Entropy solutions of an associated scalar balance law select a unique weak solution to the 1D pressureless Euler-Poisson-alignment system and reveal qualitative differences between attractive and repulsive interactions.
Equivalence of entropy solutions and gradient flows for pressureless 1D Euler systems
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abstract
We study distributional solutions of pressureless Euler systems on the line. In particular we show that Lagrangian solutions, introduced by Brenier, Gangbo, Savar\'{e} and Westdickenberg, and entropy solutions, studied by Nguyen and Tudorascu for the Euler--Poisson system, are equivalent. For the Euler--Poisson system this can be seen as a generalization to second-order systems of the equivalence between $L^2$-gradient flows and entropy solutions for a first-order aggregation equation proved by Bonaschi, Carrillo, Di Francesco and Peletier. The key observation is an equivalence between Ole\u{\i}nik's E-condition for conservation laws and a characterization due to Natile and Savar\'{e} of the normal cone for $L^2$-gradient flows. This new equivalence allows us to define unique solutions after blow-up for classical solutions of the Euler--Poisson system with quadratic confinement due to Carrillo, Choi and Zatorska, as well as to describe their asymptotic behavior.
fields
math.AP 3years
2026 3verdicts
UNVERDICTED 3representative citing papers
Global well-posedness and quantitative flocking are shown for Lagrangian p-alignment dynamics; Eulerian variables are constructed via pushforward and disintegration, with defect terms vanishing asymptotically under heavy-tailed kernels to give mono-kinetic closure and mean-field convergence.
Existence, uniqueness, and stability are established for unidirectional entropic solutions of the pressureless Euler alignment system via transverse-then-longitudinal discretization of sticky particle Cucker-Smale dynamics, plus long-time flocking results even when communication vanishes near the fl
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Entropic solutions to the 1D pressureless Euler system with nonlocal interactions
Entropy solutions of an associated scalar balance law select a unique weak solution to the 1D pressureless Euler-Poisson-alignment system and reveal qualitative differences between attractive and repulsive interactions.
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Lagrangian formulation and Eulerian closure in alignment dynamics
Global well-posedness and quantitative flocking are shown for Lagrangian p-alignment dynamics; Eulerian variables are constructed via pushforward and disintegration, with defect terms vanishing asymptotically under heavy-tailed kernels to give mono-kinetic closure and mean-field convergence.
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Unidirectional Entropic Solutions of the Pressureless Euler Alignment System
Existence, uniqueness, and stability are established for unidirectional entropic solutions of the pressureless Euler alignment system via transverse-then-longitudinal discretization of sticky particle Cucker-Smale dynamics, plus long-time flocking results even when communication vanishes near the fl