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

Binbin Shi; Weike Wang

doi:10.3934/dcdsb.2019215

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2020, Volume 25, Issue 3: 1169-1192. Doi: 10.3934/dcdsb.2019215

This issue Previous Article Global stability of the predator-prey model with a sigmoid functional response Next Article

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

Binbin Shi^1, , and
Weike Wang^2,

1.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
2.
School of Mathematical Sciences and Institute of Natural Science, Shanghai Jiao Tong University, Shanghai 200240, China

^* Corresponding author: binbinshi@sjtu.edu.cn(Binbin Shi)
^* Corresponding author: binbinshi@sjtu.edu.cn(Binbin Shi)

Received: November 30, 2018

Revised: April 30, 2019

Early access: September 2019

Published: March 2020

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Abstract

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.

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

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