Generic regularity and Lipschitz metric for the Hunter-Saxton type equations

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• 作者：Hong Cai^a ; ^b ; ^{caihong19890418@163.com" class="auth_mail" title="E-mail the corresponding author} ; Geng Chen^c ; ^{gengchen@ku.edu" class="auth_mail" title="E-mail the corresponding author} ; Yannan Shen^d ; ^{yannan.shen@csun.edu" class="auth_mail" title="E-mail the corresponding author} ; Zhong Tan^e ; ^{tan85@xmu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35Q53 ; 35B35 ; 37C20
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：262
• 期：2
• 页码：1023-1063
• 全文大小：514 K

文摘

The Hunter–Saxton equation determines a flow of conservative solutions taking values in the space c4627dab3319a4b3316ec4ed74cc60" title="Click to view the MathML source">H¹(R⁺)$H^1({\mathbb{R}}^{+})$. However, the solution typically includes finite time gradient blowups, which make the solution flow not continuous w.r.t. the natural 4e1fc1b6" title="Click to view the MathML source">H¹$H^1$ distance. The aim of this paper is to first study the generic properties of conservative solutions of some initial boundary value problems to the Hunter–Saxton type equations. Then using these properties, we give a new way to construct a Finsler type metric which renders the flow uniformly Lipschitz continuous on bounded subsets of c4627dab3319a4b3316ec4ed74cc60" title="Click to view the MathML source">H¹(R⁺)$H^1({\mathbb{R}}^{+})$.