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Register a new userOn the steady axisymmetric vortex rings for 3-D incompressible Euler flows
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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.
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This paper concerns with the existence of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle, which are governed by the Euler equations with the slip boundary condition on the wall of the nozzle and a receiver pressure at the exit. Mathematically, it can be formulated as a free boundary problem with the shock front being the free boundary to be determined. In dealing with the free boundary problem, one of the key points is determining the position of the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution gives an initial approximating position of the shock front. Compared with 2-D case, new difficulties arise due to the additional 0-order terms and singularities along the symmetric axis. New observation and careful analysis will be done to overcome these difficulties. Once the initial approximation is obtained, a nonlinear iteration scheme can be carried out, which converges to a transonic shock solution to the problem.
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 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.
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.
We construct co-rotating and traveling vortex sheets for 2D incompressible Euler equation, which are supported on several small closed curves. These examples represent a new type of vortex sheet solutions other than two known classes. The construction is based on Birkhoff-Rott operator, and accomplished by using implicit function theorem at point vortex solutions with suitably chosen function spaces.
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