The two-dimensional Euler equation in Yudovich and bmo-type spaces
Qionglei Chen
Institute of Applied Physics and Computational Mathematics, Beijing, ChinaChangxing Miao
Institute of Applied Physics and Computational Mathematics, Beijing, ChinaXiaoxin Zheng
Beihang University, Beijing, China
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Abstract
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.
Cite this article
Qionglei Chen, Changxing Miao, Xiaoxin Zheng, The two-dimensional Euler equation in Yudovich and bmo-type spaces. Rev. Mat. Iberoam. 35 (2019), no. 1, pp. 195–240
DOI 10.4171/RMI/1053