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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

${u}_{txx}+2\sigma {u}_{x}{u}_{xx}+\sigma u{u}_{xxx}-\rho {\rho }_{x}+A{u}_{x}=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\rho }_{t}+{\left(\rho u\right)}_{x}=0·$

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:
 35G61 Initial-boundary value problems for nonlinear higher-order systems of PDE 35Q53 KdV-like (Korteweg-de Vries) equations 35Q35 PDEs in connection with fluid mechanics 35B10 Periodic solutions of PDE 35B44 Blow-up (PDE)