Related papers: Bounds on a singular attractor in Euler using vort…
We consider the motion of several solids in a bounded cavity filled with a perfect incompressible fluid, in two dimensions. The solids move according to Newton's law, under the influence of the fluid's pressure, and the fluid dynamics is…
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…
We investigate the large time behavior of an axisymmetric model for the 3D Euler equations. In \cite{HL09}, Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier-Stokes equations with swirl. This model shares…
The 2D Euler system, which governs inviscid incompressible fluid flow, can admit infinitely many steady solutions in a given domain with slip boundary conditions. To select physical classical solutions, we investigate the vanishing…
Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse length scales for weak solutions of the three-dimensional, incompressible Navier-Stokes equations on a periodic box. The estimate for the smallest…
We consider the inviscid limit for the two-dimensional incompressible Navier-Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations…
We prove that some time Euler schemes for the 3D Navier-Stokes equations modified by adding a Brinkman-Forchheimer term and a random perturbation converge in $L^2(\Omega)$. This extends previous results concerning the strong rate of…
Does three-dimensional incompressible Euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out…
We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting…
In this paper, we modified the three dimensional Navier-Stokes equations by adding a l-Laplacian. We provide upper bounds on the two-dimensional Hausdorff measure the level sets of the vorticity of solutions. We express them in terms of the…
In this paper we consider the incompressible 3D Euler and Navier-Stokes equations in a smooth bounded domain. First, we study the 3D Euler equations endowed with slip boundary conditions and we prove the same criteria for energy…
A higher dimensional modified gravity theory with an action that includes dimensionally continued Euler-Poincar\'e forms up to second order in curvatures is considered. The variational field equations are derived. Matter in the universe at…
We consider the question whether starting from a smooth initial condition 3D inviscid Euler flows on a periodic domain $\mathbb{T}^3$ may develop singularities in a finite time. Our point of departure is the well-known result by Kato…
We construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in a suitable H\"older space and the…
The forced 2D Euler equations exhibit non-unique solutions with vorticity in $L^p$, $p > 1$, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit $\nu \to 0^+$ from the forced 2D…
The dynamics along the particle trajectories for the 3D axisymmetric Euler equations are considered. It is shown that if the inflow is rapidly increasing (pushy) in time, the corresponding laminar profile of the incompressible Euler flow is…
We establish the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for three-dimensional compressible isentropic flow in the whole space. It is shown that there exists a unique regular solution of compressible…
The goal of this article is to present -- in a cohesive, and somewhat self-contained fashion -- several recent results revealing an experimentally, numerically, and mathematical analysis-supported \emph{geometric scenario} manifesting…
In fairly general conditions we give explicit (smooth) solutions for the potential flow. We show that, rigorously speaking, the equations of the fluid mechanics have not rotational solutions. However, within the usual approximations of an…
This paper introduces a novel class of initial data for which the three-dimensional incompressible Navier--Stokes equations yield unique global-in-time solutions. Building on a logarithmically improved regularity criterion, we impose a…