One-dimensional pressureless flow with a nonlinear velocity law: Bregman variational construction and well-posedness

Abstract

We establish global existence, uniqueness and quantitative stability for one-dimensional pressureless flow governed by \[ \partial_tρ+\partial_x\bigl(ρf(u)\bigr)=0, \qquad \partial_t(ρu)+\partial_x\bigl(ρu f(u)\bigr)=0, \] where $\rho$ is the mass density, possibly measure-valued, $u$ is the physical velocity, and $f\in C^1(\mathbb{R})$, with $f^{\prime }>0$, determines the transport velocity $f(u)$. The initial mass has finite positive mass and finite second moment, allowing atoms and unbounded support, and the initial physical velocity is bounded. The difficulty at a cluster collision is that mass and momentum conservation determine a mass-weighted average of physical velocity, whereas transport is governed by the nonlinear law $f$ applied to that average. We construct global entropy solutions by a Bregman variational time discretization whose cost is chosen to reproduce the transport velocity determined by this mass-weighted average, coupled with conditional-expectation updates of physical velocity. For every entropy solution satisfying the Oleinik condition on the transport velocity and initial weak continuity of the kinetic energy measure, we establish a Lagrangian representation and prove uniqueness within this class by comparing flows with possibly different cluster intervals. For two such solutions with the same velocity law and common mass, we obtain a $W_{\infty }$ estimate for the normalized mass measures in terms of the initial position and physical-velocity discrepancies in mass coordinates, whenever the initial $W_{\infty }$ distance is finite. The Lagrangian evolution has the semigroup property.

arXiv comment49 pages