Related papers: Global regularity for some axisymmetric Euler flow…
We prove finite-time vorticity blowup for smooth solutions of the 2D compressible Euler equations with smooth, localized, and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity, and is accompanied by…
The global existence of weak solutions for the three-dimensional axisymmetric Euler-$\alpha$ (also known as Lagrangian-averaged Euler-$\alpha$) equations, without swirl, is established, whenever the initial unfiltered velocity $v_0$…
Axially symmetric solutions to the Navier-Stokes equations in a bounded cylinder are considered. On the boundary the normal component of the velocity and he angular components of the velocity and vorticity are assumed to vanish. If the norm…
The nonlinear asymptotic stability of shear flows in the 2D Euler equations has traditionally been linked to inviscid damping in the periodic setting. Since Gevrey regularity is required to suppress the ``echo'' phenomenon, asymptotic…
The purpose of this paper is to study the incompressible non-resistive MHD equations in $\mathbb{R}^3$. We establish the global well-posedness of classical solutions if the initial data is axially symmetric and the swirl components of the…
Steady fluid flows have very special topology. In this paper we describe necessary and sufficient conditions on the vorticity function of a 2D ideal flow on a surface with or without boundary, for which there exists a steady flow among…
We consider the pressureless Euler-Poisson equations with quadratic confinement. For spatial dimension $d\ge 2,\,d\ne 4$, we give a necessary and sufficient condition for the existence of radial global smooth solutions, which is formulated…
We derive a priori estimates for the compressible free boundary Euler equations in the case of a liquid without surface tension. We provide a new weighted functional framework which leads to the improved regularity of the flow map by using…
We present a systematic approach to regularity theory of the multi-dimensional Euler alignment systems with topological diffusion introduced in \cite{STtopo}. While these systems exhibit flocking behavior emerging from purely local…
A new important relation between fluid mechanics and differential geometry is established. We study smooth steady solutions to the Euler equations with the additional property: the velocity vector is orthogonal to the gradient of the…
We prove that the solutions to the 3D Navier-Stokes equation with constant rotation exist globally for small axisymmetric initial data, where the smallness is uniform with respect to the viscosity $\nu \in [0,\infty)$. This expands the work…
It is shown that if the system of the Euler equations has a special global in time smooth solution with the linear profile of velocity, then another solutions with Cauchy data, close in the Sobolev norm to the initial data of the given…
Global regular axially symmetric solutions with large swirl in a cylinder with periodic conditions on the top and on the bottom are proved. On the lateral part of its boundary the boundary slip conditions are assumed. The proof is obtained…
We study vanishing viscosity solutions to the axisymmetric Euler equations with (relative) vorticity in $L^p$ with $p>1$. We show that these solutions satisfy the corresponding vorticity equations in the sense of renormalized solutions.…
In this paper, we investigate Childress's conjecture proposed in [Phys.D 237(14-17):1921-1925, 2008] on the growth rate of the vorticity maximum for axisymmetric swirl-free Euler flows in three and higher dimensions. We consider the setting…
The motion of a compressible inviscid radiative flow can be described by the radiative Euler equations, which consists of the Euler system coupled with a Poisson equation for the radiative heat flux through the energy equation. Although…
We consider the barotropic Euler equations in dimension d>1 with decaying density at spatial infinity. The phase portrait of the nonlinear ode governing the equation for spherically symmetric self-similar solutions has been introduced in…
We introduce a notion of stability for non-autonomous Hamiltonian flows on two-dimensional annular surfaces. This notion of stability is designed to capture the sustained twisting of particle trajectories. The main Theorem is applied to…
We consider an initial-boundary value problem for the 4D Navier-Stokes equations posed on bounded smooth domains. We prove the existence and uniqiueness of regular solutions as well as their exponential decay and additional regularity…
Spiral structure is one of the most common structures in the nature flows. A general steady spiral solution of incompressible inviscid axisymmetric flow was obtained analytically by applying separation of variables to the 3D Euler…