Related papers: Vorticity blowup in 2D compressible Euler equation…
We perturb the 3D Euler equations by a particular non-linear Stratonovich noise. We show the existence and uniqueness of a global-in-time (i.e. no blow-up) smooth solution. The result is a corollary of a more general theorem valid in an…
In this paper, we present strong numerical evidences that the incompressible axisymmetric Euler equations with degenerate viscosity coefficients and smooth initial data of finite energy develop a potential finite-time locally self-similar…
We establish the first finite-time blow-up results for generalized 3D stochastic fractional Navier-Stokes equations \[ \Caputo \mathbf{u} = -(\mathbf{u} \cdot \nabla)\mathbf{u} - \nabla p + \nu \fLaplacian \mathbf{u} +…
We establish finite-time singularity formation for $C^{1,\alpha}$ solutions to the Boussinesq system that are compactly supported on $\mathbb{R}^2$ and infinitely smooth except in the radial direction at the origin. The solutions are smooth…
In this paper, we study the stability two-dimensional (2D) steady Euler flows with sharply concentrated vorticity in a simply-connected bounded domain. These flows are obtained as maximizers of the kinetic energy subject to the constraint…
We study the formation of singularity for the isothermal Euler-Poisson system arising from plasma physics. Contrast to the previous studies yielding only limited information on the blow-up solutions, for instance, sufficient conditions for…
Consider a random initial vorticity $\omega_0(x) = \sum_{n\in \mathbb{Z}^2} a_n \phi(x-n)$, where $\phi$ is bounded and compactly supported and $\{a_n\}$ are independent, uniformly bounded, mean $0$, variance $1$ random variables (i.e.…
We consider the Euler-Poincar\'e equation on $\mathbb R^d$, $d\ge 2$. For a large class of smooth initial data we prove that the corresponding solution blows up in finite time. This settles an open problem raised by Chae and Liu \cite{Chae…
We study the Cauchy problem for the compressible Euler equations in two spatial dimensions under any physical barotropic equation of state except that of a Chaplygin gas. We prove that the well-known phenomenon of shock formation in simple…
In this paper we consider the Cauchy problem for the 3D Navier-Stokes equations for incompressible flows. The initial data are assumed to be smooth and rapidly decaying at infinity. A famous open problem is whether classical solutions can…
In the paper, the shock formation for the two-dimensional rotating shallow water system is established. We construct a large class of initial data which leads to the finite-time blow-up for the solutions. Moreover, the solutions are allowed…
We consider axisymmetric Euler flows without swirl in $\mathbb{R}^{d}$ with $d\geq 4$, for which the global regularity of smooth solutions is an open problem. When $d = 4$, we obtain global regularity under the assumption that the initial…
Recent works have demonstrated that continuous self-similar radial Euler flows can drive primary (non-differentiated) flow variables to infinity at the center of motion. Among the variables that blow up at collapse is the pressure, and it…
We introduce a local-in-time existence and uniqueness class for solutions to the 2d Euler equation with unbounded vorticity. Furthermore, we show that solutions belonging to this class can develop stronger singularities in finite time,…
This paper concerns the finite-time blow-up and asymptotic behaviour of solutions to nonlinear Volterra integrodifferential equations. Our main contribution is to determine sharp estimates on the growth rates of both explosive and…
In this paper, we are concerned with the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping \begin{equation*} \partial_t\rho+\operatorname{div}(\rho u)=0, \quad…
We construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in a suitable H\"older space and the…
We establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely…
We consider the classical Cauchy problem for a system of equations describing 3D arbitrary electrostatic oscillations of the cold plasma and introduce an iteration procedure that allows estimating the blow-up time from below. This procedure…
A simplified form of the vorticity equation is derived for arbitrary coordinate systems. The present work unifies and extends the previous findings that vorticity is conserved in planar Euler flow, while in axisymmetric Euler rings it is…