In this talk, we discuss how to define the quasi-local conserved
quantities, the mass, angular momentum and center of mass, for a
finitely extended region in a spacetime satisfying the Einstein
equation. We start with the quasi-local mass and its properties and
then use the results to define other conserved quantities. As a
further application, we use the limit of the quasi-local conserved
quantities to define total conserved quantities of asymptotically flat
spacetimes at both the spatial and the null infinity and study the
variation of these quantities under the Einstein equation.

In 1985, J. Ericksen derived a model for uniaxial liquid crystals to allow for disclinations (i.e. line defects or curve singularities). It involved not only a unit orientation vectorfield on a region of R^3  but also a scalar order parmeter quantify- ing the expected inner product between this vector and the molecular orientation. FH.Lin, in several papers, related this model, for certain material constants, to harmonic maps to a metric cone over S^2. He showed that a minimizer would be continuous everywhere but would have higher regularity fail on the singular de- fect set s^{-1}(0). The optimal partial regularity result of R.Hardt-FH.Lin in 1993, for this model, led to regularity away from isolated points, which unfortunately still excluded line singularities. This paper accordingly also introduced a modified model involving maps to a metric cone over RP^2, the real projective plane. Here the nontrivial homotopy leads to the optimal estimate of the singular set being 1 dimensional. In 2010, J. Ball and A.Zarnescu discussed a derivation from the de Gennes Q tensor and interesting orientability questions using RP2. In recent ongo- ing work with FH.Lin and O. Alper, we see that the singular set with this model necessarily consists of Holder continuous curves. We will also survey some of the many more elaborate liquid crystal PDE’s involving a general director functional, the full Q tensor model, and possible coupling with fluid velocity. 

Conference website: http://www.math.uci.edu/~scgas/scgas-2016/2016.php

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Hydrodynamical evolution in a gravitational field arises in many astrophysical and atmospheric problems. Improper treatment of the gravitational force can lead to a solution which oscillates around the equilibrium. In this presentation, we propose a recently developed well-balanced discontinuous Galerkin method for the Euler equations under gravitational fields, which can maintain the hydrostatic equilibrium state exactly. Some numerical tests are performed to verify the well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.