On the Cauchy Problem for a KdV-Type Equation on the Circle

We shall consider the periodic Cauchy problem for a modified Camassa-Holm (mCH) equation. We begin by proving well-posedness in Bourgain spaces for sufficiently small size initial data in the Sobolev space $H^s(mathbb{T})$, $s=1/2$, by using appropriate bilinear estimates. Also we show that these bilinear estimates do not hold if $s<1/2$. Well-posedness of the mCH for $s>1/2$ has been established by Himonas and Misiol ek in [HM1]. These results indicate that $s=1/2$ may be the critical Sobolev exponent for well-posedness. In the second part of this work we show that the periodic Cauchy problem for the mCH equation with analytic initial data is analytic in the space variable $x$ for time near zero. By differentiating the equation and the initial condition with respect to $x$ we obtain a sequence of initial value problems of KdV-type equations. These, written in the form of integral equations, define a mapping on a Banach space whose elements are sequences of functions equipped with a norm expressing the Cauchy estimates in terms of the KdV norms of the components introduced in the works of Bourgain, Kenig, Ponce, Vega and others. By proving appropriate bilinear estimates we show that this mapping is a contraction, and therefore we obtain a solution whose derivatives in the space variable satisfy the Cauchy estimates.