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

Abstract

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 .

Keywords:

• Steady Navier–Stokes

Mathematics Subject Classification:

• Primary 76D05
• Secondary 35B40

Acknowledgments

Theorem 2.2 is a development of an insightful remark made to the first and the last author by an anonymous referee of their paper [3]. The authors gratefully acknowledge him. The authors would like to thank also the referees for their many useful suggestions. The work of L. Brandolese, C. Bjorland, D. Iftimie and M. Schonbek were partially supported by FBF GrantSC-08-34. The work of M. Schonbek was also partially supported by NSF Grant DMS-0600692.