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Journal of Mathematical Fluid Mechanics

, Volume 15, Issue 1, pp 1–40 | Cite as

Generalized Resolvent Estimates of the Stokes Equations with First Order Boundary Condition in a General Domain

  • Yoshihiro ShibataEmail author
Article

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 \({\Omega}\) of the N-dimensional Eulidean space \({\mathbb{R}^N, N \geq 2}\). 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 \({\lambda}\) varying in a sector \({\Sigma_{\sigma, \lambda_0} = \{\lambda \in \mathbb{C} \mid |\arg \lambda| < \pi-\sigma, \enskip |\lambda| \geq \lambda_0\}}\), where \({0 < \sigma < \pi/2}\) and \({\lambda_0 \geq 1}\). 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 \({p \in \hat{W}^1_{q, \Gamma}(\Omega)}\) to the variational problem: \({(\nabla p, \nabla \varphi) = (f, \nabla \varphi)}\) for any \({\varphi \in \hat W^1_{q', \Gamma}(\Omega)}\). Here, \({1 < q < \infty, q' = q/(q-1), \hat W^1_{q, \Gamma}(\Omega)}\) is the closure of \({W^1_{q, \Gamma}(\Omega) = \{ p \in W^1_q(\Omega) \mid p|_\Gamma = 0\}}\) by the semi-norm \({\|\nabla \cdot \|_{L_q(\Omega)}}\), and \({\Gamma}\) is the boundary of \({\Omega}\). 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 \({(\lambda_0, \infty)}\). Our assumption is satisfied for any \({q \in (1, \infty)}\) 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.

Mathematics Subject Classification (2000)

35Q35 76D07 

Keywords

Resolvent estimate Lq framework Stokes equation First order boundary condition General domain 

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© Springer Basel 2013

Authors and Affiliations

  1. 1.Research Institute of Science and Engineering, Department of MathematicsWaseda UniversityShinjuku-kuJapan

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