Integral geometry of Euler equations

Nikolai Nadirashvili (1) , Serge Vlăduţ (1, 2)

1 I2M - Institut de Mathématiques de Marseille
2 IITP - Institute for Information Transmission Problems

We develop an integral geometry of stationary Euler equations defining some function $w$ on the Grassmannian of affine lines in the space. This function depends on a putative compactly supported solution $v$ of the system, and we deduce a linear differential equation for $w$. Using the $X$-ray transform for quadratic tensor fieds and its plane version, we deduce that $w=0$ everywhere, which implies that there is no non-zero compactly supported solution of the steady Euler equations in ${\mathbb R}^3$.

Domaines: Equations aux dérivées partielles [math.AP]; Géométrie différentielle [math.DG]

Nikolai Nadirashvili, Serge Vlăduţ. Integral geometry of Euler equations. 2016. ⟨hal-01478907⟩