On the properties in solutions of the Shrira equation

Abstract

In this paper we establish the propagation of regularity phenomena for solutions of the initial-value problem (IVP) associated to the Shrira equation, a two-dimensional model appearing in shear flows. We prove that if initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In other words, the extra regularity on the data propagates in the solutions in the direction of the dispersion. A similar result is also obtained for a model arising in the study of capillary-gravity flows.

Keywords:

• Local well-posedness
• propagation of regularity
• Shrira equation

2010 Mathematics Subject Classifications:

• 35xx
• 35B65
• 35Qxx
• 35Q35

Acknowledgments

The author appreciates the careful reading of the manuscript done by Gleison N. Santos, Roger P. Moura and Marcelo Nogueira.

closureNo potential conflict of interest was reported by the author.

Additional information

Funding

This work was partially supported by CNPq and FAPEPI.