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 ∂tρ+∂x(ρf(u))=0, ∂t(ρu)+∂x(ρu f(u))=0, where ρ is the mass density, possibly measure-valued, u is the physical velocity, and f∈ C1(ℝ), with f'>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∞ estimate for the normalized mass measures in terms of the initial position and physical-velocity discrepancies in mass coordinates, whenever the initial W∞ distance is finite. The Lagrangian evolution has the semigroup property.
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BibTeX
@misc{cao2026onedimensional,
title = {{One-dimensional pressureless flow with a nonlinear velocity law: Bregman variational construction and well-posedness}},
author = {Gaowei Cao and Feimin Huang and Guirong Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16341},
eprint = {2610.16341},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16341},
}
RIS
TY - UNPB
TI - One-dimensional pressureless flow with a nonlinear velocity law: Bregman variational construction and well-posedness
AU - Gaowei Cao
AU - Feimin Huang
AU - Guirong Tang
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16341
UR - https://arxiv.org/abs/2610.16341
AB - We establish global existence, uniqueness and quantitative stability for one-dimensional pressureless flow governed by ∂_tρ+∂_x(ρf(u))=0, ∂_t(ρu)+∂_x(ρu f(u))=0, where ρ is the mass density, possibly measure-valued, u is the physical velocity, and f∈ C^1(ℝ), with f'>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_∞ estimate for the normalized mass measures in terms of the initial position and physical-velocity discrepancies in mass coordinates, whenever the initial W_∞ distance is finite. The Lagrangian evolution has the semigroup property.
ER -