We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field, which could be either vacuum or nonvacuum. The initial density is allowed to vanish, and the spatial measure of the set of vacuum can be arbitrarily large; in particular, the initial density can even have compact support. These results generalize previous results on classical solutions for initial densities being strictly away from vacuum and are the first for global classical solutions that may have large oscillations and can contain vacuum states. © 2012 Wiley Periodicals, Inc

We study the rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvaturelike quantities for polyhedral surfaces are introduced and are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law. They can be considered as 2-dimensional counterparts of the Schlaefli formula.

In this article, we study the Hamiltonian non-displaceability of Gauss images of isopara- metric hypersurfaces in the spheres as Lagrangian submanifolds embedded in complex hyperquadrics.

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We consider the following mean field equation: $$\Delta_{g}v+\rho\left(\frac{h^*e^v}{\int_Mh^*e^v}-1\right)=4\pi\sum_{j=1}^N\alpha_j(\delta_{q_j}-1)$$ on M, where M is a compact Riemann surface with volume 1, $h^*$ is a positive $C^1$ function on M, and  $\alpha_j$ are constants satisfying $\alpha_j>-1$. In this paper, we derive the topological degree counting formula for noncritical values of $\rho$. We also give several applications of this formula, including existence of curvature +1 metric with conic singularities, doubly periodic solutions of electroweak theory, and a special case of self-gravitating strings.