Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids

Abstract

This paper analyzes an initial/boundary value problem for a system of equations modelling the nonstationary flow of a nonhomogeneous incompressible asymmetric (polar) fluid. Under conditions similar to those usually imposed to the nonhomogeneous 3D Navier-Stokes equations, by using a spectral semi-Galerkin method, we prove the existence of a local in time strong solution. We also prove the uniqueness of the strong solution and some global existence results. Several estimates for the solutions and their approximations are given. These can be used to find useful error bounds of the Galerkin approximations.Dans ce papier, on analyse un problème de valeurs initiales et valeurs aux limites pour un système d’équations aux dérivées partielles qui modélise le flux instationnaire d’un fluide asymmétrique incompressible non homogène. Sous des conditions similaires aux conditions usuellement imposées aux équations tridimensionelles de Navier-Stokes non homogènes, à l’aide d’une méthode de type semi-Galerkin, nous démontrons l’éxistence d’une solution forte locale en temps. On établit aussi l’unicité de solution forte et quelques résultats d’éxistence globale. Tous ces résultats reposent sur des estimations appropriées pour les solutions et leurs approximations qui permettent d’ailleurs d´eduire des estimations de l’erreur.Conselho Nacional de Desenvolvimento Científico e Tecnológico (Brasil)Fundação de Amparo à Pesquisa do Estado de São PauloDirección General de Enseñanza Superio

On Classical Solutions to 2D Shallow Water Equations with Degenerate Viscosities

“…The following lemma is important in the derivation of uniqueness in Section 3, which can be found in Remark 1 of [1].…”

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

“…In case that ρ 0 has a positive lower bound and u 0 has the additional integrability condition u 0 ∈ L 2 , Theorem 1.1 can be proved applying the method of successive approximations or a fixed point argument as in [1,13,18,24,29]. Our proof of the theorem is based on the method of successive approximations, whose general strategy may be described as follows.…”

On the variational inequalities related to viscous density-dependent incompressible fluids

“…The existence of an m-th approximated semi-Galerkin solution can be obtained arguing as in [17] (see also [4]): First, one rewrites (20)-(21) as a fixed point equation; then, one deduces appropriate estimates and, finally, one applies Schauder's theorem. By taking V m = U m (t) in (20) and applying Lemma 1, we get the so-called energy equality:…”