Solutions to the 2D Euler Equations Satisfying the Serfati Condition With Relaxed Constraints on the Initial Vorticity
We prove finite-time existence of solutions to the 2D Euler equations where the velocity grows slower than the square root of the distance from the origin and with vorticity up to linear growth provided the initial vorticity is quasibounded, or with growth in the vorticity related to the growth in the velocity provided the initial vorticity is stable. In the quasibounded case, we provide an example with unbounded vorticity and velocity. We also prove uniqueness for solutions with stable initial vorticity and with a certain bound on the modulus of continuity of the initial velocity. This thesis expands on the recent work of Cozzi and Kelliher, which substitutes Serfati’s identity in place of the Biot-Savart law to demonstrate uniqueness and short-time existence of bounded vorticity solutions with slow growing velocity.