An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces

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).


Official URL: https://edoc.unibas.ch/84985/

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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 $\Omega\subset\mathbb{R}^2$ or on the torus $\Omega=\mathbb{T}^2$. We construct global-in-time weak solutions with vorticity in $L^1\cap L^p_{\mathrm{ul}}$ and in $L^1\cap Y^\Theta_{\mathrm{ul}}$, where $L^p_{\mathrm{ul}}$ and $Y^\Theta_{\mathrm{ul}}$ are suitable uniformly-localized versions of the Lebesgue space $L^p$ and of the Yudovich space $Y^\Theta$ respectively, with no condition at infinity for the growth function $\Theta$. We also provide an explicit modulus of continuity for the velocity depending on the growth function $\Theta$. We prove uniqueness of weak solutions in $L^1\cap Y^\Theta_{\mathrm{ul}}$ under the assumption that $\Theta$ 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
UniBasel Contributors:Crippa, Gianluca and Stefani, Giorgio
Item Type:Preprint
Publisher:Universität Basel
edoc DOI:
Last Modified:03 Nov 2021 08:01
Deposited On:03 Nov 2021 08:01

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