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Differentical equations, dynamical systems and optimal control
A Radon type transform related to the Euler equations for ideal fluid
V. A. Sharafutdinov Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia
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 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.
Keywords:
Euler equations, Nadirashvili – Vladuts transform, tensor tomography.
Received May 27, 2023, published October 26, 2023
Citation:
V. A. Sharafutdinov, “A Radon type transform related to the Euler equations for ideal fluid”, Sib. Èlektron. Mat. Izv., 20:2 (2023), 880–912
Linking options:
https://www.mathnet.ru/eng/semr1617 https://www.mathnet.ru/eng/semr/v20/i2/p880
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