Stabilities for Nonisentropic Euler-Poisson Equations

We establish the stabilities and blowup results for the nonisentropic Euler-Poisson equations by the energy method. By analysing the second inertia, we show that the classical solutions of the system with attractive forces blow up in finite time in some special dimensions when the energy is negative. Moreover, we obtain the stabilities results for the system in the cases of attractive and repulsive forces

Perturbational Blowup Solutions to the Two-Component Dullin-Gottwald-Holm System

We construct a family of nonradially symmetric exact solutions for the two-component DGH system by the perturbational method. Depending on the parameters, the class of solutions includes both blowup type and global existence type

Blowup Phenomenon of Solutions for the IBVP of the Compressible Euler Equations in Spherical Symmetry

The blowup phenomenon of solutions is investigated for the initial-boundary value problem (IBVP) of the N-dimensional Euler equations with spherical symmetry. We first show that there are only trivial solutions when the velocity is of the form c(t)xα-1x+b(t)(x/x) for any value of α≠1 or any positive integer N≠1. Then, we show that blowup phenomenon occurs when α=N=1 and c2(0)+c˙(0)<0. As a corollary, the blowup properties of solutions with velocity of the form (a˙t/at)x+b(t)(x/x) are obtained. Our analysis includes both the isentropic case (γ>1) and the isothermal case (γ=1)

Exact Self-similar Perturbational Solutions of Whitham-Broer-Kaup Equations

Abstract In this paper, by employing the perturbational method, we obtain a class of exact self-similar solutions of the Whitham-Broer-Kaup equations. These solutions are of polynomials-type whose forms, remarkably, coincident with that given by Yuen for the other physical models, such as the compressible Euler or Navier-Stokes equations, two-component Camassa-Holm equations and viscoelastic Burgers equations

The van Hiele Phases of Learning in studying Cube Dissection

Spatial sense is an important ability in mathematics. Formula application is very different from spatial concept acquisition. But it is often observed that in schools students learn spatial concepts by memorizing instead of understanding. In the past academic year we had tried out and developed a series of learning activities based on van Hiele’s model for guiding learners to explore the cube and its cut sections. The ideas in origami, and mathematical modelling by manipulative as well as mathematical software are integrated into our study. This paper gives a brief account on our works. We start by presenting a sequence of math-rich learning tasks, followed by some related folding ideas and mathematical background analysis. Finally we round up our paper with a concise discussion on some major elements of our design based on the van Hiele learning phases