Very weak solutions of the stationary Navier–Stokes equations for an incompressible fluid past obstacles. (December 2016)

• Record Type: Journal Article
• Authors: Kim, Dugyu; Kim, Hyunseok
• Is Part Of: Nonlinear analysis. Volume 147(2016)
• Page Start: 145
• Page End: 168
• Publication Date: 2016-12
• Subjects: 35Q10; Navier–Stokes equations -- Weak solutions -- Very weak solutions -- Exterior domains; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.08.017
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the stationary motion of an incompressible Navier–Stokes fluid past obstacles in R 3, subject to the given boundary velocity v b, external force f = div F and nonzero constant vector k e 1 at infinity. Our main result is the existence of at least one very weak solution v in k e 1 + L 3 ( Ω ) for arbitrary large F ∈ L 3 / 2 ( Ω ) + L 12 / 7 ( Ω ) provided that the flux of v b − k e 1 on the boundary of each body is sufficiently small with respect to the viscosity ν . The uniqueness of very weak solutions is proved by assuming that F and v b − k e 1 are suitably small. Moreover, we establish weak and strong regularity results for very weak solutions. In particular, our existence and regularity results enable us to prove the existence of a weak solution v satisfying ∇ v ∈ L 3 / 2 ( Ω ) .