L^q‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

Abstract

We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate L q -estimates. The uniqueness of very weak solutions is shown by the method of cut-off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D 1,r -result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D 1,r is the homogeneous Sobolev space.

Oseen System With Coriolis Term

“…(See Lemma 4.1 in D. Kim (2018).) Lemma 4.2: Let �� � � 3 be an unbounded domain with compact Lipschitz boundary, � = (�, 0, 0) and q, s � (1, �).…”

“…Galdi (2011), G.P. Galdi et al (2007), D. Kim (2018) and M. Kyed (2014) studied the problem in homogeneous Sobolev spaces D 2,q (�; � 3 ) × D 1,q (�). The papers �.…”

“…, G.P. Galdi (2011), T. Hishida (2006), D. Kim (2018), D. Kim (2019) and S. Kracmar et al (QAM 2010) are devoted to solutions of the problem in D 1,q (�; � 3 ) × L q loc (�). Very weak solutions of the problem with u � L q (�; � 3 ) are in D. Kim (2018).…”

“…Let us gather the known results. D. Kim writes in D. Kim (2018): Let W be an unbounded domain with compact smooth boundary. If g � H 1/2 (W; R 3 ) and f � 0, then there exists a unique solution (u, p) � [D 1,2 (W; R 3 ) � L 4 (W; R 3 )] × L 2 (W) of ( 1)-(3).…”