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Solution formula for the vorticity equations in the half plane with application to high vorticity creation at zero viscosity limit. (English) Zbl 1261.35111

Summary: 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 \(\nu \to 0\) 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.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
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Full Text: Euclid