文摘
The Hunter–Saxton equation determines a flow of conservative solutions taking values in the space class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961630328X&_mathId=si1.gif&_user=111111111&_pii=S002203961630328X&_rdoc=1&_issn=00220396&md5=6dc4627dab3319a4b3316ec4ed74cc60" title="Click to view the MathML source">H1(R+)class="mathContainer hidden">class="mathCode">. However, the solution typically includes finite time gradient blowups, which make the solution flow not continuous w.r.t. the natural class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961630328X&_mathId=si2.gif&_user=111111111&_pii=S002203961630328X&_rdoc=1&_issn=00220396&md5=2cd85fd2fcf3215dd1ccf7c04e1fc1b6" title="Click to view the MathML source">H1class="mathContainer hidden">class="mathCode"> 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961630328X&_mathId=si1.gif&_user=111111111&_pii=S002203961630328X&_rdoc=1&_issn=00220396&md5=6dc4627dab3319a4b3316ec4ed74cc60" title="Click to view the MathML source">H1(R+)class="mathContainer hidden">class="mathCode">.