By using the weak form of the governing equations for sectorial bimaterial domains and assuming that the displacement field is proportional to the (λ+1)-th power of the distance from the singular stress point, a second order characteristic matrix equation on λ is derived by a one-dimensional finite element formulation that only discretizes the domain circumferentially. Numerical examples covering a variety of interfacial singularities are presented to demonstrate the efficacy of the formulation. Accurate solutions are yielded by very few elements whereas convergence can be attained by either h-or p-refinement. The related procedures are programmed in a short MAPLE worksheet given in the appendix.

In this paper, we investigate the pointwise behavior of the solution for the compressible Navier-Stokes equations with mixed boundary condition in half space. Our results show that the leading order of Green's function for the linear system in half space are heat kernels propagating with sound speed in two opposite directions and reflected heat kernel (due to the boundary effect) propagating with positive sound speed. With the strong wave interactions, the nonlinear analysis exhibits the rich wave structure: the diffusion waves interact with each other and consequently, the solution decays with algebraic rate.

We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad's angular cutoff assumption. More precisely, for solutions inside the finite Mach number region, we obtain the pointwise fluid structure for hard potentials and Maxwellian molecules, and optimal time decay in the fluid part and sub-exponential time decay in the non-fluid part for soft potentials. For solutions outside the finite Mach number region, we obtain sub-exponential decay in the space variable. The singular wave estimate, regularization estimate and refined weighted energy estimate play important roles in this paper. Our results largely extend the classical results of 12,13] and Lee-Liu-Yu [10] to hard and soft potentials by imposing suitable exponential velocity weight on the initial condition.2000 Mathematics Subject Classification. 35Q20; 82C40.

We study the pointwise (in the space and time variables) behavior of the Fokker-Planck Equation with potential. An explicit description of the solution is given, including the large time behavior, initial layer and spatially asymptotic behavior. Moreover, it is shown that the structure of the solution sensitively depends on the potential function.2010 Mathematics Subject Classification. 35Q84; 82C40.