Crippa, Gianluca and Stefani, Giorgio. (2021) An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces. Preprints Fachbereich Mathematik, 2021 (20).
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Abstract
We revisit Yudovich's well-posedness result for the 2-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular (not necessarily bounded) open set Ω⊂R2 or on the torus Ω=T2. We construct global-in-time weak solutions with vorticity in L1∩Lpul and in L1∩YΘul, where Lpul and YΘul are suitable uniformly-localized versions of the Lebesgue space Lp and of the Yudovich space YΘ respectively, with no condition at infinity for the growth function Θ. We also provide an explicit modulus of continuity for the velocity depending on the growth function Θ. We prove uniqueness of weak solutions in L1∩YΘul under the assumption that Θ grows moderately at infinity. In contrast to Yudovich's energy method, we employ a Lagrangian strategy to show uniqueness. Our entire argument relies on elementary real-variable techniques, with no use of either Sobolev spaces, Calder\'on-Zygmund theory or Littlewood-Paley decomposition, and actually applies not only to the Biot-Savart law, but also to more general operators whose kernels obey some natural structural assumptions.
Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa) 12 Special Collections > Preprints Fachbereich Mathematik |
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UniBasel Contributors: | Crippa, Gianluca and Stefani, Giorgio |
Item Type: | Preprint |
Publisher: | Universität Basel |
Language: | English |
edoc DOI: | |
Last Modified: | 03 Nov 2021 08:01 |
Deposited On: | 03 Nov 2021 08:01 |
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