Related papers: Existence of Vortex Patch Equilibria for Active Sc…
We rigorously construct the first steady traveling wave solutions of the 2D incompressible Euler equation that take the form of a contiguous vortex-patch dipole, which can be viewed as the vortex-patch counterpart of the well-known…
We consider the vortex patch problem for both the 2-D and 3-D incompressible Euler equations. In 2-D, we prove that for vortex patches with $H^{k-0.5}$ Sobolev-class contour regularity, $k \ge 4$, the velocity field on both sides of the…
We prove the existence of non-constant time periodic vortex solutions to the Gross-Pitaevskii equations for small but \textit{fixed} $\varepsilon > 0.$ The vortices of these solutions follow periodic orbits to the point vortex system of…
We derive a relationship for the vortex aspect ratio $\alpha$ (vertical half-thickness over horizontal length scale) for steady and slowly evolving vortices in rotating stratified fluids, as a function of the Brunt-Vaisala frequencies…
For the class of quasi-periodic solutions of the vortex filament equation, we study connections between the algebro-geometric data used for their explicit construction and the geometry of the evolving curves. We give a complete description…
Motivated by the recent work of Hassainia and Hmidi [Z. Hassainia, T. Hmidi - On the {V}-states for the generalized quasi-geostrophic equations,arXiv preprint arXiv:1405.0858], we close the question of the existence of convex global…
Vortices represent a class of topological solitons arising in gauge theories coupled with complex scalar fields, holding significant importance across various domains of modern physics. In this paper we establish the existence of vortex…
We consider concentrated vorticities for the Euler equation on a smooth domain $\Omega \subset \mathbf{R}^2$ in the form of \[ \omega = \sum_{j=1}^N \omega_j \chi_{\Omega_j}, \quad |\Omega_j| = \pi r_j^2, \quad \int_{\Omega_j} \omega_j d\mu…
Inspired by recent developments in Berdina-like models for turbulence, we propose an inviscid regularization for the surface quasi-geostrophic (SQG) equations. We are particularly interested in the celebrated question of blowup in finite…
We investigate a steady planar flow of an ideal fluid in a (bounded or unbounded) domain $\Omega\subset \mathbb{R}^2$. Let $\kappa_i\not=0$, $i=1,\ldots, m$, be $m$ arbitrary fixed constants. For any given non-degenerate critical point…
In this paper we derive the equations of motion for two-layer point vortex motion on the upper half plane. We study the invariants using symmetry, including the Hamiltonian and show that the two vortex problem is integrable. We characterize…
The motion of a two-dimensional buoyant vortex patch, i.e. a vortex patch with a uniform density different from the uniform density of the surrounding fluid, is analyzed in terms of evolution equations for the motion of its centroid,…
We study the patch dynamics on the whole plane and on the half-plane for a family of active scalars called modified SQG equations. These involve a parameter $\alpha$ which appears in the power of the kernel in their Biot-Savart laws and…
There are a number of publications on relativistic objects dealing either with black holes or naked singularities in the center. Here we show that there exist static spherically symmetric solutions of Einstein equations with a strongly…
A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a perfect fluid; it can therefore be interpreted as a spinning bubble of air in water. This paper gives a general method for desingularizing…
In this paper, we obtain uniformly rotating vorticity with sufficiently large angular velocity in the unit disk. The solution consists of either a small nearly-ellipse vortex patch which is highly concentrated near the origin or a $2+1$…
A general exact weak solution to the nonlinear equation of the conservation of the absolute vorticity in a thin layer of an incompressible medium on a rotating sphere is proposed. It takes into account the helicity of the point vortices and…
In the present study, we find that the surface quasi-geostrophic equation admits exact solutions, which evolve with time in quasi-stationary states. The solutions presented are available for any dissipation effect $\kappa (-\Delta)^\alpha$…
We prove the existence of smooth solutions to the Gross-Pitaevskii equation on $\mathbf{R}^3$ that feature arbitrarily complex quantum vortex reconnections. We can track the evolution of the vortices during the whole process. This permits…
In this paper, we prove the existence of relative equilibria with holes for quasi-geostrophic shallow-water equations. More precisely, using bifurcation techniques, we establish for any m large enough the existence of two branches of m-fold…