Existence and blow up of solutions to the \(2D\) Burgers equation with supercritical dissipation. (English) Zbl 1430.35001
Summary: This paper is concerned with the Cauchy problem for a fractal Burgers equation in two dimensions. When \(\alpha \in (0, 1)\), the same problem has been studied in one dimensions, we can refer to [N. Alibaud et al., J. Hyperbolic Differ. Equ. 4, No. 3, 479–499 (2007; Zbl 1144.35038); H. Dong et al., Indiana Univ. Math. J. 58, No. 2, 807–822 (2009; Zbl 1166.35030); A. Kiselev et al., Dyn. Partial Differ. Equ. 5, No. 3, 211–240 (2008; Zbl 1186.35020)]. In this paper, we study well-posedness of solutions to the Burgers equation with supercritical dissipation. We prove the local existence with large initial data and global existence with small initial data in critical Besov space by energy method. Furthermore, we show that solutions can blow up in finite time if initial data is not small by contradiction method.
MSC:
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35B44 | Blow-up in context of PDEs |
| 35R11 | Fractional partial differential equations |
| 35S10 | Initial value problems for PDEs with pseudodifferential operators |
Keywords:
fractional dissipationReferences:
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