RECONSTRUCTION OF THE VELOCITY FROM THE VORTICITY IN 3-DIMENSIONAL FLUID-FLOWS

This paper describes variational methods for finding the stationary ve locity fields of given vorticity which obey either no-flux or no-slip boundary conditions. The density of the fluid is assumed to be known, continuous and bounded away from zero and the domain is bounded, conne cted and has a C-1,C-1 boundary. Necessary conditions on the vorticity for there to be velocity fields with finite kinetic energy are derive d. The existence of velocity fields is proven subject to these necessa ry conditions and the vorticity being 6/5 th power integrable in the n o-flux case. For no-slip boundary conditions, with square integrable v orticity, existence is proved when the density is uniform. The solutio ns are obtained by using potential representations for the velocity he ld, variational principles and weak formulations of the problems. Thes e formulations are suitable for direct numerical simulation and comput ation. Continuous dependence results are given.