Ka-Luen Cheung

On the stability of single and multiple droplets for equations of thin film type

Some general results on the energy stability/instability of droplets with zero contact angle for thin film type equations including those governed by the power laws are established. Energy instability of droplets with nonzero contact angle and configurations of droplets is also obtained. As an application, an asymptotic stability result for droplets with zero contact angle is established.

Blowup phenomenon for the initial-boundary value problem of the non-isentropic compressible Euler equations

The blowup phenomenon for the initial-boundary value problem of the non-isentropic compressible Euler equations is investigated. More precisely, we consider a functional F(t) associated with the momentum and weighted by a general test function f and show that if F(0) is sufficiently large, then the finite time blowup of the solutions of the non-isentropic compressible Euler equations occurs. As the test function f is a general function with only mild conditions imposed, a class of blowup conditions is established.

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)|x|α−1 x + 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).

Perturbational blowup solutions to the compressible Euler equations with damping

BackgroundThe N-dimensional isentropic compressible Euler system with a damping term is one of the most fundamental equations in fluid dynamics. Since it does not have a general solution in a closed form for arbitrary well-posed initial value problems. Constructing exact solutions to the system is a useful way to obtain important information on the properties of its solutions.Method In this article, we construct two families of exact solutions for the one-dimensional isentropic compressible Euler equations with damping by the perturbational method. The two families of exact solutions found include the cases

Stabilities for Nonisentropic Euler-Poisson Equations

\gamma >1

A symmetry result for an elliptic problem arising from the 2-D thin film equation

γ>1 and