In this paper, we give a condition on the bicharacteristic which guarantees the global existence of the mild solution to the Boltzmann equation with an external force for the hard-sphere model and potentials with angular cutoff in infinite vacuum. This generalizes the previous results to the case when the force can have arbitrary strength. The constructive condition on the bicharacteristic is used to obtain the pointwise estimates on the collision operator so that the global existence comes from the contraction mapping theorem.

The global existence of weak solutions to the compressible NavierStokes equations with vacuum attracts many research interests nowadays. For the isentropic gas, the viscosity coefficient depends on density function from physical point of view. When the density function connects to vacuum continuously, the vacuum degeneracy gives some analytic difficulties in proving global existence. In this paper, we consider this case with gravitational force and fixed boundary condition. By giving a series of <i>a priori</i> estimates on the solution coping with the degeneracy of vacuum, gravitational force and boundary effect, we give global existence and uniqueness results similar to the case without force and boundary. Copyright 2006 John Wiley &amp; Sons, Ltd.

We study the vanishing viscosity of the Navier-Stokes equations for interacting shocks. Given an entropy solution to <i>p</i>-system which consists of two different families of shocks interacting at some positive time, we show that such entropy solution is the vanishing viscosity limit of a family of global smooth solutions to the isentropic Navier-Stokes equations. The key point of the proofs is to derive the estimates separately before and after the interaction time and connect the incoming and outgoing viscous shock profiles.

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Given any suitably small, localized, and smooth initial data, in this paper, we prove global regularity for the 3D finite depth gravity water wave system. As a byproduct, we rule out the small, localized traveling waves in 3D, which do exist for the same system in 2D.