The uniqueness of positive solutions to some semilinear elliptic systems with variational structure arising from mathematical physics is proved. The key ingredient of the proof is the oscillatory behavior of solutions to linearized equations for cooperative semilinear elliptic systems of two equations on one-dimensional domains, and it is shown that the stability of the positive solutions for such semilinear system is closely related to the oscillatory behavior.

As a continuation of [28], the paper aims to justify the high Reynolds numbers limit for the MHD system with Prandtl boundary layer expansion when no-slip boundary condition is imposed on the velocity field and the perfect conducting boundary condition is given for the magnetic field. Under an assumption that the viscosity and resistivity coefficients are of the same order and the initial tangential magnetic field on the boundary is not degenerate, we justify the validity of the Prandtl boundary layer expansion and give a L8 estimate on the error by multi-scale analysis.

We consider the problem of the random fluctuations in the solutions to elliptic PDEs with highly oscillatory random coefficients. In our setting, as the correlation length of the fluctuations tends to zero, the heterogeneous solution converges to a deterministic solution obtained by averaging. When the Green’s function to the unperturbed operator is sufficiently singular (i.e., not square integrable locally), the leading corrector to the averaged solution may be either deterministic or random, or both in a sense we shall explain. Our main application is the solution of an elliptic problem with random Robin boundary condition that may be used to model diffusion of signaling molecules through a layer of cells into a bulk of extracellular medium. The problem is then described by an elliptic pseudo-differential operator (a Dirichlet-to-Neumann operator) on the boundary of the domain with random potential. In the physical setting of a three dimensional extracellular medium on top of a two-dimensional surface of cells forming a layer of epithelium, we show that the approximate corrector to averaging consists of a deterministic correction plus a Gaussian field of amplitude proportional to the correlation length of the random medium. The result is obtained under some assumptions on the four-point correlation function in the medium. We provide examples of such random media based on Gaussian and Poisson statistics.

The sine-Gordon (SG) equation and perturbed nonlinear Schrdinger (NLS) equations are studied numerically for modeling the propagation of two space dimensional (2D) localized pulses (the so-called <i>light bullets</i>) in nonlinear dispersive optical media. We begin with the (2+1) SG equation obtained as an asymptotic reduction in the two level dissipationless MaxwellBloch system, followed by the review on the perturbed NLS equation in 2D for SG pulse envelopes, which is globally well posed and has all the relevant higher order terms to regularize the collapse of standard critical (cubic focusing) NLS. The perturbed NLS is approximated by truncating the nonlinearity into finite higher order terms undergoing focusingdefocusing cycles. Efficient semi-implicit sine pseudospectral discretizations for SG and perturbed NLS are proposed with rigorous error estimates. Numerical comparison results between light bullet

For strong solutions of the incompressible Navier-Stokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure. These schemes require no solution of stationary Stokes systems, nor any compatibility between velocity and pressure spaces to ensure an inf-sup condition, and are representative of a class of highly efficient computational methods that have recently emerged. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the Navier-Stokes equations as a perturbed diffusion equation. © 2007 Wiley Periodicals, Inc.