Solution formula for the vorticity equations in the half plane with application to high vorticity creation at zero viscosity limit

Abstract

We consider the Navier--Stokes equations for viscous incompressible flows in the half plane under the no-slip boundary condition. In this paper we first establish a solution formula for the vorticity equations through the appropriate vorticity formulation. The formula is then applied to establish the asymptotic expansion of vorticity fields at $\nu\rightarrow 0$ that holds at least up to the time $c\nu^{1/3}$, where $\nu$ is the viscosity coefficient and $c$ is a constant. As a consequence, we get a natural sufficient condition on the initial data for the vorticity to blow up at the inviscid limit, together with explicit estimates.