Related papers: Blow-up solutions to 3D Euler are hydrodynamically…

It is still not known whether a solution to the incompressible Euler equation, endowed with a smooth initial value, can blow-up in finite time. In [{\em Comm. Math. Phys.}, 378:557--568, 2020] it has been shown that, if it exists, such a…

Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging. In [18], Elgindi proved that the 3D axisymmetric Euler equations with no swirl and $C^{1,\alpha}$ initial velocity develops a finite time…

Whether singularities can form in fluids remains a foundational unanswered question in mathematics. This phenomenon occurs when solutions to governing equations, such as the 3D Euler equations, develop infinite gradients from smooth initial…

The question of the global regularity vs finite time blow up in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the…

Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to…

In this paper we prove that for a certain class of initial data, smooth solutions of the hydrostatic Euler equations blow up in finite time.

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…

In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that…

The question of spontaneous apparition of singularity in the 3D incompressible Euler equations is one of the most important and challenging open problems in mathematical fluid mechanics. In this survey article we review some of recent…

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 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 study the singularity formation of smooth solutions of the relativistic Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any…

In recent work we have developed a renormalization framework for stabilizing reduced order models for time-dependent partial differential equations. We have applied this framework to the open problem of finite-time singularity formation…

The evolution of a pair of point vortices in whole space, subject to the inviscid Euler equations for incompressible fluid flow, is solved exactly for rotationally symmetric initial conditions. This exact solution shows that the vortex…

Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D…

This paper concerns the study of the incompressible Euler equations with variable density, in the case of space dimension $d=2$. Contrarily to their homogeneous (constant density) counterpart, those equations are not known to be well-posed…

In this paper, we consider some blow-up problems for the 1D Euler equation with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping…

Inspired by the numerical evidence of a potential 3D Euler singularity \cite{luo2014potentially,luo2013potentially-2}, we prove finite time blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite…

In this paper we study the finite time blow-up problem for the axisymmetric 3D incompressible Euler equations with swirl. The evolution equations for the deformation tensor and the vorticity are reduced considerably in this case. Under the…

The phenomenon of finite time blow-up in hydrodynamic partial differential equations is central in analysis and mathematical physics. While numerical studies have guided theoretical breakthroughs, it is challenging to determine if the…