Weak Density of Smooth Maps in W 1, 1(M,N) for Non-Abelian π1(N)

We prove that smooth maps are dense in the sense of biting convergence in W ^{1, 1}(M, N) when M and Nare compact Riemannian manifolds and N is closed.     

Singularities of Euler Flow? Not Out of the Blue!

Does three-dimensional incompressible Euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out some of the difficulties, we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations. We present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup.     

平面C-B样条的奇拐点分析

平面C-B样条曲线是三次均匀B样条的推广.通过移动C-B样条曲线段的一个控制点而固定其余三个控制点的方法,讨论了在曲线上形成零曲率点的移动控制点的轨迹,得到了C-B样条曲线段的尖点判别曲线、拐点判别区域,同时也给出了在曲线段上生成重结点的移动控制点的轨迹区域.     

Proof of the Fukui conjecture via resolution of singularities and related methods. II $$^{\star}$$

The present article is a direct continuation of the first part of this series. We reduce a proof of the Fukui conjecture (concerning the additivity problem of the zero-point vibrational energies of hydrocarbons) to that of a proposition related to the theory of algebraic curves, so that we can focus on the key mechanism of the additivity phenomena. Namely, by establishing what is called the Basic Piecewise Monotone Theorem (BPMT), we reduce a proof of the Fukui conjecture to that of a proposition, called the Local Analyticity Proposition, Version 1 (LAP1), which admits a proof via resolution of singularities. By LAP1, the essential part of the mechanism of the asymptotic linearity phenomena is extracted and is elucidated by using tools from the mathematical theory of algebraic curves, whose language is of vital importance in analyzing the crux of the additivity mechanism. Dedicated to the memory of Prof. Kenichi Fukui (1918–1998).     

Finite-time singularity formation at a magnetic neutral line in Hall magnetohydrodynamics

The formation of a current sheet in a weakly collisional plasma can be modelled as a finite-time singularity solution of magnetohydrodynamic equations. We use an exact self-similar solution to confirm and generalise a previous finding that, in sharp contrast to two-dimensional solutions in standard MHD, a finite-time collapse to a current sheet can occur in Hall MHD. We derive a criterion for the finite-time singularity in terms of initial conditions, and we use an intermediate asymptotic solution for the evolution of an axial magnetic field to obtain a general expression for the singularity formation time. We illustrate the analytical results by numerical solutions.     

Fine singularity analysis of solutions to the Laplace equation

We present here a fine singularity analysis of solutions to the Laplace equation in special polygonal domains in the plane. We assume piecewise constant Neumann data on one component of the boundary. Our motivation is to study the so‐called Berg effect, which is explained in the introduction. Copyright © 2014 John Wiley & Sons, Ltd.     

带有加权Hardy-Sobolev临界指数的非齐次Neumann边界奇异的多解问题

主要研究了非齐次Neumann边界奇异的问题,利用Ekeland变分原理、山路引理和一些分析技巧,证明了正解的存在性.     

On the generalized self-similar singularities for the Euler and the Navier-Stokes equations

In this paper we study blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the Euler and the Navier-Stokes equations, where the sense of convergence and self-similarity are considered in various generalized senses. We improve substantially, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities. Generalization of the self-similar transforms is also considered, and by appropriate choice of the parameterized transform we obtain new a priori estimates for the Euler and the Navier-Stokes equations depending on a free parameter.     

On a new 3D model for incompressible euler and navier-stokes equations

In this paper, we investigate some new properties of the incompressible Euler and Navier-Stokes equations by studying a 3D model for axisymmetric 3D incompressible Euler and Navier-Stokes equations with swirl. The 3D model is derived by reformulating the axisymmetric 3D incompressible Euler and Navier-Stokes equations and then neglecting the convection term of the resulting equations. Some properties of this 3D model are reviewed. Finally, some potential features of the incompressible Euler and Navier-Stokes equations such as the stabilizing effect of the convection are presented.     

Models of thermodiffusion in 1D

The goal of the paper is to study the Cauchy problem for 1D models of thermodiffusion. We explain qualitative properties of solutions. In particular, we show which part of the model has a dominant influence on well‐posedness, propagation of singularities, L^p ? L^q decay estimates on the conjugate line, and on the diffusion phenomenon. Copyright © 2013 John Wiley & Sons, Ltd.