Geodesic
A Radon type transform related to the Euler equations for ideal fluid
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 880-912.

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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: tensor tomography.
Mots-clés : Euler equations, Nadirashvili – Vladuts transform 
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V. A. Sharafutdinov. A Radon type transform related to the Euler equations for ideal fluid. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 880-912. http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a45/

[1] P. Constantin, J. La, V. Vicol, “Remarks on a paper by Gavrilov: Grad-Shafranov equations, sready solutions of the three dimensional incompressible Euler equations with compactly supported velocities, and applications”, Geom. Funct. Anal., 29:6 (2019), 1773–1793 | DOI | MR | Zbl

[2] A. Enciso, D. Peralta-Salas, F. Torres de Lizaur, Quasi-periodic solutions to the incompressible Euler equations in dimensions two and higher, 2022, arXiv: 2209.09812 | MR

[3] A.V. Gavrilov, “A steady Euler flow with compact support”, Geom. Funct. Anal., 29:1 (2019), 190–197 | DOI | MR | Zbl

[4] V. Krishnan, V. Sharafutdinov, “Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas”, Inverse Probl. Imaging, 16:4 (2022), 787–826 | DOI | MR | Zbl

[5] N. Nadirashvili, S. Vlǎduts, “Integral geometry of Euler equations”, Arnold Math. J., 3:3 (2017), 397–421, arXiv: 1608.08850 | DOI | MR | Zbl

[6] N. Nadirashvili, V. Sharafutdinov, S. Vlǎduts, “The John equation in tensor tomography in three-dimensions”, Inverse Probl., 32:10 (2016), 105013, 15 pp. | DOI | MR | Zbl

[7] V.Yu. Rovenski, V.A. Sharafutdinov, “Steady-state flows of ideal incompressible fluid with velocity pointwise orthogonal to the pressure gradient”, Arnold Math. J., 2023 (to appear) , arXiv: 2209.14572

[8] V.A. Sharafutdinov, Integral geometry for tensor fields, VSP, Utrecht, 1994 | MR | Zbl

[9] V. Sharafutdinov, “Orthogonality relations for a stationary flow of ideal fluid”, Sib. Math. J., 59:4 (2018), 731–752 | DOI | MR | Zbl