On the shallow water equations with periodicity in one direction

Abstract

The linearized 2D shallow water equations, supplemented with suitable boundary conditions in one direction and periodicity in the other direction are considered. The well-posedness of this mixed boundary value problem is proven, using the linear semi-group theory. The well-posedness of the totally periodic problem is also proven.

Keywords:

• Shallow water equations
• mixed boundary-value problems
• semi-group theory
• well-posedness
• periodicity

Keywords:

• Mathematics Subject Classifications: 35F15
• 3SL50
• 35Q35
• 47D06
• 34K13

Acknowledgements

This work was partially supported by the NSF grant DMS-0305110, and by the Research Fund of Indiana University. The author is very thankful to Professor Roger Temam for suggesting this problem and for his helpful advice in its resolution.

Notes

¹ Here denotes the Schwartz space of C ^∞ functions with compact support in .