◽  
2D Euler Equations

scholarly journals Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey

2020 ◽  
Author(s):  
Klas Modin ◽  
Milo Viviani
Keyword(s):  
Euler Equations ◽  
Vortex Dynamics ◽  
Point Vortex ◽  
Point Vortices ◽  
Symplectic Reduction ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Unified Framework ◽  
Long Time ◽  
2D Euler Equations

Abstract Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and techniques scattered in the literature. Here, we give a unified framework for proving integrability results for $$N=2$$ N = 2 , 3, or 4 point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any two-dimensional manifold with a symmetry group; we illustrate it on the sphere, the plane, the hyperbolic plane, and the flat torus. A systematic study of integrability is prompted by advances in two-dimensional turbulence, bridging the long-time behaviour of 2D Euler equations with questions of point-vortex integrability. A gallery of solutions is given in the appendix.

scholarly journals Well-posedness of the 2D Euler equations when velocity grows at infinity

2019 ◽  
Vol 39 (5) ◽  
pp. 2361-2392
Author(s):  
Elaine Cozzi ◽  
◽  
James P. Kelliher ◽  
Keyword(s):  
Euler Equations ◽  
Well Posedness ◽  
2D Euler Equations

scholarly journals The Concepts of Well-Posedness and Stability in Different Function Spaces for the 1D Linearized Euler Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Stefan Balint ◽  
Agneta M. Balint
Keyword(s):  
Euler Equation ◽  
Function Space ◽  
Euler Equations ◽  
Small Perturbation ◽  
Function Spaces ◽  
Well Posedness ◽  
Small Solution ◽  
Null Solution ◽  
Linearized Euler Equations ◽  
The Stability

This paper considers the stability of constant solutions to the 1D Euler equation. The idea is to investigate the effect of different function spaces on the well-posedness and stability of the null solution of the 1D linearized Euler equations. It is shown that the mathematical tools and results depend on the meaning of the concepts “perturbation,” “small perturbation,” “solution of the propagation problem,” and “small solution, that is, solution close to zero,” which are specific for each function space.

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