The Growth Of Vorticity Gradient Of Euler Flow In A Two-dimensional Angular Region And Some Explicit Finite Energy Solution Of Axisymmetric Fluid

We consider weak solutions of the incompressible Euler equation on a two-dimensional symmetric domain with a corner: the quarter unit disk,and investigate their vorticity“gradient”growth.By finding the explicit expression for the green’s function and the Biot-Savart kernel,one can estimate the fluid velocity on the boundary near the corner point,and hence get a lower bound estimate of the growth rate of the Lipschitz quotient of the vorticity.For the three-dimensional axisymmetric Euler equation,we find some z-periodic explicit finite energy(in one period)solutions.