L3-solutions for the stationary Navier–Stokes equations in the exterior of a rotating obstacle. (April 2020)

• Record Type: Journal Article
• Authors: Kim, Dugyu
• Is Part Of: Nonlinear analysis. Volume 52(2020)
• Publication Date: 2020-04
• Subjects: Stationary Navier–Stokes equations -- Weak solutions -- Very weak solutions -- Exterior domains -- Rotating obstacle; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2019.103020
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: Consider the stationary motion of an incompressible Navier–Stokes fluid around a rotating body R 3 ∖ Ω which is also moving in the direction of the axis of rotation with nonzero constant velocity − k e 1 . We assume that the angular velocity ω = | ω | e 1 is also constant and the external force is given by f = div F . Then the motion is described by a variant of the stationary Navier–Stokes equations with the velocity k e 1 at infinity. Our main result is the existence of at least one solution u satisfying u − k e 1 ∈ L 3 ( Ω ) for arbitrarily large F ∈ L 3 ∕ 2 ( Ω ) . The uniqueness is also proved by assuming that | ω | + | k | + ‖ F ‖ L 3 ∕ 2 ( Ω ) is sufficiently small in comparison with the viscosity ν . Moreover, we establish several regularity results to obtain an existence theorem for weak solutions u satisfying ∇ u ∈ L 3 ∕ 2 ( Ω ) and u − k e 1 ∈ L 3 ( Ω ) .