Nonexistence of self-similar singularities in the ideal magnetohydrodynamics

Nonexistence of self-similar singularities for the 3D incompressible Euler equations

We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations. By similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in $\Bbb R^n$. This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions.

Nonexistence results for compressible non-Newtonian fluid with magnetic effects in the whole space

We consider a generalization of the compressible barotropic Navier-Stokes equations to the case of non-Newtonian fluid in the whole space. The viscosity tensor is assumed to be coercive with an exponent $q>1.$ We prove that if the total mass and momentum of the system are conserved, then one can find a constant $q_0>1$ depending on the dimension of space $n$ and the heat ratio $γ$ such that for $q\in [q_0,n)$ there exists no global in time smooth solution to the Cauchy problem. We prove also an analogous result for solutions to equations of magnetohydrodynamic non-Newtonian fluid in 3D space.

Quasilinear elliptic problems with cylindrical singularities and multiple critical nonlinearities: existence, regularity, nonexistence

This work deals with existence of solutions for the class of quasilinear elliptic problems with cylindrical singularities and multiple critical nonlinearities that can be written in the form \begin{align*} -\operatorname{div}\left[\frac{|\nabla u|^{p-2}}{|y|^{ap}}\nabla u\right] -μ\,\frac{u^{p-1}}{|y|^{p(a+1)}} = \frac{u^{p^*(a,b)-1}}{|y|^{bp^*(a,b)}} + \frac{u^{p^*(a,c)-1}}{|y|^{cp^*(a,c)}}, \qquad (x,y) \in \mathbb{R}^{N-k}\times\mathbb{R}^k. \end{align*} The existence of a positive, weak solution $u \in \mathcal{D}_a^{1,p}(\mathbb{R}^N\backslash\{|y|=0\})$ is proved with the help of the mou…

On partial regularity for the steady Hall magnetohydrodynamics system

We study partial regularity of suitable weak solutions of the steady Hall magnetohydrodynamics equations in a domain $Ω\subset \Bbb R^3$. In particular we prove that the set of possible singularities of the suitable weak solution has Hausdorff dimension at most one. Moreover, in the case $Ω=\Bbb R^3$, we show that the set of possible singularities is compact.

On partial regularity for the $3D$ non-stationary Hall magnetohydrodynamics equations on the plane

We study partial regularity of weak solutions of the 3D valued non-stationary Hall magnetohydrodynamics equations on $ \Bbb R^2$. In particular we prove the existence of a weak solution whose set of possible singularities has the space-time Hausdorff dimension at most two.

Nonexistence of self-similar blowup for the nonlinear Dirac equations in (1+1) dimensions

We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexistence of classical self-similar blowup solutions in the space of bounded functions. While this argument does not exclude the possibility of finite-time blowup, it still suggests that smooth solutions to the nonlinear Dirac equations in (1+1) dimensions do not develop self-similar singularities in a finite time. In the particular case of the cubic Dirac equations, we characterize (unbounded) self-similar solutions in the closed analytical form.

电子磁流体力学中的自相似奇异性

Self-similar singularities for electron MHD

本文研究无电阻率电子MHD的自相似解，证明在一定条件下可排除局部自相似与伪自相似爆破解。

Isolated singularities of positive solutions for Choquard equations in sublinear case

Our purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case $q \in (0,1)$ $$\displaystyle \ \ -Δu+ u =I_α[u^p] u^q\;\; {\rm in}\; \mathbb{R}^N\setminus\{0\}, % [2mm] \phantom{ } \;\; \displaystyle \lim_{|x|\to+\infty}u(x)=0, $$ where $p >0, N \geq 3, α\in (0,N)$ and $I_α[u^p](x) = \int_{\mathbb{R}^N} \frac{u^p(y)}{|x-y|^{N-α}}dy$ is the Riesz potential, which appears as a nonlocal term in the equation. We investigate the nonexistence and existence of isolated singular solutions of Choquard equation under different range of t…

三维磁流体力学的一维模型

1D Model for the 3D Magnetohydrodynamics

本文提出三维不可压缩理想磁流体力学的1D模型，建立Beale-Kato-Majda型正则性准则，证明无拉伸效应时模型具有全局强解，并通过数值模拟表明光滑周期初值下解在有限时间内不产生奇点。

带表面张力的自由边界理想磁流体力学中的低正则性先验估计、爆破准则和自交奇点

Low-regularity a priori estimates, blow-up criterion, and self-intersection singularities for free-boundary ideal magnetohydrodynamics with surface tension

针对带表面张力的三维不可压缩自由边界理想MHD方程，在一般有界区域中建立了H^3先验估计，将门槛从H^6降低；提出了区分拓扑自交、边界正则性丧失等的有限时间爆破准则，揭示了磁场不对称性；构造了正则初始数据，证明表面张力和理想磁耦合不能阻止自由边界拓扑自交。

Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation

The problem we are concerned with is whether singularities form in finite time in incompressible fluid flows. It is well known that the answer is ``no'' in the case of Euler and Navier-Stokes equations in dimension two. In dimension three it is still an open problem for these equations. In this paper we focus on a two-dimensional active scalar model for the 3D Euler vorticity equation. Constantin, Majda and Tabak suggested, by studying rigorous theorems and detailed numerical experiments, a general principle: ``If the level set topology in the temperature field for the 2D quasi-geostrophic act…

On the blow-up problem and new a priori estimates for the 3D Euler and the Navier-Stokes equations

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 particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities obtained in \cite{cha1,cha2}. Some implications the notions for the 3D Navier-Stokes equations are also deduced. Generalization of the self-similar transforms is also considered, and by appropriate choice of the transform we obtain new \textit{a priori} es…

On the apparition of singularities of vector fields transported by volume preserving diffeomorphisms

We consider possible generation of singularities of a vector field transported by diffeomorphisms with derivatives of uniformly bounded determinants. A particular case of volume preserving diffeomrphism is the most important, since it has direct applications to the incompressible, inviscid hydrodynamics. We find relations between the directions of the vector field and the eigenvectors of the derivative of the back-to-label map near the singularity. We also find an invariant when we follow the motion of the integral curves of the vector field. For the 3D Euler equations these results have immed…

Homogeneous Euler equation: blow-ups, gradient catastrophes and singularity of mappings

The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes and dynamics of mappings of $\mathbb{R}^n \to \mathbb{R}^n$ associated with the $n$-dimensional homogeneous Euler equation. Several characteristic features of the multi-dimensional case ($n>1$) are described. Existence or nonexistence of blow-ups in different dimensions, boundness of certain linear combinations of blow-up derivatives and the first occurrence of the gradient catastrophe are among of them. It is shown that the potential solutions of the Euler equations exhibit blow-up derivatives in any di…