On the Lagrangian Trajectories for the One-Dimensional Euler Alignment Model without Vacuum Velocity

Trevor M. Leslie

doi:10.5802/crmath.56

Équations aux dérivées partielles

On the Lagrangian Trajectories for the One-Dimensional Euler Alignment Model without Vacuum Velocity

Trevor M. Leslie ¹

¹ Department of Mathematics, University of Wisconsin, Madison

Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 421-433.

Résumé

A well-known result of Carrillo, Choi, Tadmor, and Tan [1] states that the 1D Euler Alignment model with smooth interaction kernels possesses a “critical threshold” criterion for the global existence or finite-time blowup of solutions, depending on the global nonnegativity (or lack thereof) of the quantity $e_{0} = \partial_{x} u_{0} + ϕ * ρ_{0}$ . In this note, we rewrite the 1D Euler Alignment model as a first-order system for the particle trajectories in terms of a certain primitive $ψ_{0}$ of $e_{0}$ ; using the resulting structure, we give a complete characterization of global-in-time existence versus finite-time blowup of regular solutions that does not require a velocity to be defined in the vacuum. We also prove certain upper and lower bounds on the separation of particle trajectories, valid for smooth and weakly singular kernels, and we use them to weaken the hypotheses of Tan [25] sufficient for the global-in-time existence of a solution in the weakly singular case, when the order of the singularity lies in the range $s \in (0, \frac{1}{2})$ and the initial data is critical.

Reçu le : 2019-09-29
Révisé le : 2020-04-21
Accepté le : 2020-04-23
Publié le : 2020-07-28

DOI : 10.5802/crmath.56

Classification : 92D25, 35Q35, 76N10

Affiliations des auteurs :

Trevor M. Leslie ¹

¹ Department of Mathematics, University of Wisconsin, Madison

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2020__358_4_421_0,
     author = {Trevor M. Leslie},
     title = {On the {Lagrangian} {Trajectories} for the {One-Dimensional} {Euler} {Alignment} {Model} without {Vacuum} {Velocity}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {421--433},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {4},
     year = {2020},
     doi = {10.5802/crmath.56},
     language = {en},
}

TY  - JOUR
AU  - Trevor M. Leslie
TI  - On the Lagrangian Trajectories for the One-Dimensional Euler Alignment Model without Vacuum Velocity
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 421
EP  - 433
VL  - 358
IS  - 4
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.56
LA  - en
ID  - CRMATH_2020__358_4_421_0
ER  -

%0 Journal Article
%A Trevor M. Leslie
%T On the Lagrangian Trajectories for the One-Dimensional Euler Alignment Model without Vacuum Velocity
%J Comptes Rendus. Mathématique
%D 2020
%P 421-433
%V 358
%N 4
%I Académie des sciences, Paris
%R 10.5802/crmath.56
%G en
%F CRMATH_2020__358_4_421_0

Trevor M. Leslie. On the Lagrangian Trajectories for the One-Dimensional Euler Alignment Model without Vacuum Velocity. Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 421-433. doi : 10.5802/crmath.56. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.56/

Bibliographie
Cité par

[1] José A. Carrillo; Young-Pil Choi; Eitan Tadmor; Changhui Tan Critical thresholds in 1D Euler equations with non-local forcesi, Math. Models Methods Appl. Sci., Volume 26 (2016) no. 1, pp. 185-206 | DOI | Zbl

[2] José A. Carrillo; Massimo Fornasier; Jesús Rosado; Giuseppe Toscani Asymptotic flocking dynamics for the kinetic Cucker–Smale model, SIAM J. Math. Anal., Volume 42 (2010) no. 1, pp. 218-236 | DOI | MR | Zbl

[3] Felipe Cucker; Steve Smale Emergent behavior in flocks, IEEE Trans. Autom. Control, Volume 52 (2007) no. 5, pp. 852-862 | DOI | MR | Zbl

[4] Felipe Cucker; Steve Smale On the mathematics of emergence, Jpn. J. Math., Volume 3 (2007) no. 2, pp. 197-227 | DOI | MR | Zbl

[5] Raphaël Danchin; Piotr B. Mucha; Jan Peszek; Bartosz Wróblewski Regular solutions to the fractional Euler alignment system in the Besov spaces framework, Math. Models Methods Appl. Sci., Volume 29 (2019) no. 1, pp. 89-119 | DOI | MR | Zbl

