The Cauchy Problem for the Fifth Order Shallow Water Equation

Abstract

The local well-posedness of the Cauchy problem for the fifth order shallow water equation $$ \partial _{t} u + \alpha \partial ^{5}_{x} u + \beta \partial ^{3}_{x} u + \gamma \partial _{x} u + \mu u\partial _{x} u = 0,\;x,t \in \mathbb{R}, $$ is established for low regularity data in Sobolev spaces $$ H^{s} {\left( {s \geqslant - \frac{3} {8}} \right)} $$ by the Fourier restriction norm method. Moreover, the global well-posedness for L 2 data follows from the local well-posedness and the conserved quantity. For data in H s (s > 0), the global well-posedness is also proved, where the main idea is to use the generalized bilinear estimates associated with the Fourier restriction norm method to prove that the existence time of the solution only depends on the L 2 norm of initial data.