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arXiv · 2610.29462

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 $B^s_{p,r}(\mathbb{R}^d)$ with $s>1+\frac{d}{p}$ (resp., $s=1+\frac{d}{p}$, $r=1$), including the endpoint space $B^1_{\infty,1}(\mathbb{R}^d)$. 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 $B^{1+\frac{d}{p}}_{p,r}(\mathbb{R}^d)$ for $1\leq p\leq\infty$ and $r>1$ in the sense of norm inflation.

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Jian Chen, Zhaonan Luo, Yuchen Yang, Zhaoyang Yin. 2026-10-21. Local well-posedness, blow-up phenomenon and ill-posedness for a model system of the radiating gas. https://arxiv.org/abs/2610.29462

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