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Existence and blow up of solutions to the $ 2D $ Burgers equation with supercritical dissipation

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  • 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 [1, 17, 24]. 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.

    Mathematics Subject Classification: Primary: 35A01, 35B44, 35R11; Secondary: 35S10.


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