Bounded vorticity, unbounded velocity solutions to the 2D Euler equations

The pioneering work on bounded vorticity solutions to the 2D Euler equations was done by Yudovich in the early 1960s, working in a bounded domain. He proved existence and uniqueness of such solutions.  Subsequently, existence was shown to hold in much weaker settings (vorticity lying in any Lebesgue space), but uniqueness has only ever been extended to an incrementally larger class of initial data.

Yudovich's theory extends easily to the full plane (indeed it is slightly less technical there) as long as the velocity is assumed to decay sufficiently rapidly at infinity that the Biot-Savart law holds. In 1995, Ph. Serfati established the existence and uniqueness of bounded vorticity solutions having no decay at infinity. Such solutions very much violate the Biot-Savart law, but Serfati discovered an identity that the solutions hold that can be used as a kind of substitute for that law.

The boundedness of the velocity was very important in Serfati's argument, yet there is room in his identity to accommodate some growth of the velocity at infinity. I will speak on ongoing joint work with Elaine Cozzi in which we exploit Serfati's identity to obtain existence and uniqueness classes allowing growth at infinity as large as possible (without assuming any special symmetry of the initial data). Roughly speaking, we show that existence can be achieved only for very slowly growing velocities, but that uniqueness holds for velocities growing slower than the square root of the distance from the origin. We also consider the issue of continuous dependence on initial data, which is already an interesting problem even in Yudovich's original setting.