In this paper we prove a well-posedness result for the Cauchy problem. We study a class of first order hyperbolic differential [2] operators of rank zero on an involutive submanifold of$T$^{*}$R$^{$n$+1}-{0} and prove that under suitable assumptions on the symmetrizability of the lifting of the principal symbol to a natural blow up of the “singular part” of the characteristic set, the operator is strongly hyperbolic.

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This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton-Cabrera-Frank (BCF) type model following the work [Xiang, SIAM J. Appl. Math. 2002]. We prove that as the lattice parameter goes to zero, for a finite time interval, a modified discrete model converges to the strong solution of the limiting PDE with first order convergence rate.

Texture mapping and texture synthesis are two popular methods for the decoration of surfaces with visual detail. Here, an existing challenge is to preserve, or at least balance, two competing metrics: scale and angle. In this paper we present two methods for this, both based on global conformal parameterization. First, we describe a texture synthesis algorithm for surfaces with arbitrary topology. By using the conformal parameterization, the 3D surface texture synthesis problem can be converted to a 2D image synthesis problem, which is more intuitive, easier, and conceptually simpler. While the conformality of the parameterization naturally preserves the angles of the texture, in this paper we provide a multi-scale technique to also maintain a more uniform area scaling factor. A second novel contribution is to employ the global parameterization to simultaneously preserve orthogonality and size in texture

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.