Related papers: On Singularity Formation of a Nonlinear Nonlocal S…
We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei in [16] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the…
We investigate the large time behavior of an axisymmetric model for the 3D Euler equations. In \cite{HL09}, Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier-Stokes equations with swirl. This model shares…
Inspired by the numerical evidence of a potential 3D Euler singularity \cite{luo2014potentially,luo2013potentially-2}, we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in…
We study the singularity formation of a quasi-exact 1D model proposed by Hou-Li in \cite{hou2008dynamic}. This model is based on an approximation of the axisymmetric Navier-Stokes equations in the $r$ direction. The solution of the 1D model…
We present a formal, approximate model for singularity formation in classical fluid dynamics in three dimensions. The construction utilizes an approximation of local two-dimensionality to study an anti-parallel hairpin vortex structure with…
We demonstrate finite-time blow-up in a simple, realistic shell model of the 3D Navier-Stokes equations, equipped with "smooth" (i.e., rapidly decaying in frequency) initial data and forcing. Previously studied models either exhibit a…
We investigate the role of convection on its large time behavior of 3D incompressible Euler equations. In \cite{HL09a}, we constructed a new 3D model by neglecting the convection term from the reformulated axisymmetric Navier-Stokes…
We present a 5D-lifted analytic-profile program for finite-time singularity formation in the 3D incompressible Navier--Stokes equations on the periodic torus $\T^3$. The core of the construction is a stationary rescaled profile $\Ombar$…
Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D…
We study a 1D fluid mechanics model with nonlocal velocity. The equation can be viewed as a fractional porous medium flow, a 1D model of the quasi-geostrophic equation, and also a special case of Euler-Alignment system. For strictly…
Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the…
Here we consider a Cahn-Hilliard-Navier-Stokes system characterized by a nonlocal Cahn-Hilliard equation with a singular (e.g., logarithmic) potential. This system originates from a diffuse interface model for incompressible isothermal…
The question of the global regularity vs finite time blow up in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the…
It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularities in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup…
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 +…
The motion of two contiguous incompressible and viscous fluids is described within the diffuse interface theory by the so-called Model H. The system consists of the Navier-Stokes equations, which are coupled with the Cahn-Hilliard equation…
This review article offers a survey of the research program focused on a systematic computational search for extreme and potentially singular behavior in hydrodynamic models motivated by open questions concerning the possibility of a…
In light of the question of finite-time blow-up vs. global well-posedness of solutions to problems involving nonlinear partial differential equations, we provide several cautionary examples which indicate that modifications to the boundary…
Cahn-Hilliard-Navier-Stokes system describes the evolution of two isothermal, incompressible, immiscible fluids in a bounded domain. In this work, we consider the stationary nonlocal Cahn-Hilliard-Navier-Stokes system in two and three…
Under the genuinely nonlinear assumption for 1-D $n\times n$ strictly hyperbolic conservation laws, we investigate the geometric blowup of smooth solutions and the development of singularities when the small initial data fulfill the generic…