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Register a new userGluing methods for vortex dynamics in Euler flows
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A classical problem for the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain. is that of finding regular solutions with highly concentrated vorticities around $N$ moving {em vortices}. The formal dynamic law for such objects was first derived in the 19th century by Kirkhoff and Routh. In this paper we devise a {em gluing approach} for the construction of smooth $N$-vortex solutions. We capture in high precision the core of each vortex as a scaled finite mass solution of Liouvilles equation plus small, more regular terms. Gluing methods have been a powerful tool in geometric constructions by {em desingularization}. We succeed in applying those ideas in this highly challenging setting.
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In this paper, we study nonlinear desingularization of steady vortex rings of three-dimensional incompressible Euler flows. We construct a family of steady vortex rings (with and without swirl) which constitutes a desingularization of the classical circular vortex filament in $mathbb{R}^3$. The construction is based on a study of solutions to the similinear elliptic problem begin{equation*} -frac{1}{r}frac{partial}{partial r}Big(frac{1}{r}frac{partialpsi^varepsilon}{partial r}Big)-frac{1}{r^2}frac{partial^2psi^varepsilon}{partial z^2}=frac{1}{varepsilon^2}left(g(psi^varepsilon)+frac{f(psi^varepsilon)}{r^2}right), end{equation*} where $f$ and $g$ are two given functions of the Stokes stream function $psi^varepsilon$, and $varepsilon>0$ is a small parameter.
In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of steady vortex pairs for the Euler equations with a general vorticity function, which constitutes a desingularization of a pair of point vortices with equal magnitude and opposite signs. The results are obtained by using an improved vorticity method.
Global regularity of axisymmetric incompressible Euler flows with non-trivial swirl in 3d is an outstanding open question. This work establishes that in the presence of uniform rotation, suitably small, localized and axisymmetric initial data lead to global strong solutions to the rotating 3d Euler equations. The solutions constructed are of Sobolev regularity, have non-vanishing swirl and scatter linearly, thanks to the dispersive effect induced by the rotation. To establish this, we introduce a framework that builds on the symmetries of the problem and precisely captures the anisotropic, dispersive mechanism due to rotation. This enables a fine analysis of the geometry of nonlinear interactions and allows us to propagate sharp decay bounds, which is crucial for the construction of global flows.
In this paper, we consider steady Euler flows in two-dimensional bounded annuli, as well as in exterior circular domains, in punctured disks and in the punctured plane. We always assume rigid wall boundary conditions. We prove that, if the flow does not have any stagnation point, and if it satisfies further conditions at infinity in the case of an exterior domain or at the center in the case of a punctured disk or the punctured plane, then the flow is circular, namely the streamlines are concentric circles. In other words, the flow then inherits the radial symmetry of the domain. The proofs are based on the study of the trajectories of the flow and the orthogonal trajectories of the gradient of the stream function, which is shown to satisfy a semilinear elliptic equation in the whole domain. In exterior or punctured domains, the method of moving planes is applied to some almost circular domains located between some streamlines of the flow, and the radial symmetry of the stream function is shown by a limiting argument. The paper also contains two Serrin-type results in simply or doubly connected bounded domains with free boundaries. Here, the flows are further assumed to have constant norm on each connected component of the boundary and the domains are then proved to be disks or annuli.
We investigate a steady planar flow of an ideal fluid in a (bounded or unbounded) domain $Omegasubset mathbb{R}^2$. Let $kappa_i ot=0$, $i=1,ldots, m$, be $m$ arbitrary fixed constants. For any given non-degenerate critical point $mathbf{x}_0=(x_{0,1},ldots,x_{0,m})$ of the Kirchhoff-Routh function defined on $Omega^m$ corresponding to $(kappa_1,ldots, kappa_m)$, we construct a family of stationary planar flows with vortex sheets that have large vorticity amplitude and are perturbations of small circles centered near $x_i$, $i=1,ldots,m$. The proof is accomplished via the implicit function theorem with suitable choice of function spaces. This seems to be the first nontrivial result on the existence of stationary vortex sheets in domains.
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