### Abstract

In this paper, we prove unique existence of solutions to the generalized resolvent problem of the Stokes operator with first order boundary condition in a general domain Ω of the N-dimensional Eulidean space ℝ^{N}, N ≥ 2. This type of problem arises in the mathematical study of the flow of a viscous incompressible one-phase fluid with free surface. Moreover, we prove uniform estimates of solutions with respect to resolvent parameter Λ varying in a sector ∑_{σ}, _{λ0} = {λ ∈ ℂ |arg λ| < π-σ, |λ| ≥ λ_{0}}, where 0 < σ < π/2 and λ_{0} ≥ 1. The essential assumption of this paper is the existence of a unique solution to a suitable weak Dirichlet problem, namely it is assumed the unique existence of solution p ∈ Ŵ^{1}
_{q,Γ}, (Ω) to the variational problem: (∇_{p}, ∇ _{φ}) = (f, ∇_{φ}) for any φ ∈ Ŵ^{1}
_{q',Γ}(Ω). Here, 1 < q < ∞, q' = q/(q-1), Ŵ^{1}
_{q,Γ}(Ω) is the closure of Ŵ^{1}
_{q,Γ}(Ω) = {p ∈ Ŵ^{1}
_{q}(Ω) |p|Γ = 0} by the semi-norm ||∇ ̇ ||L_{q}(Ω), and Γ is the boundary of Ω. In fact, we show that the unique solvability of such a Dirichlet problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to resolvent parameter varying in (λ_{0}, ∞). Our assumption is satisfied for any q ∈ (1, ∞) by the following domains: whole space, half space, layer, bounded domains, exterior domains, perturbed half space, perturbed layer, but for a general domain, we do not know any result about the unique existence of solutions to the weak Dirichlet problem except for q = 2.

Original language | English |
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Pages (from-to) | 1-40 |

Number of pages | 40 |

Journal | Journal of Mathematical Fluid Mechanics |

Volume | 15 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 Mar |

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### Keywords

- 76D07
- Mathematics Subject Classification (2000): 35Q35

### ASJC Scopus subject areas

- Applied Mathematics
- Mathematical Physics
- Computational Mathematics
- Condensed Matter Physics