Blow-up Properties Of The Strong Solutions Of Some Shallow Water Equations

In this paper, we consider the blow-up properties of the strong solutions ofthe Degasperis-Procesi equation and the Camassa-Holm equation in fnite time.We fnd that if a higher derivative of their strong solutions blow-up in the L2sense, then the L2norms of those derivatives blow up fast. Our method is tofnd an appropriate substitution of variables, and does energy estimate for thetransformed equation.It is well known that strong solutions of the Degasperis-Procesi equation andthe Camassa-Holm equation can blow up in fnite time. They occur in the formof wave-breaking. That is, the strong solutions remain bounded, but the L∞-norm of their frst derivative tends to infnity. For the Camassa-Holm equation,it follows easily that the L2-norm of the second derivative of a blow-up strongsolution blows up fast. This thesis gives a further description of the blow-up.Our method is similar to that used by Chae in obtaining various estimatesfor the strong solutions of the three-dimensional incompressible Euler equation.Since the Degasperis-Procesi equation, the Camassa-Holm equation and the Eulerequation are similar in some ways, the method can be applied to the Degasperis-Procesi equation and the Camassa-Holm equation.