Critical Thresholds in 2D Restricted Euler-Poisson Equations

Tadmor, Eitan; Liu, Hailiang

doi:10.1137/s0036139902416986

Cited by 66 publications

(12 citation statements)

References 30 publications

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“…On the other hand, there exists a set of super-critical initial configurations which lead to the loss of regularity at finite time, u x (·, t) L ∞ → ∞ as t ↑ T c . The so-called critical threshold phenomenon for Eulerian dynamics was first systematically studied in [17] in Euler-Poisson equations, followed by a series of related studies [18][19][20][21]. In particular, Liu & Tadmor [22] We remark in passing that the same phenomenon of flocking hydrodynamics for subcritical initial data occurs in the presence of a pressure term, ∇p(ρ), added to the left-hand side of (1.1b) [21]; this issue is left for future work.…”

Section: Fractional Dissipationmentioning

confidence: 99%

“…On the other hand, there exists a set of super-critical initial configurations which lead to the loss of regularity at finite time, u x (•, t) L ∞ → ∞ as t ↑ T c . The so-called critical threshold phenomenon for Eulerian dynamics was first systematically studied in [17] in Euler-Poisson equations, followed by a series of related studies [18][19][20][21]. In particular, Liu & Tadmor [22] studied the critical threshold phenomenon of the one-dimensional Burgers equation with non-local convolution source term,…”

Section: Introductionmentioning

confidence: 99%

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Critical thresholds in flocking hydrodynamics with non-local alignment

Tadmor

Tan

2014

Phil. Trans. R. Soc. A.

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129

158

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We study the large-time behaviour of Eulerian systems augmented with non-local alignment. Such systems arise as hydrodynamic descriptions of agent-based models for self-organized dynamics, e.g. Cucker & Smale (2007 IEEE Trans. Autom. Control 52 , 852–862. (doi: 10.1109/TAC.2007.895842 )) and Motsch & Tadmor (2011 J. Stat. Phys. 144 , 923–947. (doi: 10.1007/s10955-011-0285-9 )) models. We prove that, in analogy with the agent-based models, the presence of non-local alignment enforces strong solutions to self-organize into a macroscopic flock. This then raises the question of existence of such strong solutions. We address this question in one- and two-dimensional set-ups, proving global regularity for subcritical initial data. Indeed, we show that there exist critical thresholds in the phase space of the initial configuration which dictate the global regularity versus a finite-time blow-up. In particular, we explore the regularity of non-local alignment in the presence of vacuum.

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Section: Fractional Dissipationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Critical thresholds in flocking hydrodynamics with non-local alignment

Tadmor

Tan

2014

Phil. Trans. R. Soc. A.

Self Cite

129

158

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“…For u 0 = 0 or u 0 = 1, finite time blow up can be easily obtain by the Ricatti-type dynamics (24) and (25). Moreover, as 0 ≤ u ≤ 1, we must have d 0 = 0 when u 0 = 0 or 1.…”

Section: 3mentioning

confidence: 99%

Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics

Lee¹,

Liu²

2015

DCDS

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“…We are concerned with both global in time regularity and finite time singularity in solutions to such a relaxation system. As is known, the typical well-posedness result of a one-dimensional system of quasi-linear hyperbolic balance laws asserts that either a solution exists for all time or else there is a finite time such that slopes of the solution become unbounded as the life span is approached; see, e.g., Lax [17], John [15], Liu [33], Nishida [37], Dafermos and Hsiao [5], Wang and Chen [40], Engelberg, Liu and Tadmor [7]. In [23], we identified one lower threshold for global existence of smooth solutions and one upper threshold for the finite time breakdown.…”

Section: Introductionmentioning

confidence: 99%

Critical thresholds in a relaxation model for traffic flows

Li¹,

Liu²

2008

Indiana Univ. Math. J.

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In this paper, we consider a hyperbolic relaxation system arising from a dynamic continuum traffic flow model. The equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system in this model. Thus the usual sub-characteristic condition only holds marginally. In spite of this obstacle, we prove global in time regularity and finite time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the underlying relaxation system. We identify five upper thresholds for finite time singularity in solutions and three lower thresholds for global existence of smooth solutions. The set of initial data leading to global smooth solutions is large, in particular allowing initial velocity of negative slope. Our results show that the shorter the drivers' responding time to the traffic, the larger the set of initial conditions leading to global smooth solutions which correctly predicts the empirical findings for traffic flows.

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Critical Thresholds in 2D Restricted Euler-Poisson Equations

Cited by 66 publications

References 30 publications

Critical thresholds in flocking hydrodynamics with non-local alignment

Critical thresholds in flocking hydrodynamics with non-local alignment

Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics

Critical thresholds in a relaxation model for traffic flows

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