On Volterra Boundary Integral Equations of the First Kind for Nonstationary Stokes Equations

Abstract

We are concerned here with slow viscous incompressible flow in a container or around a rigid body. The physical situation is described mathematically by the initial- boundary value problem for the nonstationary Stokes equations:

$$\upsilon _t - \nu \Delta \upsilon + \nabla _p = f\,\,{\rm in}\,\Omega _T \,{\rm or}\,\Omega _T^c ,$$

((1.1))

$${\rm div}\,\upsilon = 0\,\,{\rm in}\,\Omega _T \,{\rm or}\,\Omega _T^c ,$$

((1.2))

$$\upsilon = - \upsilon _\infty \,\,{\rm on}\,\partial \Omega _T ,$$

((1.3))

$$\upsilon ,p \to 0\,\,{\rm as}\left| x \right| \to \infty ,t > 0,$$

((1.4))

$$\upsilon = \upsilon _0 \,\,{\rm as}\,t = 0,x \in \Omega \,\,{\rm or}\,x \in \Omega ^c $$

((1.5))

Here denotes υ_∞(t) the vector of uniform onset flow at infinity, w = υ_∞ + υ the velocity field, and p the scalar pressure field of the flow, which evolves from an initial state υ₀. Further, $\nu \sim {1 \over { R}e}$ is the dynamic viscosity of the medium. In a body-fixed frame, the flow region is Ω ⊂ ℝ ³ or $\Omega ^C = \mathbb{R}^3 /\bar \Omega$, the boundary ∂Ω of which is assumed as sufficiently smooth (∂Ω ∈ C ^∞, e.g.). Further, Ω _T = Ω × (0, T) (and so Ω ^c _T , ∂Ω_T) where (0, T) denotes a fixed finite time interval. The vector field ƒ includes the exterior forces (and virtually some small convective terms).