GROUNDWATER MASS TRANSPORT AND HOMOGENEOUS EQUILIBRIUM CHEMISTRY IN THE PRESENCE OF FLUX BOUNDARY CONDITIONS

Considering a solute transport problem deseribed by some algebraic and partial differentialequations with the presence of flux boundary conditions, we reduce the problem to a fixed point oneand use a priori estimates to prove the existence and uniqueness of the global solutions.

General semi-analytical solutions to one-dimensional consolidation for unsaturated soils

This paper presents general semi-analytical solutions to Fredlund and Hasan＇s one-dimensional （1D） consolidation equations for unsaturated soils subject to different initial conditions, homogeneous boundaries and time-dependent loadings. Two variables are introduced to transform the two-coupled governing equations of pore-water and poreair pressures into an equivalent set of partial differential equations （PDEs）, which are solved with the Laplace transform method. The pore-water and pore-air pressures and settlement are obtained in the Laplace transform domMn. The Crump＇s method is used to perform inverse Laplace transform to obtain the solutions in the time domain. The present solutions are more general in practical applications and show good agreement with the previous solutions in the literature.

Existence theory for Rosseland equation and its homogenized equation

The global boundness and existence are presented for the kind of the Rosseland equation with a general growth condition. A linearized map in a closed convex set is defined. The image set is precompact, and thus a fixed point exists. A multi-scale expansion method is used to obtain the homogenized equation. This equation satisfies a similar growth condition.

The Initial Boundary Value Problem for Navier-Stokes Equations

By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equations in arbitrary three dimensional domain with uniformly C3 boundary, under the assumption that ‖a‖L^2(Ω)+‖f‖L^1(o,∞;L^2(Ω)) or‖▽a‖L^2(Ω)+‖f‖L^2(o,∞;L^2(Ω)) small or viscosity, large. Here a is a given initial velocity and f is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed.

Numerical Study of Quantized Vortex Interaction in theGinzburg-Landau Equation on Bounded Domains

In this paper,we study numerically quantized vortex dynamics and their interaction in the two-dimensional(2D)Ginzburg-Landau equation(GLE)with a dimensionless parameter#>0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition.We begin with a reviewof the reduced dynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically.Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition.Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws,we simulate quantized vortex interaction of GLE with different#and under different initial setups including single vortex,vortex pair,vortex dipole and vortex lattice,compare them with those obtained from the corresponding reduced dynamical laws,and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction.Finally,we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.