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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

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It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the “curse of dimensionality" in some degree. In this note, we give the error estimate of hyperbolic cross (HC) approximations with all sorts of Askey polynomials. These polynomials are useful in generalized polynomial chaos (gPC) in the field of uncertainty quantification. The exponential convergences in both regular and optimized HC approximations have been shown under the condition that the random variable depends on the random inputs smoothly in some degree. Moreover, we apply gPC to numerically solve the ordinary differential equations with slightly higher dimensional random inputs. Both regular and optimized HC have been investigated with Laguerre-chaos, Charlier-chaos and Hermite-chaos in the numerical experiment. The discussion of the connection between the standard ANOVA approximation and Galerkin approximation is in the appendix.