SUT J. Math. 33 (1), 11-45, (January 1997) DOI: 10.55937/sut/1262184427
Kenji Shirota, Kazuhisa Minowa, Kazuei Onishi
KEYWORDS: Nonstationary Navier-Stokes equations, boundary Volterra integral equation of the first kind, piecewise Lyapunov surface, Dirichlet problem, weak singularity, coercivity, 45D05, 47G10
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 is assumed piecewise Lyapunov surface and the interior solid angle at the non-smooth boundary point 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 is derived.
where are components of the Stokes fundamental solution tensor and can be regarded as given functions. The integral is the single layer potential. The integral involved in the definition of (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 , the operator
with a constant , .