BOUNDARY INTEGRAL EQUATION FOR NAVIER-STOKES EQUATIONS IN A NON-SMOOTH DOMAIN

Abstract

Boundary integral equations corresponding to the differential equations describing a transient flow of incompressible viscous fluid in three dimensions are considered. Emphasis is put on the treatment of edges and corners. The boundary Γ is assumed piecewise Lyapunov surface and the interior solid angle Θ(x) at the non-smooth boundary point x must satisfy the inequality lim⁡δ→0 sup⁡x∈Γ12π{ ∫0<|y−x|≤δ|dΘx(y)|+|2π−Θ(x)| }<1. Corresponding to the Dirichlet problem of the Navier-Stokes equations, the following series of Volterra integral equations of the first kind for unknown tractions σj(n)(j=1,2,3:n=0,1,2,…) is derived. Gσj(n)(x,t)=∫t0∫Γσi(n)(y,τ)Uij*(y,τ;x,t)dS(y)dτ=bj(n)(x,t), where Uij* are components of the Stokes fundamental solution tensor and bj(n) can be regarded as given functions. The integral Gσj(n) is the single layer potential. The integral involved in the definition of bj(n) (see the text) is the double layer potential. Those integrals are shown to be weakly singular for the non-smooth domain under consideration. It is proved that, with ∑​=Γ×[0,T], the operator G:H−12,−14(∑)→H12,14(∑) is coercive; ((Gσ,σ))L2(∑)≥β|||σ|||H−12,−14(∑)2 with a constant β>0, σ=(σ1,σ2,σ3).