Equivalence of entropy solutions and gradient flows for pressureless 1D Euler systems
- 1Mathematical Institute, University of Oxford, Oxford, UK
- 2Department of Mathematical Sciences, NTNU–Norwegian University of Science and Technology, Trondheim, Norway
- 3Department of Mathematics and Cybernetics, SINTEF Digital, Oslo, Norway
Abstract
We study distributional solutions of pressureless Euler systems on the line. In particular we show that Lagrangian solutions [7], introduced by Brenier, Gangbo, Savaré and Westdickenberg, and entropy solutions [37], 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 L2 L 2 -gradient flows and entropy solutions for a first-order aggregation equation proved by Bonaschi, Carrillo, Di Francesco and Peletier [4]. The key observation is an equivalence between Oleĭnik’s E-condition for conservation laws and a characterization due to Natile and Savaré of the normal cone for L2 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 [14], as well as to describe their asymptotic behavior.
Related
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- On one-dimensional pressureless Euler--Poisson systems: existence for non-semiconvex interaction potentials
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- The equivalence of solutions to fractional and logarithmic Helmholtz equations
BibTeX
@article{carrillo2026equivalence,
title = {{Equivalence of entropy solutions and gradient flows for pressureless 1D Euler systems}},
author = {José Antonio Carrillo and Sondre Tesdal Galtung},
journal = {Mathematische Annalen},
year = {2026},
month = jun,
doi = {10.1007/s00208-026-03522-5},
url = {https://doi.org/10.1007/s00208-026-03522-5},
note = {CC BY},
}
RIS
TY - JOUR
TI - Equivalence of entropy solutions and gradient flows for pressureless 1D Euler systems
AU - José Antonio Carrillo
AU - Sondre Tesdal Galtung
PY - 2026
DA - 2026/06/24
JO - Mathematische Annalen
DO - 10.1007/s00208-026-03522-5
UR - https://doi.org/10.1007/s00208-026-03522-5
AB - We study distributional solutions of pressureless Euler systems on the line. In particular we show that Lagrangian solutions [7], introduced by Brenier, Gangbo, Savaré and Westdickenberg, and entropy solutions [37], 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 L 2 -gradient flows and entropy solutions for a first-order aggregation equation proved by Bonaschi, Carrillo, Di Francesco and Peletier [4]. The key observation is an equivalence between Oleĭnik’s E-condition for conservation laws and a characterization due to Natile and Savaré of the normal cone for L^2 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 [14], as well as to describe their asymptotic behavior.
SN - 0025-5831
SN - 1432-1807
ER -