Eulerian Dynamics in Multidimensions with Radial Symmetry
Abstract
We study the global wellposedness of pressureless Eulerian dynamics in multidimensions, with radially symmetric data. Compared with the one-dimensional system, a major difference in multidimensional Eulerian dynamics is the presence of the spectral gap, which is difficult to control in general. We propose a new pair of scalar quantities that provides significantly better control of the spectral gap. Two applications are presented: (i) the Euler--Poisson equations: we show a sharp threshold condition on initial data that distinguish global regularity and finite time blowup; (ii) the Euler-alignment equations: we show a large subcritical region of initial data that leads to global smooth solutions.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
J. A. Carrillo, Y.-P. Choi, E. Tadmor, and C. Tan, Critical thresholds in 1D Euler equations with non-local forces, Math. Models Methods Appl. Sci., 26 (2016), pp. 185--206.
2.
G. Chen, Optimal time-dependent lower bound on density for classical solutions of 1-D compressible Euler equations, Indiana Univ. Math. J., 66 (2017), pp. 725--740.
3.
F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), pp. 852--862.
4.
T. Do, A. Kiselev, L. Ryzhik, and C. Tan, Global regularity for the fractional Euler alignment system, Arch. Ration. Mech. Anal., 228 (2018), pp. 1--37.
5.
S. Engelberg, H. Liu, and E. Tadmor, Critical thresholds in Euler-Poisson equations, Indiana Univ. Math. J., 50 (2001), pp. 109--157.
6.
A. Figalli and M.-J. Kang, A rigorous derivation from the kinetic Cucker--Smale model to the pressureless euler system with nonlocal alignment, Anal. PDE, 12 (2018), pp. 843--866.
7.
Y. Guo, Smooth irrotational flows in the large to the Euler--Poisson system in $\mathbb{R}^{3+ 1}$, Comm. Math. Phys., 195 (1998), pp. 249--265.
8.
S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), pp. 415--435.
9.
S. He and E. Tadmor, Global regularity of two-dimensional flocking hydrodynamics, C. R. Math., 355 (2017), pp. 795--805.
10.
J. Jang, The two-dimensional Euler-Poisson system with spherical symmetry, J. Math. Phys., 53 (2012), 023701.
11.
J. Jang, D. Li, and X. Zhang, Smooth global solutions for the two-dimensional Euler Poisson system, Forum Math. 26, De Gruyter, Berlin, 2014, pp. 645--701.
12.
D. Lear and R. Shvydkoy, Existence and Stability of Unidirectional Flocks in Hydrodynamic Euler Alignment Systems, preprint, https://arxiv.org/abs/1911.10661, 2019.
13.
H. Liu and E. Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Comm. Math. Phys., 228 (2002), pp. 435--466.
14.
H. Liu and E. Tadmor, Rotation prevents finite-time breakdown, Phys. D, 188 (2004), pp. 262--276.
15.
Q. Miao, C. Tan, and L. Xue, Global Regularity for a 1D Euler-Alignment System with Misalignment, Math. Models Methods Appl. Sci., 31 (2021), pp. 473--524.
16.
R. Shvydkoy, Global existence and stability of nearly aligned flocks, J. Dynam. Differential Equations, 31 (2019), pp. 2165--2175.
17.
R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing, Trans. Math. Appl., 1 (2017), tnx001.
18.
E. Tadmor and H. Liu, Critical thresholds in 2D restricted Euler-Poisson equations, SIAM J. Appl. Math., 63 (2003), pp. 1889--1910.
19.
E. Tadmor and C. Tan, Critical thresholds in flocking hydrodynamics with non-local alignment, Philos. Trans. A, 372 (2014), 20130401.
20.
E. Tadmor and D. Wei, On the global regularity of subcritical Euler--Poisson equations with pressure, J. Eur. Math. Soc. (JEMS), 10 (2008), pp. 757--769.
21.
C. Tan, On the Euler-alignment system with weakly singular communication weights, Nonlinearity, 33 (2020), 1907.
22.
D. Wang, Global solutions and relaxation limits of Euler-Poisson equations, Z. Angew. Math. Phys., 52 (2001), pp. 620--630.
23.
D. Wei, E. Tadmor, and H. Bae, Critical thresholds in multi-dimensional Euler-Poisson equations with radial symmetry, Commun. Math. Sci., 10 (2012), pp. 75--86.
24.
M. Yuen, Blowup for the Euler and Euler--Poisson equations with repulsive forces, Nonlinear Anal., 74 (2011), pp. 1465--1470.
Information & Authors
Information
Published In
SIAM Journal on Mathematical Analysis
Pages: 3040 - 3071
DOI: 10.1137/20M1358682
ISSN (online): 1095-7154
Copyright
© 2021, Society for Industrial and Applied Mathematics.
History
Submitted: 10 August 2020
Accepted: 19 February 2021
Published online: 25 May 2021
Keywords
MSC codes
Authors
Funding Information
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1853001
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- The Riemann Problem for the Main Model Cases of the Euler–Poisson EquationsJournal of Mathematical Sciences, Vol. 45 | 29 November 2024
- On radially symmetric oscillations of a collisional cold plasmaMathematical Methods in the Applied Sciences, Vol. 47, No. 11 | 11 March 2024
- The Riemann problem for the main model cases of the Euler-Poisson equationsContemporary Mathematics. Fundamental Directions, Vol. 70, No. 1 | 15 March 2024
- Global well-posedness and asymptotic behavior in critical spaces for the compressible Euler system with velocity alignmentNonlinearity, Vol. 37, No. 2 | 4 January 2024
- Asymptotic behaviors for the compressible Euler system with nonlinear velocity alignmentJournal of Differential Equations, Vol. 380 | 1 Jan 2024
- Critical thresholds in the Euler-Poisson-alignment systemJournal of Differential Equations, Vol. 375 | 1 Dec 2023
- Sharp critical thresholds for a class of nonlocal traffic flow modelsNonlinear Analysis: Real World Applications, Vol. 73 | 1 Oct 2023
- Existence of Radial Global Smooth Solutions to the Pressureless Euler–Poisson Equations with Quadratic ConfinementArchive for Rational Mechanics and Analysis, Vol. 247, No. 4 | 1 August 2023
- Swarming: hydrodynamic alignment with pressureBulletin of the American Mathematical Society, Vol. 60, No. 3 | 25 April 2023
- Sticky particle Cucker–Smale dynamics and the entropic selection principle for the 1D Euler-alignment systemCommunications in Partial Differential Equations, Vol. 48, No. 5 | 20 April 2023
- On the behavior of multidimensional radially symmetric solutions of the repulsive Euler–Poisson equationsPhysica D: Nonlinear Phenomena, Vol. 443 | 1 Jan 2023
- Critical Threshold for Global Regularity of the Euler--Monge--Ampère System with Radial SymmetrySIAM Journal on Mathematical Analysis, Vol. 54, No. 4 | 14 July 2022
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].