Related papers: Removing Type II singularities off the axis for th…
In this paper we investigate analytically the formation of finite time singularities in the three dimensional incompressible Euler equations under the model of Gibbon, Fokas, and Doering for vorticity stretching within a bounded cylindrical…
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…
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 consider a one-parameter family of 1D models for the 3D axisymmetric incompressible Euler equation with $C^{\alpha}$ vorticity and without swirl near the symmetry axis. For $\alpha = \frac13$, we impose a crucial normalization and…
This work is a companion to [EJE1] and its purpose is threefold: first, we will establish local well-posedness for the axi-symmetric $3D$ Euler equation in the domains $\{(x_1,x_2,x_3) \in \mathbb{R}^3 : x_3^2 \le \mathfrak{c}(x_1^2 +…
We find a smooth solution of the 2D Euler equation on a bounded domain which exists and is unique in a natural class locally in time, but blows up in finite time in the sense of its vorticity losing continuity. The domain's boundary is…
We establish a new a priori estimate on solutions to the space-inhomogeneous Landau and Boltzmann equations. As a consequence, we prove a new continuation criterion, based on a weighted $L^\infty$-norm, without requiring bounds on the…
The search of finite-time singularity solutions of Euler equations is considered for the case of an incompressible and inviscid fluid. Under the assumption that a finite-time blow-up solution may be spatially anisotropic as time goes by…
Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in \cite{MR673830} that they could only blow up on…
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 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…
We study stationary homogeneous solutions to the 3D Euler equation. The problem is motivated be recent exclusions of self-similar blowup for Euler and its relation to Onsager conjecture and intermittency. We reveal several new classes of…
We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an…
We establish the first complete classification of finite-time blow-up scenarios for strong solutions to the three-dimensional incompressible Euler equations with surface tension in a bounded domain possessing a closed, moving free boundary.…
We exclude Type I blow-up, which occurs in the form of atomic concentrations of the $L^2$ norm for the solution of the 3D incompressible Euler equations. As a corollary we prove nonexistence of discretely self-similar blow-up in the energy…
We investigate the axisymmetric incompressible Euler equations without swirl in $\mathbb R^d$ with $d\geq 3$. For any $\alpha\in(0, \alpha_d)$, where $\alpha_d=1-2/d$, we construct a self-similar blow-up solution whose initial velocity…
For any $\alpha \in (0,1/3)$, we construct exact $C^{\alpha}$ self-similar blowup profiles for the vorticity of the 3D incompressible Euler equation without swirl, and build on them to prove asymptotically self-similar blowup from…
Blowups of vorticity for the three- and two- dimensional homogeneous Euler equations are studied. Two regimes of approaching a blowup points, respectively, with variable or fixed time are analysed. It is shown that in the $n$-dimensional…
The 3D incompressible Euler equations with a passive scalar $\theta$ are considered in a smooth domain $\Omega\subset \mathbb{R}^{3}$ with no-normal-flow boundary conditions $\bu\cdot\bhn|_{\partial\Omega} = 0$. It is shown that smooth…
A sufficient condition is derived for a finite-time $L_2$ singularity of the 3d incompressible Euler equations, making appropriate assumptions on eigenvalues of the Hessian of pressure. Under this condition $\lim_{t \to T_*} \sup | \frac{D…