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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.

We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole,and denoted sysπ1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). The Fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field. We generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces. The key tool is a new trace estimate valid for an arbitrary ideal in a quaternion algebra. We obtain a particularly sharp bound for a principal congruence tower of Hurwitz surfaces (PCH), namely the 4/3bound sysπ1(XPCH) ≥ 43 log(g(XPCH)). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.

where p > 1. 1 < p < 2 is called the fast p-Laplacian diffusion, while p > 2 is called the slow p-Laplacian diffusion. Especially, the p-Laplacian Keller-Segel model turns to the original model when p = 2.The Keller-Segel model was firstly presented in 1970 to describe the chemotaxis of cellular slime molds [13, 14]. The original model was considered in 2D,

Peskin's Immersed Boundary (IB) method is one of the most popular numerical methods for many years and has been applied to problems in mathematical biology, fluid mechanics, material sciences, and many other areas. Peskiness IB method is associated with discrete delta functions. It is believed that the IB method is first order accurate in the $L^{\infty}$ norm. But almost no rigorous proof could be found in the literature until recently [Mori, Comm. Pure. Appl. Math: 61:2008] in which the author showed that the velocity is indeed first order accurate for the Stokes equations with a periodic boundary condition. In this paper, we show first order convergence with a $\log h$ factor of the IB method for elliptic interface problems with Dirichlet boundary conditions.