Speaker: Tai-Peng Tsai, University of British Columbia.

Title: Boundary conditions and derivative behavior for incompressible viscous fluids

Abstract: Incompressible viscous fluids are described by the Navier-Stokes equations, first written down in 1822. However, it had been in constant debate which boundary condition (BC) should be used. The assumption that liquid molecules adjacent to a solid boundary are stationary relative to the solid leads to the no-slip BC which has been applied successfully to model many macroscopic experiments. In contrast, Navier's slip BC states that the tangential fluid velocity is proportional to the tangential stress force the fluid exerts on the boundary. Although physically convincing, it could not be verified experimentally until recently and is now considered the appropriate BC when the boundary is rough. Mathematically, no-slip BC gives a stronger constraint on the velocity than slip BC. For example, in the inviscid limits (when the viscosity constant goes to 0), there is a strong Prandtl boundary layer under no-slip BC which remains order one in the limits, but only a weak boundary layer under Navier BC. We are motivated by the regularity problem of the Navier-Stokes equations: Would the presence of the boundary make it easier to obtain singularities? There are examples in the study of inviscid Euler equations: Kiselev and Sverak constructed 2D Euler flows with optimal double exponential growth near the boundary, and Tom Hou obtained numerical and theoretical results on blow up of 3D axisymmetric Euler flows. For Navier-Stokes equations, when the velocity is bounded in the interior, so are its derivatives. However, near the boundary, the gradient may blow up under no-slip BC. Under the Navier BC, the gradient remains bounded, but the second derivative may blow up. I hope to explain these phenomena and provide some heuristics in this colloquium.