Two-Dimensional Point Vortex Dynamics in Bounded Domains: Global Existence for Almost Every Initial Data

2022 ◽  
Vol 54 (1) ◽  
pp. 79-113
Author(s):  
Donati Martin
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Vortex Dynamics ◽  
Point Vortex ◽  
Bounded Domains ◽  
Two Dimensional

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.

Analogous formulation of electrodynamics and two-dimensional fluid dynamics

2014 ◽  
Vol 761 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Fluid Dynamics ◽  
Charge Density ◽  
Compressible Flow ◽  
Vortex Dynamics ◽  
Point Vortex ◽  
Point Vortices ◽  
Classical Electrodynamics ◽  
Two Dimensional ◽  
Electric Charge Density ◽  
Limiting Case

AbstractA single, simply stated approximation transforms the equations for a two-dimensional perfect fluid into a form that is closely analogous to Maxwell’s equations in classical electrodynamics. All the fluid conservation laws are retained in some form. Waves in the fluid interact only with vorticity and not with themselves. The vorticity is analogous to electric charge density, and point vortices are the analogues of point charges. The dynamics is equivalent to an action principle in which a set of fields and the locations of the point vortices are varied independently. We recover classical, incompressible, point vortex dynamics as a limiting case. Our full formulation represents the generalization of point vortex dynamics to the case of compressible flow.

scholarly journals Well-Posedness of the Two-Dimensional Fractional Quasigeostrophic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Yongqiang Xu
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Well Posedness ◽  
Two Dimensional ◽  
Small Initial Data ◽  
General Initial Data ◽  
Local Existence Of Solutions

This paper is concerned with the fractional quasigeostrophic equation with modified dissipativity. We prove the local existence of solutions in Sobolev spaces for the general initial data and the global existence for the small initial data when1/2≤α<1.

scholarly journals On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
David Massatt
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Well Posedness ◽  
Two Dimensional ◽  
Periodic Domain ◽  
Uniqueness Of Solutions ◽  
Global Existence And Uniqueness ◽  
Sivashinsky Equation

<p style='text-indent:20px;'>We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data <inline-formula><tex-math id="M1">\begin{document}$ u_{01} \in L^2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ u_{02} \in H^{-1 + \eta} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M3">\begin{document}$ \eta &gt; 0 $\end{document}</tex-math></inline-formula>.</p>

scholarly journals A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data

2020 ◽  
Vol 19 (2) ◽  
pp. 835-882
Author(s):  
Seung-Yeal Ha ◽  
◽  
Bingkang Huang ◽  
Qinghua Xiao ◽  
Xiongtao Zhang ◽  
...  
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Fluid Model ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Large Initial Data

Global Existence of Solution for Cauchy Problem of Two-Dimensional Boussinesq-Type Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Qingying Hu ◽  
Chenxia Zhang ◽  
Hongwei Zhang
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Initial Data ◽  
Global Existence ◽  
Exponential Growth ◽  
Type Equation ◽  
Global Weak Solution ◽  
Existence Of Solution ◽  
Two Dimensional ◽  
The Cauchy Problem

In this paper, we consider the Cauchy problem of two-dimensional Boussinesq-type equations utt-Δu-Δutt+Δ2u=Δfu. Under the assumptions that fu is a function with exponential growth at infinity and under some assumptions on the initial data, we prove the existence of global weak solution.

Two-dimensional vortex dynamics in a stratified barotropic fluid

1996 ◽  
Vol 314 ◽  
pp. 139-161 ◽  
Author(s):  
Steve C. Arendt
Keyword(s):  
Gravitational Field ◽  
Vortex Dynamics ◽  
Point Vortex ◽  
The Self ◽  
Uniform Density ◽  
Hamiltonian Form ◽  
Two Dimensional ◽  
Barotropic Fluid ◽  
Isothermal Fluid ◽  
Uniform Fluid

We show that two-dimensional ‘point’ vortex dynamics in both a polytropic fluid of γ = 3/2 and an isothermal fluid stratified by a constant gravitational field can be written in Hamiltonian form. We find that the formulation admits only one constant of the motion in addition to the Hamiltonian, so that two vortices are the most for which the motion is generally integrable. We study in detail the two-vortex problem and find a rich collection of behaviour: closed trajectories analogous to the circular orbits of the uniform-fluid two-vortex problem, open trajectories for which the self-propelled vortices scatter off each other, and both unstable and stable steadily translating pairs of vortices. Comparison is made to the case of two vortices in a uniform-density fluid bounded by a wall.

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