Abstract: We study the Nadirashvili -- Vladuts transform $\mathcal N$ that integrates second rank tensor fields $f$ on ${\mathbb R}^n$ over hyperplanes. More precisely, for a hyperplane $P$ and vector $\eta$ parallel to $P$, ${\mathcal N}f(P,\eta)$ is the integral of the function $f_{ij}(x)\xi^i\eta^j$ over $P$, where $\xi$ is the unit normal vector to $P$. We prove that, given a divergence-free vector field $v$, the tensor field $f=v\otimes v$ belongs to the kernel of $\mathcal N$ if and only if there exists a function $p$ such that $(v,p)$ is a solution to the Euler equations. Then we study the Nadirashvili -- Vladuts potential $w(x,\xi)$ determined by a solution to the Euler equations. The function $w$ solves some 4th order PDE. We describe all solutions to the latter equation. The work was performed according to the Government research assignment for IM SB RAS, project FWNF-2022-0006.

Cite: Sharafutdinov V.A.
A Radon type transform related to the Euler equations for ideal fluid
In compilation Сборник тезисов. Международная научная конференция Современные проблемы обратных задач посвященная 85-летию академика РАН В.Г. Романова. Новосибирск, Академгородок, 6 – 9 ноября 2023 года. 2023. – C.27.

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