This paper concerns the well-posedness theory of the motion of a physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in (Coutand et al., Commun Math Phys 296:559–587, 2010; Coutand and Shkoller, Arch Ration Mech Anal 206:515–616, 2012; Jang andMasmoudi,Well-posedness of compressible Euler equations in a physical vacuum, 2008) by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary.

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.

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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.

A new augmented method is proposed for elliptic interface problems with a piecewise variable coefficient that has a finite jump across a smooth interface. The main motivation is to get not only a second order accurate solution but also a second order accurate gradient from each side of the interface. Key to the new method is introducing the jump in the normal derivative of the solution as an augmented variable and rewriting the interface problem as a new PDE that consists of a leading Laplacian operator plus lower order derivative terms near the interface. In this way, the leading second order derivative jump relations are independent of the jump in the coefficient that appears only in the lower order terms after the scaling. An upwind type discretization is used for the finite difference discretization at the irregular grid points on or near the interface so that the resulting coefficient matrix is an M-matrix. A multigrid solver is used to solve the linear system of equations, and the GMRES iterative method is used to solve the augmented variable. Second order convergence for the solution and the gradient from each side of the interface is proved in this paper. Numerical examples for general elliptic interface problems confirm the theoretical analysis and efficiency of the new method.