In this paper we study the zero-viscosity limit of 2-D Boussinesq equations with vertical viscosity and zero diffusivity, which is a nonlinear system with partial dissipation arising in atmospheric sciences and oceanic circulation. The domain is taken as R2+ with Navier-type boundary. We prove the nonlinear stability of the approximate solution constructed by boundary layer expansion in conormal Sobolev space. The optimal expansion order and convergence rates for the inviscid limit are also identified in this paper. Our paper extends the partial zero-dissipation limit results of Boussinesq system with full dissipation by Chae D. [Adv.Math.203,no.2,2006] in the whole space to the case with partial dissipation and Navier boundary in the half plane.

Quasilocal mass in general relativity is a notion defined for a closed space like 2-surface in spacetime. In this note, we explain the definition in [23] and [24] from a mathematical viewpoint, emphasizing the connection to differential geometry and nonlinear partial differential equations. We also discuss a minimax interpretation of the definition and compare with other notions of quasilocal mass.

Necessary and sufficient conditions for the discreteness of the Laplacian on a noncompact Riemannian manifold M are established in terms of the isocapacitary function of M. The relevant capacity takes a different form according to whether M has finite or infinite volume. Conditions involving the more standard isoperimetric function of M can also be derived, but they are only sufficient in general, as we demonstrate by concrete examples.

In this paper, we extend the high order finite-difference method with subcell resolution (SR) for two-species stiff one-reaction models to multispecies and multireaction chemical reactive flows, which are significantly more difficult because of the multiple scales generated by different reactions. For reaction problems, when the reaction time scale is very small, the reaction zone scale is also small and the governing equations become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present SR method for reactive Euler system is a fractional step method. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with certain computed flow variables in the shock region modified by the Harten subcell resolution idea. Several numerical examples of multispecies and multireaction reactive flows are performed in both one and two dimensions. Studies demonstrate that the SR method can capture the correct propagation speed of discontinuities in very coarse meshes.

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