Related papers: Long time dynamics for helical vortex filament in …
It is well-known that the dynamics of vortices in an ideal incompressible two-dimensional fluid contained in a bounded not necessarily simply connected smooth domain is described by the Kirchhoff--Routh point vortex system. In this paper,…
For a $C^1_{t,x}$ solution $u$ to the incompressible 3D Euler equations, the helicity $H(u(t))=\int_{\mathbb{T}^3} u \cdot \textrm{curl}\, u$ is constant in time. For general low-regularity weak solutions, it is not always clear how to…
We consider a nonlinear model equation describing the motion of a vortex filament immersed in an incompressible and inviscid fluid. In the present problem setting, we also take into account the effect of external flow. We prove the unique…
We consider the dynamics of a two-dimensional incompressible perfect fluid on a M\"obius strip embedded in $\mathbb{R}^3$. The vorticity-streamfunction formulation of the Euler equations is derived from an exterior-calculus form of the…
In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as \begin{align*} \partial_t \omega + {\mathcal L}_u \omega &= 0\\ u &= \delta \tilde \eta^{-1}…
We consider the Cauchy problem for the full free boundary Euler equations in $3$d with an initial small velocity of size $O(\epsilon_0)$, in a moving domain which is initially an $O(\epsilon_0)$ perturbation of a flat interface. We assume…
A simplified form of the vorticity equation is derived for arbitrary coordinate systems. The present work unifies and extends the previous findings that vorticity is conserved in planar Euler flow, while in axisymmetric Euler rings it is…
We study the time evolution of an incompressible fluid with axial symmetry without swirl when the vorticity is sharply concentrated on $N$ annuli of radii of the order of $r_0$ and thickness $\varepsilon$. We prove that when $r_0= |\log…
Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and showed to tend asymptotically to singular vortex filaments. The construction is based on the…
The evolution of highly concentrated vorticity around rings in the three-dimensional axisymmetric Euler equations is studied in a regime for which the leapfrogging dynamics predicted by Helmholtz is expected to occur. We provide in this…
It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equations $$\mathrm{St} \partial_t F + v\cdot…
We study the Euler equations on a rotating unit sphere, focusing on the dynamics of vortex caps. Leveraging the $L^1$-stability of monotone, longitude-independent profiles, we demonstrate that certain ill-prepared initial data within the…
A {\em vortex pair} solution of the incompressible $2d$ Euler equation in vorticity form $$ \omega_t + \nabla^\perp \Psi\cdot \nabla \omega = 0 , \quad \Psi = (-\Delta)^{-1} \omega, \quad \hbox{in } \mathbb{R}^2 \times (0,\infty)$$ is a…
The dynamics along the particle trajectories for the 3D axisymmetric Euler equations in an infinite cylinder are considered. It is shown that if the inflow-outflow is highly oscillating in time, the corresponding Euler flow cannot keep the…
By exploring a local geometric property of the vorticity field along a vortex filament, we establish a sharp relationship between the geometric properties of the vorticity field and the maximum vortex stretching. This new understanding…
The Euler-$\alpha$ equations model the averaged motion of an ideal incompressible fluid when filtering over spatial scales smaller than $\alpha$. We show that there exists $\beta>1$ such that weak solutions to the two and three dimensional…
We consider the following coupled Ginzburg-Landau system in ${\mathbb R}^3$ \begin{align*} \begin{cases} -\epsilon^2 \Delta w^+ +\Big[A_+\big(|w^+|^2-{t^+}^2\big)+B\big(|w^-|^2-{t^-}^2\big)\Big]w^+=0, \\[3mm] -\epsilon^2 \Delta w^-…
We consider a nonlinear third order dispersive equation which models the motion of a vortex filament immersed in an incompressible and inviscid fluid occupying the three dimensional half space. We prove the unique solvability of…
In this paper we examine two opposite scenarios of energy behavior for solutions of the Euler equation. We show that if $u$ is a regular solution on a time interval $[0,T)$ and if $u \in L^rL^\infty$ for some $r\geq \frac{2}{N}+1$, where…
We prove the existence of critical points of the $N$-vortex Hamiltonian $H_\Omega (x_1,\ldots, x_N) =\sum\limits^N_{i=1}\Gamma^2_i h(x_i) + \sum\limits_{i,j=1\atop j\not= k}^N…