On the existence of Burgers vortices for high Reynolds numbers

Axisymmetric or non-axisymmetric Burgers vortices have been studied numerically as a model of concentrated vorticity fields. Recently, Th. Gallay and C. E. Wayne rigorously proved that non-axisymmetric Burgers vortices exist for all values of the vortex Reynolds number if an asymmetric parameter is sufficiently small. Several numerical results suggest that Burgers vortices have simpler structures if the vortex Reynolds number is large, even when the asymmetric parameter is not small. In this paper, we give a rigorous explanation for this numerical observation and extend the existence results obtained by Th. Gallay and C. E. Wayne for high vortex Reynolds numbers

Large time behavior of derivatives of the vorticity for the two dimensional Navier-Stokes flow

This paper studies the large time asymptotic behavior of derivatives of the vorticity solving the two-dimensional vorticity equations equivalent to the Navier-Stokes equations. It is well-known by now that the vorticity behaves asymptotically as the Oseen vortex provided that the initial vorticity is integrable. This paper shows that each derivative of the vorticity also behave asymptotically as that of the Oseen vortex. For the proof new spatial decay estimates for derivatives are established. These estimates control behavior at the space infinity. The convergence result follows from a rescaling and compactness argument

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

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 ! 0 that holds at least up to the time c 1=3, where 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 in the inviscid limit, together with explicit estimates

On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half plane

We consider the Navier-Stokes equations for viscous incompressible flows in the half plane under the no-slip boundary condition. By using the vorticity formulation we prove the (local in time) convergence of the Navier-Stokes flows to the Euler flows outside a boundary layer and to the Prandtl flows in the boundary layer at the inviscid limit when the initial vorticity is located away from the boundary

A lower bound for fundamental solutions of the heat convection equations

This paper studies the heat convection equations when the convection term has some singularities at time zero. We shall establish the pointwise estimates for fundamental solutions from below by the Gaussian-like functions. As an application, we prove the existence and uniqueness of the mild solutions of the equations