The periodic Cauchy problem of the modified Hunter-Saxton equation

Abstract

We prove that the periodic initial value problem for the modified Hunter-Saxton equation is locally well-posed for initial data in the space of continuously differentiable functions on the circle and in Sobolev spaces $$H^s (\mathbb{T})$$ when s > 3/2. We also study the analytic regularity (both in space and time variables) of this problem and prove a Cauchy-Kowalevski type theorem. Our approach is to rewrite the equation and derive the estimates which permit application of o.d.e. techniques in Banach spaces. For the analytic regularity we use a contraction argument on an appropriate scale of Banach spaces to obtain analyticity in both time and space variables.