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On the weak solutions to a shallow water equation
Author(s) -
Xin Zhouping,
Zhang Ping
Publication year - 2000
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/1097-0312(200011)53:11<1411::aid-cpa4>3.0.co;2-5
Subject(s) - mathematics , a priori and a posteriori , camassa–holm equation , integrable system , norm (philosophy) , mathematical analysis , limit (mathematics) , shallow water equations , waves and shallow water , initial value problem , euler equations , cauchy problem , hamiltonian (control theory) , mathematical optimization , physics , philosophy , epistemology , political science , law , thermodynamics
We obtain the existence of global‐in‐time weak solutions to the Cauchy problem for a one‐dimensional shallow‐water equation that is formally integrable and can be obtained by approximating directly the Hamiltonian for Euler's equation in the shallow‐water regime. The solution is obtained as a limit of viscous approximation. The key elements in our analysis are some new a priori one‐sided supernorm and space‐time higher‐norm estimates on the first‐order derivatives. © 2000 John Wiley & Sons, Inc.