L<SUP>q</SUP>-estimates for the stationary Oseen equations on exterior domains

초록

We study the Dirichlet problem for the stationary Oseen equations on exterior smooth domains Omega in R-n, n >= 2. Our main results are the existence and uniqueness of weak and very weak solutions of the Oseen equations satisfying appropriate L-q -estimates. The uniqueness of very weak solutions is shown by utilizing recent results for very weak solutions of the Stokes equations on bounded domains. Then for (n + 1)/(n - 1) < q < infinity, the existence and L-q-estimates of very weak solutions in L-q (Omega)(n) is deduced by a duality argument from our existence result and D-2,D-r-estimates for strong solutions. These results, combined with some regularity results for very weak solutions, enable us to prove the existence, uniqueness and D-1,D-r-estimates of weak solutions in D-1,D-r (Omega)(n) boolean AND L(n+1)r/(n+1-r) (Omega)(n), where (n + 1)/n < r < n + 1. (C) 2014 Elsevier Inc. All rights reserved.

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