On a Failure to Extend Yudovich’s Uniqueness Theorem for 2d Euler Equations

In 1995, Yudovich extended his own 1963 uniqueness result for solutions to the 2D Euler equations with bounded initial vorticity to allow a certain class of initial vorticities whose L-norms grow no faster than roughly log p. Yudovich’s argument involves estimating part of the difference between two velocities in terms of the L∞-norm of each velocity. Because the two velocities have a (common) modulus of continuity, however, the L∞-norm of the difference can be bounded by a function of its L-norm, which allows an improvement of this estimate. We show that, though this does, indeed, improve the bound on the difference at time t of the L-norm of two solutions having different initial vorticities, it nonetheless does not result in a larger uniqueness class for solutions to the 2D Euler equations.