Dissipative Euler flows for vortex sheet initial data without distinguished sign

Convergence of filtered weak solutions to the 2D Euler equations with vortex sheet initial data

We study weak solutions of the two-dimensional (2D) filtered Euler equations whose vorticity is a finite Radon measure and velocity has locally finite kinetic energy, which is called the vortex sheet solution. The filtered Euler equations are a regularized model based on a spatial filtering to the Euler equations. The 2D filtered Euler equations have a unique global weak solution for measure valued initial vorticity, while the 2D Euler equations require initial vorticity to be in the vortex sheet class with a distinguished sign for the existence of global solutions. In this paper, we prove tha…

关于具有涡片初值的二维理想流体中刚体运动的研究

On the motion of a rigid body in a two-dimensional ideal flow with vortex sheet initial data

本文研究二维不可压缩理想流体中刚体在涡片初值下的运动问题，在初涡量为定号弥散有界Radon测度的特定情形下证明了à la Delort弱解的存在性，且解满足能量不等式、刚体加速度有界。

Non-autonomous scalar linear-dissipative and purely dissipative parabolic PDEs over a compact base flow

In this paper a family of non-autonomous scalar parabolic PDEs over a general compact and connected flow is considered. The existence or not of a neighbourhood of zero where the problems are linear has an influence on the methods used and on the dynamics of the induced skew-product semiflow. That is why two cases are distinguished: linear-dissipative and purely dissipative problems. In both cases, the structure of the global and pullback attractors is studied using principal spectral theory. Besides, in the purely dissipative setting, a simple condition is given, involving both the underlying …

On bounded two-dimensional globally dissipative Euler flows

We examine the two-dimensional Euler equations including the local energy (in)equality as a differential inclusion and show that the associated relaxation essentially reduces to the known relaxation for the Euler equations considered without local energy (im)balance. Concerning bounded solutions we provide a sufficient criterion for a globally dissipative subsolution to induce infinitely many globally dissipative solutions having the same initial data, pressure and dissipation measure as the subsolution. The criterion can easily be verified in the case of a flat vortex sheet giving rise to the…

On concentration in vortex sheets

The question of energy concentration in approximate solution sequences $u^ε$, as $ε\to 0$, of the two-dimensional incompressible Euler equations with vortex-sheet initial data is revisited. Building on a novel identity for the structure function in terms of vorticity, the vorticity maximal function is proposed as a quantitative tool to detect concentration effects in approximate solution sequences. This tool is applied to numerical experiments based on the vortex-blob method, where vortex sheet initial data without distinguished sign are considered, as introduced in \emph{[R.~Krasny, J. Fluid …

二维欧拉方程DiPerna-Majda间隙问题的稀疏分解

A sparse resolution of the DiPerna-Majda gap problem for $2$D Euler equations

本文解决了二维欧拉方程中DiPerna-Majda间隙问题，证明混合符号涡片排除集中现象的最优衰减为$f(r)=[\log(1/r)]^{-1}$，并提出基于稀疏立方体几何的显式构造方法。