Very weak solutions of the stationary Navier–Stokes equations for an incompressible fluid past obstacles

• Published in: Nonlinear analysis Vol. 147; pp. 145 - 168
• Main Authors: Kim, Dugyu, Kim, Hyunseok
• Format: Journal Article
• Language: English
• Published: Elmsford Elsevier Ltd 01.12.2016; Elsevier BV
• Subjects: Barriers; Exterior domains; Fluid dynamics; Fluid flow; Incompressible flow; Infinity; Navier-Stokes equations; Nonlinear equations; Regularity; Uniqueness; Very weak solutions; Weak solutions; Navier–Stokes equations; 35Q10; Weak solutions; Very weak solutions; Exterior domains
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2016.08.017

We consider the stationary motion of an incompressible Navier–Stokes fluid past obstacles in R3, subject to the given boundary velocity vb, external force f=divF and nonzero constant vector ke1 at infinity. Our main result is the existence of at least one very weak solution v in ke1+L3(Ω) for arbitrary large F∈L3/2(Ω)+L12/7(Ω) provided that the flux of vb−ke1 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 vb−ke1 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∈L3/2(Ω).