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 wellposedness 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 globalintime 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 uniformlylocalized 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 realvariable techniques, with no use of either Sobolev spaces, Calder\'onZygmund theory or LittlewoodPaley decomposition, and actually applies not only to the BiotSavart 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 
Language:  English 
edoc DOI:  
Last Modified:  03 Nov 2021 08:01 
Deposited On:  03 Nov 2021 08:01 
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