Well-posedness for a transport equation with nonlocal velocity

Hongjie Dong

Analysis of PDEs mathscidoc:1912.431025

Journal of Functional Analysis, 255, (11), 3070-3097, 2008.12
We study a one-dimensional transport equation with nonlocal velocity which was recently considered in the work of Crdoba, Crdoba and Fontelos [A. Crdoba, D. Crdoba, MA Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math.(2) 162 (3)(2005) 13771389]. We show that in the subcritical and critical cases the problem is globally well-posed with arbitrary initial data in H max {3/2 , 0}. While in the supercritical case, the problem is locally well-posed with initial data in H 3/2 , and is globally well-posed under a smallness assumption. Some polynomial-in-time decay estimates are also discussed. These results improve some previous results in [A. Crdoba, D. Crdoba, MA Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math.(2) 162 (3)(2005) 13771389].
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@inproceedings{hongjie2008well-posedness,
  title={Well-posedness for a transport equation with nonlocal velocity},
  author={Hongjie Dong},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224211318957344589},
  booktitle={Journal of Functional Analysis},
  volume={255},
  number={11},
  pages={3070-3097},
  year={2008},
}
Hongjie Dong. Well-posedness for a transport equation with nonlocal velocity. 2008. Vol. 255. In Journal of Functional Analysis. pp.3070-3097. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224211318957344589.
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