• Corpus ID: 119560945

Vortex filament solutions of the Navier-Stokes equations

@article{Bedrossian2018VortexFS,
  title={Vortex filament solutions of the Navier-Stokes equations},
  author={Jacob Bedrossian and Pierre Germain and Benjamin Harrop-Griffiths},
  journal={arXiv: Analysis of PDEs},
  year={2018}
}
We consider solutions of the Navier-Stokes equations in \(3d\) with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary mass supported on a smooth curve. First, we prove global well-posedness for perturbations of the Oseen vortex column in scaling-critical spaces. Second, we prove local well-posedness (in a sense to be made precise) when the filament is a smooth, closed, non-self-intersecting curve… 

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References

SHOWING 1-10 OF 73 REFERENCES
Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation
Both experimental and numerical studies of fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing
Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations
Motivated by applications to vortex rings, we study the Cauchy problem for the three-dimensional axisymmetric Navier-Stokes equations without swirl, using scale invariant function spaces. If the
Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity
Abstract.We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in L1(R2) for positive times is entirely determined by the trace of
Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space?
An important open problem in the theory of the Navier-Stokes equations is the uniqueness of the Leray-Hopf weak solutions with $L^2$ initial data. In this paper we give sufficient conditions for
Uniqueness of axisymmetric viscous flows originating from circular vortex filaments
The incompressible Navier-Stokes equations in R^3 are shown to admit a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large
On the Vortex Filament Conjecture for Euler Flows
In this paper, we study the evolution of a vortex filament in an incompressible ideal fluid. Under the assumption that the vorticity is concentrated along a smooth curve in $${\mathbb{R}^{3}}$$R3, we
Stability of the Self-similar Dynamics of a Vortex Filament
In this paper we continue our investigation of self-similar solutions of the vortex filament equation, also known as the binormal flow or the localized induction equation. Our main result is the
Stability and Interaction of Vortices in Two-Dimensional Viscous Flows
The aim of these notes is to present in a comprehensive and relatively self-contained way some recent developments in the mathematical analysis of two-dimensional viscous flows. We consider the
Vorticity and incompressible flow
Preface 1. An introduction to vortex dynamics for incompressible fluid flows 2. The vorticity-stream formulation of the Euler and the Navier-Stokes equations 3. Energy methods for the Euler and the
On the uniqueness of the solution of the two‐dimensional Navier–Stokes equation with a Dirac mass as initial vorticity
We propose two different proofs of the fact that Oseen's vortex is the unique solution of the two-dimensional Navier–Stokes equation with a Dirac mass as initial vorticity. The first argument, due to
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5
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