The lifespan of spherically symmetric solutions of the compressible Euler equations outside an impermeable sphere

We consider smooth three-dimensional spherically symmetric Eulerian flows of ideal polytropic gases outside an impermeable sphere, with initial data equal to the sum of a constant flow with zero velocity and a smooth perturbation with compact support. Under a natural assumption on the form of the perturbation, we obtain precise information on the asymptotic behavior of the lifespan as the size of the perturbation tends to 0. When there is no sphere, so that the flow is defined in all space, corresponding results have been obtained in [P. Godin, The lifespan of a class of smooth spherically symmetric solutions of the compressible Euler equations with variable entropy in three space dimensions, Arch. Ration. Mech. Anal. 177 (2005) 479–511].     

Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains

We consider solutions of some nonlinear parabolic boundary value problems in radial bounded domains whose initial profile satisfies a reflection inequality with respect to a hyperplane containing the origin. We show that, under rather general assumptions, these solutions are asymptotically (in time) foliated Schwarz symmetric, that is, all elements in the associated omega limit set are axially symmetric with respect to a common axis passing through the origin and nonincreasing in the polar angle from this axis. In this form, the result is new even for equilibria (i.e., solutions of the corresponding elliptic problem) and time periodic solutions.     

Radially symmetric wave maps from (1 + 2)-dimensional Minkowski space to the sphere

By a blow-up analysis as in [8] for a related problem we rule out concentration of energy for radially symmetric wave maps from the (1+ 2)-dimensional Minkowski space to the sphere. When combined with the local existence and regularity results of Christodoulou and Tahvildar-Zadeh for this problem, our result implies global existence of smooth solutions to the Cauchy problem for radially symmetric wave maps for smooth radially symmetric data. Received: 1 November 2000; in final form: 12 April 2001 / Published online: 1 February 2002     

带自由边界的可压缩欧拉与欧拉-泊松方程组径向对称解的爆破北大核心CSCD

本文在R^(N)(N=2,3)中研究描述流向外部真空的可压缩流体的欧拉与欧拉-泊松方程组径向对称解的爆破.在分离流体与真空的连续自由边界条件下考虑其自由边值问题.对于径向对称的欧拉方程组,证明若初始流平均向外流动,则其光滑解将在有限时刻爆破.对于带有斥力与弛豫项的单极与双极径向对称欧拉-泊松方程组,证明若某个与初始动量有关的加权泛函适当大,则其光滑解将在有限时刻爆破。     

On a Characteristic Initial Value Problem in Plasma Physics

The relativistic Vlasov-Maxwell system of plasma physics is considered with initial data on a past light cone. This characteristic initial value problem arises in a natural way as a mathematical framework to study the existence of solutions isolated from incoming radiation. Various consequences of the mass-energy conservation and of the absence of incoming radiation condition are first derived assuming the existence of global smooth solutions. In the spherically symmetric case, the existence of a unique classical solution in the future of the initial cone follows by arguments similar to the case of initial data at time t=0. The total mass-energy of spherically symmetric solutions equals the (properly defined) mass-energy on backward and forward light cones. Communicated by Sergiu Klainerman submitted 8/03/05, accepted 26/05/05     

Interior penalty discontinuous Galerkin method for Maxwell's equations: Energy norm error estimates

We develop the symmetric interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell equations in second-order form. We derive optimal a priori error estimates in the energy norm for smooth solutions. We also consider the case of low-regularity solutions that have singularities in space.     

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.     

Smooth solutions for one‐dimensional relativistic radiation hydrodynamic equations

In this research article, we investigated the existence of local smooth solutions for relativistic radiation hydrodynamic equations in one spatial variable. The proof is based on a classical iteration method and the Banach contraction mapping principle. However, because of the complexity of relativistic radiation hydrodynamics equations, we first rewrite this system into a semilinear form to construct the iteration scheme and then use left eigenvectors to decouple the system instead of applying standard energy method on symmetric hyperbolic systems. Different from multidimensional case, we just use the characteristic method, which can keep the properties of the initial data. Copyright © 2015 John Wiley & Sons, Ltd.     

A simple application of the homotopy method to symmetric eigenvalue problems

The homotopy method is used to find all eigenpairs of symmetric matrices. A special homotopy is constructed for Jacobi matrices. It is shown that there are exactly n distinct smooth curves connecting trivial solutions to desired eigenpairs. These curves are solutions of a certain ordinary differential equation with different initial values. Hence, they can be followed numerically. Incorporated with sparse matrix techniques, this method might be used to solve eigenvalue problems for large scale matrices.     

Symmetric Euler and Navier-Stokes shocks in stationary barotropic flow on a bounded domain

We construct stationary solutions to the barotropic, compressible Euler and Navier-Stokes equations in several space dimensions with spherical or cylindrical symmetry. For given Dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the Euler equations in the exterior domain. On the other hand, stationary smooth solutions in the interior domain necessarily become sonic and cannot be continued beyond a critical inner radius. We then use these solutions to construct entropy-satisfying shocks for the Euler equations in the region between two concentric spheres or cylinders. Next we construct smooth Navier-Stokes solutions converging to the previously constructed Euler shocks in the small viscosity limit. In the process we introduce a new technique for constructing smooth solutions, which exhibit a fast transition in the interior, to a class of two-point boundary problems.