Related papers: Lower bounds on the blowup rate of vorticity in th…

Given that a solution to the 3D incompressible Euler equations on a bounded domain blows up at a time $T_\ast$ and that $T_\ast$ is the first such time, we provide pointwise-in-time lower bounds on $\|D^k\omega\|_{L^\infty(\Omega)}$ for $k…

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…

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 establish a new BKM-type blow-up criterion for solutions of the incompressible Euler equations that belong to Sobolev or H\" older spaces. Our criterion involves the $L^2$ norm in time of the $L^\infty$ norm of the first order tangential…

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…

In the present note, we address the question about behavior of $L_3$-norm of the velocity field as time $t$ approaches blow-up time $T$. It is known that the upper limit of the above norm must be equal to infinity. We show that, for…

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…

This paper studies the non-implosion mechanism for the 3D incompressible Euler equations. We prove that vorticity blows up in finite time, whereas the $L^p_T L^\infty_{loc}$ $(p\in[1,\infty))$ norm of the velocity field remains bounded.…

Under assumption that $T^{\ast}$ is the maximal time of existence of smooth solution of the 3D Navier-Stokes equations in the Sobolev space $H^{s}$, we establish lower bounds for the blow-up rate of the type$\ \left( T^{\ast }-t\right)…

In this paper we give optimal lower bounds for the blow-up rate of the $\dot{H}^{s}\left(\mathbb{T}^3\right)$-norm, $\frac{1}{2}<s<\frac{5}{2}$, of a putative singular solution of the Navier-Stokes equations, and we also present an…

We establish local-in-time existence for the Euler equations on a bounded domain with space-time dependent variable coefficients, given initial data $v_0 \in H^r$ under the optimal regularity condition $r > 2.5$. In the case $r = 3$, we…

We prove a Beale-Kato-Majda type criterion for the loss of regularity for solutions of the incompressible Euler equations in $H^{s}({\mathbb R}^3)$, for $s>\frac52$. Instead of double exponential estimates of Beale-Kato-Majda type, we…

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…

Let $v$ be a solution of the axially symmetric Euler equations (ASE) in a finite cylinder in $\mathbb{R}^3$. We show that suitable blow-up limits of possible velocity singularity and most self similar vorticity singularity near maximal…

This paper presents a novel approach to establish a blow-up mechanism for the forced 3D incompressible Euler equations, with a specific focus on non-axisymmetric solutions. We construct solutions on $\mathbb{R}^3$ within the function space…

This paper studies the heat equation $u_t=\Delta u$ in a bounded domain $\Omega\subset\mathbb{R}^{n}(n\geq 2)$ with positive initial data and a local nonlinear Neumann boundary condition: the normal derivative $\partial u/\partial n=u^{q}$…

We prove local non blow-up theorems for the 3D incompressible Euler equations under local Type I conditions. More specifically, for a classical solution $v\in L^\infty (-1,0; L^2 ( B(x_0,r)))\cap L^\infty_{\rm loc} (-1,0; W^{1, \infty}…

In a previous work with Tai-Peng Tsai, the author studied the dynamics of axisymmetric, swirl-free Euler equation in four and higher dimensions. One conclusion of this analysis is that the dynamics become dramatically more singular as the…

We propose a new blow-up criterion for the 3D Euler equations of incompressible fluid flows, based on the 3D Euler-Voigt inviscid regularization. This criterion is similar in character to a criterion proposed in a previous work by the…

In this paper we use maximum principle in the far field region for the time dependent self-similar Euler equations to exclude discretely self-similar blow-up for the Euler equations of the incompressible fluid flows. Our decay conditions…