We show that in a tubular domain with sufficiently small width, the normal and tangential gradients of a harmonic function have almost the same L 2 norm. This estimate yields a sharp estimate of the pressure in terms of the viscosity term in the Navier-Stokes equation with no-slip boundary condition. By consequence, one can analyze the Navier-Stokes equations simply as a perturbed vector diffusion equation instead of as a perturbed Stokes system. As an application, we describe a rather easy approach to establish a new isomorphism theorem for the non-homogeneous Stokes system.

We present novel algorithms for compressible flows that\n\nare efficient for all Mach numbers. The approach is based on several\n\ningredients: semi-implicit schemes, the gauge decomposition\n\nof the velocity field and a second order formulation of the density\n\nequation (in the isentropic case) and of the energy equation (in\n\nthe full Navier-Stokes case). Additionally, we show that our approach\n\ncorresponds to a micro-macro decomposition of the model,\n\nwhere the macro field corresponds to the incompressible component\n\nsatisfying a perturbed low Mach number limit equation and\n\nthe micro field is the potential component of the velocity. Finally,\n\nwe also use the conservative variables in order to obtain a proper\n\nconservative formulation of the equations when the Mach number\n\nis order unity. We successively consider the isentropic case, the\n\nfull Navier-Stokes case, and the isentropic Navier-Stokes-Poisson\n\ncase. In this work, we only concentrate on the question of the\n\ntime discretization and show that the proposed method leads to\n\nAsymptotic Preserving schemes for compressible flows in the low\n\nMach number limit.

The first part of this paper describes some important underlying themes in the mathematical theory of continuum mechanics that are distinct from formulating and analyzing governing equations. The main part of this paper is devoted to a survey of some concrete, conceptually simple, pretty problems that help illuminate the underlying themes. The paper concludes with a discussion of the crucial role of invariant constitutive equations in computation.

For strong solutions of the incompressible Navier-Stokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure. These schemes require no solution of stationary Stokes systems, nor any compatibility between velocity and pressure spaces to ensure an inf-sup condition, and are representative of a class of highly efficient computational methods that have recently emerged. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the Navier-Stokes equations as a perturbed diffusion equation. © 2007 Wiley Periodicals, Inc.

In this paper, we study the ill posed Perona-Malik equation of image processing [14], and the regularized P-M model i.e. C-model proposed by Catte et al. [4]. The authors present the convex compound of these two models in the form of the system of partial differential equations. The weak solution for the equations is proved in detail. The additive operator splitting (AOS) algorithm for the proposed model is also given. Finally, we show some numeric experiment results on images.