The proper orthogonal decomposition method for the Navier-Stokes equations

The proper orthogonai decomposition (POD) method for the instatiouary Navier-Stokes equations is considered. Several numerical approaches to evaluating the POD eigenfunctions are presented. The POD eigenfunctions are applied as a basis for a Galerkin projection of the instationary Navier-Stokes equations. And a low-dimensional ordinary differential models for fluid flows governed by the instationary Navier-Stokes equations are constructed. The numerical examples show that the method is feasible and efficient for optimal control of fluids.     

Improvement of two inequalities in the stationary Navier-Stokes equations

Two inequalities in the book Navier-Stokes Equations Theory and Numerical Analysis written by Roger Temam are improved via Schwarz＇s inequality and Young＇s inequality, and the coefficients of them are simplified from 2^1/4 · 3^1/2 and 2^1/2 · 3^3/4to 2^1/4 and 2^3/4 , respectively. Therefore, to some extent the approximating error can be reduced and the accuracy of approximation can be improved.