For the momentum equation with unbounded pressure gradient

Abstract

We study the momentum equation with unbounded pressure gradient on the interior curve starting at the non-convex vertex. The inflow boundary by the horizontal directional vector U = (1, 0)^t on the L-shaped domain is not connected. When the pressure is integrated along the streamline, it has a jump across the interior curve. Hence the pressure gradient is not well-defined there. To handle this we construct a vector field which lifts the pressure jump value on the curve into the region. To find a precise structure for the solution we split from the solution the lifting vector field, the contact singularity, the corner singularity and the remainder part. The remainder one is shown to have the higher regularity. The contact singularity is because the lifting vector show a non-smooth behavior at the contact point where the interface curve meets the boundary. The corner singularity is due to the non-convex vertex. Finally we give some numerical examples confirming the critical roles of each part in the decomposition.