Related papers: On the blow-up problem for the axisymmetric 3D Eul…
A 1934 paper by Leray posed the question of the regularity of solutions of the dynamical equations for incompressible inviscid fluids with smooth initial data. Since there has been many attempts to answer this question. Leray examined the…
We consider the isentropic compressible Euler equations in the half-line which govern the motion of gaseous fluids in contact with stationary vacuum boundary. We construct a large class of solutions that are initially smooth and…
We revisit, both numerically and analytically, the finite-time blowup of the infinite-energy solution of 3D Euler equations of stagnation-point-type introduced by Gibbon et al. (1999). By employing the method of mapping to regular systems,…
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…
A sufficient integral criterion for a blow-up solution of the Hopf equations (the Euler equations with zero pressure) is found. This criterion shows that a certain positive integral quantity blows up in a finite time under specific initial…
In this paper, we investigate the initial boundary value problem of the following nonlinear extensible beam equation with nonlinear damping term $$u_{t t}+\Delta^2 u-M\left(\|\nabla u\|^2\right) \Delta u-\Delta u_t+\left|u_t\right|^{r-1}…
We study in this article the blow-up of solutions to a coupled semilinear wave equations which are characterized by linear damping terms in the \textit{scale-invariant regime}, time-derivative nonlinearities, mass terms and Tricomi terms.…
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…
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}…
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…
Let $G=(V,E)$ be a locally finite connected weighted graph, $\Delta$ be the usual graph Laplacian. In this paper, we study the blow-up problems for the nonlinear parabolic equation $u_t=\Delta u + f(u)$ on $G$. The blow-up phenomenons of…
This paper presents the vortical and self-similar solutions for 2D compressible Euler equations using the separation method. These solutions complement Makino's solutions in radial symmetry without rotation. The rotational solutions provide…
We study blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the 3D Euler equations, where the sense of convergence and self-similarity are considered in various sense. We extend much further, in…
In this paper, we consider the finite time blow-up results for a parabolic equation coupled with superlinear source term and local linear boundary dissipation. Using a concavity argument, we derive the sufficient conditions for the…
This paper investigates the connection between blow-up solutions of scalar reaction-diffusion equations, in particular of $u_t = u_{xx} + u^2, $ and its counterpart - eternally existing solutions like heteroclinic orbits - by complex time.…
We construct a new class of asymptotically self-similar finite-time blowups that have two collapsing spatial scales for the 1D Constantin-Lax-Majda model. The larger spatial scale measures the decreasing distance between the bulk of the…
We prove local blow-up criterion for smooth axisymmetric solutions to the 3D incompressible Euler equation. If the vorticity satisfies $ \intl_{0}^{t_*} (t_*-t) \| \omega (t)\|_{ L^\infty(B(x_{ \ast}, R_0))} dt <+\infty$ for a ball $B(x_{…
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…
The pressureless Euler equations can be used as simple models of cosmology or plasma physics. In this paper, we construct the exact solutions in non-radial symmetry to the pressureless Euler equations in $R^{N}:$% [c]{c}%…
We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u &= u_b \quad \text{on } \partial…