A LOGARITHMIC 3D EULER INEQUALITY

The magnitude of the vorticity omega=\omega\ for the 3d incompressible Euler equations on the domain Omega=[0,L](3) with boundary conditions u(n)\(partial derivative Omega)=0 is shown to satisfy the inequality parallel to log omega(t)parallel to(2)-parallel to log omega(0) parall el to(2) less than or equal to integral(0)(t) parallel to omega(tau)(2 ) d tau, for smooth initial data with no zeros in omega. The notation is parallel to omega parallel to(2)(2)=integral(Omega)omega(2) dV and t is time. The case when initial data have zeros in omega is also disc ussed. (C) 1997 American Institute of Physics.