Global existence and well-posedness of the 2D viscous shallow water system in Sobolev spaces with low regularity

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• 作者：Yanan Liu^a ; ^{babyinarm@126.com" class="auth_mail" title="E-mail the corresponding author} ; Zhaoyang Yin^a ; ^b ; ^{mcsyzy@mail.sysu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：The 2D viscous shallow water system ; The Littlewood&ndash ; Paley theory ; The Bony decomposition ; Sobolev spaces ; Global existence
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 June 2016
• 年：2016
• 卷：438
• 期：1
• 页码：14-28
• 全文大小：371 K

文摘

In this paper we consider the Cauchy problem for the 2D viscous shallow water system in H^s(R²)$H^s({\mathbb{R}}^2)$, s>1$s>1$. We first prove the local well-posedness of this problem by using the Littlewood–Paley theory, the Bony decomposition, and the theories of transport equations and transport diffusion equations. Then, we get the global existence of the system with small initial data in H^s(R²)$H^s({\mathbb{R}}^2)$, s>1$s>1$. Our obtained result improves considerably the recent result in [14].