Long Time Evolution of Concentrated Euler Flows with Planar Symmetry

@article{Butt2018LongTE,
  title={Long Time Evolution of Concentrated Euler Flows with Planar Symmetry},
  author={Paolo Butt{\`a} and Carlo Marchioro},
  journal={SIAM J. Math. Anal.},
  year={2018},
  volume={50},
  pages={735-760}
}
We study the time evolution of an incompressible Euler fluid with planar symmetry when the vorticity is initially concentrated in small disks. We discuss how long this concentration persists, showing that in some cases this happens for quite long times. Moreover, we analyze a toy model that shows a similar behavior and gives some hints on the original problem. 
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References

SHOWING 1-10 OF 34 REFERENCES
Vortices and localization in Euler flows
We study the time evolution of a non-viscous incompressible two-dimensional fluid when the initial vorticity is concentrated inN small disjoint regions of diameter ε. We prove that the time evolved…
Euler evolution for singular initial data and vortex theory
We study the evolution of a two dimensional, incompressible, ideal fluid in a case in which the vorticity is concentrated in small, disjoint regions of the physical space. We prove, for short times,…
Bounds on the growth of the support of a vortex patch
We study the time evolution of the support of a vortex patch evolving in ℝ2 according to the Euler Equation for an incompressible fluid and we bound its growth. Furthermore we discuss the same…
Euler evolution for singular initial data and vortex theory: A global solution
We study the evolution of a two-dimensional, incompressible, ideal fluid in a case in which the vorticity is concentrated in small disjoint regions and we prove, globally in time, its connection with…
On the motion of a vortex ring with a sharply concentrated vorticity
We study an incompressible non-viscous fluid with axial symmetry without swirl, in the case when the vorticity is supported in an annulus. It is well known that there exist particular initial data…
Vanishing viscosity limit for an incompressible fluid with concentrated vorticity
We study an incompressible fluid with a sharply concentrated vorticity moving in the whole space. First, we review some known results for the inviscid case, which prove, in particular situations, the…
On the Inviscid Limit for a Fluid with a Concentrated Vorticity
Abstract:We study the time evolution of a viscous incompressible fluid in ℝ2 when the initial vorticity is sharply concentrated in N regions of diameter ε. We prove that in the zero viscosity limit…
On the Growth of the Vorticity Support for an Incompressible Non-viscous Fluid in a Two-dimensional Exterior Domain
We study the time evolution of the support of a positive vorticity for a non-viscous incompressible fluid evolving in R 2 - D, where D is a compact domain with smooth boundary. We bound its growth.…
Evolution of two concentrated vortices in a two-dimensional bounded domain
We prove that two initially concentrated vortices with opposite vorticity of an incompressible ideal fluid moving in a two-dimensional bounded domain, remain concentrated during the time. The motion…
On the vanishing viscosity limit for two-dimensional Navier–Stokes equations with singlular initial data
We study the solutions of the Navier–Stokes equations when the initial vorticity is concentrated in small disjoint regions of diameter ϵ. We prove that they converge, uniformily in ϵ. for vanishing…
...
...