A Radon type transform related to the Euler equations for ideal fluid

V. A. Sharafutdinov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 880-912

Résumé

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 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.

Détail

Keywords: tensor tomography.
Mots-clés : Euler equations, Nadirashvili – Vladuts transform