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We construct global-in-time, unique solutions of the two-dimensional Eulerequations in a Yudovich type space and a $\rm bmo$-type space. First, we showthe regularity of solutions for the two-dimensional Euler equations in theSpanne space involving an unbounded and non-decaying vorticity. Next, weestablish an estimate with a logarithmic loss of regularity for the transportequation in a bmo-type space by developing classical analysis tool such as theJohn-Nirenberg inequality. We also optimize estimates of solutions to thevorticity-stream formulation of the two-dimensional Euler equations with abi-Lipschitz vector field in bmo-type space by combining an observationintroduced by Yodovich with the so-called "quasi-conformal property" of theincompressible.

The two-dimensional Euler equations in Yudovich type space and $\mathrm{\textbf{bmo}}$-type space