### 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 1 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

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*Communications in Partial Differential Equations*, vol. 36, no. 11, pp. 1919-1944. https://doi.org/10.1080/03605302.2011.613079

**Dirichlet Problem for the Stationary Navier-Stokes System on Lipschitz Domains.** / Choe, Hi Jun; Kim, Hyunseok.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Dirichlet Problem for the Stationary Navier-Stokes System on Lipschitz Domains

AU - Choe, Hi Jun

AU - Kim, Hyunseok

PY - 2011/11/1

Y1 - 2011/11/1

N2 - 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.

AB - 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.

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U2 - 10.1080/03605302.2011.613079

DO - 10.1080/03605302.2011.613079

M3 - Article

VL - 36

SP - 1919

EP - 1944

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 11

ER -