On Global Finite Energy Solutions of the Camassa-Holm Equation

We consider the Camassa-Holm equation with data in the energy norm H¹(R¹). Global solutions are constructed by the small viscosity method for the frequency localized equations. The solutions are classical, unique and energy conservative. For finite band data, we show that global solutions for CH exist, satisfy the equation pointwise in time and satisfy the energy conservation law. We show that blow-up for higher Sobolev norms generally occurs in finite time and it might be of power type even for data in H^3/2−.