Journal
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
Volume 69, Issue 4, Pages -Publisher
SPRINGER BASEL AG
DOI: 10.1007/s00033-018-0992-z
Keywords
Modified two-component Camassa-Holm system; Lipschitz metric; Generic regularity; Conservative solution
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Funding
- Shandong Provincial Natural Science Foundation, China [ZR2018BA008]
- Qingdao University of Science and Technology [010022896]
- National Natural Science Foundation of China [11531010]
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In this paper, we study the Lipschitz continuous dependence of conservative solutions to a modified two-component Camassa-Holm system. Even for smooth initial data, it is known that the gradient of these solutions can blow up in finite time. When this happens, the flow generated by the equations fails to be Lipschitz continuous w.r.t. the usual distance in the Sobolev space. To cope with this issue, we construct a new distance, generated by a Finsler structure, which renders Lipschitz continuous the semigroup of conservative solutions. Our distance is constructed first on a subset of sufficiently regular solutions, then extended by continuity to the entire space. This is possible thanks to a generic regularity result, which has independent interest. Roughly speaking, we show that, for generic smooth initial data, the solution remains piecewise smooth.
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