Global conservative solutions for a model equation for shallow water waves of moderate amplitude |
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Authors: | Shouming Zhou Chunlai Mu |
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Institution: | 1. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China;2. College of Mathematics Science, Chongqing Normal University, Chongqing 400047, China |
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Abstract: | In this paper, we study the continuation of solutions to an equation for surface water waves of moderate amplitude in the shallow water regime beyond wave breaking (in 11], Constantin and Lannes proved that this equation accommodates wave breaking phenomena). Our approach is based on a method proposed by Bressan and Constantin 2]. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, the evolution problem is rewritten as a semilinear system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows one to continue the solution after collision time, giving a global conservative solution where the energy is conserved for almost all times. Finally, returning to the original variables, we obtain a semigroup of global conservative solutions, which depend continuously on the initial data. |
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Keywords: | 65M06 65M12 35B10 35Q53 |
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