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 $\text{Γ}$ is assumed piecewise Lyapunov surface and the interior solid angle $\text{Θ}(x)$ at the non-smooth boundary point $x$ must satisfy the inequality

Corresponding to the Dirichlet problem of the Navier-Stokes equations, the following series of Volterra integral equations of the first kind for unknown tractions ${\sigma }_j^{(n)}(j=1,2,3:n=0,1,2,\dots )$ is derived.

$$G{\sigma }_j^{(n)}(x,t)={\displaystyle {\int }_t^0{\displaystyle {\int }_{\mathrm{\Gamma }}{\sigma }_i^{(n)}(y,\tau )U_{ij}^{*}(y,\tau ;x,t)dS(y)d\tau =b_j^{(n)}(x,t)}},$$

where $U_{ij}^{*}$ are components of the Stokes fundamental solution tensor and $b_j^{(n)}$ can be regarded as given functions. The integral $G{\sigma }_j^{(n)}$ is the single layer potential. The integral involved in the definition of $b_j^{(n)}$ (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 ${\sum }^{\text{}}=\mathrm{\Gamma }\times [0,T]$, the operator

$$G:H^{-\frac{1}{2},-\frac{1}{4}}(\sum )\to H^{\frac{1}{2},\frac{1}{4}}(\sum )$$

is coercive;

$${((G\sigma ,\sigma ))}_{L^2(\sum )}\ge \beta |||\sigma ||{|}_{H^{-\frac{1}{2},-{\frac{1}{4}}_{(\sum )}}}^2$$

with a constant , $\sigma =({\sigma }_1,{\sigma }_2,{\sigma }_3)$.