Existence and weak-strong uniqueness of measure solutions to Euler-alignment/Aw--Rascle--Zhang model of collective behaviour
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| Format: | Preprint |
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2025
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| _version_ | 1866912693668020224 |
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| author | Woźnicki, Jakub Zatorska, Ewelina |
| author_facet | Woźnicki, Jakub Zatorska, Ewelina |
| contents | We study the multi-dimensional Euler-alignment system with a matrix-valued communication kernel, motivated by models of anticipation dynamics in collective behaviour. A key feature of this system is its formal equivalence to a nonlocal variant of the Aw--Rascle--Zhang (ARZ) traffic model, in which the desired velocity is modified by a nonlocal gradient interaction. We prove the global-in-time existence of measure solutions to both formulations, obtained via a single degenerate pressureless Navier--Stokes approximation. Furthermore, we establish a weak-strong uniqueness principle adapted to the pressureless setting and to nonlocal alignment forces. As a consequence, we rigorously justify the formal correspondence between the nonlocal ARZ and Euler-alignment models: they arise from the same inviscid limit, and the weak-strong uniqueness property ensures that, whenever a classical solution exists, both formulations coincide with it. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_05326 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and weak-strong uniqueness of measure solutions to Euler-alignment/Aw--Rascle--Zhang model of collective behaviour Woźnicki, Jakub Zatorska, Ewelina Analysis of PDEs 35Q35, 35Q70, 35A02, 35B35 We study the multi-dimensional Euler-alignment system with a matrix-valued communication kernel, motivated by models of anticipation dynamics in collective behaviour. A key feature of this system is its formal equivalence to a nonlocal variant of the Aw--Rascle--Zhang (ARZ) traffic model, in which the desired velocity is modified by a nonlocal gradient interaction. We prove the global-in-time existence of measure solutions to both formulations, obtained via a single degenerate pressureless Navier--Stokes approximation. Furthermore, we establish a weak-strong uniqueness principle adapted to the pressureless setting and to nonlocal alignment forces. As a consequence, we rigorously justify the formal correspondence between the nonlocal ARZ and Euler-alignment models: they arise from the same inviscid limit, and the weak-strong uniqueness property ensures that, whenever a classical solution exists, both formulations coincide with it. |
| title | Existence and weak-strong uniqueness of measure solutions to Euler-alignment/Aw--Rascle--Zhang model of collective behaviour |
| topic | Analysis of PDEs 35Q35, 35Q70, 35A02, 35B35 |
| url | https://arxiv.org/abs/2511.05326 |