For a bipolar hydrodynamic model of semiconductors in the form of EulerPoisson equations with Dirichlet or Neumann boundary conditions, in this paper we first heuristically analyze the most probable asymptotic profile (the so-called diffusion waves) and then prove this long-time behavior rigorously. For this, we construct correction functions to show the convergence of the original solution to the diffusion wave with optimal convergence rates by the energy method. Moreover, in the case with Dirichlet boundary condition, when the initial perturbation is in some weighted L^1 space, a faster and optimal convergence rate is also given.

In this paper, we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary condition. We use the idea of volume constraint to enforce the no-slip boundary condition. We prove that the nonlocal model is well-posed and the solution of the nonlocal system converges to the solution of the original Stokes system when the nonlocality vanishes.

We study the averaged behavior of interfaces moving with oscillatory normal velocity that is periodic in time and stationary ergodic in space. This problem can be interpreted as a homogenization problem of a Hamilton-Jacobi equation with a positively homogeneous and time dependent Hamiltonian. In an earlier work, we studied the setting when the environment is periodic in space and random in time, and established homogenization results through the averaging of reachable sets. In the present setting, we show that the minimal travel time between two spatial points, which now depends also on a starting time, has a deterministic averaging limit that is independent of the starting time. This averaged travel time function is then shown to determine the effective behavior of the moving interfaces.

We estimate the heat conducted by a cluster of many small cavities. We show that the dominating heat is a sum, over the number of cavities, of the heat generated by each cavity after interacting with each other. This interaction is described through densities computable as solutions of a closed, and invertible, system of time domain integral equations of second kind. As an application of these expansions, we derive the effective heat conductivity which generates approximately the same heat as the cluster of cavities, distributed in a three-dimensional bounded domain, with explicit error estimates in terms of that cluster. At the analysis level, we use time domain integral equations. Doing that, we have two choices. First, we can favor the space variable by reducing the heat potentials to the ones related to the Laplace operator (avoiding Laplace transform). Second, we can favor the time variable by reducing the representation to the Abel integral operator. As the model under investigation has time-independent parameters, we follow the first approach here.

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