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PreprintarXiv licencearXiv

Local well-posedness, blow-up phenomenon and ill-posedness for a model system of the radiating gas

  • Jian Chen
  • Zhaonan Luo
  • Yuchen Yang
  • Zhaoyang Yin
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Abstract

In this paper, we investigate the Cauchy problem for a model system of radiating gas in multi-dimensional space. Firstly, we prove that the model is locally well-posed in Besov spaces Bsp,r(ℝd) with s>1+d/p (resp., s=1+d/p, r=1), including the endpoint space B1∞,1(ℝd). Secondly, we show that the data-to-solution map is not uniformly continuous on bounded subsets of these Besov spaces. Moreover, we construct a blow-up scenario and derive two sufficient blow-up conditions, which improve the previous blow-up condition obtained in \cite{MR3016436}. Finally, we prove that the Cauchy problem is ill-posed in the critical space B1+d/pp,r(ℝd) for 1≤ p≤∞ and r>1 in the sense of norm inflation.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026local,
  title = {{Local well-posedness, blow-up phenomenon and ill-posedness for a model system of the radiating gas}},
  author = {Jian Chen and Zhaonan Luo and Yuchen Yang and Zhaoyang Yin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29462},
  eprint = {2610.29462},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29462},
}
RIS
TY  - UNPB
TI  - Local well-posedness, blow-up phenomenon and ill-posedness for a model system of the radiating gas
AU  - Jian Chen
AU  - Zhaonan Luo
AU  - Yuchen Yang
AU  - Zhaoyang Yin
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29462
UR  - https://arxiv.org/abs/2610.29462
AB  - In this paper, we investigate the Cauchy problem for a model system of radiating gas in multi-dimensional space. Firstly, we prove that the model is locally well-posed in Besov spaces B^s_p,r(ℝ^d) with s>1+d/p (resp., s=1+d/p, r=1), including the endpoint space B^1_∞,1(ℝ^d). Secondly, we show that the data-to-solution map is not uniformly continuous on bounded subsets of these Besov spaces. Moreover, we construct a blow-up scenario and derive two sufficient blow-up conditions, which improve the previous blow-up condition obtained in \cite{MR3016436}. Finally, we prove that the Cauchy problem is ill-posed in the critical space B^1+d/p_p,r(ℝ^d) for 1≤ p≤∞ and r>1 in the sense of norm inflation.
ER  -