Displaying items by tag: fractional diffusion
Global well-posedness and asymptotic behavior for the Euler-alignment system with pressure
Xiang Bai, Changhui Tan and Liutang Xue
Journal of Differential Equations, Volume 407, pp. 269-310 (2024).
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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doi:10.1016/j.jde.2024.06.020 |
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This work is supported by NSF grants DMS #2108264 and DMS #2238219 |
Global well-posedness and asymptotic behavior in critical spaces for the compressible Euler system with velocity alignment
Xiang Bai, Qianyun Miao, Changhui Tan and Liutang Xue
Nonlinearity, Volume 37, 025007, 46pp. (2024).
Abstract
In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behavior and optimal decay estimates of the solutions as \(t\to\infty\).
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doi:10.1088/1361-6544/ad140b |
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This work is supported by NSF grant DMS #1853001 and DMS #2108264 |