Global well-posedness and asymptotic behavior for the Euler-alignment system with pressure

Xiang Bai; Changhui Tan; Liutang Xue

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arXiv · 2402.07686

Global well-posedness and asymptotic behavior for the Euler-alignment system with pressure

Xiang Bai·Changhui Tan·Liutang Xue

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Abstract

We study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We establish a global well-posedness theory for the system with small smooth initial data. Additionally, we derive asymptotic emergent behaviors for the system, providing time decay estimates with optimal decay rates. Notably, the optimal decay rate we obtain does not align with the corresponding fractional heat equation within our considered range, where the parameter $\alpha\in(0,1)$. This highlights the distinct feature of the alignment operator.

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Xiang Bai, Changhui Tan, Liutang Xue. 2024-02-12. Global well-posedness and asymptotic behavior for the Euler-alignment system with pressure. https://arxiv.org/abs/2402.07686

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