Georgia Institute of Technology College of Sciences

In this talk we will present several results concerning the behavior of the Laplace operator with Neumann boundary conditions in a thin domain where its boundary presents a highly oscillatory behavior. Using homogenization and domain perturbation techniques, we obtain the asymptotic limit as the thickness of the domain goes to zero even for the case where the oscillations are not necessarily periodic. We will also indicate how this result can be applied to analyze the asymptotic dynamics of reaction diffusion equations in these domains.

In a bounded domain with smooth boundary (which can be considered as a smooth sub-manifold of R3), we consider the Boltzmann equation with general Maxwell boundary condition---linear combination of specular reflection and diffusive absorption. We analyze the kinetic (Knudsen layer) and fluid (viscous layer) coupled boundary layers in both acoustic and incompressible regimes, in which the boundary layers behave significantly different. The existence and damping properties of these kinetic-fluid layers depends on the relative size of accommodation number and Kundsen number, and the differential geometric property of the boundary (the second fundamental form.) As applications, first we justify the incompressible Navier-Stokes-Fourier limit of the Boltzmann equation with Dirichlet, Navier, and diffusive boundary conditions respectively, depending on the relative size of accommodation number and Kundsen number. Using the damping property of the boundary layer in acoustic regime, we proved the convergence is strong. The second application is that we derive and justified the higher order acoustic approximation of the Boltzmann equation. This is a joint work with Nader Masmoudi.

We prove a conjecture of Bryant, Griffiths, and Yang concerning the characteristic variety for the determined isometric embedding system. In particular, we show that the characteristic variety is not smooth for any dimension greater than 3. This is accomplished by introducing a smaller yet equivalent linearized system, in an appropriate way, which facilitates analysis of the characteristic variety.

Under the classical small-amplitude, long wave-length assumptions in which the Stokes number is of order one, so featuring a balance between nonlinear and dispersive effects, the KdV-equation u_t+ u_x + uu_x + u_xxx = 0 (1) and the regularized long wave equation, or BBM-equation u_t + u_x + uu_x-u_xxt = 0 (2) are formal reductions of the full, two-dimensional Euler equations for free surface flow. This talk is concerned with the two-point boundary value problem for (1) and (2) wherein the wave motion is specified at both ends of a finite stretch of length L of the media of propagation. After ascertaining natural boundary specifications that constitute well posed problems, it is shown that the solution of the two-point boundary value problem, posed on the interval [0;L], say, converges as L converges to infinity, to the solution of the quarter-plane boundary value problem in which a semi-infinite stretch [0;1) of the medium is disturbed at its finite end (the so-called wavemaker problem). In addition to its intrinsic interest, our results provide justification for the use of the two-point boundary-value problem in numerical studies of the quarter plane problem for both the KdV-equation and the BBM-equation.