The two-dimensional Euler equation in Yudovich and BMO-type spaces (Q1733575)

Summary: We construct global-in-time, unique solutions of the two-dimensional Euler equations in a Yudovich type space and a bmo-type space. First, we show the regularity of solutions for the two-dimensional Euler equations in the Spanne space involving an unbounded and non-decaying vorticity. Next, we establish an estimate with a logarithmic loss of regularity for the transport equation in a bmo-type space by developing classical analysis tool such as the John-Nirenberg inequality. We also optimize estimates of solutions to the vorticity-stream formulation of the two-dimensional Euler equations with a bi-Lipschitz vector field in bmo-type space by combining an observation introduced by Yodovich with the so-called ``quasi-conformal property'' of the incompressible flow.