The Cauchy problem for an integrable shallow-water equation

Abstract

We prove that the periodic initial value problem for a completely integrable shallow-water equation is not locally well-posed for initial data in the Sobolev space H s (T) whenever s < 3/2. Since on the other hand this problem is locally well-posed in the sense of Hadamard for s > 3/2 our result suggests that s = 3/2 is the critical Sobolev index for well-posedness. We also show that the nonperiodic initial value problem is not locally well-posed in H s (R) for s < 3/2.