On the global classical solution to compressible Euler system with singular velocity alignment

Chen, Li; Tan, C.Y.; Tong, Lining

doi:10.48550/arxiv.2007.08356

Cited by 1 publication

(3 citation statements)

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“…Bouchut and James have developed a related but alternative theory of duality solutions [3,4] for solutions to (10). Their theory relies on properties of monotone solutions to (13), and they prove uniqueness under an assumption similar to (but stronger than) (12). The framework of [7] has been successfully applied to the 1D Euler-Poisson equations [6,48,49].…”

Section: 3mentioning

confidence: 99%

“…The hydrodynamic analog of ( 44) is the one-sided Lipschitz condition (12), which serves as a uniqueness criterion for the 1D pressureless Euler equations, c.f. [30, Theorem 2].…”

Section: 4mentioning

confidence: 99%

“…The resulting isothermal Euler-alignment system was investigated in [8,13]. A more general class of isentropic Euler-alignment system with a pressure term ∇ x (ρ γ ), γ ≥ 1 was considered in [14,12]. The regularity theories of these cases are less well-developed than that of their pressureless cousin.…”

Section: Introductionmentioning

confidence: 99%

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Sticky particle Cucker-Smale dynamics and the entropic selection principle for the 1D Euler-alignment system

Leslie¹,

Tan²

2021

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We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density and bounded velocity. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as an entropic selection principle that determines a unique weak solution of the Euler-alignment system. Moreover, the distinguished weak solution of the system can be approximated by the sticky particle Cucker-Smale dynamics. Our approach is largely inspired by the work of Brenier and Grenier [7] on the pressureless Euler equations. CONTENTS 1. Introduction 1.1. A brief derivation of the Euler-alignment system 1.2. Existing wellposedness theory on the pressureless Euler-alignment system 1.3. Weak solutions and the non-uniqueness issue 1.4. Sticky particle dynamics and selection principles 1.5. Scalar balance laws for the Euler-alignment system 1.6. Main results and structure of the paper 2. Derivation of the scalar balance law and entropy conditions 2.1. The scalar balance law 2.2. Entropy conditions to the scalar balance law 3. The sticky particle Cucker-Smale dynamics 3.1. Definitions and notation 3.2. Basic properties for the sticky particle Cucker-Smale dynamics 3.3. Collision analysis 3.4. A discrete one-sided Lipschitz condition 4. Entropy solutions to the discretized balance law 5. The scalar balance law 5.1. Existence and approximability 5.2. Uniqueness and L 1 stability 6. The entropic selection principle for the Euler-alignment System 6.1. Construction of the solution 6.2. Approximation by sticky particle Cucker-Smale dynamics 7. Asymptotic behavior of the solution 7.1. Uniform flocking estimates on the sticky particle approximations 7.2. Flocking for the 1D Euler-alignment system References

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Section: 3mentioning

confidence: 99%

“…The hydrodynamic analog of ( 44) is the one-sided Lipschitz condition (12), which serves as a uniqueness criterion for the 1D pressureless Euler equations, c.f. [30, Theorem 2].…”

Section: 4mentioning

confidence: 99%