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On the weak solutions to a shallow water equation. (English) Zbl 1048.35092
The authors obtain the existence of global-in-time weak solutions to the Cauchy problem for a one-dimensional formally integrable shallow-water equation of Camassa-Holm type. First, for an approximate viscous problem they establish uniform in viscosity energy estimates for the solution and its gradient. This allows to derive a priori one-sided supernorm and space-time higher-norm estimates for first-order derivatives for viscous solutions, and thus, as the solution of the original problem is obtained as the limit of viscous approximation, these estimates are also valid for the solution of the original problem. The results are applied to the construction of large-time asymptotics.

MSC:
35Q35PDEs in connection with fluid mechanics
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B40Asymptotic behavior of solutions of PDE