Multilinear estimates for periodic KdV equations, and applications
The authors deal with the Cauchy problem for periodic generalized Korteweg-de Vries (KdV) equations of the form \[ \begin{cases} \partial_t u+\frac{1}{4\pi^2} \partial^3_x u+ (F(u))_x= 0,\quad & u:\mathbb{T}\times [0,T]\to \mathbb{R},\\ u(x,0)= u_0(x),\quad & x\in\mathbb{T},\end{cases}\tag{1} \] where \(F\) is a polynomial of degree \(k+ 1\), the initial data \(u_0\) is in a Sobolev space \(H^s(\mathbb{T})\), and \(\mathbb{T}= \mathbb{R}/\mathbb{Z}\) is the torus. The main result is a sharp multilinear estimate which allows the authors to prove that (1) is locally well-posed in \(H^s(\mathbb{T})\) for \(s> 1/2\).
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