Search

The $m$-waves of Kelvin are uniformly rotating patch solutions of the 2D Euler equations with $m$-fold rotational symmetry for $m\geq 2$. For Kelvin waves sufficiently close to the disc, we prove a nonlinear stability result up to an arbitrarily long time in the $L^1$ norm of the vorticity, for $m$-fold symmetric perturbations. To obtain this result, we first prove that the Kelvin wave is a strict local maximizer of the energy functional in some admissible class of patches, which had been claimed by Wan in 1986. This gives an orbital stability result with a support condition on the evolution of perturbations, but using a Lagrangian bootstrap argument which traces the particle trajectories of the perturbation, we are able to drop the condition on the evolution. Based on this unconditional stability result, we establish that long time filamentation, or formation of long arms, occurs near the Kelvin waves, which have been observed in various numerical simulations. Additionally, we discuss stability of annular patches in the same variational framework., 37 pages, 2 figures, presentation updated, typos corrected

We consider the incompressible Euler equations in the half cylinder $ \mathbb{R}_{>0}\times\mathbb{T}$. In this domain, any vorticity which is independent of $x_2$ defines a stationary solution. We prove that such a stationary solution is nonlinearly stable in a weighted $L^{1}$ norm involving the horizontal impulse, if the vorticity is non-negative and non-increasing in $x_1$. This includes stability of cylindrical patches $\{x_{1}0$. The stability result is based on the fact that such a profile is the unique minimizer of the horizontal impulse among all functions with the same distribution function. Based on stability, we prove existence of vortex patches in the half cylinder that exhibit infinite perimeter growth in infinite time., Comment: 17 pages, 2 figures, changed title to specify our main idea

We prove linear in time filamentation for perturbations of the Lamb dipole, which is a traveling wave solution to the incompressible Euler equations in $\mathbb{R}^2$. The main ingredient is a recent nonlinear orbital stability result by Abe-Choi. As a consequence, we obtain linear in time growth for the vorticity gradient for all times, for certain smooth and compactly supported initial vorticity in $\mathbb{R}^2$. The construction carries over to some generalized SQG equations., Comment: 24 pages

In connection with the recent proposal for possible singularity formation at the boundary for solutions of three-dimensional axisymmetric incompressible Euler's equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite-time blowup from smooth data.

We consider the two-dimensional incompressible Euler equations. We construct vortex patches with smooth boundary on $T^2$ and $R^2$ whose perimeter grows with time. More precisely, for any constant $M > 0$, we construct a vortex patch in $T^2$ whose smooth boundary has length of order 1 at the initial time such that the perimeter grows up to the given constant $M$ within finite time. The construction is done by cutting a thin stick out of an almost square patch. A similar result holds for an almost round patch with a thin handle in $R^2$., 6 pages, 2 figures

We consider the incompressible 2D Euler equation in an infinite cylinder $\mathbb{R}\times \mathbb{T}$ in the case when the initial vorticity is non-negative, bounded, and compactly supported. We study $d(t)$, the diameter of the support of vorticity, and prove that it allows the following bound: $d(t)\leq Ct^{1/3}\log^2 t$ when $t\rightarrow\infty$., Comment: 14 pages

The 2D conservative Boussinesq system describes inviscid, incompressible, buoyant fluid flow in gravity field. The possibility of finite time blow up for solutions of this system is a classical problem of mathematical hydrodynamics. We consider a 1D model of 2D Boussinesq system motivated by a particular finite time blow up scenario. We prove that finite time blow up is possible for the solutions to the model system., Comment: 13 pages, 2 figures