FREE BOUNDARY VALUE PROBLEM FOR THE CYLINDRICALLY SYMMETRIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

In this paper, we investigate the free boundary value problem(FBVP) for the cylindrically symmetric isentropic compressible Navier-Stokes equations(CNS) with densitydependent viscosity coefficients in the case that across the free surface stress tensor is balanced by a constant exterior pressure. Under certain assumptions imposed on the initial data, we prove that there exists a unique global strong solution which tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity.     

Cylindrically symmetric gravitational-wavelike space–times

We present Noether symmetries of a geodetic Lagrangian for a time-conformal cylindrically symmetric space–time. We introduce a time-conformal factor in the general cylindrically symmetric space–time to make it nonstatic and then find approximate Noether symmetries of the action of the corresponding Lagrangian. Taking the perturbation up to the first order, we find all Lagrangians for cylindrically symmetric space–times for which approximate Noether symmetries exist.     

Approximate solutions to one-dimensional nonlinear heat conduction problems with a given flux

One-dimensional (planar, cylindrically symmetric, and spherically symmetric) nonlinear heat conduction problems with the heat flux at the origin specified in the form of a power time dependence are considered. The initial temperature of the medium is assumed to be zero. Approximate solutions to the problems are obtained. The convergence of the resulting solutions is discussed.     

Remarks on Exponential Stability of Solutions for the Compressible p-th Power Newtonian Fluid with Large Initial Data

In this paper, we establish the exponential stability of the global spherically and cylindrically symmetric solutions in H^i (i=1,2,4) for the p-th power Newtonian fluid in multi-dimension with large initial data. The key point is that the smallness of initial data is not needed if the initial data are cylindrically symmetric.     

Multiple symmetric solutions for a Neumann problem with lack of compactness

The existence of multiple cylindrically symmetric solutions for a class of non-autonomous elliptic Neumann problems in a strip-like domain of the Euclidean space is investigated. The proof combines a recent compactness result and the Palais symmetric critically principle. A concrete application illustrates the main abstract result of this Note.     

Study of cylindrically symmetric solutions in an $$f(R)$$ gravity background

Theoretical and Mathematical Physics - We investigate static cylindrically symmetric solutions of the Weyl and Gödel space–times in the framework of modified $$f(R)$$ gravity. With this...     

An exact solution of a scalar-tensor theory of gravitation

A class of exact solutions of the cylindrically symmetric space-time with two degrees of freedom corresponding to the vacuum field equations of a scalar-tensor theory, formally similar to Brans-Dicke theory, is obtained. The solutions possess wave-like character.     

The Self-Focusing Singularity in the Nonlinear Schrödinger Equation

Under certain circumstances, solutions of the cylindrically symmetric nonlinear Schrödinger equation collapse to a singularity in a finite time. An asymptotic series for the solution near the singularity is derived here. At leading order, the central amplitude of the spike grows like[(log Δt)/Δt]^1/2, where Δt is the time remaining to the appearance of the singularity.     

Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations

We prove nondegeneracy of extremals for some Hardy-Sobolev-Maz'ya inequalities and present applications to scalar curvature-type problems, including the Webster scalar curvature equation in a cylindrically symmetric setting. The main theme is hyperbolic symmetry.     

A cylindrically symmetric universe in general relativity

A cylindrically symmetric universe with two degrees of freedom which is of Petrov Type I degenerate, has been derived. Various physical and geometrical properties of the model have also been discussed.     

Symmetry analysis and some exact solutions of cylindrically symmetric null fields in general relativity

The symmetry reduction method based on the Fréchet derivative of the differential operators is applied to investigate symmetries of the Field equations in general relativity corresponding to cylindrically symmetric space–time, that is a coupled system of nonlinear partial differential equations of second order. More specifically, this technique yields invariant transformation that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied for exact solutions.     

On the Webster Scalar Curvature Problem on the CR Sphere with a Cylindrical-Type Symmetry

By variational methods, for a kind of Webster scalar curvature problems on the CR sphere with cylindrically symmetric curvature, we construct some multi-peak solutions as the parameter is sufficiently small under certain assumptions. We also obtain the asymptotic behaviors of the solutions.     

On the Cylindrically Symmetric Einstein–Vlasov System

In this article we begin to study the question of global existence for the cylindrically symmetric Einstein–Vlasov system with general (in size) data and we show that if a singularity occurs at all, the first one occurs at the axis of symmetry. This is done by a combination of light cone estimates and a careful analysis of the matter terms in the “exterior” region, together with Sobolev methods for t he analysis in the “interior” region.     

Nonlinear degenerate curvature flows for weakly convex hypersurfaces

We study the evolution of closed, weakly convex hypersurfaces in in direction of their normal vector, where the speed equals a quotient of successive elementary symmetric polynomials of the principal curvatures. We show that there exists a solution for these weakly convex surfaces at least for some short time if the elementary symmetric polynomial in the denominator of the quotient is positive. The results for this nonlinear, degenerate flow are obtained by a cylindrically symmetric barrier construction.Received: 10 November 2003, Accepted: 5 April 2004, Published online: 16 July 2004     

Remarks for the axisymmetric Navier-Stokes equations

We are concerned with 3-D incompressible Navier-Stokes equations when the initial data and the domain are cylindrically symmetric. We show that there exists a solution in a weighted space and certain weighted norms of vorticity of the solution remain finite if they are finite initially. Consequently, we can estimate the growth rate of the solution both spatially and temporally.     

可允许度量下有限能量辛涡旋的渐近行为献给钱敏教授90华诞

假设(X,ω)是一个具有紧致单连通Lie群G Hamilton作用的紧致光滑辛流形.本文证明只要Riemann面的柱形端口具有一个比标准柱形度量增长速度快的线性度量,那么任何一个有限能量辛涡旋将以指数衰减的速度收敛到辛流形X在正则值辛约化的扭曲分支或非扭曲分支上.本文结果无需假设群G在正则水平集上的作用是自由的.因此,它直接推广了Ziltener在群作用自由的假设下得出的相关结果.本文结果在作者关于量子化Kirwan同态的系列工作中有重要应用.     

Asymptotic behavior of exact solutions for the Cauchy problem to the 3D cylindrically symmetric Navier–Stokes equations

In this paper, we establish exact solutions of the Cauchy problem for the 3D cylindrically symmetric incompressible Navier–Stokes equations and further study the global existence and asymptotic behavior of solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Axially symmetric volume constrained anisotropic mean curvature flow

We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume in comparison with their initial surface energy, we show that the flow exists for all time. Numerical simulations of the curvature flow are presented.