An explicit potential theory for the stokes resolvent boundary value problems in three dimensions

Asymptotic formula for the solution of the Stokes problem with a small perturbation of the domain in two and three dimensions

“…Additionally, from (2.10) and (2.11), we conclude that the components S ijk (x, y) ν k (y) of S(x, y)ν(y) can be written in the following manner (see in [3], [4]). …”

“…For φ ∈ (C(∂D)) d , the following trace relation for W D holds (see in [6]): From the previous section, the solution for the interior Dirichlet problem can be presented by the pure double-layer potential (see in [4]), then taking D = Ω and D = Ω δ in the fomulae (2.13), (2.14), (2.15) and (2.17), we get the boundary integral representation of the solutions for the initial Stokes problem (1.1) and the perturbation problem (1.2) are (W Ω φ, Π Ω φ) and (W Ω δ φ, Π Ω δ φ), respectively.…”

“…We note that D ij (x, y) behaves like O(r −d+1 ) as r → 0 , which implies this kernel is weakly singular, so the integral operator is bounded on (C(∂D)) d (see in [11, p. 243] or in [3], [4], [6]). …”

“…The asymptotic expansion is derived due to the theory of layer potentials and Fredholm's alternative, and the properties of small perturbation of an interface. In connection with this, we refer to recent works in the context of interface problems [1], [2], [7], the continuity of the solution due to a small perturbation of an interface [7] and layer-potential theory for Stokes problem [3], [4], [5], [6], [8], [9]. In the work of H. Ammari, H. Kang, M. Lim, and H. Zribi [1] , they derived the asymptotic expansion of the boundary perturbations of steady-state voltage potentials, and now in our work, we derive the asymptotic expansion for the Stokes problem with Dirichlet boundary condition.…”

An indirect boundary integral method for an oscillatory Stokesflow problem

“…For our purpose, we also need the surface force H λ 2 ,Γ h, associated to the unsteady single-layer potential ᐂ s λ 2 ,Γ h and defined in a neighborhood ᐂ ⊂ R 3 of Γ as follows [5,16]: 5) where x ∈ Γ is the unique determined projection of x ∈ ᐂ onto Γ . Let Ᏻ be the free-space Green function (or the Oseen-Burgers tensor) of the steady Stokes flow and let be the associated stress tensor.…”