ABSTRACT
This work presents some results regarding the stability of local solutions for the system of equations. The method is based on damping in multi-dimensions and introduces a mixed Fourier pseu-dospectral and Hermite finite difference scheme in space, and an efficient projection method to solve stress velocity coupling. The local existence and pointwise estimates of the solutions are obtained. Furthermore, the optimal stability and convergence rate of the solution when it is a perturbation of a constant state is obtained. It is shown that for some compressible equations, the corresponding local stability result holds and the properties of the analytic semi-group are employed to show the compactness for the semi-process generated by Euler solutions. Provided that the compressible equations are strictly stabilised.
