L p -Solutions of the Steady-State Navier-Stokes Equations with Rough External Forces

• Published in: Communications in partial differential equations Vol. 36; no. 2; pp. 216 - 246
• Main Authors: Bjorland, Clayton, Brandolese, Lorenzo, Iftimie, Dragoş, Schonbek, Maria E.
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 16-12-2010
• Subjects: Primary 76D05; Secondary 35B40; Steady Navier-Stokes
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2010.485286

In this paper we address the existence, the asymptotic behavior and stability in L
p
and L
p, ∞
,
, for solutions to the steady state 3D Navier-Stokes equations with possibly very singular external forces. We show that under certain smallness conditions of the forcing term there exists solutions to the stationary Navier-Stokes equations in L
p
spaces, and we prove the stability of these solutions. Namely, we prove that such small steady state solutions attract time dependent solutions with large initial velocity driven by the same forcing. We also give non-existence results of stationary solutions in L
p
, for
.