粘流—无粘干扰流动理论广义解的存在性和唯一性条件

本文证明了粘流-无粘干扰流动理论基本控制方程—简化Navier-Stokes方程原始变量变分形式广义解的存在性,给出了广义解的唯一性条件和Re数上限估计,得到与完全N-S方程的已知结果相应的一系列结论。一、预备知识和基本假设     

部分耗散的三维不可压Hall-MHD方程组的整体正则性

该文研究了具有部分耗散的三维不可压霍尔-磁流体力学方程组,在Sobolev范数意义下,运用能量估计的一些特殊性质,得到了大初值经典解的局部存在性.另外,当初值在适当Sobolev范数意义下充分小时,该文也得到了经典解的整体存在性.     

磁场微极流方程组弱解的正则性准则

本文研究了三维空间中磁场微极流方程组弱解的正则性准则问题.利用能量估计的方法证明了如果速度场以及磁场满足一定的条件,则弱解在(0,T]是唯一的强解.     

关于Boussinesq方程组无粘极限的研究

该文主要研究三维Boussinesq方程组的无粘极限问题.为了克服Boussinesq方程组中温度和速度耦合项产生的困难,带温度的涡量方程需要与Slip边界条件匹配,通过计算得到温度更高阶的边界条件,结合迹定理和能量估计,最后得到了三维粘性Boussinesq方程组初边值问题强解的存在唯一性,并在平坦区域上得到了强解的...     

粘流—无粘干扰流动理论变分问题迭代序列的收敛性和算子方程…

本文证明了粘流－无粘干扰流动理论基本控制方程－简化的Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程变分问题几种迭代序列的收敛性，并探讨了其算子方程的性质。本文的结论对于简化Ｎ－Ｓ方程的数值计算具有指导意义。     

三维Navier-Stokes方程组关于速度梯度两个分量的正则性准则的一个改进

该文考虑三维Navier-Stokes方程组的Cauchy问题,得到了一个改进的、各向异性的、关于速度梯度的两个分量的正则性准则.此准则改进了文献[24]中的结果.     

一类非线性椭圆方程组很弱解的正则性

本文在可控增长条件（１．２）－（１．４）下,对一类非线性椭圆方程组（１．１）改进其很弱解偏微商的可积性,使其为经典意义下的弱解。     

粘性依赖于温度的MHD方程组整体经典解的正则性

本文主要研究可压缩非等熵平面磁流体动力学方程组的Cauchy问题整体经典解的正则性,其中方程组的粘性系数λ,μ,磁扩散系数η和热传导系数κ都是比容v和温度0的函数,正比于h(v)θα,h是满足一定条件的非退化光滑函数.在正则性准则∫+∞0 ||b||2L∞ds ＜+∞的条件下,当α适当小时,我们证明了大初值整体经典解的...     

三维空间中的Navier-Stokes方程部分正则性问题

本文研究了Navier-Stokes方程的部分正则性问题.推进了L.Caffarelli.R.Kohn和L.Nirenberg、Hermann Schr和Wolf von Wahl及Paolo Maremontic中的结论,并且还给出了非正则点集的估计.     

粘流－无粘干扰流动理论变分问题迭代序列的收敛性和算子方程性质

本文证明了粘流－无粘干扰流动理论［１，２］基本控制方程─—简化的Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程变分问题几种迭代序列的收敛性，并探讨了其算子方程的性质。本文的结论对于简化Ｎ－Ｓ方程的数值计算具有指导意义。     

Boussinesq方程解的部分正则性

本文考虑Boussinesq方程一类合适弱解的部分正则性.我们先运用广义能量不等式和奇异积分理论得到一些无维量的估计;再通过合适弱解满足的等式,运用迭代技巧,推导出温度场的小性估计;最后由尺度分析(scaling arguments)得到了一类合适弱解的部分正则性.     

Partial Regularity for Stationary Solutions to Liouville-Type Equation in Dimension 3

In dimension n = 3, we prove that the singular set of any stationary solution to the Liouville equation ? Δ u = e ^u , which belongs to W ^1,2, has Hausdorff dimension at most 1.     

Regularity criterion of weak solutions to the 3D Navier-Stokes equations via two velocity components

This paper is concerned with the regularity criterion of Leray-Hopf weak solutions to the 3D Navier-Stokes equations with respect to Serrin type condition on two velocity filed components. It is shown that the weak solution u=(u₁,u₂,u₃) is regular on (0,T] if there exist two solution components, for example, u₂ and u₃, satisfying the condition

     

Partial regularity for the stochastic Navier-Stokes equations

The effects of random forces on the emergence of singularities in the Navier-Stokes equations are investigated. In spite of the presence of white noise, the paths of a martingale suitable weak solution have a set of singular points of one-dimensional Hausdorff measure zero. Furthermore statistically stationary solutions with finite mean dissipation rate are analysed. For these stationary solutions it is proved that at any time the set of singular points is empty. The same result holds true for every martingale solution starting from -a.e. initial condition , where is the law at time zero of a stationary solution. Finally, the previous result is non-trivial when the noise is sufficiently non-degenerate, since for any stationary solution, the measure is supported on the whole space of initial conditions.

     

Global existence of weak solutions for the anisotropic compressible Stokes system

In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key idea is a new identity that we obtain by comparing the limit of the equations of the energies associated to a sequence of weak-solutions with the energy equation associated to the system verified by the limit of the sequence of weak-solutions. In the context of stability of weak solutions, this allows us to construct a defect measure which is used to prove compactness for the density and therefore allowing us to identify the pressure in the limiting model. By doing so we avoid the use of the so-called effective flux. Using this new tool, we solve an open problem namely global existence of solutions à la Leray for such a system without assuming any restriction on the anisotropy amplitude. This provides a flexible and natural method to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime.     

A Theory of Exact Solutions for Annular Viscous Blobs

Summary. A new theory of exact solutions is presented for the problem of the slow viscous Stokes flow of a plane, doubly connected annular viscous blob driven by surface tension. The formulation reveals the existence of an infinite number of conserved quantities associated with the flow for a certain general class of initial conditions. These conserved quantities are associated with a class of exact solutions. This work is believed to provide the first exact solutions for the evolution of a doubly connected fluid region evolving under Stokes flow with surface tension. Received December 19, 1996; revised September 22, 1997, and accepted October 13, 1997     

Regularity of Pressure in the Neighbourhood of Regular Points of Weak Solutions of the Navier-Stokes Equations

In the context of the weak solutions of the Navier-Stokes equations we study the regularity of the pressure and its derivatives in the space-time neighbourhood of regular points. We present some global and local conditions under which the regularity is further improved.