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

Qionglei Chen ^[1] ; Changxing Miao ^[1] ; Xiaoxin Zheng ^[2]
1. [1] Institute of Applied Physics and Computational Mathematics
  
  Institute of Applied Physics and Computational Mathematics
  
  China
2. [2] Beihang University
  
  Beihang University
  
  China
Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 35, Nº 1, 2019, págs. 195-240
Idioma: inglés
DOI: 10.4171/rmi/1053
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Resumen
- 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.