Revista Matemática Iberoamericana


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Volume 35, Issue 1, 2019, pp. 195–240
DOI: 10.4171/rmi/1053

Published online: 2019-01-08

The two-dimensional Euler equation in Yudovich and bmo-type spaces

Qionglei Chen[1], Changxing Miao[2] and Xiaoxin Zheng[3]

(1) Institute of Applied Physics and Computational Mathematics, Beijing, China
(2) Institute of Applied Physics and Computational Mathematics, Beijing, China
(3) Beihang University, Beijing, China

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.

Keywords: Two-dimensional incompressible Euler equations, Yudovich type data, John–Nirenberg inequality, global existence and uniqueness of solutions

Chen Qionglei, Miao Changxing, Zheng Xiaoxin: The two-dimensional Euler equation in Yudovich and bmo-type spaces. Rev. Mat. Iberoam. 35 (2019), 195-240. doi: 10.4171/rmi/1053