*Submitted Paper*

**Inserted:** 29 oct 2021

**Last Updated:** 15 dec 2021

**Year:** 2021

**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 $\Omega\subset\mathbb{R}^2$ 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, CalderÃ³n-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.

**Keywords:**
Euler equations, Osgood condition, weak solution, Global existence, incompressible inviscid fluid, vorticity equation, Biot-Savart law, Lagrangian solution, Yudovich uniqueness theorem, uniformly-localized Lebesgue space, uniformly-localized Yudovich space, Aubin-Lions Lemma

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