Error Bounds for Semi-Galerkin Approximations of Nonhomogeneous Incompressible Fluids

Abstract

We consider spectral semi-Galerkin approximations for the strong solutions of the nonhomogeneous Navier-Stokes equations. We derive an optimal uniform in time error bound in the H 1 norm for approximations of the velocity. We also derive an error estimate for approximations of the density in some spaces L r . (2000). 35Q30, 76M22, 65M15, 65M60. Mathematics Subject Classification

Uniform-in-time error estimates for spectral Galerkin approximations of a mass diffusion model

“…Let us introduce here the concept of perturbations of solutions to (1) analogous to [15] and [7]. The difference in definition of perturbations among these two works lies in the decay rate as time goes to infinity.…”

“…The difference in definition of perturbations among these two works lies in the decay rate as time goes to infinity. To be more precise, in [15], an exponential decay rate is assumed, whereas, in [7], any decay rate is considered.…”

“…In [7] error estimates were derived by assuming that the solution (u, ρ) is p 0conditionally asymptotically stable. The number p 0 is required to satisfy 6 ≤ p 0 ≤ ∞, and is related to the regularity of allowed perturbations of the density equation.…”

“…respectively. Clearly, ρ k satisfies equation (7). The details of existence of approximate solutions to (6)-(7) will be given in the appendix.…”

A uniform error estimate in time for spectral Galerkin approximations of the magneto‐micropolar fluid equations

“…Although this is not an interesting case from practical point of view, we hope that the techniques that we developed here could be adapted in the important case where the full discretization is used. It would be interesting to extend the works [18,22,23] for the model studied in this article.…”