Radial pressureless Euler flow: structure and stability of absorption at the origin

Abstract

We study absorption at the origin in radial pressureless Euler flow in $\mathbb{R}^n$, $n\geq2$. For rotation-invariant nonnegative locally finite initial mass and bounded radial initial velocity vanishing at the origin, we prove global existence and uniqueness in a full-space radial entropy class with a one-sided Lipschitz bound and an initial kinetic-energy trace. Absorption is permanent, including retention of any initial origin atom. For finite positive mass with finite second moment, a Lagrangian projection representation characterizes the absorbed mass and its left limits at positive times through extreme minimizers of a free-position primitive. We derive a radial scalar momentum balance with a nonnegative Stieltjes boundary impulse measure. The minimum value of the same primitive yields a weighted Stieltjes identity equating accumulated time-weighted impulse with the integral of initial radii over absorbed labels, valid also for jumps and singular continuous absorption; the boundary impulse measure and the absorption measure are mutually absolutely continuous on $(0,\infty)$. For such finite-mass data with common total mass, strong $L^2$ convergence of initial quantiles and Lagrangian velocities, together with convergence of initial origin masses, yields convergence of absorbed mass at continuity times of the limiting absorption and local weak convergence of absorption measures on $(0,\infty)$. Finally, even with a fixed smooth initial radial probability density, every probability measure compactly supported in $(0,\infty)$ can be realized as the absorption measure by choosing a bounded radial initial velocity; in particular, purely singular continuous absorption occurs.