Finite-time singularity formation for $C^{1,\alpha}$ solutions to the incompressible Euler equations on $\mathbb{R}^3$

Singularity formation for complex solutions of the 3D incompressible Euler equations

Moore 's approximation method, first formulated for vortex sheets, is generalized and applied to axi-symmetric flow with swirl and with smooth initial data. The approximation preserves the forward cascade of energy but neglects any backflow of energy. It splits the Euler equations into two sets of equations: one for u . = u+(r, z, t) containing all non-negative wavenumbers (in z) and the second...

Finite Time Blow-up for the 3D Incompressible Euler Equations

We prove the finite time blow-up for solutions of the 3D incompressible Euler equations, which happens along the fluid particle trajectories starting from a set of points. This set is specified by the relation between the deformation tensor and the Hessian of pressure both coupled with the vorticity directions, associated with the initial data. As a corollary of this result we prove the finite ...

Singularity formation for compressible Euler equations

In this paper, for the p-system and full compressible Euler equations in one space dimension, we provide an equivalent and a sharp condition on initial data, respectively, under which the classical solution must break down in finite time. Moreover, we provide time-dependent lower bounds on density for arbitrary classical solutions for these two equations. Our results have no restriction on the ...

Weak Solutions to the Stationary Incompressible Euler Equations

On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations

We investigate the large time behavior of an axisymmetric model for the 3D Euler equations. In [22], Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier-Stokes equations with swirl. This model shares many properties of the 3D incompressible Euler and Navier-Stokes equations. The main difference between the 3D model of Hou and Lei and the reformulated 3D Euler an...