Special Issue of the Proceedings of 1999 International Conference on Nonlinear Analysis (October 16–20, 1999, Academia Sinica, Taipei, Taiwan) in honor of the 60th birthday of Fon-Che Liu

We survey the qualitative shock wave theory. The survey is meant to explain some of the key issues close to the interest of the author and raise open questions.

Some recent results on the equation $$ \triangle_g u+\rho\left(\displaystyle\frac{he^u}{\int_M he^u}-1\right )=0 \text{ on } M $$ are reviewed. We focus on the variational structure and the blowup behavior of the solutions.

In this paper, we will survey some recent results on the study of the viscous and invisid compressible flow with vacuum. It is wellknown that the study on vacuum has significance in the investigation on some important physical phenomena. However, most of the important questions about vacuum are still open due to the singularities caused by vacuum which need new mathematical tools and techniques to handle.

In this paper, we survey some recent results concerning the fractal structure of Brownian sample paths. The following aspects are discussed: (1) average densities of Brownian trails and intersections; (2) dimension spectra of Brownian zeroes; (3) multifractal properties of Brownian substitutions.

In previous works, we have established discrete versions of the Aleksandrov -Bakelman maximum principle for elliptic operators, on general meshes in Euclidean space. In this paper, we prove a variant of these estimates in terms of a discrete analogue of the determinant of the coefficient matrix in the differential operator case. Our treatment depends on an interesting connection between the determinant and volumes of cells in the underlying mesh.

We consider a semilinear equation arising from the Navier- Stokes equations – the governing equations of viscous fluid motion – and related model equations. The solutions of the semilinear equation represent a certain class of exact solutions of the Navier-Stokes equations. Both the equation and our models have nonlocal terms. We will show that the nonlocality will play an intriguing role for the blow-up and/or global existence of the solutions and that the convection term, which is often neglected in the study of the blow-up problems, plays a very decisive role. In addition to our new contributions, open problems and known facts are surveyed.

We study the Cauchy problem for a $3\times 3$-system of conservation laws describing the phase transition: $u_t-v_x=0$, $v_t-\sigma(u)_x=0$, $(e+\frac{1}{2}v^2)_t-(\sigma v)_x=0$. A phase boundary is said to be admissible if it satisfies the Abeyaratne-Knowles kinetic condition. We give a physical account of the kinetic condition by means of the $Gibbs function$. We also obtain a useful description of the entropy function using the Gibbs function.

We consider Couette-Taylor problems of the perturbation to Couette flow between two rotating cylinders, and show that the stationary bifurcation or Hopf bifurcation occurs when Taylor number increases. We make precise analysis of the eigenvalue problems