Cauchy problem for quasilinear hyperbolic systems of shallow water equations

Abstract

This article investigates the Cauchy problem for two different models (modified and classical), governed by quasilinear hyperbolic systems that arise in shallow water theory. Under certain reasonable hypotheses on the initial data, we obtain the global smooth solutions for both the systems. The bounds on simple wave solutions of the modified system are shown to depend on the parameter H characterizing the advective transport of impulse. Similarly the bounds on simple wave solutions of the classical system describing the flow over a sloping bottom with profile b(x) are shown to depend on the bottom topography. On the other hand, if the initial data are specified differently, then it is shown that solutions for both the systems exhibit finite time blow-up from specific smooth initial data. Moreover, we show that an increase in H and convexity of b would reduce the time taken for the solutions to blow up.

Keywords:

• quasilinear hyperbolic system
• Cauchy problem
• smooth solution
• shallow water waves

AMS Subject Classifications::

• 35L45
• 35L60

Acknowledgements

Xiaoyu Fu gratefully acknowledges the facilities and support received from IIT Bombay, the NSF of China under grant 10901114, the Doctoral Fund for New Teachers of Ministry of Education of China under grant 20090181120084 and the National Basic Research Program of China (973 program) under grant 2011CB808002 for carrying out this work.