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

Abstract

We construct global-in-time, unique solutions of the two-dimensional Euler equations in a Yudovich type space and a $\mathrm{bmo}$-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.

39pages. Thanks Philippe Serfati for providing us with his paper: Pertes de régularité le laplacien et l'équation d'Euler sui ${\mathbb{R}}^n$, priprint.15,pp., 1994."