## Abstract

We consider the stationary Navier-Stokes system on a bounded Lipschitz domain Ω in R ^{3} with connected boundary ∂Ω. The main concern is the solvability of the Dirichlet problem with external force and boundary data having minimal regularity. Here L ^{q} _{s+1/q-2}(ω) denotes the standard Sobolev space with the pair (s, q) being admissible for the unique solvability in L ^{q} _{s+1/q} (ω) of the Stokes system. We show that if 1+s≥2/q in addition, then for any and satisfying the necessary compatibility condition, there exists at least one solution in L ^{q} _{s+1/q} (ω) + L ^{2} _{1/2} (ω) of the Dirichlet problem and this solution has a complete regularity property. The uniqueness of solutions is also shown under the smallness condition on the corresponding norms of the data.

Original language | English |
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Pages (from-to) | 1919-1944 |

Number of pages | 26 |

Journal | Communications in Partial Differential Equations |

Volume | 36 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2011 Nov |

### Bibliographical note

Funding Information:The first author is supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund; KRF-2007-314-C00020). The second author is supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund; KRF-2007-313-C00049).

## All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics