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Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system. (English) Zbl 1248.35052

The work is dealing with spatially-periodic solutions to the generalized (two-component) Hunter-Saxton (HS) equation, in the form of

u txx +2σu x u xx +σuu xxx -ρρ x +Au x =0,ρ t +(ρu) x =0·

This system is a short-wave (high-frequency) limit of the Camassa-Holm two-component system, which, in turn, is a model for non-small water waves, that admits wave breaking. On the other hand, the single-component HS equation, which has other physical realizations, in addition to the water waves, is integrable. The paper at first addresses the well-posedness of the initial-value problem for the two-component HS system in the spatially-periodic setting. Then, the wave-breaking criterion for particular initial profiles is established. The blow-up rate for strong wave-breaking solutions is considered too. Finally, sufficient conditions for the global existence of the solutions are given.

MSC:
35G61Initial-boundary value problems for nonlinear higher-order systems of PDE
35Q53KdV-like (Korteweg-de Vries) equations
35Q35PDEs in connection with fluid mechanics
35B10Periodic solutions of PDE
35B44Blow-up (PDE)
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