[6] Tam Do; Alexander Kiselev; Lenya Ryzhik; Changhui Tan Global regularity for the fractional Euler alignment system, Arch. Ration. Mech. Anal., Volume 228 (2018) no. 1, pp. 1-37 | MR | Zbl

[7] Alessio Figalli; Moon-Jin Kang A rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignment, Anal. PDE, Volume 12 (2019) no. 3, pp. 843-866 | DOI | MR | Zbl

[8] Seung-Yeal Ha; Moon-Jin Kang; Bongsuk Kwon Emergent dynamics for the hydrodynamic Cucker–Smale system in a moving domain, SIAM J. Math. Anal., Volume 47 (2015) no. 5, pp. 3813-3831 | MR | Zbl

[9] Seung-Yeal Ha; Eitan Tadmor From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, Volume 1 (2008) no. 3, pp. 415-435 | MR | Zbl

[10] Siming He; Eitan Tadmor Global regularity of two-dimensional flocking hydrodynamics, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 7, pp. 795-805 | MR | Zbl

[11] Trygve K. Karper; Antoine Mellet; Konstantina Trivisa Hydrodynamic limit of the kinetic Cucker–Smale flocking model, Math. Models Methods Appl. Sci., Volume 25 (2015) no. 1, pp. 131-163 | DOI | MR | Zbl

[12] Alexander Kiselev; Changhui Tan Global regularity for 1D Eulerian dynamics with singular interaction forces, SIAM J. Math. Anal., Volume 50 (2018) no. 6, pp. 6208-6229 | DOI | MR | Zbl

[13] Daniel Lear; Roman Shvydkoy Existence and stability of unidirectional flocks in hydrodynamic euler alignment systems (2019) (https://arxiv.org/abs/1911.10661)

[14] Trevor M. Leslie Weak and strong solutions to the forced fractional Euler alignment system, Nonlinearity, Volume 32 (2019) no. 1, pp. 46-87 | DOI | MR | Zbl

[15] Piotr Minakowski; Piotr B. Mucha; Jan Peszek; Ewelina Zatorska Singular Cucker–Smale Dynamics, Active Particles, Volume 2 (Modeling and Simulation in Science, Engineering and Technology), Birkhäuser, 2019, pp. 201-243 | DOI

[16] Javier Morales; Jan Peszek; Eitan Tadmor Flocking with short-range interactions, J. Stat. Phys., Volume 176 (2019) no. 2, pp. 382-397 | DOI | MR | Zbl

[17] Piotr B. Mucha; Jan Peszek The Cucker–Smale equation: singular communication weight, measure-valued solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal., Volume 227 (2018) no. 1, pp. 273-308 | DOI | MR | Zbl

[18] Jan Peszek Existence of piecewise weak solutions of a discrete Cucker–Smale’s flocking model with a singular communication weight, J. Differ. Equations, Volume 257 (2014) no. 8, pp. 2900-2925 | DOI | MR | Zbl

[19] Jan Peszek Discrete Cucker–Smale flocking model with a weakly singular weight, SIAM J. Math. Anal., Volume 47 (2015) no. 5, pp. 3671-3686 | DOI | MR | Zbl

[20] Roman Shvydkoy Global existence and stability of nearly aligned flocks, J. Dyn. Differ. Equations, Volume 31 (2019) no. 4, pp. 2165-2175 | DOI | MR | Zbl

[21] Roman Shvydkoy; Eitan Tadmor Eulerian dynamics with a commutator forcing, Trans. Math. Appl., Volume 1 (2017) no. 1, tnx001, 26 pages | MR | Zbl

[22] Roman Shvydkoy; Eitan Tadmor Eulerian dynamics with a commutator forcing. II: Flocking, Discrete Contin. Dyn. Syst., Volume 37 (2017) no. 11, pp. 5503-5520 | DOI | MR | Zbl

[23] Roman Shvydkoy; Eitan Tadmor Eulerian dynamics with a commutator forcing III. Fractional diffusion of order $0 < α < 1$ , Physica D, Volume 376-377 (2018), pp. 131-137 | DOI | MR | Zbl

[24] Eitan Tadmor; Changhui Tan Critical thresholds in flocking hydrodynamics with non-local alignment, Philos. Trans. R. Soc. Lond., Ser. A, Volume 372 (2014) no. 2028, 20130401, 22 pages | MR | Zbl

[25] Changhui Tan On the Euler-alignment system with weakly singular communication weights, Nonlinearity, Volume 33 (2020) no. 4, pp. 1907-1924 | Zbl

Cité par Sources :

Commentaires - Politique