Large time decay of the Oseen flow in exterior domains subject to the Navier slip-with-friction boundary condition
Toshiaki Hishida
TL;DR
This work analyzes the linearized Oseen problem in a 3D exterior domain under the Navier slip-with-friction boundary condition with nonzero outflow at infinity. By constructing a resolvent parametrix that couples the whole-space Oseen resolvent with an interior problem and imposing a boundary–geometry–friction compatibility condition $oxed{\alpha(x)+\kappa(x)\ge \frac{\eta\cdot\nu(x)}{2}}$, the authors establish that the resolvent set contains the right half-plane (modulo the exceptional line $S_\eta$) and derive robust $L^q$–$L^r$ decay estimates for the Oseen semigroup. The approach hinges on a Weingarten-map framework, a cut-off/Bogovskii-based exterior construction, and a local energy decay analysis, extending known no-slip results to Navier boundaries and providing decay rates that are essential for stability and global well-posedness questions for the nonlinear Navier–Stokes system in exterior domains. The results unify and generalize prior work on the Stokes and Oseen problems, including the full slip and no-slip limits, and deliver quantitative time-decay information under realistic boundary conditions.
Abstract
Consider the motion of a viscous incompressible fluid filling a 3D exterior domain $Ω$ subject to the Navier slip-with-friction boundary condition as well as outflow at infinity. For the Oseen system as the linearization, we discuss the resolvent set under a certain relationship among the geometry of the boundary $\partialΩ$, friction coefficient $α(x)$ and the outflow $u_\infty$. We then study the regularity of the resolvent near the origin in the complex plane to develop $L^q$-$L^r$ decay estimates of the Oseen semigroup provided that $α(x)+u_\infty\cdotν(x)/2\geq 0$ for every $x\in\partialΩ$, where $ν(x)$ stands for the outward unit normal to the boundary $\partialΩ$.