An inverse problem of Newtonian aerodynamics

Abstract

We consider a rarefied medium in $\mathbb{R}^d,\, d \ge 2$ consisting of non-interacting point masses moving at unit velocity in all directions. Given the density of velocity distribution, one easily calculates the pressure created by the medium in any direction. We then consider the inverse problem: given the pressure distribution $f: S^{d-1} \to \mathbb{R}_+$, determine the density $\rho: S^{d-1} \to \mathbb{R}_+$. Assuming that the reflection of medium particles by obstacles is elastic, we show that the solution for the inverse problem is generally non-unique, derive exact inversion formulas, and state necessary and sufficient conditions for existence of a solution. We also present arguments indicating that the inversion is typically unique in the case of non-elastic reflection, and derive exact inversion formulas in a special case of such reflection.