Преобразование типа Радона, связанное с уравнениями Эйлера для идеальной жидкости

Аннотация

We study the Nadirashvili -- Vladuts transform $\N$ that integrates second rank symmetric tensor fields $f$ on ${\R}^n$ over hyperplanes. More precisely, for a hyperplane $P$ and vector $\eta$ parallel to $P$, ${\N}f(P,\eta)$ 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 $\N$ 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.